Research Article Frequency-Domain Assessment of ...downloads.hindawi.com/journals/mpe/2015/259530.pdf · Research Article Frequency-Domain Assessment of Integration Schemes for Earthquake

Research ArticleFrequency-Domain Assessment of Integration Schemes forEarthquake Engineering Problems

Juana Arias-Trujillo1 Rafael Blaacutezquez2 and Susana Loacutepez-Querol3

1Department of Construction School of Engineering University of Extremadura Avenida de la Universidad sn 10003 Caceres Spain2Department of Civil Engineering Technical University of Cartagena Paseo Alfonso XIII 52 30203 Cartagena Spain3Department of Civil Environmental and Geomatic Engineering University College London Gower Street London WC1E 6BT UK

Correspondence should be addressed to Susana Lopez-Querol susanalopezquerolgmailcom

Received 19 April 2015 Revised 15 June 2015 Accepted 22 June 2015

Academic Editor Xiao-Qiao He

Copyright copy 2015 Juana Arias-Trujillo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Although numerical integration is a technique commonly employed in many time-dependent problems usually its accuracyrelied on a time interval small enough However taking into account that time integration formulae can be considered to berecursive digital filters in this research a criterion based on transfer functions has been employed to characterize a wide rangeof integration algorithms from a frequency approach both in amplitude and in phase By adopting Nyquistrsquos criterion to avoid thealiasing phenomena a total of seven integration schemes have been reviewed in terms of accuracy and distortion effects on thefrequency content of the signal Some of these schemes are very well-known polynomial approximations with different degrees ofinterpolation but others have been especially defined for solving earthquake engineering problems or have been extracted fromthe digital signal processing methodology Finally five examples have been developed to validate this frequency approach and toinvestigate its influence on practical dynamic problems This research focused on earthquake and structural engineering revealsthat numerical integration formulae are clearly frequency-dependent a conclusion that obviously has a relevant interest in alldynamic engineering problems even when they are formulated and solved in the time-domain

1 Introduction

In earthquake engineering problems integration algorithmsare daily used especially to approach time-dependentproblems Some of the most employed among them arethe well-known closed Newton-Cotes formulae with differ-ent degrees of interpolation polynomial functions whichincludes the endpoints of the integration domain (119886 119887) asnodes of the problem [1] In recent years many otherformulae have been developed ldquoad hocrdquo for earthquake engi-neering problems or have been borrowed from other areasof knowledge in particular from digital signal processingUsually the choice of an integration algorithm depends on itssimplicity its computational cost and its accuracy The laterfactor is difficult to assess and very often it is entrusted toan integration interval small enough However knowing thecomputational error associated with a numerical integrationscheme is a crucial task in dynamic problems and has been

and it is nowadays a topic of growing interest in manyresearches [2ndash4]

Although the main advantages and disadvantages ofthe most common methods are well-known in this studythe performance of these schemes is analyzed attendingto its applicability to earthquake engineering and dynamicproblems by means of a frequency-domain assessment inspite of the fact that time is the usual integration variableThis research extends a frequency approach typical of the fieldof digital signal processing to various numerical algorithmswhich can be considered to be recursive digital filters [5ndash9]This methodology allows us to analyze not only the accuracyof the algorithms but also the distortion introduced by themin the frequency content of the signal being integrated orhow the frequency content of the signal affects the accuracyof each algorithm This new way of approaching the topicof the accuracy of the algorithms using the frequencydomain can be of great usefulness in earthquake engineering

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 259530 18 pageshttpdxdoiorg1011552015259530

2 Mathematical Problems in Engineering

problems Several researchers have also employed similarmethodologies to examine the performance of other time-dependent numerical algorithms within the area of Soil andstructural dynamics problems [10ndash13]

In this paper seven integration formulae have beenanalyzed by comparing the corresponding numerical transferfunctions to the analytical one since all of themare frequency-dependent By means of this comparison a rational criterionis proposed to predict in advance the performance of a givennumerical scheme depending on time interval (ℎ) and thefrequency content of the input signal (120596) Two examples areshown to check the performance of the different methodsand to verify the proposed criterion An additional exampleis included to highlight these findings applied to a practicalearthquake engineering problem

2 Brief Revision of Integration Formulae

In general any numerical integration formula tries to approx-imate the function to be integrated 119883(119905) (1) by an interpola-tion polynomial function of degree 119899 through known valuesof it 119883

119894= 119883(119905

119894) which are defined at the integration points

119905119894placed at equal time interval ℎ = 119905

119899+1 minus 119905119899

119884 (119905) = int119887

119886

119883 (119905) 119889119905 (1)

In this research seven formulae are includedThefirst fourare used in plenty of disciplines and are defined as the closedNewton-Cotes formulae They have different degree (119899) ofinterpolation polynomial function [1] and they are as followstrapezoidal rule (119899 = 1) Simpsonrsquos rule (119899 = 2) Simpsonthree-eights rule (119899 = 3) and Newton-Cotes scheme with119899 = 4 which is labeled as NC 119899 = 4 along this documentThe fifth algorithm is the rectangle rule which is widely useddue to its simplicityThe sixth formula is called Tickrsquos rule [5]which is well-known in the field of digital signal processingThis formula was designed to have a transfer function asclose to unity as possible throughout the lower half of theNyquist interval involving only three consecutive points [5]Transfer functions and Nyquistrsquos criterion are also used inthis research and they are explained in Section 3 Finally theseventh method is the digital integration filter defined bySchuessler and Ibler [14] The mathematical definitions of allthe aforementioned algorithms are listed as follows

(i) Trapezoidal rule

119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (2)

(ii) Simpsonrsquos rule

119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (3)

(iii) Simpson three-eights rule

119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (4)

(iv) Newton-Cotes 119899 = 4

119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(5)

(v) Rectangle rule

119884119899+1 = 119884

119899+ ℎ119883119899 (6)

(vi) Tickrsquos rule

119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (7)

where 1198870 = 1198872 = 03584 and 1198871 = 12832 It must behighlighted that if 1198870 = 1198872 = 13 and 1198871 = 43 Tickrsquosrule (7) becomes identical to Simpsonrsquos rule (3)

(vii) Schuessler-Ibler rule

119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (8)

where 1198870 = 1198877 = minus001330 and 1198871 = 1198876 = 007760 and1198872 = 1198875 = minus028279 and 1198873 = 1198874 = 171822

3 Methodology

Normally integration algorithms are defined as a linearcombination of the data or sampled points of the functionto be integrated When these are data equally spaced theintegration algorithms can be regarded as digital filters [5] Infact integration formulae have been a typical application offiltering techniques in digital signal processing methodologybut only someof themost classical rules have been consideredin practice [5ndash7]

The performance of any numerical integration algorithmcan be analyzed in the frequency domain by means of itstransfer function (TF) The transfer function of a system canbe defined as a mathematical operator which connects theinput signal to the output response In this particular casethe integration algorithm is the system itself To obtain thetransfer function it is assumed a complex harmonic functionis input signal 119890119894120596119905 where 120596 is the angular frequency of thesignal and 119905 is the time Since all of the above integration algo-rithms (2) to (8) are linear and time-invariant systems theoutput signal (after integration) can be defined as119867(119894120596)119890

(119894120596119905)where 119867(119894120596) is the complex transfer function (TF) of thecorresponding integration scheme which depends on 120596 and119905 [5] Also since 119867(119894120596) is a complex fuction it is possibleto derive the corresponding expressions for amplitude andphase for the function

In the following sections it is shown how the exact andnumerical transfer functions formulae can be obtained Sub-sequently comparing the corresponding numerical transferfunction to the exact one it is possible to develop a criterionto analyze the performance of a particular integration schemeusing a frequency approach

Mathematical Problems in Engineering 3

31 Exact Transfer Function For the input signal119883(119905) = 119890119894120596119905the exact integration is shown in the first part of (9) Applyingto (9) the definition of transfer function given above thetransfer function for the exact integrator (named as 119867lowast1 (119894120596)along this research) can be obtained (10) The symbol lowast isreferring to the exact integrator and the subindex 1 is referringto the single integral The exact transfer function can beexpressed in different forms according to the properties ofcomplex numbers

int119883 (119905) 119889119905 = int 119890(119894120596119905)

119889119905 =119890(119894120596119905)

119894120596= 119867lowast

1 (119894120596) 119890(119894120596119905)

997888rarr (9)

119867lowast

1 (119894120596) =1119894120596

=minus1120596119894 =

1120596119890minus1198941205872

(10)

Now if (9) is integrated again it is possible to obtain theexact transfer function for the double integral 119867lowast2 (119894120596) asdeveloped in (11)

∬119883(119905) 119889119905 = int119867lowast

1 (119894120596) 119890(119894120596119905)

119889119905 = 119867lowast

1 (119894120596) int 119890(119894120596119905)

119889119905

= [119867lowast

1 (119894120596)]2119890(119894120596119905)

= 119867lowast

2 (119894120596) 119890(119894120596119905)

997888rarr

119867lowast

2 (119894120596) = [119867lowast

1 (119894120596)]2

(11)

32 Numerical Transfer Functions Similarly to the exact caseit is possible to compute the transfer functions correspondingto each numerical integration method when the input signalis 119890119894120596119905 [5 8ndash10] For the sake of conciseness hereafter thetransfer functions of the seventh numerical algorithms pre-viously mentioned are listed while the complete derivationof every formula can be found in the Appendix


1198671 (119894120596) =ℎ

2(1 + 119890119894120596ℎ)(119890119894120596ℎ minus 1)

(12)


1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1) (13)


1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1) (14)


1198671 (119894120596)

=2ℎ45

(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)(1198901198941205964ℎ minus 1)

(15)

(v) Rectangle rule

1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1) (16)


1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1) (17)

where 1198870 = 1198872 = 03584 and 1198871 = 12832


1198671 (119894120596) =(ℎ3) (1198870119890

119894120596ℎ + 1198871 + 1198872119890minus119894120596ℎ + 1198873119890

minus2119894120596ℎ + 1198874119890minus3119894120596ℎ + 1198875119890

minus4119894120596ℎ + 1198876119890minus5119894120596ℎ + 1198877119890

minus6119894120596ℎ)

(119890119894120596ℎ minus 1) (18)


The procedure followed in the exact case to obtain thetransfer function for the double integral (11) is also analogousfor the numerical case1198672(119894120596) (19)

1198672 (119894120596) = [1198671 (119894120596)]2 (19)

4 Frequency Performance ofIntegration Algorithms

Since the exact and numerical transfer functions have beenrespectively calculated in (10) and in (12) to (18) it becomespossible to assess the performance of each algorithm relativeto the exact solution by comparing the corresponding numer-ical transfer function with the exact one In order to simplifythis task the ratio119877 has been defined as the quotient betweenthe module of the numerical and exact transfer functionsThis ratio can be obtained both for the single integral 1198771 asshown in (20) and for the double integral1198772 as shown in (21)


Table 1 Maximum input frequencies for different values of ℎ inorder to avoid aliasing phenomena according to Nyquistrsquos criterion

ℎ (sec) 119891119873(Hz) 120596 (radsec)

002 25 50120587001 50 1001205870005 100 200120587

In both cases the closer the ratio to unity the more accuratethe numerical solution

1198771 =

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816 (20)

1198772 =

10038161003816100381610038161198672 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

2 (119894120596)1003816100381610038161003816=[10038161003816100381610038161198671 (119894120596)

1003816100381610038161003816]2

[1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]2 = [

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]

2

= 11987721

(21)

To define the valid range of frequencies avoiding thealiasing phenomena Nyquistrsquos criterion has been taken intoaccount [5] This criterion defines the maximum frequencyvalue which can be achieved by an input signal depending onthe time interval (ℎ) used to sample the signal Owing to thatthe maximum frequency on the input signal (119891input

max or 120596inputmax )

must be less than Nyquistrsquos frequency (119891119873or 120596119873) (22) which

implies that (23) must be guaranteed

119891119873=

12ℎ

997888rarr 120596119873=21205872ℎ

=120587

ℎ (22)

119891inputmax lt 119891

119873997888rarr 120596

inputmax lt 120596

119873=120587

ℎ (23)

In this research the most common values for timeintervals employed for strong ground motion discretization(ℎ = 002 001 and 0005 sec) have been adopted Thecorresponding maximum values of the input frequencieswhich can be reached to avoid aliasing phenomena are shownin Table 1 As it can be observed the lower the value of ℎthe higher the allowed input frequency For simplicity thedimensionless parameter 120582 defined as the quotient betweenthe input frequency (120596) and Nyquistrsquos frequency (120596

119873) is

adopted as shown in (24)

120582 =120596

120596119873

=120596ℎ

120587 (24)

Figure 1 shows the variations with respect to 120582 of theratios 1198771 (20) and 1198772 (21) for the single and double integralrespectively and the phase diagram (120601) for the seven integra-tion algorithms selected previously As previouslymentionedthe ideal numerical algorithm should maintain the ratios 1198771and 1198772 as close to unity as possible for a large range of 120582 andalso the phase value should coincide with the exact one

Since the transfer function is defined by a complex expres-sion it is convenient to analyze not only its amplitude butalso its phase diagram From Figure 1 relevant conclusionsare extracted which ratify the earlier conclusions reportedby other researchers [5ndash9] The main conclusions which are

later on addressed by the examples included in Section 5 arelisted next

(1) Although the parameter 120582 guarantees that the aliasingphenomena are avoided it can be observed that ingeneral all algorithms show values of 1198771 and 1198772 faraway from unity for the upper-half of 120582 This effect iseven more critical for the 1198772 case

(2) Trapezoidal and Schuessler-Ibler rules tend to deam-plify the output signal (after integration) while therectangle Tick and Simpson algorithms tend toamplify it although in different quantity according to120582 On the other hand the Simpson 3-8 and Newton-Cotes 119899 = 4 rules display a singular behaviourdepending on 120582 they can amplify or deamplify theoutput signal

(3) For the single integral case (1198771) Tickrsquos rule slightlydeamplifies the output signal for intermediate valuesof 120582 while it tends to amplify from 120582 gt 05 On theother hand Simpsonrsquos rule consistently amplifies theoutput signal especially for 120582 above 03The rectanglerule amplifies for values of 120582 higher than 01 in otherwords for almost thewhole range of120582 By contrast thetrapezoidal rule progressively attenuates the outputsignal for almost the entire range of frequencieswhereas Schuessler-Ibler strongly deamplifies beyond120582 gt 06 For Simpson 3-8 rule 1198771 is close to unityuntil 120582 asymp 03 tending to amplify the signal forhigher values of 120582 (with a discontinuity around 120582 =

066) until reaching values around 072 deamplifyingsignificantly after that Finally the Newton-Cotes 119899 =4 formula is close to unity up to values of120582 around 04deamplifying quickly for 120582 between 04 and 05 For120582 = 05 the transfer function of this scheme presentsalso a discontinuity tending to amplify the outputsignal in a nonuniform manner for higher values of120582

(4) For the double integral case (1198772) all methods showthe same performance (amplification or deamplifica-tion) compared to the single integration (1198771) but inmost of the formulae the ratio 1198772 is further fromunity for lower values of120582 than1198771 except for Tick andSchuessler-Iblerrsquos rules In general this implies thatgiven a signal and a value of ℎ the error introducedinto the solution by the numerical procedures growsfor upper integration orders although in differentmagnitude for each method

(5) Those algorithms tending to amplify for the upper-half of 120582 may introduce undesirable errors specifi-cally high-frequency terms into the integrated signalObviously these algorithms should be avoided ifthe input signal is contaminated by spurious high-frequency noise

(6) Regarding phase values the exact transfer functionhas a phase value equal to minus1205872 for the wholerange of 120582 Only the trapezoidal Simpson and Tickrules show the same phase behaviour while the rest


0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

TrapezoidalSimpsonSimpson 3-8

RectangleTickSchuessler-Ibler

NC n = 4

120582 = 120596120596N

R1=|H

1(i120596)||H

lowast 1(i120596)|

(a) 1198771 single integration

0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

TrapezoidalSimpson

RectangleTickSchuessler-IblerSimpson 3-8

NC n = 4

120582 = 120596120596N

R2=|H

2(i120596)||H

lowast 2(i120596)|

(b) 1198772 double integration

0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05

minus1

minus15

minus2

Φ(r

ad)

120582 = 120596120596N

Trapezoidal (120601 = minus1205872)Simpson (120601 = minus1205872)Simpson 3-8NC n = 4

RectangleTick (120601 = minus1205872)Schuessler-IblerExact (120601 = minus1205872)

(c) Phase diagram

Figure 1 Ratios between numerical and exact amplitude transfer functions (1198771-first integration 119877

2-second integration) and phase diagrams

for tested algorithms

of numerical algorithms have a phase performanceabsolutely different The Simpson 3-8 and Newton-Cotes 119899 = 4 alternatively display phase values equal tominus1205872 or equal to 1205872 However the phase of rectanglerule linearly varies from Φ = 1205872 for 120582 = 0 toΦ = 0 for 120582 = 1 The phase of Schuessler-Iblerrsquosrule periodically varies along the whole range of 120582and in a linear way between Φ = 1205872 and Φ = minus1205872in the intervals of 120582 between 0ndash13 13ndash23 and 23ndash1 Thus both rules introduce a frequency-dependentphase error in the integration scheme for any value of120582

5 Numerical Examples

Five common examples from earthquake engineering havebeenworked out to illustrate the applicability of the frequency

criterion derived in this research as well as the validity ofthe above listed conclusions concerning the performance ofthe seven selected integration algorithms In the first threeexamples the analytical solutions are known in advance andtherefore the accuracy of the numerical results can be clearlyassessed In the fourth one the permanent displacement ofa slope due to an earthquake is computed by the Newmarksliding block method and in the last one the error in thedisplacement record of a strong ground motion is obtainedso that the frequency-effects of the integration algorithms canbe analyzed in practical engineering problems

51 Example 1 Variable Frequency Input In order to assessthe accuracy of the integration schemes using the proposedfrequency approach a sine function with variable frequencyis considered input signal (acceleration) to be integratedTheacceleration is defined in (25) where 119860

119888is the amplitude


0 01 02 03 04 05 06 07 08 09 1minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100



120582 = 120596120596N

NC n = 4

Relat

ive e

rror

()|

max|

(a) Relative error () for |Vmax|

minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Relat

ive e

rror

()|d

max|

0 01 02 03 04 05 06 07 08 09 1

120582 = 120596120596N



NC n = 4

(b) Relative error () for |119889max|

Figure 2 Relative error (in percentage) reached by each algorithm along the whole range of 120582 for a sinusoidal signal (both velocity anddisplacement records are shown)

equal to 3ms2 120596 is the frequency of the input signal and119905 denotes time whereas zero initial conditions have beenassumed The acceleration has a duration of 15 s and thetime interval (ℎ) is equal to 0005 s meaning that the inputfrequency ranges from2120587 rads to 200120587 rads reaching a valueof 120582 between 001 and 1 respectively The input signal isintegrated twice to calculate the velocity and displacementrecords and after that the maximum velocity (Vmax) and themaximum displacement (119889max) are calculated and comparedwith the analytical solutions which can be obtained from (26)and (27) Consider

119886 (119905) = 119860119888sin (120596119905) (25)

V (119905) = int119905

0119860119888sin (120596119905) 119889119905 =

119860119888

120596(1minus cos (120596119905)) (26)

119889 (119905) = int119905

0

119860119888

120596(1minus cos (120596119905)) 119889119905 =

119860119888

120596(119905 minus

sin (120596119905)120596

) (27)

Figure 2 shows the relative error (in percentage) forboth |Vmax| and |119889max| for the whole range of values of120582 A positive relative error means that the correspondingnumerical algorithm amplifies the numerical solution andvice versa From Figure 2 it is concluded that the ratios 1198771and 1198772 plotted in Figure 1 properly predict the performanceof each numerical integration algorithm since the errorcurves shown in Figure 2 follow exactly the trend plottedin Figure 1 It means that the higher the value of 120582 thehigher the error except for the rectangle and Schuessler-Iblerformulae In the case of the rectangle rule 1198771 and 1198772 predictan amplification in the numerical response whereas the errorplotted in Figure 2 shows a deamplification tendency whichis even closer to the trapezoidal rule for the displacementcase On the other hand this rule presents a positive linearvariation of the phase depending on 120582 while the exactphase has a negative constant value This contradictory

phase performance could explain the abnormal response ofthis algorithm As concerns the Schuessler-Ibler formula itdeamplifies the output record according to 1198771 and 1198772 but ina nonuniform way since the error curve shows an evidentperiodic behaviour according to its phase diagram It must behighlighted that the highest errors take place when the phasecurve is equal to zero and the lowest errors take place whenthe phase reaches any of the extreme values

Although this example has been computed with aconstant value of ℎ it can be observed that numerical integra-tion accuracy is frequency-dependent amatter of paramountinterest in earthquake engineering problems Moreover ifthis example is computed for other values of time interval ℎwith120582 ranging between 0 and 1 these algorithms show similartrends

52 Example 2 Influence of Noise Content of a Signal Inearthquake engineering problems it is necessary to employdifferent types of signal as accelerograms or recordsmeasuredfrom the response of a structure Any kind of record couldbe contaminated by different noise sources (internal charac-teristics of the measurement apparatus the placement wherethe instrument is located ambient noise etc) [15 16] In thisexample the sensitivity of the different integration algorithmsto the noise content in a signal is analyzed A sine signalhas been modified by three types of noise and then has beenintegrated twice to compute the corresponding displacementrecord Thanks to the simplicity of a sine function and sinceits analytical solution is known in advance ((26) and (27)) itis possible to verify the influence of the noise components ofa signal in the final results for each integration algorithm

The original sine function (labeled as original signal isthis research) is defined in (25) where 119860

119888= 3ms2 and

120596 = 10120587 rads with a duration of 15 s and ℎ = 0005 s (120582 =

005) Three different types of noise have been chosen high-frequency low-frequency and phase-noise In the three cases


0 05 1 15

High-frequency noise420

minus2minus4a

(ms2)

t (s)

Low-frequency noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise

Figure 3 Original function and contaminated signals by three typesof noise high-frequency noise low-frequency noise and phase-noise

the contaminated signal is defined as a sine function toowhich has been added to the original oneThe characteristicsof the high-frequency noise are 119860high = 03ms2 and 120596high =150120587 rads while for the low-frequency noise are 119860 low =

06ms2 and 120596low = 01120587 rads For the phase-noise case aconstant variation in the phase equal to 015120587 rads has beenadded to the frequency of the original sine functionThe foursignals considered in this example are plotted in Figure 3as can be observed the noise-signals are very close to theoriginal one

The three contaminated signals from Figure 3 have beenintegrated twice to obtain the velocity and displacementrecords Figure 4 In all cases the analytical result from theoriginal sine function has been included for comparativepurposes Moreover in all cases the solutions computedby the Schuessler-Ibler rule show an anomalous behaviourrelative to the rest of methods both in velocity and indisplacement In the cases of high-frequency noise andphase-noise the computed results are very close to the analyticalsolution even when this has been computed for the originalsignal without any kind of noise However in the case oflow-frequency noise there can be observed a drift tendencywith respect to the analytical solution both in velocity and indisplacement records for all numerical integration schemesThis behaviour has been already reported by many authors[2 15 17] when a strong ground motion is integrated twicefor obtaining the ground displacement The numerical phe-nomenon observed in this example is in agreement with therecommendation postulated by Wang et al [15] of applyinglow-cut filtering to a record before integrating it to reduce theeffects of baseline offsets

For analyzing in detail the sensitivity of each integrationalgorithm to the noise content of a particular signal inFigure 5 the difference between the result computed by eachalgorithm for every contaminated signal with respect to thesolution obtained by each algorithm for the original signalwithout noise has been plotted both in velocity and indisplacement From this figure the following observationscan be drawn

(i) All integration algorithms are sensitive to high-frequency noise content NC 119899 = 4 Simpson Tickand Trapezoidal rules perceive the phase of the noisealthough with different magnitud between themwhereas Schuessler-Ibler Rectangule and especialySimpson 3-8 detect the noise with a different value ofphase All methods display a drift in the displacementresult although the most sensitive ones are NC 119899 = 4which also shows the highest differences in amplitudein the velocity records and Simpson 3-8 which alsoshows a phase absolutely different to the rest ofschemes in the corresponding velocity records

(ii) With respect to the case of low-frequency noiseintegration algorithms do not exhibit a particular sen-sitivity to this type of noise neither in velocity nor indisplacement since all of them are overlapping withthe exact solution (Figure 5) Asmentioned above forthis type of noise an important drift is observed in thecomputed result which could probably be induced bythe presence of the low-frequency noise but does notdepend on the integration schemes employed

(iii) Finally as concerns the case of phase-frequency noisemost of the integration algorithms show a very similartrend among them except Schuessler-Ibler rule Allof them detect the phase-noise included in the signalalthough with a little variation in the amplitudecompared with the exact result Again in most of theintegration schemes a slight drift is also induced inthe displacement record lower for the rectangle ruleand exaggerated for Schuessler-Ibler method

53 Example 3 Bogdanoff rsquos Accelerogram In this examplethe velocity records computed from Bogdanoff rsquos artificialaccelerogram [18] are analyzed through Fourier amplitudespectra This accelerogram is defined as the amplitude-modulated sum of 119899 harmonic terms with uniformly dis-tributed random phase angles which allows us tomanage thefrequency content of the accelerogramMoreover it becomespossible to know the analytical solution of the integrationand therefore to calibrate the accuracy of each numericalintegration algorithm

Bogdanoff rsquos accelerogram is defined by (28) where 119861 and120572 are two constants equal to 0292 and minus0333 respectively119899 is the number of terms of the summation 120596

119895and 120601

119895are

the frequency and phase angles of each harmonic term 119905 is


Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04

High-frequency noise

t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)

Exact original signalTrapezoidalSimpson


NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Low-frequency noise

d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)

Figure 4 Velocity (a) and displacement (b) records obtained for the three types of contaminated signal computed by different integrationschemes (several numerical solutions are overlapping)

the time in seconds and 119886 is the acceleration expressed inmsec2 Consider

119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)

For this example the input record has a duration of20 s and it is sampled with a time interval equal to 001 s

The accelerogram is composed by 22 terms with frequenciesranging from 6 rads (095Hz) to 8025 rads (1277Hz) andhence the parameter 120582 varies between 00191 and 02554The resulting accelerogram is plotted in Figure 6(a) and itsfrequency content can be visualized by means of the Fourieramplitude diagram shown in Figure 6(b)

The analytical and numerical integration velocity recordsof this accelerogram and also the differences between both


0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise

ExactTrapezoidalSimpson


t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)

Figure 5 Differences between computed results for signals with and without noise obtained from each algorithm Both velocity (a) anddisplacement (b) results for the analytical and numerical integration are shown (several numerical solutions are overlapping)

are plotted in Figure 7 for each numerical scheme As canbe observed all numerical solutions are very close to theanalytical one especially those obtained using the Simpsonand Tick rules while Simpson 3-8 and NC 119899 = 4 truncatethe peaks of the signal and the Schuessler-Ibler record isslightly shifted with respect to the analytical solution Themaximum positive and negative velocity computed by eachalgorithm are shown in Table 2 and they are compared with

their analytical values to calculate the relative error of eachmethod (expressed in percentage) A positive error meansthat the corresponding numerical algorithm amplifies thenumerical solution and vice versa For this example whileSimpson and Tick show the lowest error other algorithmslike NC 119899 = 4 have lower error for Vminusmax case than for theV+max case By contrast the trapezoidal rule always deamplifiesthe output whereas the rectangle and Schuessler-Ibler rules


0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)

Figure 6 Artificially generated Bogdanoff rsquos accelerogram and corresponding Fourier amplitude spectrum

Table 2 Maximum positive and negative velocities computed analytically and numerically for Bogdanoff rsquos accelerogram and correspondingrelative errors expressed in percentage

V+max (ms) Relative error () Vminusmax (ms) Relative error ()Analytical 01501 mdash minus01811 mdashTrapezoidal 01493 minus05543 minus01806 minus02823

Simpson 01502 00141 minus01811 00058

Simpson 3-8 01456 minus30392 minus01782 minus16175

Newton-Cotes 119899 = 4 01455 minus30666 minus01811 minus00016

Rectangle 01513 08061 minus01809 minus01429

Tick 015 minus00714 minus01811 minus00375

Schuessler-Ibler 01511 06599 minus01801 minus05667

amplify in one case and deamplify in the other Finally theSimpson 3-8 gives the highest error values for this example

To further understand the frequency performance of eachalgorithm after integration of Bogdanoff rsquos accelerogram thecorresponding Fourier amplitude spectra for the numericalvelocity records have been computed and they have beensubtracted from the analytical Fourier amplitude spectrumThe resulting deviation Fourier amplitude spectra are plottedin Figure 8 where Fourier amplitude spectra have beenpreviously divided by the number of intervals of each recordAccording to this figure the trapezoidal rule shows a clearlyincreasing deamplification for upper values of frequencythat is for higher 120582 By contrast rectangle formula has theinverse tendency although the magnitude of the deviationis smaller than in the trapezoidal case The strong influenceof rectangle-phase in its own numerical solution must betaken into account On the other hand Simpson and Tickrsquosformulae are practically close to zero throughout the wholerange of frequency Newton-Cotes 119899 = 4 and Schuesller-Ibler schemes do not display a clear trend since both amplifyand deamplify the Fourier amplitude spectra As for theSimpson 3-8 rule it has an evident periodic deviation of highmagnitude and as indicated above this formula gives thehighest relative error for this example Notice that these threealgorithms have also periodic phase diagrams

54 Example 4 Newmark Sliding Block Analysis The influ-ence of each numerical algorithm is directly analyzed over a

practical earthquake engineering problem precisely the well-known Newmark sliding block method [17] This methodgives an estimation for the permanent displacement of aslope under a seismic ground motion When the groundacceleration exceeds a yield acceleration threshold the blockstarts to move and slides until the velocity (first integrationof the acceleration) becomes zero To compute the relativedisplacement it is necessary to integrate the correspondingrelative velocity record (second integration of the inputacceleration) The permanent slope displacement dependsthen on the relationship between the yield acceleration (119886

119910)

and the maximum acceleration of the ground motion (119886max)if 119886119910is greater than 119886max no displacement will occur whereas

if 119886119910decreases the slope displacement increasesIn this example two strong groundmotionswith different

frequency contents have been employed namely the E-Wcomponents of theGilroy number 1 recorded at Bedrock andthe Gilroy number 2 recorded at Soil The acceleration time-histories and the corresponding Fourier amplitude spectraare plotted in Figure 9 It can be observed that the Gilroynumber 1 (Bedrock) has a high-frequency content higherthan the Gilroy number 2 (Soil) the mean periods (119879

119898)

being 0387 s and 0735 s respectively [19] These recordshave been sampled with a time interval equal to 0005 s andthe maximum accelerations are 04731 g in Gilroy number 1(Bedrock) and 03223 g in Gilroy number 2 (Soil) Moreoverthree values of yield acceleration have been employed for eachaccelerogram specifically 14 12 and 34 of 119886maxThe higher


0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)

Newton-Cotes n = 4 rule

0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)


Analytical solutionNumerical solution

Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20

Schuessler-Ibler rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2

Figure 7 Analytical and numerical solutions for the first integration of Bogdanoff rsquos accelerogram computed by each scheme Comparisonbetween both types of velocity records (a) and numerical integration errors ((b) for plotting purpose different scales have been used)

119886119910 the higher influence of the proper integration of the peaks

of the record in the computed permanent displacementThe permanent slope displacements for each accelero-

gram and for each value of 119886119910computed by different algo-

rithms are listed in Table 3 For comparison purposes thoughno exact solution is available the numerical deviation ofeach integration formula has been calculated with respect tothe trapezoidal scheme since this is the usual algorithm inthe normal practice These results (in percentage) are alsoincluded in Table 3 From these results it can be concluded

that the deviations between schemes are significantly higherin the case of Gilroy number 1 (Bedrock) which has aricher high-frequency content compared to Gilroy number 2(Soil) Moreover normally deviations are higher for the case119886119910= 34119886max where the peaks of the records have a greater

influence than for the other yield accelerations valuesWith regard to integration formulae it can be concluded

that normally trapezoidal rule gives lower permanent dis-placements while Simpson and Tick are always very closeones to each other and give values greater than trapezoidal


Table 3 Permanent slope displacements computed by Newmark sliding method for the strong ground motion Gilroy number 1 (Bedrock)and Gilroy number 2 (Soil) applying different integration formulae and several values of 119886

119910 The corresponding deviations (in percentage)

with respect to the trapezoidal rule are also listed for comparison purposes

119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max

119889 (cm) Deviation () 119889 (cm) Deviation () 119889 (cm) Deviation ()Gilroy number 1 (Bedrock)

Trapezoidal 115954 mdash 37081 mdash 0663 mdashSimpson 11597 00143 36869 minus05725 06861 34829Simpson 3-8 116022 00585 37256 04719 0622 minus6185

Newton-Cotes 119899 = 4 113222 minus23556 36338 minus20041 0466 minus297149

Rectangle 116881 07998 37432 09445 06769 21014Tick 115924 minus00256 36832 minus06722 06837 31203Schuessler-Ibler 117432 12745 37292 05693 05402 minus185223

Gilroy number 2 (Soil)Trapezoidal 170186 mdash 28038 mdash 03117 mdashSimpson 170514 01932 28074 0126 0312 00936Simpson 3-8 170043 minus0084 27929 minus03912 0301 minus34428

Newton-Cotes 119899 = 4 173194 1768 27726 minus11153 02968 minus47808

Rectangle 170362 01038 2812 02921 03142 08138Tick 170496 01826 28063 0089 03119 00596Schuessler-Ibler 174455 25086 28077 01364 03119 00585

1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5

TrapezoidalSimpsonSimpson 3-8Newton-Cotes n = 4


Figure 8 Deviations of numerical from analytical Fourier ampli-tude spectra for first integration records of Bogdanoff rsquos accelero-gram (Simpson 3-8 curve has been cut off for plotting purposes)

formula On the other hand Newton-Cotes 119899 = 4 yields thehighest deviation results in any case compared to the rest ofalgorithms followed by the Schuessler-Ibler scheme

In spite of the fact that the time interval is constantfor all cases and equal to the time discretization of theaccelerograms the influence of the frequency content of theinput signal which is related to ℎ by means of 120582 is veryrelevant in the accuracy of the integration results as well asthe proper selection of the integration algorithm

55 Example 5 Error inDisplacement Record from Integrationof an Accelerogram In this example the results obtainedby each integration algorithm are compared for computingthe velocity and displacement histories after integration ofa particular accelerogram [17] The Gilroy strong groundmotions employed in Example 4 have been also used forthis goal These accelerograms with different frequencycontent have been well characterized both in time and infrequency domain in Figure 9 In both cases the duration ofthe accelerogram is 40 seconds and the sampling frequencyis ℎ = 0005 sec

Velocity records are obtained after one integration whereno differences between schemes have been observed Onthe contrary the most relevant differences appear for thedisplacement histories Figure 10 A drift can be observedin displacement when an accelerogram is integrated twiceand the reasons why this phenomenon occurs are currentlyunder research [2 15 17] However the focus of this exampleis not to clarify the reasons for this phenomenon but toanalyze the influence of each integration algorithm in thecomputed result The errors in displacement (drift) for thefinal time 119905 = 40 sec obtained for each algorithm and forboth accelerograms are summarized in Table 4 In generalbetter results (closer to zero) are obtained for the case ofGilroy number 2 (Soil) than for the case of Gilroy number 1(Bedrock) which has a high-frequency content that is higherFor these particular accelerograms minimal differences canbe observed between algorithms except for the Schuessler-Ibler rule although the results obtained by TrapezoidalSimpson and Tick rules are better than for the rest ofmethods As remarked above this is an issue in continuousdevelopment Therefore a deeper research is neccesary andthe above conclusions should be taken with caution


0 10 20 30 40Time (s)

Gilroy number 1 (Bedrock)

a(g

)

05040302010

minus01minus02minus03minus04minus05

ay = 34amaxay = 12amaxay = 14amax

0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)

Gilroy number 2 (Soil)

a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

Figure 9 Acceleration records and corresponding Fourier amplitude spectra for the E-W components of the strong ground motions Gilroynumber 1 (Bedrock) and Gilroy number 2 (Soil) The three values of yield acceleration 119886

119910are indicated in the plots

0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson


0 5 10 15 20 25 30 35 40minus006minus004minus002

0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)

Newton-Cotes n = 4 Newton-Cotes n = 4

Figure 10 Displacement records obtained after integrating twice the strong ground motions Gilroy number 1 (Bedrock) and Gilroy number2 (Soil) using different integration algorithms (several numerical solutions are overlapping)

6 Conclusions

In this research the frequency performance of seven numeri-cal integration algorithms which behave as recursive digitalfilters has been explored The methodology employed isbased on comparing for each algorithm the analytical andnumerical transfer functions For the sake of simplicitythe ratios 1198771 for single integration and 1198772 for doubleintegration have been defined for 120582 ranging between 0 and

1 where the parameter 120582 links the time interval ℎ with theinput frequency120596 through Nyquistrsquos criterion In general theerror of any algorithm increases for higher values of 120582 but itmust be highlighted that 120582 depends on both the time intervaland the frequency content of the input signal

In this frequency-domain approach both amplitude andphase components have been taken into account as opposedto the normal practice where phase is scarcely investigatedThis methodology has been validated by means of five


Table 4 Displacement in cm for time 119905 = 40 sec computed afterdouble integration of the strong ground motions Gilroy number 1(Bedrock) and Gilroy number 2 (Soil) using different integrationschemes

Gilroy number 1(Bedrock)

Gilroy number 2(Soil)

Trapezoidal 000167 minus000035

Simpson 00168 000570Simpson 3-8 minus0028 006352Newton-Cotes 119899 = 4 minus0408 minus000049

Rectangle minus0219 minus010834

Tick 00146 000479Schuessler-Ibler 1485 117585

examples from the field of earthquake engineering From thisresearch the following general conclusions can be drawn

(i) The numerical integration methods investigated inthis research are clearly frequency-dependent sinceno one of them maintains the ratio 1198771 close to onefor the whole range of 120582 displaying in all cases anabnormal behaviour for120582higher than 12 whereas theaccuracy of each scheme improves for lower values of120582

(ii) In general the trapezoidal rule tends to deamplify thesolution along the whole range of 120582 On the contrarySimpsonrsquos formula tends to amplify the results whileTickrsquos formula slightly deamplifies for intermediatevalues of 120582 and tends to amplify for 120582 gt 12 Henceboth algorithms should be used with caution if theinput signal has a relevant high-frequency contentespecially if it is due to noise and they should beavoided if high-frequencies are wanted to be dampedout Normally the results computed by Simpsonrsquosformula are very close to Tickrsquos algorithm althoughthe latter is more accurate than the former No one ofthese algorithms exhibits phase problems

(iii) Algorithms like Simpson 3-8 and Newton-Cotes 119899 =

4 in spite of being defined with a higher degreeof interpolation function give worse results thanthe previous ones and show an evident undesirableperiodic phase dependency

(iv) The rest of algorithms rectangle and Schuessler-Iblernot always follow the corresponding trends predictedby 1198771 and 1198772 and sometimes give acceptable resultswhile others do not Probably these unexpectedbehaviours are due to their particular phase diagramsIt must be highlighted that the accuracy of Schuesller-Iblerrsquos formula oscillates according to its own phasebehaviour

Unfortunately there is no general algorithm which canbe used successfully for all cases at least from the list ofschemes analyzed in this research From the cases analyzedthe Trapezoidal Simpson and Tick rules do not exhibitanomalous behaviours and generally give acceptable results

whereas the Schuessler-Ibler rule shows strange tendencies insome of the examples proposed so it is not recommendedto use it The clues to choose a particular integration for-mula depend on the frequency content of the input signalthe computational cost due to the size of the interval ofintegration the required accuracy and mainly the furtheruse of the results Careful attention should be paid to theproper selection of the integration algorithm not only fromthe accuracy point of view but from frequency-dependentbehaviour of the algorithm

Appendix

If it is assumed that the function to be integrated is 119883(119905) =119890119894120596119905 where 120596 is the angular frequency of the input signal and119905 is the time the integral119884(119905) = int

119887

119886119883(119905)119889119905 can be numerically

calculated following the definition of each quadrature ruleConsequently the corresponding transfer function of eachnumerical integration algorithm can be obtained as follows

(i) Trapezoidal Rule The trapezoidal formula is defined as

119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)

According to the transfer function definition every termof (A1) can be expressed as

119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)

Substituting the above equations into (A1) it is obtainedthat

1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)

(ii) Simpsonrsquos Rule Simpsonrsquos formula is defined as

119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)

(iii) SimpsonThree-Eights Points The Simpson 3-8 formula isdefined as

119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)

(iv) Newton-Cotes 119899 = 4The Newton-Cotes 119899 = 4 formula isdefined as

119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)

(v) Rectangle Rule The rectangle method is defined as

119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ 997888rarr

1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)

(vi) Tickrsquos Rule Tickrsquos method is defined as

119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)

where the parameters 119887119894are 1198870 = 1198872 = 03584 and 1198871 = 12832


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)

(vii) Schuessler-Iblerrsquos Formula Schuessler-Iblerrsquos formula isdefined as

119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)

where 1198870 = 1198877 = minus001330 and 1198871 = 1198876 = 007760 and 1198872 =

1198875 = minus028279 and 1198873 = 1198874 = 171822According to the transfer function definition every term

of (A19) can be expressed as

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study has been partially funded by Spanish Ministryof Science and Innovation through Project BIA2007-67401and scholarship BES-2008-002770 The financial supportto complete this research is gratefully appreciated by theauthors Finally the authors would also like to thank theconstructive suggestions and comments provided by theanonymous reviewers during the reviewing process

References

[1] R Burden and J D Faires Numerical Analysis BrooksColeCengage Learning 2011

[2] S C Stiros ldquoErrors in velocities and displacements deducedfrom accelerographs an approach based on the theory of errorpropagationrdquo Soil Dynamics and Earthquake Engineering vol28 no 5 pp 415ndash420 2008

[3] J Yang J B Li and G Lin ldquoA simple approach to integrationof acceleration data for dynamic soil-structure interactionanalysisrdquo Soil Dynamics and Earthquake Engineering vol 26no 8 pp 725ndash734 2006

[4] K-T Park S-H Kim H-S Park and K-W Lee ldquoThe deter-mination of bridge displacement using measured accelerationrdquoEngineering Structures vol 27 no 3 pp 371ndash378 2005

[5] RWHammingDigital Filters PrenticeHall EnglewoodCliffsNJ USA 1977

[6] J H Karl An Introduction to Digital Signal Processing Elsevier2012

[7] R G Lyons Understanding Digital Signal Processing PrenticeHall 2010

[8] R Blazquez ldquoTime-domain response of simple elastic sys-tems to seismic motion error assessment and engineeringimplicationsrdquo in Proceedings of the 14th World Conference onEarthquake Engineering (K001 rsquo08) Keynote Lecture BeijingChina 2008

[9] J Arias-Trujillo and R Blazquez ldquoAccuracy of time integra-tion procedures for seismic response of linear oscillatorsrdquoin Proceedings of the 14th World Conference on EarthquakeEngineering Beijing China 2008

[10] A Preumont ldquoFrequency domain analysis of time integrationoperatorsrdquo Earthquake Engineering and Structural Dynamicsvol 10 no 5 pp 691ndash697 1982

[11] J Arias-Trujillo R Blazquez and S Lopez-Querol ldquoA method-ology based on a transfer function criterion to evaluate timeintegration algorithmsrdquo Soil Dynamics and Earthquake Engi-neering vol 37 pp 1ndash23 2012

[12] R Blazquez and J Arias-Trujillo ldquoNon-stationary frequencyresponse functionrdquo Bulletin of Earthquake Engineering vol 11no 6 pp 1895ndash1908 2013

[13] P Pegon ldquoAlternative characterization of time integrationschemesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 20-21 pp 2707ndash2727 2001

[14] HW Schuessler andW Ibler ldquoDigital filters for integrationrdquo inProceedings of the European Conference on Circuit Theory and

Design Conference Publication (116) Institution of ElectricalEngineers July 1974

[15] G Q Wang D M Boore H Igel and X Y Zhou ldquoSomeobservation on colocated and closely spaced strong ground-motion records of the 1999 Chi-Chi Taiwan earthquakerdquoBulletin of the Seismological Society of America vol 93 no 2pp 674ndash693 2003

[16] F Moschas and S Stiros ldquoPhase effect in time-stampedaccelerometer measurementsmdashan experimental approachrdquoInternational Journal of Metrology and Quality Engineering vol3 no 3 pp 161ndash167 2012

[17] S L Kramer Geotechnical Earthquake Engineering Prentice-Hall New York NY USA 1996

[18] J L Bogdanoff J E Goldberg and M C Bernard ldquoResponseof a simple structure to a random earthquake-type disturbancerdquoBulletin of the Seismological Society of America vol 51 no 2 pp293ndash310 1961

[19] E M Rathje N A Abrahamson and J D Bray ldquoSimplifiedfrequency content estimates of earthquake ground motionsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol124 no 2 pp 150ndash158 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


problems Several researchers have also employed similarmethodologies to examine the performance of other time-dependent numerical algorithms within the area of Soil andstructural dynamics problems [10ndash13]

In this paper seven integration formulae have beenanalyzed by comparing the corresponding numerical transferfunctions to the analytical one since all of themare frequency-dependent By means of this comparison a rational criterionis proposed to predict in advance the performance of a givennumerical scheme depending on time interval (ℎ) and thefrequency content of the input signal (120596) Two examples areshown to check the performance of the different methodsand to verify the proposed criterion An additional exampleis included to highlight these findings applied to a practicalearthquake engineering problem

2 Brief Revision of Integration Formulae

In general any numerical integration formula tries to approx-imate the function to be integrated 119883(119905) (1) by an interpola-tion polynomial function of degree 119899 through known valuesof it 119883

119894= 119883(119905

119894) which are defined at the integration points

119905119894placed at equal time interval ℎ = 119905

119899+1 minus 119905119899

119884 (119905) = int119887

119886

119883 (119905) 119889119905 (1)

In this research seven formulae are includedThefirst fourare used in plenty of disciplines and are defined as the closedNewton-Cotes formulae They have different degree (119899) ofinterpolation polynomial function [1] and they are as followstrapezoidal rule (119899 = 1) Simpsonrsquos rule (119899 = 2) Simpsonthree-eights rule (119899 = 3) and Newton-Cotes scheme with119899 = 4 which is labeled as NC 119899 = 4 along this documentThe fifth algorithm is the rectangle rule which is widely useddue to its simplicityThe sixth formula is called Tickrsquos rule [5]which is well-known in the field of digital signal processingThis formula was designed to have a transfer function asclose to unity as possible throughout the lower half of theNyquist interval involving only three consecutive points [5]Transfer functions and Nyquistrsquos criterion are also used inthis research and they are explained in Section 3 Finally theseventh method is the digital integration filter defined bySchuessler and Ibler [14] The mathematical definitions of allthe aforementioned algorithms are listed as follows


119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (2)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (3)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (4)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(5)

(v) Rectangle rule

119884119899+1 = 119884

119899+ ℎ119883119899 (6)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (7)

where 1198870 = 1198872 = 03584 and 1198871 = 12832 It must behighlighted that if 1198870 = 1198872 = 13 and 1198871 = 43 Tickrsquosrule (7) becomes identical to Simpsonrsquos rule (3)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (8)


3 Methodology

Normally integration algorithms are defined as a linearcombination of the data or sampled points of the functionto be integrated When these are data equally spaced theintegration algorithms can be regarded as digital filters [5] Infact integration formulae have been a typical application offiltering techniques in digital signal processing methodologybut only someof themost classical rules have been consideredin practice [5ndash7]

The performance of any numerical integration algorithmcan be analyzed in the frequency domain by means of itstransfer function (TF) The transfer function of a system canbe defined as a mathematical operator which connects theinput signal to the output response In this particular casethe integration algorithm is the system itself To obtain thetransfer function it is assumed a complex harmonic functionis input signal 119890119894120596119905 where 120596 is the angular frequency of thesignal and 119905 is the time Since all of the above integration algo-rithms (2) to (8) are linear and time-invariant systems theoutput signal (after integration) can be defined as119867(119894120596)119890

(119894120596119905)where 119867(119894120596) is the complex transfer function (TF) of thecorresponding integration scheme which depends on 120596 and119905 [5] Also since 119867(119894120596) is a complex fuction it is possibleto derive the corresponding expressions for amplitude andphase for the function

In the following sections it is shown how the exact andnumerical transfer functions formulae can be obtained Sub-sequently comparing the corresponding numerical transferfunction to the exact one it is possible to develop a criterionto analyze the performance of a particular integration schemeusing a frequency approach



int119883 (119905) 119889119905 = int 119890(119894120596119905)

119889119905 =119890(119894120596119905)

119894120596= 119867lowast

1 (119894120596) 119890(119894120596119905)

997888rarr (9)

119867lowast

1 (119894120596) =1119894120596

=minus1120596119894 =

1120596119890minus1198941205872

(10)


∬119883(119905) 119889119905 = int119867lowast

1 (119894120596) 119890(119894120596119905)

119889119905 = 119867lowast

1 (119894120596) int 119890(119894120596119905)

119889119905

= [119867lowast

1 (119894120596)]2119890(119894120596119905)

= 119867lowast

2 (119894120596) 119890(119894120596119905)

997888rarr

119867lowast

2 (119894120596) = [119867lowast

1 (119894120596)]2

(11)



1198671 (119894120596) =ℎ

2(1 + 119890119894120596ℎ)(119890119894120596ℎ minus 1)

(12)


1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1) (13)


1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1) (14)


1198671 (119894120596)

=2ℎ45

(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)(1198901198941205964ℎ minus 1)

(15)

(v) Rectangle rule

1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1) (16)


1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1) (17)

where 1198870 = 1198872 = 03584 and 1198871 = 12832


1198671 (119894120596) =(ℎ3) (1198870119890

119894120596ℎ + 1198871 + 1198872119890minus119894120596ℎ + 1198873119890

minus2119894120596ℎ + 1198874119890minus3119894120596ℎ + 1198875119890

minus4119894120596ℎ + 1198876119890minus5119894120596ℎ + 1198877119890

minus6119894120596ℎ)

(119890119894120596ℎ minus 1) (18)



1198672 (119894120596) = [1198671 (119894120596)]2 (19)





ℎ (sec) 119891119873(Hz) 120596 (radsec)

002 25 50120587001 50 1001205870005 100 200120587


1198771 =

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816 (20)

1198772 =

10038161003816100381610038161198672 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

2 (119894120596)1003816100381610038161003816=[10038161003816100381610038161198671 (119894120596)

1003816100381610038161003816]2

[1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]2 = [

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]

2

= 11987721

(21)





119891119873=

12ℎ

997888rarr 120596119873=21205872ℎ

=120587

ℎ (22)


119873997888rarr 120596

inputmax lt 120596

119873=120587

ℎ (23)


119873) is


120582 =120596

120596119873

=120596ℎ

120587 (24)












0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25



NC n = 4

120582 = 120596120596N

R1=|H

1(i120596)||H

lowast 1(i120596)|


0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

TrapezoidalSimpson


NC n = 4

120582 = 120596120596N

R2=|H

2(i120596)||H

lowast 2(i120596)|


0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05

minus1

minus15

minus2

Φ(r

ad)

120582 = 120596120596N



(c) Phase diagram











0 01 02 03 04 05 06 07 08 09 1minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100



120582 = 120596120596N

NC n = 4

Relat

ive e

rror

()|

max|


minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Relat

ive e

rror

()|d

max|

0 01 02 03 04 05 06 07 08 09 1

120582 = 120596120596N



NC n = 4




119886 (119905) = 119860119888sin (120596119905) (25)

V (119905) = int119905

0119860119888sin (120596119905) 119889119905 =

119860119888

120596(1minus cos (120596119905)) (26)

119889 (119905) = int119905

0

119860119888

120596(1minus cos (120596119905)) 119889119905 =

119860119888

120596(119905 minus

sin (120596119905)120596

) (27)






119888= 3ms2 and




0 05 1 15


minus2minus4a

(ms2)

t (s)


minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise











119895and 120601

119895are



Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int119883 (119905) 119889119905 = int 119890(119894120596119905)

119889119905 =119890(119894120596119905)

119894120596= 119867lowast

1 (119894120596) 119890(119894120596119905)

997888rarr (9)

119867lowast

1 (119894120596) =1119894120596

=minus1120596119894 =

1120596119890minus1198941205872

(10)


∬119883(119905) 119889119905 = int119867lowast

1 (119894120596) 119890(119894120596119905)

119889119905 = 119867lowast

1 (119894120596) int 119890(119894120596119905)

119889119905

= [119867lowast

1 (119894120596)]2119890(119894120596119905)

= 119867lowast

2 (119894120596) 119890(119894120596119905)

997888rarr

119867lowast

2 (119894120596) = [119867lowast

1 (119894120596)]2

(11)



1198671 (119894120596) =ℎ

2(1 + 119890119894120596ℎ)(119890119894120596ℎ minus 1)

(12)


1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1) (13)


1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1) (14)


1198671 (119894120596)

=2ℎ45

(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)(1198901198941205964ℎ minus 1)

(15)

(v) Rectangle rule

1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1) (16)


1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1) (17)

where 1198870 = 1198872 = 03584 and 1198871 = 12832


1198671 (119894120596) =(ℎ3) (1198870119890

119894120596ℎ + 1198871 + 1198872119890minus119894120596ℎ + 1198873119890

minus2119894120596ℎ + 1198874119890minus3119894120596ℎ + 1198875119890

minus4119894120596ℎ + 1198876119890minus5119894120596ℎ + 1198877119890

minus6119894120596ℎ)

(119890119894120596ℎ minus 1) (18)



1198672 (119894120596) = [1198671 (119894120596)]2 (19)





ℎ (sec) 119891119873(Hz) 120596 (radsec)

002 25 50120587001 50 1001205870005 100 200120587


1198771 =

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816 (20)

1198772 =

10038161003816100381610038161198672 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

2 (119894120596)1003816100381610038161003816=[10038161003816100381610038161198671 (119894120596)

1003816100381610038161003816]2

[1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]2 = [

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]

2

= 11987721

(21)





119891119873=

12ℎ

997888rarr 120596119873=21205872ℎ

=120587

ℎ (22)


119873997888rarr 120596

inputmax lt 120596

119873=120587

ℎ (23)


119873) is


120582 =120596

120596119873

=120596ℎ

120587 (24)












0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25



NC n = 4

120582 = 120596120596N

R1=|H

1(i120596)||H

lowast 1(i120596)|


0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

TrapezoidalSimpson


NC n = 4

120582 = 120596120596N

R2=|H

2(i120596)||H

lowast 2(i120596)|


0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05

minus1

minus15

minus2

Φ(r

ad)

120582 = 120596120596N



(c) Phase diagram











0 01 02 03 04 05 06 07 08 09 1minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100



120582 = 120596120596N

NC n = 4

Relat

ive e

rror

()|

max|


minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Relat

ive e

rror

()|d

max|

0 01 02 03 04 05 06 07 08 09 1

120582 = 120596120596N



NC n = 4




119886 (119905) = 119860119888sin (120596119905) (25)

V (119905) = int119905

0119860119888sin (120596119905) 119889119905 =

119860119888

120596(1minus cos (120596119905)) (26)

119889 (119905) = int119905

0

119860119888

120596(1minus cos (120596119905)) 119889119905 =

119860119888

120596(119905 minus

sin (120596119905)120596

) (27)






119888= 3ms2 and




0 05 1 15


minus2minus4a

(ms2)

t (s)


minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise











119895and 120601

119895are



Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ (sec) 119891119873(Hz) 120596 (radsec)

002 25 50120587001 50 1001205870005 100 200120587


1198771 =

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816 (20)

1198772 =

10038161003816100381610038161198672 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

2 (119894120596)1003816100381610038161003816=[10038161003816100381610038161198671 (119894120596)

1003816100381610038161003816]2

[1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]2 = [

10038161003816100381610038161198671 (119894120596)1003816100381610038161003816

1003816100381610038161003816119867lowast

1 (119894120596)1003816100381610038161003816]

2

= 11987721

(21)





119891119873=

12ℎ

997888rarr 120596119873=21205872ℎ

=120587

ℎ (22)


119873997888rarr 120596

inputmax lt 120596

119873=120587

ℎ (23)


119873) is


120582 =120596

120596119873

=120596ℎ

120587 (24)












0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25



NC n = 4

120582 = 120596120596N

R1=|H

1(i120596)||H

lowast 1(i120596)|


0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

TrapezoidalSimpson


NC n = 4

120582 = 120596120596N

R2=|H

2(i120596)||H

lowast 2(i120596)|


0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05

minus1

minus15

minus2

Φ(r

ad)

120582 = 120596120596N



(c) Phase diagram











0 01 02 03 04 05 06 07 08 09 1minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100



120582 = 120596120596N

NC n = 4

Relat

ive e

rror

()|

max|


minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Relat

ive e

rror

()|d

max|

0 01 02 03 04 05 06 07 08 09 1

120582 = 120596120596N



NC n = 4




119886 (119905) = 119860119888sin (120596119905) (25)

V (119905) = int119905

0119860119888sin (120596119905) 119889119905 =

119860119888

120596(1minus cos (120596119905)) (26)

119889 (119905) = int119905

0

119860119888

120596(1minus cos (120596119905)) 119889119905 =

119860119888

120596(119905 minus

sin (120596119905)120596

) (27)






119888= 3ms2 and




0 05 1 15


minus2minus4a

(ms2)

t (s)


minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise











119895and 120601

119895are



Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25



NC n = 4

120582 = 120596120596N

R1=|H

1(i120596)||H

lowast 1(i120596)|


0 01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

TrapezoidalSimpson


NC n = 4

120582 = 120596120596N

R2=|H

2(i120596)||H

lowast 2(i120596)|


0 01 02 03 04 05 06 07 08 09 1

2

15

1

05

0

minus05

minus1

minus15

minus2

Φ(r

ad)

120582 = 120596120596N



(c) Phase diagram











0 01 02 03 04 05 06 07 08 09 1minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100



120582 = 120596120596N

NC n = 4

Relat

ive e

rror

()|

max|


minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Relat

ive e

rror

()|d

max|

0 01 02 03 04 05 06 07 08 09 1

120582 = 120596120596N



NC n = 4




119886 (119905) = 119860119888sin (120596119905) (25)

V (119905) = int119905

0119860119888sin (120596119905) 119889119905 =

119860119888

120596(1minus cos (120596119905)) (26)

119889 (119905) = int119905

0

119860119888

120596(1minus cos (120596119905)) 119889119905 =

119860119888

120596(119905 minus

sin (120596119905)120596

) (27)






119888= 3ms2 and




0 05 1 15


minus2minus4a

(ms2)

t (s)


minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise











119895and 120601

119895are



Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 1minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100



120582 = 120596120596N

NC n = 4

Relat

ive e

rror

()|

max|


minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Relat

ive e

rror

()|d

max|

0 01 02 03 04 05 06 07 08 09 1

120582 = 120596120596N



NC n = 4




119886 (119905) = 119860119888sin (120596119905) (25)

V (119905) = int119905

0119860119888sin (120596119905) 119889119905 =

119860119888

120596(1minus cos (120596119905)) (26)

119889 (119905) = int119905

0

119860119888

120596(1minus cos (120596119905)) 119889119905 =

119860119888

120596(119905 minus

sin (120596119905)120596

) (27)






119888= 3ms2 and




0 05 1 15


minus2minus4a

(ms2)

t (s)


minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise











119895and 120601

119895are



Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15


minus2minus4a

(ms2)

t (s)


minus2minus4a

(ms2)

0 05 1 15

t (s)

Phase-noise420

minus2minus4a

(ms2)

0 05 1 15

t (s)

OriginalPhase-noise











119895and 120601

119895are



Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Low-frequency noise

0 05 1 15

0

01

02

03

04

t (s)

(m

s)

minus01

0 05 1 15

0

01

02

03

04


t (s)

(m

s)

0 05 1 15

0

01

02

03

04Phase-noise

(m

s)

t (s)



NC n = 4

(a)

0 05 1 150

005

01

015

02

025


d(m

)

t (s)

0 05 1 150

005

01

015

02


d(m

)

t (s)

0 05 1 150

005

01

015

02

025Phase-noise

d(m

)

t (s)



NC n = 4

(b)



119886 (119905) = 119861119905119890120572119905

119899

sum119895=1

cos (120596119895119905 + 120601119895) (28)





0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

noise

(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

002

0

minus002

minus004

t (s)

orig

inalminus p

hase

-noi

se(m

s)

(a)

0 05 1 15


0

minus05

minus1

minus15

minus2

minus25

times10minus3

t (s)

dor

igin

alminusd

high

-noi

se(m

)

0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

)

002

0

minus002

minus004

minus006

minus008

minus01

minus012

t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

minus1

minus2

minus3

minus4

a(m

s2)

Time (s)

(a)

0 5 10 15 200

002

004

006

008

Frequency (Hz)

Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15


05

0

minus05

minus1

minus15

minus2

minus25

orig

inalminus h

igh-

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(ms

)

times10minus3

t (s)

0 05 1 15

Low-frequency noise



t (s)

0

minus005

minus01

minus015

minus02

minus025

Simpson 3-8

NC n = 4

orig

inalminus l

ow-n

oise

(m)

0 05 1 15

Phase-noise006

004

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minus004

t (s)

orig

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hase

-noi

se(m

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(a)

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0

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minus1

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minus2

minus25

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t (s)

dor

igin

alminusd

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-noi

se(m

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0 05 1 15

Low-frequency noise

dor

igin

alminusd

low

-noi

se(m

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002

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minus006

minus008

minus01

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t (s)

0 05 1 15minus0005

0

0005

001

0015

002

0025Phase-noise

t (s)

dor

igin

alminusd

phas

e-no

ise(m

)



NC n = 4

(b)





0 2 4 6 8 10 12 14 16 18 20

4

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minus2

minus3

minus4

a(m

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Time (s)

(a)

0 5 10 15 200

002

004

006

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Four

ier a

mpl

itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

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plitu

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5

4

3

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minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20

4

3

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1

0

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minus2

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(a)

0 5 10 15 200

002

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006

008

Frequency (Hz)

Four

ier a

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itude

(b)




Simpson 01502 00141 minus01811 00058










119910)




119898)



0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

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(m

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01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20

Trapezoidal rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule02

01

0

minus01

minus02

(m

s)

t (s)

0 2 4 6 8 10 12 14 16 18 20minus02

minus01

0

01

02

(m

s)

t (s)


0 2 4 6 8 10 12 14 16 18 20minus1

minus05

0

05

1Trapezoidal rule

num

minus e

xact

(ms

)

times10minus3

t (s)

0 2 4 6 8 10 12 14 16 18 20

Simpson rule2

1

0

minus1

minus2

times10minus4

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

Simpson 3-8 rule01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)

0 2 4 6 8 10 12 14 16 18 20

01

005

0

minus005

minus01

t (s)

num

minus e

xact

(ms

)



Figure 7 Continued


0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20

Rectangle rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Rectangle rule

t (s)

num

minus e

xact

(ms

)

002

001

0

minus001

minus002

0 2 4 6 8 10 12 14 16 18 20

Tick rule02

01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20

Tick rule

t (s)

num

minus e

xact

(ms

)

times10minus4

0 2 4 6 8 10 12 14 16 18 20


01

0

minus01

minus02

(m

s)

t (s)0 2 4 6 8 10 12 14 16 18 20


005

0

minus005

minus01

num

minus e

xact

(ms

)

t (s)


(a) (b)

2

1

0

minus1

minus2













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119886119910=14119886max 119886

119910=12119886max 119886

119910=34119886max








1 2 3 4 5 6 7 8 9 10 11 12

Frequency (Hz)

Dev

iatio

n Fo

urie

r am

plitu

de

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

times10minus5









0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3

0 10 20 30 40Time (s)


a(g

)

05040302010



0 10 20 300

2

4

6

8

Frequency (Hz)

Four

ier a

mpl

itude

times10minus3



0 5 10 15 20 25 30 35 40


01

008

006

004

002

0

minus002

minus004

d(m

)

t (s)

TrapezoidalSimpson



0002004006008

01012014


t (s)

TrapezoidalSimpson


d(m

)



6 Conclusions




















Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Appendix


119887




119884119899+1 = 119884

119899+ℎ

2(119883119899+119883119899+1) (A1)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A2)


1198671 (119894120596) 119890119894120596(119905+ℎ)

= 1198671 (119894120596) 119890119894120596119905

+ℎ

2(119890119894120596119905

+ 119890119894120596(119905+ℎ)

)

997888rarr 1198671 (119894120596) 119890119894120596ℎ

= 1198671 (119894120596) +ℎ

2(1+ 119890119894120596ℎ)

1198671 (119894120596) =ℎ (1 + 119890119894120596ℎ)2 (119890119894120596ℎ minus 1)

(A3)


119884119899+2 = 119884

119899+ℎ

3(119883119899+ 4119883119899+1 +119883119899+2) (A4)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A5)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

ℎ

3(119890119894120596119905119899 + 4119890119894120596(119905119899+ℎ) + 119890119894120596(119905119899+2ℎ)) 997888rarr

1198671 (119894120596) 1198901198941205962ℎ

= 1198671 (119894120596) +ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

997888rarr 1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ

3(1+ 4119890119894120596ℎ + 1198901198941205962ℎ)

1198671 (119894120596) =ℎ

3(1 + 4119890119894120596ℎ + 1198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A6)


119884119899+3 = 119884

119899+3ℎ8(119883119899+ 3119883119899+1 + 3119883119899+2 +119883119899+3) (A7)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+3 = 1198671 (119894120596) 119890

119894120596(119905119899+3ℎ)

(A8)


1198671 (119894120596) 119890119894120596(119905119899+3ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+3ℎ8(119890119894120596119905119899 + 3119890119894120596(119905119899+ℎ) + 3119890119894120596(119905119899+2ℎ) + 119890119894120596(119905119899+3ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205963ℎ

minus 1)

=3ℎ8(1+ 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

1198671 (119894120596) =3ℎ8(1 + 3119890119894120596ℎ + 31198901198941205962ℎ + 1198901198941205963ℎ)

(1198901198941205963ℎ minus 1)

(A9)


119884119899+4 = 119884

119899+2ℎ45

(7119883119899+ 32119883

119899+1 + 12119883119899+2 + 32119883119899+3

+ 7119883119899+4)

(A10)


119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119883119899+3 = 119890

119894120596(119905119899+3ℎ)

119883119899+4 = 119890

119894120596(119905119899+4ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+4 = 1198671 (119894120596) 119890

119894120596(119905119899+4ℎ)

(A11)


1198671 (119894120596) 119890119894120596(119905119899+4ℎ)

= 1198671 (119894120596) 119890119894120596119905119899 +

2ℎ45

(7119890119894120596119905119899

+ 32119890119894120596(119905119899+ℎ) + 12119890119894120596(119905119899+2ℎ) + 32119890119894120596(119905119899+3ℎ)

+ 7119890119894120596(119905119899+4ℎ)) 997888rarr

1198671 (119894120596) (1198901198941205964ℎ

minus 1) = 2ℎ45

(7+ 32119890119894120596ℎ + 121198901198941205962ℎ

+ 321198901198941205963ℎ + 71198901198941205964ℎ)

1198671 (119894120596) =2ℎ45

sdot(7 + 32119890119894120596ℎ + 121198901198941205962ℎ + 321198901198941205963ℎ + 71198901198941205964ℎ)

(1198901198941205964ℎ minus 1)

(A12)


119884119899+1 = 119884

119899+ ℎ119883119899 (A13)


119883119899= 119890119894120596119905119899

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A14)



1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198671 (119894120596) 119890119894120596119905119899119890119894120596ℎ

= 1198671 (119894120596) 119890119894120596119905119899 + ℎ119890119894120596119905119899

997888rarr 1198671 (119894120596) (119890119894120596ℎ


1198671 (119894120596) =ℎ

(119890119894120596ℎ minus 1)

(A15)


119884119899+2 = 119884

119899+ ℎ (1198870119883119899 + 1198871119883119899+1 + 1198872119883119899+2) (A16)



119883119899= 119890119894120596119905119899

119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899+2 = 119890

119894120596(119905119899+2ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+2 = 1198671 (119894120596) 119890

119894120596(119905119899+2ℎ)

(A17)


1198671 (119894120596) 119890119894120596(119905119899+2ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ ℎ (1198870119890119894120596119905119899 + 1198871119890

119894120596(119905119899+ℎ)

+ 1198872119890119894120596(119905119899+2ℎ)

) 997888rarr

1198671 (119894120596) (1198901198941205962ℎ

minus 1) = ℎ (1198870 + 1198871119890119894120596ℎ

+ 11988721198901198941205962ℎ

)

1198671 (119894120596) = ℎ(1198870 + 1198871119890

119894120596ℎ + 11988721198901198941205962ℎ)

(1198901198941205962ℎ minus 1)

(A18)


119884119899+1 = 119884

119899+ℎ

3

119896=7sum119896=0

119887119896119883119899+1minus119896 119899 = 0 1 2 (A19)




119883119899+1 = 119890

119894120596(119905119899+ℎ)

119883119899= 119890119894120596119905119899

119883119899minus1 = 119890

119894120596(119905119899minusℎ)

119883119899minus2 = 119890

119894120596(119905119899minus2ℎ)

119883119899minus3 = 119890

119894120596(119905119899minus3ℎ)

119883119899minus4 = 119890

119894120596(119905119899minus4ℎ)

119883119899minus5 = 119890

119894120596(119905119899minus5ℎ)

119883119899minus6 = 119890

119894120596(119905119899minus6ℎ)

119884119899= 1198671 (119894120596) 119890

119894120596119905119899

119884119899+1 = 1198671 (119894120596) 119890

119894120596(119905119899+ℎ)

(A20)


1198671 (119894120596) 119890119894120596(119905119899+ℎ)

= 1198671 (119894120596) 119890119894120596119905119899

+ℎ

3(1198870119890(119894120596119905119899+ℎ)

+ 1198871119890119894120596119905119899 + 1198872119890

119894120596(119905119899minusℎ)

+ 1198873119890119894120596(119905119899minus2ℎ)

+ 1198874119890119894120596(119905119899minus3ℎ)

+ 1198875119890119894120596(119905119899minus4ℎ)

+ 1198876119890119894120596(119905119899minus5ℎ)

+ 1198877119890119894120596(119905119899minus6ℎ)

) 997888rarr

1198671 (119894120596) (119890119894120596ℎ

minus 1) = ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ+ 1198873119890minus2119894120596ℎ

+ 1198874119890minus3119894120596ℎ

+ 1198875119890minus4119894120596ℎ

+ 1198876119890minus5119894120596ℎ

+ 1198877119890minus6119894120596ℎ

)

1198671 (119894120596) =ℎ

3(1198870119890119894120596119905119899 + 1198871 + 1198872119890

minus119894120596ℎ + 1198873119890minus2119894120596ℎ + 1198874119890

minus3119894120596ℎ + 1198875119890minus4119894120596ℎ + 1198876119890

minus5119894120596ℎ + 1198877119890minus6119894120596ℎ)

(119890119894120596ℎ minus 1)

(A21)




Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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