The Unbiased Characteristic of Doppler Frequency in GNSS ... · ResearchArticle The Unbiased Characteristic of Doppler Frequency in GNSS Antenna Array Processing YuchenXie ,ZhengrongLi

Research ArticleThe Unbiased Characteristic of Doppler Frequency in GNSSAntenna Array Processing

Yuchen Xie Zhengrong Li Feiqiang Chen Huaming Chen and FeixueWang

College of Electronic Science National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Yuchen Xie olien x163com and Zhengrong Li zr linudteducn

Received 10 December 2018 Accepted 7 March 2019 Published 24 April 2019

Academic Editor Ikmo Park

Copyright copy 2019 Yuchen Xie et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The antenna array technology especially the spaced-time array processing (STAP) is one of the effective methods used in GlobalNavigation Satellite System (GNSS) receivers to refrain the power of jamming and enhance the performance of receivers in thecircumstance of interference However biases induced to the receiver because ofmany reasons including characteristic of antennasfront-end channel electronics and space-time filtering are extremely harmful to the high precise positioning of receivers Althoughplenty of works have been done to calibrate the antenna and tomitigate these biases achieving a good performance of antijamminghigh accuracy and low complexity at the same time still remains challenging Different from existing works this paper leverages thecharacteristic of GNSS signalrsquos Doppler frequency in STAP which is proven to remain unbiased to solve the problem even whenthe nonideal antennas are used and the interference circumstance changes Since the integration of frequency is carrier phasethe unbiased Doppler frequency leads to an accurate estimation of carrier phase which can be used to calibrate the antenna arraywithout extra apparatus or complicating algorithms Therefore a simple Doppler-aid strategy may be developed in the future tosolve the difficulty of STAP bias mitigation

1 Introduction

Array processing is one of the most effective ways to refrainthe jamming aimed at GNSS receiver and among thosearray processing methods the use of controlled receptionpattern antenna (CRPA) arrays has attracted more andmore attention CRPA arrays can provide beam formingnullsteering in specific direction by adaptively adjusting theweight of each antenna element Typically an adaptive finiteimpulse response (FIR) filter placed behind each elementallows a better performance of antijamming which is knownas the space-time array processing [1] CRPA arrays especiallySTAP-based antenna arrays are so popular that numerousadaptive algorithms have been researched to provide a highperformance of interference suppression [2]

Unfortunately despite of the antijamming benefit thatSTAP can provide undesirable and unpredictable biases arealso induced to the GNSS receiver [2] which harms theaccuracy of navigation solution [3ndash9] There are several factscontributing to the STAP bias including the overall gain and

phase response of antennas mutual coupling of antennasfront-end channel electronics and space-time filtering Pre-vious research has focused on the code and carrier phasemeasurement bias using STAP For space only processing(SOP) a code phase bias on the order of meters can beobserved in simulation [3] and the phase bias of real datais discussed in [4] The measurements are distorted furtherin STAP because of the applied FIR filters [5 7] In [1]the dependency of bias on interference circumstances isanalyzed which leads to the unpredictable characteristic ofSTAP-based bias

Therefore to fulfill the ability of adaptive arraysresearchers have made great efforts to mitigate these biasesOne solution is to use an adaptive antenna array that exhibitssmall biases [8] or to do the precalibration to correct theantenna based bias but it may be impractical to some lowcost receivers [10 11] Another way is to apply some specificalgorithms that constrain the distortion of signal phasebut this method causes the loss of antijamming freedomdegree Some software based algorithms are also proposed to

HindawiInternational Journal of Antennas and PropagationVolume 2019 Article ID 5302401 10 pageshttpsdoiorg10115520195302401

2 International Journal of Antennas and Propagation

calibrate the bias [12 13] but the computational complexityprevents them from the real time application It seems thathigh accuracy low complexity and real time response areincompatible in the bias mitigation of STAP

Different from previous works this paper analyzes theperformance of Doppler frequency during the processing ofantenna arrays Logically neither the response of antennasand channels nor the adaptive weights will change the carrierfrequency of signal In this paper the unbiased characteristicof Doppler frequency of GNSS signal in STAP is proved byusing the auto correlation function (ACF) and the simulationresults also show that the changing of interference leads tothe uncertain variation of phase but it will not affect the esti-mation of Doppler frequency of receiver When the responseof channel is not ideal the unpredictable distortion of phaseis more obvious while the estimation of Doppler frequencyis still unbiased Since the integration of frequency is carrierphase the unbiased characteristic of Doppler frequency canbe critical to high precision GNSS applications Althoughthe deduction is based on the GNSS signal the unbiasedcharacteristic is common in array processing It means thatwe can correct the phase error induced by STAP or calibratethe antenna instantaneously using the Doppler frequencywithout any extra complexity of algorithm or hardware

The rest of this paper is organized as follows First thearray model is established and a brief description of theantenna induced bias in phase is given in Section 2 Thenthe unbiased characteristic of theDoppler frequency ofGNSSsignal is proved in Section 3with the help ofACF In Section 4simulation results are presented to compare the phase errorand the frequency error which further proves the unbiasedcharacteristic of Doppler frequency Finally the conclusion ismade in Section 5

2 Array Model

Although STAP is an effective way to mitigate the inter-ference in GNSS signal [14] biases induced by antennasand algorithms are not negligible for precision applicationsespecially when a large antenna array with complicatedfiltering is considered [3] This section presents a model ofarray processing for describing errors induced by STAP andproving their dependency on interference circumstances

The antenna model is depicted in Figure 1 In this modelK individual antenna element with an M-tap FIR filterfollowing is considered 119860119896(119891 120579 120593) represents the systemresponse of the kth element in (120579 120593) direction (120579 and 120593stand for azimuth and elevation angles respectively) Akis also affected by the frequency of received signals (theeffect of mutual coupling is not considered) 119865119896(119891) stands forthe electronic which downconverts the signal to basebandand perform analog-to-digital conversion The output digitalsignal of each element is then filtered by an M-tap adaptivefilter and the frequency response of each FIR filter is denotedby 119882119896(119891) Then the outputs of the filters are summed forpostprocessing

The complex weight in STAP filter can be represented bya stack vector

w =[[[[[[[

w1w2

w119870

]]]]]]]

(1)

w119896 is an 119872 times 1 vector corresponding to the kth filter

w119896 = [1199081198961 1199081198962 sdot sdot sdot 119908119896119872]T (2)

where 119908119896119898 is each adaptive weightThe instantaneous digital output snapshot on the taps of

the kth front-end channel is denoted by

x119896 [119899] = [119909119896 [119899] 119909119896 [119899 minus 1] sdot sdot sdot 119909119896 [119899 minus 119872 + 1]]T (3)

119909119896[119899] is the received signal on the kth element which containsdesired signal 119904119896[119899] undesired interference signal 119895119896[119899] andnoise 120578119896[119899]

119909119896 [119899] = 119904119896 [119899] + 119895119896 [119899] + 120578119896 [119899] (4)

The snapshot on each filter is combined into a received signalstack vector

x [119899] =[[[[[[[

x1 [119899]x2 [119899]

x119870 [119899]

]]]]]]]

(5)

The STAP makes the weighted sum of signal vector bymultiplying it with the weight vector

119910 [119899] = wTx [119899] (6)

By applying some criterions [14] such as powerminimization[15 16] multiple constrained minimum variance [17] andminimum mean square error [18] the power of interferencesignal can be effectively restrained after the adaptive weightedsummation

However characteristics of antennas and channels as wellas STAP algorithms cause biases to the output signal Tomeasure them it is reasonable to analyze their effects on theGNSS receiver cross-correlation

The GNSS receiver correlators perform the cross-correlation by multiplying local CA code replicas 119903[119899] withthe received signal 119904[119899] and then doing the average of thecorrelation time 119873119903 [19]

119877 (120591) = 1119873119903119873119903sum119899=1

119904 [119899 + 120591] 119903 [119899] (7)

119877(120591) named as the auto correlation function is the output ofthe correlator where 120591 is the code delay between the receivedsignal and the local replica

In STAP the correlation process can be treated as thecorrelation between the signal of each FIR filter tap and the

International Journal of Antennas and Propagation 3

sum

A1(f ) A2(f ) AK(f )

F1(f) F2(f) FK(f)

W1(f) W2(f) WK(f)

Figure 1 STAP-based adaptive antenna array model

local CA code replica and then all of these119870times119872ACFs sumup after multiplying their weight

119877119910119889 (120591) = 1119873119903119873119903sum119899=1

119910 [119899 + 120591] 119903 [119899]

=119870119872

sum119897=1

1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 119877 (120591 + 120591119897) + 1198951015840

+ 1205781015840

(8)

|119886119897(119891 120579 120593)| |119891119897(119891)| and |119908119897| are the amplitude effects ofantenna receiving pattern channel response and adaptiveweight respectively while 120591119897 is the total code delay inducedby antenna channel and weight for each tap 1198951015840 and 1205781015840 arethe remaining interference and noise after correlation

It can be known from (8) that the output ACF of STAPis a combination of 119870 times 119872 different ACFs whose amplitudeand time delay vary from one to another In fact the antennareceiving pattern depends on the received signalrsquos directionand frequency the channel response changes with timeand temperature and adaptive weights are also affected byinterference circumstance Therefore the bias induced bySTAP in ACF changes with the GNSS signal the interferenceand the environment which is consequently unpredictable

Simulation results in different interference circumstancessupport this conclusion which will be explained in detail inSection 4 In this case even if STAP has successfully refrainedthe power of interferences the navigation output which isbased on code phase measuring and carrier phase aid maynot be accurate However it can be proved that Dopplerfrequency is unbiased after array processing which can befurther used to enhance the measuring accuracy of receiverThe details of deduction will be presented in the next section

3 Doppler Frequency Estimation in STAP

In the single array receiver the IQorthogonal demodulationis applied to the received signal to move the carrier of it [19]

After orthogonal demodulation and correlation the output ofcorrelator can be denoted by

119894 [119899] = 119886119863 [119899] 119877 (120591) cos [2120587119891119890119899 + 120601119890] (9)

119902 [119899] = 119886119863 [119899] 119877 (120591) sin [2120587119891119890119899 + 120601119890] (10)

119886 is the amplitude of received signal and 119863[119899] is the data bitboth ofwhich can be regarded as 1 for the sake of convenience119877(120591) is the ACF which has been defined in Section 2 119891119890 and120601119890 are the frequency and phase discrepancies between localcarrier and received signalrsquos carrier respectively

The relation among the local carrier119891119897119900119888119886119897 the receivedsignalrsquos carrier 119891119888119886119903119903 the frequency error 119891119890 the Dopplerfrequency of signal 119891119889 and the standard carrier frequency 1198910can be written as

119891119888119886119903119903 = 119891119897119900119888119886119897 + 119891119890 (11)

119891119889 = 119891119888119886119903119903 minus 1198910 = 119891119897119900119888119886119897 + 119891119890 minus 1198910 (12)

Ideally 119891119890 is zero the Doppler frequency will exactly be thediscrepancy between local generated carrier frequency andthe standard frequency ie119891119897119900119888119886119897ndash1198910 But in real situation119891119890 isnonzero and includes theDoppler frequency estimation errorand some other noise errors Although they are difficult to beseparated to get the exact Doppler frequency in simulationtest with setting Doppler frequency and nonsignificantestimation error the Doppler frequency can be reasonablyestimated by calculating 119891119890

Therefore the119891119890 calculating process of the GNSS receiverin its fine acquisition is firstly introduced By simplifying (9)and (10) and doing the square we get

1198942 [119899] = 1198772 (120591) cos2 [2120587119891119890119899 + 120601119890] (13)

1199022 [119899] = 1198772 (120591) sin2 [2120587119891119890119899 + 120601119890] (14)

Further we set

119911119903 [119899] = 1198942 [119899] minus 1199022 [119899] = 1198772 (120591) cos [4120587119891119890119899 + 2120601119890] (15)

119911119894 [119899] = 2119894 [119899] 119902 [119899] = 1198772 (120591) sin [4120587119891119890119899 + 2120601119890] (16)


Combining 119911119903[119899] and 119911119894[119899] into a complex signal we get

119911 [119899] = 119911119903 [119899] + 119895119911119894 [119899]= 1198772 (120591) cos [4120587119891119890119899 + 2120601119890]

+ 1198951198772 (120591) sin [4120587119891119890119899 + 2120601119890](17)

It is noticed that z[n] is a single frequency complex signalwiththe amplitude of 1198772(120591) the frequency of 2119891119890 and the phase of2120601119890 After doing the Fast Fourier Transform (FFT) to z[n] themaximum in its frequency domain is located at 2119891119890 which isunrelated to its phase error 2120601119890

Therefore 119891119890 can be achieved by

119891119890 = 12findmax (FFT (119911 [119899])) (18)

where findmax means searching for the frequency maximiz-ing |FFT(119911[119899])|

In the STAP receiver the IQorthogonal demodulation isalso applied to the received signal and derived from (8) (9)and (10) it can be rewritten as

119894119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) cos [2120587119891119890119899 + 120601119890 + 120601119897] (19)

119902119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) sin [2120587119891119890119899 + 120601119890 + 120601119897] (20)

where

119887119897 = 1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 (21)

is the coefficient containing all amplitude effects 119891119890 has beendefined in (10) and jammer induced Doppler shift at thereceiver is assumed to be small compared to satellite Doppler120591119897 has been defined in (8) and120601119897 is the total carrier phase errorinduced by STAPAsmentioned before 119887119897 120591119897 and120601119897 vary fromone to another

Similarly the square and multiplication of (19) and (20)are

1198942119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotcos [2120587119891119890119899 + 120601119890 + 120601119901]

(22)

1199022119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(23)

119894119886119903119903119886119910119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(24)

So 119911119903 119886119903119903119886119910[119899] and 119911119894 119886119903119903119886119910[119899] can be denoted as

119911119903 119886119903119903119886119910 [119899] = 1198942119886119903119903119886119910 [119899] minus 1199022119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(25)

119911119894 119886119903119903119886119910 [119899] = 2119894119886119903119903119886119910 [119899] 119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(26)

and

119911119886119903119903119886119910 [119899] = 119911119903 119886119903119903119886119910 [119899] + 119895119911119894 119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

+ j119870119872

sum119894=1

119870119872

sum119895=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(27)

Therefore 119911119886119903119903119886119910[119899] is the sum of119870119872times119870119872 single frequencycomplex signals Although they are different in amplitude andcarrier phase as

10038161003816100381610038161003816119911119886119903119903119886119910 119897119901 [119899]10038161003816100381610038161003816 = 119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) (28)

120601119886119903119903119886119910 119897119901 = 2120601119890 + 120601119897 + 120601119901 (29)

they have the same frequency 2119891119890 In this case (18) is stilleffective in estimating the Doppler frequency

Based on these deductions it is reasonable to say thatunlike code and carrier phase which will be unpredictablyshifted because of the changing of interferences the Dopplerfrequency of the received signal remains unbiased in arrayprocessing

In fact even in the situation that the bias of phase istoo severe to calibrate or in the situation that the electroniccharacteristic of analog element changes which makes theprevious calibration ineffective the Doppler frequency stillremains unbiased because interference and STAP only affectthe received signalrsquos phase rather than its frequency

Although the deduction is based onGNSS signal it is alsotrue to other signals of array processing As the integrationof frequency through time leads to phase the phase errorcan also be corrected with the help of Doppler frequencyTherefore the unbiased and accurate Doppler frequency isespecially useful to high precision locating applications Inanother way the integration of the Doppler frequency canalso be used to calibrate the bias induced by STAP as thedifference between the integrationDoppler frequency and theoutput carrier phase is the total bias of STAP In that case thereal time calibration for STAP can be realized to significantlyenhance the performance of array processing


Table 1 Simulation parameters

Parameter ValueSignal Type BeiDouCarrier Frequency 126852MHzSignal Length 1000 msIntermediate Frequency 4652MHzSampling Frequency 61MHzCode Frequency 1023MHzCode Length 10230Signal Noise Ratio (SNR) -15dBSignal Direction (120579=85∘ 120593=70∘)Jamming Noise Ratio (JNR) Jammer1 50dB Jammer2 30dBAntenna Element 4Array Structure CircularTime Taps 3 5 7Anti Jamming Criterion PI

A1

A2

A3 A4

d

d d

x

y

Figure 2 Structure of antenna array

Simulation results in Section 4 support the conclusionabove and further prove the unbiased characteristic ofDoppler frequency

4 Simulation Results

In this section simulations of the typical BeiDou receiverrsquosperformance in different interference circumstances are pre-sented to analyze the bias induced by STAP

The simulated receiver is a 4-element circular antennaarray receiver which has one antenna at the origin point withthree others surrounded and the distance d from the originto each other antenna is half carrier wavelengthThe structureof the antenna array is shown in Figure 2

The STAP with an FIR filter back to each antenna isapplied and the power inverse (PI) [20] criterion is chosento adapt the weight of each tap which minimizes the outputof antenna array processing to mitigate the effect of jammerMeanwhile the same signal received in an interferencefree circumstance and without applying any antijammingmethod is also processed by the receiver as the referenceTheparameters for this simulation are listed in Table 1

In the simulation the circumstance of interferenceschanges with time and contains different types directions

and powers of jamming The simulation includes four stepsas follows

Step 1 Turn on the signal (120579=85∘ 120593=70∘ SNR -15dB)Step 2 Turn on the jammer1 (120579=300∘120593=5∘ JNR 50dB) at the200ms

Step 3 Turn on the jammer2 (120579=135∘ 120593=10∘ JNR 30dB) atthe 400ms

Step 4 Change the direction of jammer1 (120579=180∘ 120593=30∘) atthe 600ms

Step 5 Turn off the jammer1 and the jammer2 at the 800msThe jammer1 emits a white noise interference with the

band of 2046 MHz centered at 126852MHz while thejammer2 emits a single frequency interference closed to thesignalrsquos frequency During the whole time of simulation thesignal is combined with white noise

41 Ideal Channel Simulation In the first simulation wefocus on the bias induced by adaptive algorithm thereforeideal antennas and channels are considered Figures 3 and 4show the simulation results


3Taps5Taps7Taps

minus28minus26minus24minus22

minus2minus18minus16minus14minus12

minus1minus08minus06

Cod

e Err

or (m

)

200 400 600 800 10000Time (ms)

(a) Code phase error in simulation1

3Taps5Taps7Taps

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)

(b) Carrier phase error in simulation1

Figure 3

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus10

0

10

20

30

Freq

uenc

y Er

ror (

Hz)

(a) Tracking loop frequency error in simulation1

3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)

(b) Fine acquisition error in simulation1

Figure 4

Figure 3(a) is the difference of code phase between thesignal after STAP and the reference signal and Figure 3(b)is its counterpart carrier phase difference It can be knownfrom the figures that STAP successfully restrains the power ofinterferences and enables the receiver to keep tracking of thecode phase Nevertheless even when the jammers are turnedoff the code error is not zero because STAP induces bias tothe receiver Taking the 3-tap filter simulation in Figure 3(a)as example the average error is -0907m (1ms to 200ms)with the minimum of 0799m at the 1ms considering thatone chip corresponds to 30m in our simulation When the

jammer1 is turned on and switched the code phase errorvaries slightly as the average error is -0744mduring 200ms to400ms and -0904m during 400ms to 600ms However whenthe jammer2 is turned on as well the error of code phasesees a dramatic jump near the 600ms after which it fluctuatesseverely and the average error is -2124m during 600ms to800ms The situations for 5- and 7-tap filter are similar butthe errors are more severe than that of 3-tap filter

As for the carrier phase error it can be known fromFigure 3(b) that it strongly depends on the circumstanceof interference In detail when there is no interference the


Table 2

(a) Average phase error in simulation1

119890119903119903119888119900119889119890119890119903119903119888119886119903119903 0 - 200ms 200 ndash 400ms 400 - 600ms 600 ndash 800ms 800 ndash 1000ms total

3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘

(b) Standard deviation of phase in simulation1120590119888119900119889119890120590119888119886119903119903 0 - 200ms 200 ndash 400ms 400 - 600ms 600 ndash 800ms 800 ndash 1000ms total

3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)

1 2 3 4 5 60Frequency (Hz)

(a) Channel amplitude response

ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )


(b) Channel phase response

Figure 5

carrier phase error fluctuates around zero in the 3-tap filtersimulation with a maximal absolute value of 1003∘ at the131ms However when turning on the jammers or switchingthe direction of jammer1 the carrier phase error jumpsdramatically as can be seen at the 200ms 400ms 600ms and800ms Besides when there exist interferences the averageof error is -16227∘ (200ms to 400ms) -20362∘ (400ms to600ms) and 12110∘ (600ms to 800ms) respectively whichpresents an obvious bias from zero It can also be noticed thatthe error fluctuates more drastically during the period from600ms to 800ms when interferences are more complicatedThe number of filter taps also influences the phase error butnot significantly as can be seen from 400ms to 600ms inFigure 3(b)

The average of code phase error and carrier phase error(denoted as 119890119903119903119888119900119889119890 and 119890119903119903119888119886119903119903) as well as their standarddeviation (denoted as 120590119888119900119889119890 and 120590119888119886119903119903) is shown in Tables 2(a)and 2(b) with the maximal value of each row being bold

Figures 4(a) and 4(b) present two types of Dopplerfrequency errors in the simulation using different estimationmethods Data in Figure 4(a) is calculated from the outputcarrier frequency of the tracking loop while in Figure 4(b)fine acquisition is applied to each 10ms signal to estimate theaccurate Doppler frequency It can be noticed in Figure 4(a)that as the tracking loop calculates the frequency by using car-rier phase when the interference changes which causes dra-matic jumps to the code phase at 200ms 400ms 600ms and800ms the output frequency jumps consequently However


Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘

(b) Standard deviation of phase in simulation2


3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6

after this variation the output frequency returns back toits original value and the error fluctuates around zerofor example the average error is -0002Hz for 3-tap filtersimulation This can be further proved in Figure 4(b) wherethe accurate Doppler frequency is estimated the errors areexact zero for different taps filter simulations during thewhole simulation time

42 Imperfect Channel Simulation In the second simulationimperfect antennas and channels are taken into considera-tion The characteristics of channels are presented in Figures5(a) and 5(b) whose amplitude response waves randomlyrang from -05dB to 05dB and phase response is nonlinear

with a maximal shift of 30∘ The other parameters and stepsare exactly the same as those in the first simulation and theresults are shown in Figures 5ndash7

Comparing Figure 6(a) with Figure 3(a) it is obvious thatthe nonideal response of channels worsens the error of phaseto vary more randomly the gap between the maximum andminimum errors is about 5039m Similarly the comparisonbetween Figures 6(b) and 3(b) also shows a more drastic andrandom variation of the carrier phase and the stable states oftwo pictures are different as well which suggests new biasesare induced because of channel characteristics

119890119903119903119888119900119889119890 119890119903119903119888119886119903119903 120590119888119900119889119890 and 120590119888119886119903119903 of the second simulationare shown in Table 3


3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7

On the contrary it can be figured out in Figures 7(a) and7(b) that the frequency error remains unbiased even with theconsideration of channel effect Although the deviation inFigure 7(a) is larger than that in Figure 4(a) the error returnsto fluctuate around zero very soon and the fine acquisitionresult in Figure 7(b) is the same zero as that in Figure 4(b)

Based on the simulation results above it can be concludedthat STAP induces unpredictable bias into receivers whichcauses errors in the estimation of code and carrier phaseand the situation is even worse when antennas and channelsare nonideal However thanks to the unbiased characteristicof Doppler frequency the estimation of frequency in oursimulation remains stable no matter how the interferencecircumstance changes

5 Conclusion

This paper analyzes the bias induced by STAP of the phaseof the GNSS antenna array receiver and proves the unbiasedcharacteristic of Doppler frequency of it Simulation resultsshow that the distortion of phase is unpredictable and itwill be even worse when the nonideal antennas are usedor the interference circumstance changes On the contrarythe Doppler frequency remains unbiased in these situationswhich can be used to estimate an unbiased carrier phaseto enhance the accuracy of positioning Since a good-performance low-complexity and real-time bias mitigationis difficult to be realized by traditional methods the Doppler-aid carrier phase correctionmay be a simple and effective wayto achieve this goal

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 41604016

References

[1] Z Lu J Nie F Chen and G Ou ldquoImpact on antijammingperformance of channel mismatch in GNSS antenna arraysreceiversrdquo International Journal of Antennas and Propagationvol 2016 Article ID 1909708 9 pages 2016

[2] T Marathe S Daneshmand and G Lachapelle ldquoAssessmentof measurement distortions in GNSS antenna array space-timeprocessingrdquo International Journal of Antennas and Propagationvol 2016 Article ID 2154763 17 pages 2016

[3] A J OrsquoBrien and I J Gupta ldquoMitigation of adaptive antennainduced bias errors in GNSS receiversrdquo IEEE Transactions onAerospace andElectronic Systems vol 47 no 1 pp 524ndash538 2011

[4] Y C Chuang et al ldquoPrediction of antenna induced biases forGNSS receiversrdquo in Proceedings of the International TechnicalMeeting of the Institute of Navigation SanDiego CAUSA 2014

[5] S K Kalyanaraman and M S Braasch ldquoGPS adaptive arrayphase compensation using a software radio architecturerdquo Jour-nal of the Institute of Navigation vol 57 no 1 pp 53ndash68 2010

[6] U S Kim D S De Lorenzo D Akos J Gautier P Engeand J Orr ldquoPrecise phase calibration of a controlled receptionpattern GPS antenna for JPALSrdquo in Proceedings of the PLANS -2004 Position Location andNavigation Symposium pp 478ndash485April 2004

[7] D S De Lorenzo Navigation Accuracy and Interference Rejec-tion forGPSAdaptiveAntennaArrays StanfordUniversity 2007


[8] S Caizzone G Buchner and W Elmarissi ldquoMiniaturizeddielectric resonator antenna array for GNSS applicationsrdquoInternational Journal of Antennas and Propagation vol 2016Article ID 2564087 10 pages 2016

[9] I Sisman and K Yegin ldquoReconfigurable antenna for jammingmitigation of legacy GPS receiversrdquo International Journal ofAntennas andPropagation vol 2017 Article ID4563571 7 pages2017

[10] S Backen DM Akos andM L Nordenvaad ldquoPost-processingdynamic GNSS antenna array calibration and deterministicbeamformingrdquo in Proceedings of the 21st International TechnicalMeeting of the Satellite Division of the Institute of NavigationION GNSS 2008 vol 3 pp 1311ndash1319 September 2008

[11] C M Church and I J Gupta ldquoCalibration of GNSS adaptiveantennasrdquo in Proceedings of the 22nd International TechnicalMeeting of the Satellite Division of the Institute of Navigation2009 ION GNSS 2009 pp 2735ndash2741 2001

[12] S Daneshmand N Sokhandan M Zaeri-Amirani and GLachapelle ldquoPrecise calibration of a GNSS antenna array foradaptive beamforming applicationsrdquo Sensors vol 14 no 6 pp9669ndash9691 2014

[13] A J OrsquoBrien J Andrew and I J Gupta ldquoOptimum adaptivefiltering for GNSS antenna arraysrdquo in Proceedings of the 21stInternational Technical Meeting of the Satellite Division of theInstitute of Navigation ION GNSS 2008 pp 1301ndash1310 Septem-ber 2008

[14] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[15] G Carrie F Vincent T Deloues D Pietin and A RenardldquoA new blind adaptive antenna array for GNSS interferencecancellationrdquo in Proceedings of the 39th Asilomar Conference onSignals Systems and Computers pp 1326ndash1330 November 2005

[16] S Mehmood Z U Khan F Zaman and B Shoaib ldquoPerfor-mance analysis of the different null steering techniques in thefield of adaptive beamformingrdquo Research Journal of AppliedSciences EngineeringampTechnology vol 5 no 15 pp 4006ndash40122013

[17] M D Zoltowski and A S Gecan ldquoAdvanced adaptive nullsteering concepts for GPSrdquo in Proceedings of the 1995 MilitaryCommunications Conference (MILCOM) Part 1 (of 3) vol 3 pp1214ndash1218 November 1995

[18] A Gecan and M Zoltowski ldquoPower minimization techniquesfor GPS null steering antennardquo in Proceedings of the 8thInternational Technical Meeting of the Satellite Division of TheInstitute of Navigation (ION GPS 1995) 1995

[19] K Elliott and C Hegarty Understanding GPS Principles andApplications Artech House 2005

[20] R T Compton ldquoThe power-inversion adaptive array conceptand performancerdquo IEEE Transactions on Aerospace and Elec-tronic Systems vol 15 no 6 pp 803ndash814 1979

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Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


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wwwhindawicom Volume 2018

Advances in

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calibrate the bias [12 13] but the computational complexityprevents them from the real time application It seems thathigh accuracy low complexity and real time response areincompatible in the bias mitigation of STAP

Different from previous works this paper analyzes theperformance of Doppler frequency during the processing ofantenna arrays Logically neither the response of antennasand channels nor the adaptive weights will change the carrierfrequency of signal In this paper the unbiased characteristicof Doppler frequency of GNSS signal in STAP is proved byusing the auto correlation function (ACF) and the simulationresults also show that the changing of interference leads tothe uncertain variation of phase but it will not affect the esti-mation of Doppler frequency of receiver When the responseof channel is not ideal the unpredictable distortion of phaseis more obvious while the estimation of Doppler frequencyis still unbiased Since the integration of frequency is carrierphase the unbiased characteristic of Doppler frequency canbe critical to high precision GNSS applications Althoughthe deduction is based on the GNSS signal the unbiasedcharacteristic is common in array processing It means thatwe can correct the phase error induced by STAP or calibratethe antenna instantaneously using the Doppler frequencywithout any extra complexity of algorithm or hardware

The rest of this paper is organized as follows First thearray model is established and a brief description of theantenna induced bias in phase is given in Section 2 Thenthe unbiased characteristic of theDoppler frequency ofGNSSsignal is proved in Section 3with the help ofACF In Section 4simulation results are presented to compare the phase errorand the frequency error which further proves the unbiasedcharacteristic of Doppler frequency Finally the conclusion ismade in Section 5

2 Array Model

Although STAP is an effective way to mitigate the inter-ference in GNSS signal [14] biases induced by antennasand algorithms are not negligible for precision applicationsespecially when a large antenna array with complicatedfiltering is considered [3] This section presents a model ofarray processing for describing errors induced by STAP andproving their dependency on interference circumstances

The antenna model is depicted in Figure 1 In this modelK individual antenna element with an M-tap FIR filterfollowing is considered 119860119896(119891 120579 120593) represents the systemresponse of the kth element in (120579 120593) direction (120579 and 120593stand for azimuth and elevation angles respectively) Akis also affected by the frequency of received signals (theeffect of mutual coupling is not considered) 119865119896(119891) stands forthe electronic which downconverts the signal to basebandand perform analog-to-digital conversion The output digitalsignal of each element is then filtered by an M-tap adaptivefilter and the frequency response of each FIR filter is denotedby 119882119896(119891) Then the outputs of the filters are summed forpostprocessing

The complex weight in STAP filter can be represented bya stack vector

w =[[[[[[[

w1w2

w119870

]]]]]]]

(1)

w119896 is an 119872 times 1 vector corresponding to the kth filter

w119896 = [1199081198961 1199081198962 sdot sdot sdot 119908119896119872]T (2)

where 119908119896119898 is each adaptive weightThe instantaneous digital output snapshot on the taps of

the kth front-end channel is denoted by

x119896 [119899] = [119909119896 [119899] 119909119896 [119899 minus 1] sdot sdot sdot 119909119896 [119899 minus 119872 + 1]]T (3)

119909119896[119899] is the received signal on the kth element which containsdesired signal 119904119896[119899] undesired interference signal 119895119896[119899] andnoise 120578119896[119899]

119909119896 [119899] = 119904119896 [119899] + 119895119896 [119899] + 120578119896 [119899] (4)

The snapshot on each filter is combined into a received signalstack vector

x [119899] =[[[[[[[

x1 [119899]x2 [119899]

x119870 [119899]

]]]]]]]

(5)

The STAP makes the weighted sum of signal vector bymultiplying it with the weight vector

119910 [119899] = wTx [119899] (6)

By applying some criterions [14] such as powerminimization[15 16] multiple constrained minimum variance [17] andminimum mean square error [18] the power of interferencesignal can be effectively restrained after the adaptive weightedsummation

However characteristics of antennas and channels as wellas STAP algorithms cause biases to the output signal Tomeasure them it is reasonable to analyze their effects on theGNSS receiver cross-correlation

The GNSS receiver correlators perform the cross-correlation by multiplying local CA code replicas 119903[119899] withthe received signal 119904[119899] and then doing the average of thecorrelation time 119873119903 [19]

119877 (120591) = 1119873119903119873119903sum119899=1

119904 [119899 + 120591] 119903 [119899] (7)

119877(120591) named as the auto correlation function is the output ofthe correlator where 120591 is the code delay between the receivedsignal and the local replica

In STAP the correlation process can be treated as thecorrelation between the signal of each FIR filter tap and the


sum

A1(f ) A2(f ) AK(f )

F1(f) F2(f) FK(f)

W1(f) W2(f) WK(f)



119877119910119889 (120591) = 1119873119903119873119903sum119899=1

119910 [119899 + 120591] 119903 [119899]

=119870119872

sum119897=1

1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 119877 (120591 + 120591119897) + 1198951015840

+ 1205781015840

(8)







119894 [119899] = 119886119863 [119899] 119877 (120591) cos [2120587119891119890119899 + 120601119890] (9)

119902 [119899] = 119886119863 [119899] 119877 (120591) sin [2120587119891119890119899 + 120601119890] (10)



119891119888119886119903119903 = 119891119897119900119888119886119897 + 119891119890 (11)

119891119889 = 119891119888119886119903119903 minus 1198910 = 119891119897119900119888119886119897 + 119891119890 minus 1198910 (12)



1198942 [119899] = 1198772 (120591) cos2 [2120587119891119890119899 + 120601119890] (13)

1199022 [119899] = 1198772 (120591) sin2 [2120587119891119890119899 + 120601119890] (14)

Further we set

119911119903 [119899] = 1198942 [119899] minus 1199022 [119899] = 1198772 (120591) cos [4120587119891119890119899 + 2120601119890] (15)

119911119894 [119899] = 2119894 [119899] 119902 [119899] = 1198772 (120591) sin [4120587119891119890119899 + 2120601119890] (16)



119911 [119899] = 119911119903 [119899] + 119895119911119894 [119899]= 1198772 (120591) cos [4120587119891119890119899 + 2120601119890]

+ 1198951198772 (120591) sin [4120587119891119890119899 + 2120601119890](17)



119891119890 = 12findmax (FFT (119911 [119899])) (18)



119894119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) cos [2120587119891119890119899 + 120601119890 + 120601119897] (19)

119902119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) sin [2120587119891119890119899 + 120601119890 + 120601119897] (20)

where

119887119897 = 1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 (21)



1198942119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotcos [2120587119891119890119899 + 120601119890 + 120601119901]

(22)

1199022119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(23)

119894119886119903119903119886119910119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(24)


119911119903 119886119903119903119886119910 [119899] = 1198942119886119903119903119886119910 [119899] minus 1199022119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(25)

119911119894 119886119903119903119886119910 [119899] = 2119894119886119903119903119886119910 [119899] 119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(26)

and

119911119886119903119903119886119910 [119899] = 119911119903 119886119903119903119886119910 [119899] + 119895119911119894 119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

+ j119870119872

sum119894=1

119870119872

sum119895=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(27)


10038161003816100381610038161003816119911119886119903119903119886119910 119897119901 [119899]10038161003816100381610038161003816 = 119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) (28)

120601119886119903119903119886119910 119897119901 = 2120601119890 + 120601119897 + 120601119901 (29)








A1

A2

A3 A4

d

d d

x

y
















3Taps5Taps7Taps




Cod

e Err

or (m

)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 3

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus10

0

10

20

30

Freq

uenc

y Er

ror (

Hz)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 4





Table 2



3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘


3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)



ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )



Figure 5





Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



sum

A1(f ) A2(f ) AK(f )

F1(f) F2(f) FK(f)

W1(f) W2(f) WK(f)



119877119910119889 (120591) = 1119873119903119873119903sum119899=1

119910 [119899 + 120591] 119903 [119899]

=119870119872

sum119897=1

1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 119877 (120591 + 120591119897) + 1198951015840

+ 1205781015840

(8)







119894 [119899] = 119886119863 [119899] 119877 (120591) cos [2120587119891119890119899 + 120601119890] (9)

119902 [119899] = 119886119863 [119899] 119877 (120591) sin [2120587119891119890119899 + 120601119890] (10)



119891119888119886119903119903 = 119891119897119900119888119886119897 + 119891119890 (11)

119891119889 = 119891119888119886119903119903 minus 1198910 = 119891119897119900119888119886119897 + 119891119890 minus 1198910 (12)



1198942 [119899] = 1198772 (120591) cos2 [2120587119891119890119899 + 120601119890] (13)

1199022 [119899] = 1198772 (120591) sin2 [2120587119891119890119899 + 120601119890] (14)

Further we set

119911119903 [119899] = 1198942 [119899] minus 1199022 [119899] = 1198772 (120591) cos [4120587119891119890119899 + 2120601119890] (15)

119911119894 [119899] = 2119894 [119899] 119902 [119899] = 1198772 (120591) sin [4120587119891119890119899 + 2120601119890] (16)



119911 [119899] = 119911119903 [119899] + 119895119911119894 [119899]= 1198772 (120591) cos [4120587119891119890119899 + 2120601119890]

+ 1198951198772 (120591) sin [4120587119891119890119899 + 2120601119890](17)



119891119890 = 12findmax (FFT (119911 [119899])) (18)



119894119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) cos [2120587119891119890119899 + 120601119890 + 120601119897] (19)

119902119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) sin [2120587119891119890119899 + 120601119890 + 120601119897] (20)

where

119887119897 = 1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 (21)



1198942119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotcos [2120587119891119890119899 + 120601119890 + 120601119901]

(22)

1199022119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(23)

119894119886119903119903119886119910119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(24)


119911119903 119886119903119903119886119910 [119899] = 1198942119886119903119903119886119910 [119899] minus 1199022119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(25)

119911119894 119886119903119903119886119910 [119899] = 2119894119886119903119903119886119910 [119899] 119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(26)

and

119911119886119903119903119886119910 [119899] = 119911119903 119886119903119903119886119910 [119899] + 119895119911119894 119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

+ j119870119872

sum119894=1

119870119872

sum119895=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(27)


10038161003816100381610038161003816119911119886119903119903119886119910 119897119901 [119899]10038161003816100381610038161003816 = 119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) (28)

120601119886119903119903119886119910 119897119901 = 2120601119890 + 120601119897 + 120601119901 (29)








A1

A2

A3 A4

d

d d

x

y
















3Taps5Taps7Taps




Cod

e Err

or (m

)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 3

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus10

0

10

20

30

Freq

uenc

y Er

ror (

Hz)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 4





Table 2



3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘


3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)



ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )



Figure 5





Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119911 [119899] = 119911119903 [119899] + 119895119911119894 [119899]= 1198772 (120591) cos [4120587119891119890119899 + 2120601119890]

+ 1198951198772 (120591) sin [4120587119891119890119899 + 2120601119890](17)



119891119890 = 12findmax (FFT (119911 [119899])) (18)



119894119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) cos [2120587119891119890119899 + 120601119890 + 120601119897] (19)

119902119886119903119903119886119910 [119899] =119870119872

sum119897=1

119887119897119877 (120591 + 120591119897) sin [2120587119891119890119899 + 120601119890 + 120601119897] (20)

where

119887119897 = 1003816100381610038161003816119886119897 (119891 120579 120593)1003816100381610038161003816 1003816100381610038161003816119891119897 (119891)1003816100381610038161003816 10038161003816100381610038161199081198971003816100381610038161003816 (21)



1198942119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotcos [2120587119891119890119899 + 120601119890 + 120601119901]

(22)

1199022119886119903119903119886119910 [119899] =119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(23)

119894119886119903119903119886119910119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [2120587119891119890119899 + 120601119890 + 120601119897] sdotsin [2120587119891119890119899 + 120601119890 + 120601119901]

(24)


119911119903 119886119903119903119886119910 [119899] = 1198942119886119903119903119886119910 [119899] minus 1199022119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(25)

119911119894 119886119903119903119886119910 [119899] = 2119894119886119903119903119886119910 [119899] 119902119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(26)

and

119911119886119903119903119886119910 [119899] = 119911119903 119886119903119903119886119910 [119899] + 119895119911119894 119886119903119903119886119910 [119899]

=119870119872

sum119897=1

119870119872

sum119901=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotcos [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

+ j119870119872

sum119894=1

119870119872

sum119895=1

119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) sdotsin [4120587119891119890119899 + 2120601119890 + 120601119897 + 120601119901]

(27)


10038161003816100381610038161003816119911119886119903119903119886119910 119897119901 [119899]10038161003816100381610038161003816 = 119887119897119887119901119877 (120591 + 120591119897) 119877 (120591 + 120591119901) (28)

120601119886119903119903119886119910 119897119901 = 2120601119890 + 120601119897 + 120601119901 (29)








A1

A2

A3 A4

d

d d

x

y
















3Taps5Taps7Taps




Cod

e Err

or (m

)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 3

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus10

0

10

20

30

Freq

uenc

y Er

ror (

Hz)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 4





Table 2



3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘


3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)



ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )



Figure 5





Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





A1

A2

A3 A4

d

d d

x

y
















3Taps5Taps7Taps




Cod

e Err

or (m

)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 3

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus10

0

10

20

30

Freq

uenc

y Er

ror (

Hz)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 4





Table 2



3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘


3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)



ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )



Figure 5





Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



3Taps5Taps7Taps




Cod

e Err

or (m

)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 3

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus10

0

10

20

30

Freq

uenc

y Er

ror (

Hz)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 4





Table 2



3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘


3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)



ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )



Figure 5





Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Table 2



3 taps -0907m-0936∘

-0744m-16227∘

-0904m-20362∘

-2124m12110∘

-0877m-0266∘

-1114m-5195∘

5 taps -1233 m-0864∘

-1081m-16199∘

-1210m-18508∘

-2446m15585∘

-1211m-0094∘

-1439m-4064∘

7 taps -1247m-0773∘

-1098m-16135∘

-1216m-16948∘

-2264m17682∘

-1214m-0290∘

-1410m-3329∘


3 taps 0041m0382∘

0041m0650∘

0088m0479∘

0188m1080∘

0245m0550∘

0532m12527∘

5 taps 0036m0507∘

0036m0589∘

0086m0575∘

0200m1270∘

0228m0714∘

0530m13193∘

7 taps 0056m0660∘

0040m0667∘

0078m0610∘

0159m1499∘

0200m0817∘

0450m13543∘

ch1ch2

ch3ch4

times 104

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Am

plitu

de R

espo

nse (

dB)



ch1ch2

ch3ch4

times 104

minus30

minus20

minus10

0

10

20

30

Phas

e Res

pons

e(∘ )



Figure 5





Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Table 3



3 taps 0242m96960∘

1803m-159035∘

1645m-156356∘

3210m-83518∘

1029m1545∘

1593m-60829∘

5 taps 0108m68139∘

0901m-160019∘

0817m-158376∘

3016m-91127∘

2247m16234∘

1408m-66017∘

7 taps 0126m32475∘

0756m-159938∘

0706m-158293∘

2541m-94550∘

1805m46786∘

1179m-68082∘



3 taps 1062m146941∘

0407m0609∘

0096m14355∘

0455m19584∘

2183m2151∘

1593m119025∘

5 taps 1170m162459∘

0260m0723∘

0080m13630∘

0629m134521∘

1908m178308∘

1478m142145∘

7 taps 1193m173186∘

0227m0711∘

0079m13246∘

0552m34942∘

2121m172617∘

1403m141533∘

3Taps5Taps7Taps

200 400 600 800 10000Time (ms)

minus2

minus1

0

1

2

3

4

Cod

e Err

or (m

)


3Taps5Taps7Taps

minus200

minus150

minus100

minus50

0

50

100

150

200

Phas

e Err

or(∘)

200 400 600 800 10000Time (ms)


Figure 6







3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



3Taps5Taps7Taps

minus80

minus60

minus40

minus20

0

20

40

60

80Fr

eque

ncy

Erro

r (H

z)

200 400 600 800 10000Time (ms)


3Taps5Taps7Taps

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Freq

uenc

y Er

ror (

Hz)

10 20 30 40 50 60 70 80 900Time (ms)


Figure 7



5 Conclusion


Data Availability




Acknowledgments


References
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

The Unbiased Characteristic of Doppler Frequency in GNSS ... · ResearchArticle The Unbiased Characteristic of Doppler Frequency in GNSS Antenna Array Processing YuchenXie ,ZhengrongLi