A"D-Ai46 580 DERIVATION OF RECURSIVE DIGITAL FILTERS BV THE 1/1STEP-INVARIANT AND THE RAM..(U) DEFENCE RESEARCHESTABLISHMENT YALCARTIER (QUEBEC) A MORIN ET AL.

UNCLASSIFIED MAY 84 DREV-R-4325/84 F/6G 12/1 N

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DREV REPORT 4325/84 CRDV RAPPORT 4325/84FILE: 3621J-005 DOSSIER: 362IJ-005MAY 1984 MAI 1984

AD-A 146 500

DERIVATION OF RECURSIVE DIGITAL FILTERS BY THE

STEP-INVARIANT AND THE RAMP-INVARIANT TRANSFORMATIONS

A. Morin

P. Labb6

SELECTE f-OCT 0 4 1984

WE

BUREAU.- REC"ERCHE FT DIIVELOPPEMENT RESEARCH AND DEVELOPMENT BRANCH -'

MINISTIIRE 01 LA DIIFENSE NATIONALE DEPARTMENT OF NATIONAL DEFENCECANADA CANADA *

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UNCLASSIFIE

d1ital flesfrom continuous-tim filterswhnterioftesmqling frequency to tepl~rqec osal b ofiinsothe proposed digital filters, whc r eie rmtestep adreapinvariance of the corresponding analog filters, have been determinedfor real and complex poles. For higher-order filters realized in aparallel form, It Is demonstrated that the discrete-tim transfer func.-tion of digital filters obtained by the step and rmp invariance can bederived directly from the standard x-transfommetion or from the, partialfraction ezpansion of the coutimaous-tim transfer function. Thediscrete-tim transfer functions of the step- and raw-Invariant f 11-torn realized In a cascade form have also been derived. Finally, theperformance of these methods Is demonstrated by plotting the magnitudeand phase responses of the first- and second-order digital filters.For highorder hutterworth and elliptic filters, the magnitude re-sponses of step-invariant end rap-invariant fiters are compared withthose obtained by usual methods such as the standard s and the bilineartransformet ions.

Ce document dMerit down adthodes powr coecewoir des filtresnuafriques rkcursifs 1 partir des filtree mslogiques lasque le rap-port entre Ie taux d'dchantillonnage et In frqence doi ple et fal-ble. lee coefficients des filtree -wI ques propWW~, qul provien-nent do Itinvariance A l'dchelon et A ls ramps des filtres analogiqascorrespondants, ont U&S dterainde pour lese lk e et ce.plexes.Poir lea filtres d'ordres plus Glevs ralisb dans us rgoau paral-lale, i1 set dnontr& quo Is fonction do trensfert en s des filtresnunmriques caract~rieSs per 1' invariance A l'dcbelon at A In raw* poutGtre Oduite direatmnt do Us treneforude en s on do la dcompositionen fractions partielles de I& foaction. do transfert des filtres analo-Siques. On a asel dult la fonction do trasfert en a des I iltresinvarias & l'dchelon, et & ls raw"e po me ralieation dam. me rG6-sam en aftie. Finslesent, la perform=*c do coo xdtbodes eat dison-tide en tragant lee rponsesaen amplitude et en pbese des filtres nvt&-riques doi preeier et dui dowihae ordre. four des filtres Buttewrthet elliptiques; doidres dle~e, on compare Is idponee en amplitude desfiltres Invariants I 1' dchelon at I Is rmpe avec cello obtenue par deufthodes courentes tells quo Is transfordfe en a et I& transformG*bibindaire.

Mili

UICLASSII ID

1

1.0 IEMROUCI

As with analog filters, the approximation step in the design of

digital filters is the process whereby a realizable transfer function

that satisfies prescribed conditions is obtained. Several methods

permit the derivation of recursive digital filters from the continuous-

time transfer function of their analog counterpart when the ratio of the

sempling frequency to the pole frequency (fs/fp) is sufficiently

large. The metbods constitute textbook material and include the

impulse-invarlant, watched z and bilinear transformations. On the other

hand, they suffer from serious drawbacks such as aliasing and frequency

warping, which lead to Inappropriate approximations when f.If is

The two approximation methods described in this report, the

step-invariant and rasp-Invariant transfomations, fill this gap by

providing valid matches even wbn fs/fp Is mll. The proposed

methods are an extension of the impulse-Invarat transformation. With

this technique, which is also called the standard a-transformation, the

response of the derived digital filter to an Impulse is identical to

that of the sapled-impulse response of the continuous-time filter. In

the step-Invariant transformation, the response of analog filters Is

found by assuming that the input is approximated by a sequence of steps

whose duration is set to the smpling period and whose amplitude

corresponds to the instantaneous value of the input signal at the time

of sampling. In the ramp-Invariant transformation, the input signal

consists of a sequence of ramps that join the sampled values. This

procedure leads to the determination of the discrete-time transfer

function of digital filters, wbose response is identical to that of the

reference analog filter in relation to the input signal (step or ramp).

U3CLASSFI3D2

In this report, discrete-time transfer functions were determined

for first- and second-order filters with real and complex poles. lkp-pings from the continuous-time (Laplace transform) and discrete-time

(u-transform) transfer functions were derived for the step- and ramp-

invariant transformations. The methods were generalized for filters of

any orders with poles of any multiplicity by considering partial frac-

tion expansions in terms of first- and second-order transfer functions.

This approach leads directly to a parallel implementation. fhe discrete-

time transfer function of the step-invariant and ramp-invariant filters

realized in a cascade form has also been derived.

This theoretical approach ws validated by plotting the magnitude

and phase response of first- and second-order digital filters derived by

the step- and ramp-invariant transformations. These responses are com-

pared with those of the analog filters and the impulse-invariant digital

filters. Finally, for high-order Butterworth and elliptic filters, the

magnitude responses of the step- and remp-invariant filters are compared

with those obtained by the impulse-invariant transformation, the bilin-

ear transformation and the original analog filter.

This work was performed at DREV between September 1982 and

February 1983 under PCI 21J05, Guidance and Control Concepts.

*qti

2. ~chd Trfosom.. . . . . . . . . . . . . . .

2 .3 KInle r im ransfor mation.. . . . . . . . 0 a & 4

3.0 WTimar~IE Tansforma&tion . . . . . . . . . . . . . . . . 7

3.1 lirst-Oer Term (Taal Poles) 9 * . e . e e * 73.2 Secod-Order Tem (Colz Poles) . . . . . * . * 0 * .

4.0 RAMP-IN VARIANT TASOU ION o * 9 * # o * * * * * 9 o & a 11

5.0 GUAIZ a--a TRANSFORMATlION . . . . . . . e e * o * * o . 13

6.0 cowmiSon or 113 mnUTOD e 17

6.1 Comparison of the Notbods for Deriving FIrst-sAd -Secm -Order Filters o * 9 * e e % * * * e a * 18

6.2 Comparison of the Usthods for Deriving Nigher-Order Filters * * * e e * . * * @ * * * * * 9 * * 9 e v 27

7.0 CONCLUSION 0 . * . * * & 0 a & * 0 0 a . 0 0 0 0 * 0 0 0 0 0

8.0 33V333NCU . 0 0 0 . * 0 0 0 . 0 0 0 0 . 0 0 a 0 0 * 0 0 0 0 41

FIGMlS 1 to 15

APPINDIZ A - lepresentation of the Cotiauoms-TiaeTransfer Function of Amalol Filters * & . * e * 42

APPENDIX 3 - Sealisation of the Step- and Samp-IawarisutDiital. Filters In a Cascade form . . . . . . . 45

APPEDIX C - Coutianuius-Tias Transfer Funct ions of theButteruurtb and Elliptic Filters a . . ... 46

UNCLASSIFIED3

2.0 BACKGROUND

Filtering is a process by which the frequency spectrum of a

signal can be modified, reshaped, or manipulated according to some

desired specification. It may imply amplifying or attenuating a range

of frequency components, rejecting or isolating one specific frequency

component, etc.

The digital filter is used to process discrete-time or sampled

signals. It can be implemented by means of software or dedicated

hardware, and it can be represented by a network comprising a collec-

tion of interconnected elements. The analysis of a digital filter

determines the response of the filter network to a given excitation.

Its design, on the other hand, consists in synthetizing and implemen-

ting a filter network so that a set of prescribed excitations results

in a set of desired responses.

The design of a digital filter comprises at least two steps:

the approximation and realization ones. The approximation step is the

process of generating a transfer function satisfying a set of specifi-

cations that may concern the amplitude, phase, and possibly time-

domain response of the filter. The realization step is the process of

converting the transfer function into a filter network using inter-

connected unit delays, adders and multipliers.

In the approximation step, the wealth of knowledge acquired in

the design of analog filters may be transposed to the design of their

digital counterparts. The discrete-time transfer function (expressed

in z-transform) is derived from the continuous-time transfer function

(expressed in Laplace transform) of a reference analog filter of known

characteristics. Several methods are currently available that perform

such approximations: the invariant-impulse, matched z- and bilinear

transformations.

UNCLASS IFIED4

Once the discrete-time transfer function of the digital filter

has been obtained, this transfer function can be converted into a fil-

ter network. The realization of this filter network will not necessar-

ily operate exactly as prescribed because the previous operations were

carried out assuming that the components to be used are of infinite

precision. These effects need to be studied and corrected as neces-

sary. The realization step of the digital filter is fully described

in many textbooks such as Refs. 1 and 2. The transfer function can be

broken down into simpler transfer functions and be realized directly.

However, if the transfer function is given as a sum of partial frac-

tions or as a product of first- and second-order factors, it can be

realized either in parallel or cascade form without any further

modifications.

2.1 Impulse-Invariant Transformation

The Lmpulse-invariant transformation (Refs. 1 to 5) yields a

digital filter with an Impulse response equal to the sampled impulse

response of the continous filter. In this transformation, the input

signal x(t) is sampled at frequency fs" The response of the analog

filter is found by assuming that it is excited by a sequence of im-

pulses equally spaced at intervals T (the sampling period). The am-

plitude of impulses is equal to that of the sampled values of x(t).

Given that the analog filter has only simple poles, its transfer

function is written as:

HAc) A0 + + - + [i.

The details of the derivation of this equation can be found in

Appendix A.

N%

k, 7SII3

5

The Japls'-yarant transfovustiou I(s) of eq. 1 gives

1(s)-A + T +i Io.Aa T r2]0 1 -(2 -atco Is -1 + 241 T 2

wbov e 3 aT T being the ampling period

A,- T Ku,

Ai - 2 IF-[, coo 1 T + 2 1 OiuP1 T]

%t- C1 + C1 - 2 Bo(C 1L)

K2 1 - CC,- CI) - 2 U(C 1 )

di ai - 0

d* conjugate of d,

IMe mapping relation betwen the s plane and the z plane can bededrsced directly from eq. 2. it is written as

~ -- iT [3]

Wei mothod yields valid mtces with the corresponding analogSystem, OWLY at high empling rates and It Is satisfactory only hn the8800 asoSYStin Is sufficiently bend limited.

2.2 Nstcbei a-TransibmastIon

The mtched s-traasformstion Is a technique based on mapping the

poles and zeros, of the contismuo-l filter. It Is performed an the

cascade form of the transfer function of analog filters. As shown In

Appendix A, this form Is written as

HS

H(s) - K Iin 1j ( [4]a Nt

II (a6+ i

The discrete-time transfer function of eq. 4 Is given In Ref. 5 as

L II (z-e ia

where L Is an Integer whose valu Isis equal to the number of zeros of

H(s) at a - -- In this transformation, the poles of digital filters

are Identical to those obtained by the Impulse-Invariant transform-tics. However, the zeros do not correspond. In general, the use of

the impulse-invariant or bilinear transformation is preferable to that

of the matched u-transformation (Ref. I).

2.3 lilnear Transformation

A bilinear transformation Is obtained by substituting a in the

continuous-time transfer function H(s) by

*5 !I21 B- [6]

This transformation, which is used to circumvent the alasing

problem of the standard s-transformation, results in a nonlinear warping

1 i l i i

of the frequency scale between the coat Lnuous-tlue frequency and the

discrt-tium one, according to the relation

2 2tny 7

where *A Is the continuous-tine frequency variable sand %D to thediserete!-tm one.

3.0 STIP-INVAlIANT TIANSPONW1flO

The step-invariant transformation is an extension of theimpulse-invariant technique. 7he response of the analog system isfound by assumng that the Input x( t) Is approximated by a sequence ofsteps. The amlItud* of steps corresponds to the Instantaneous valueof x(t) at the time of sampling and their duration is equal to samplingperiod T. In this case, the Input signal Is written as

x(t) - 10 (nT) {Ia(t - uT) - pa(t - (n + 1) T))(S

where iA(t - aT) - 0 If t c WT and- 1 if t > aT.

In the following sections, an invariant-step transformation will bederived for first-order (real poles) and second-order (complex poles)term of the partial fraction expansion given by eq. 1.

3.1 First-Order Term (Real Poles)

Consider a real-partial fraction of the expansion given by eq. I

X (s) si+p(]

UNCLASSIFIED8

e an derive from eq. 9 a difference equation that will giveY[Ra + 1)?], the value of y(t) at time t + V and from x(nT) and

y(nT), the values of x(t) and y(t) at time tna. The transfer function

must be rewritten by setting the initial condition of the output

variable at time t n . Then, the filter io stimulated by a pulse whose

amplitude and duration are set to the instantaneous value of the input

signal at time tn and to the sampling period respectively. Substi-

tuting the input variable by the Laplace transformation of a pulse

function of amplitude x(nT) and iAltiplying this result with the trans-

fer function of the filter yields a new relationship. Finding its

inverse Laplace transformation and substituting t by (n+1)T gives the

value of the output at time tn + 1

If us take into account the initial condition of the variable

y(t) at tim tn(-nT), noted by y(nT) 6(t - nT) where

6(t - nT) - 1 if t -nT

- 0 elsewhere

Thus, the transfer function of eq. 9 is transformed as

y(s) , AX(s) + y(nT) a-nTs• + p a + p [o

Since the input variable at time tn is a pulse of amplitude

x(nT), its Laplace transformation Is

X(8) =(wIZ) (e-nTs - -(n + 1)T]

Substituting eq. 11 in eq. 10 and finding the inverse Laplace

transformation of this relation yields the following time function:

j i j ii i i 1"lX

nialy, let gs determine the value of y(t) at -time t a + by

replacing t by (a + 1)? In eq. 12. This. leads to

y( (a + 1)?] -Az('T) (1 - f1P) + y(nmT)e-P [13]

Equation 13 Is a finite difference equation whose discrete-time

transfer function is

jKj As) - e-PT5 1 [4

By comparing eqs. 14 and 9, It Is seen that the discrete-time transfer

function can be obtained from the continvous-time transfer function by

using the mapping relation

U/-s U [

3.2 Second-Order Ter (Complez Poles)

Now consider one comple-partial fraction of the expansion given

by eq. 1

C C

UCLBOVIN10

if we apply the substltution of eq.- 15 to a&h of the individual termsof eq. 16, we find

C -AT -1 C* -*T -1

-2 -1

L-(jaTcoaST)s -1 + [17]Ts

-2aT -aT -aTwbere KA 7- 1 0 -KIe coo 0T -K gins?0

It, (C/d) + (C*/d*)

K2 -(Ce/d* - /j

7be step-imwriant transformtion of analog filters described byeq. 1. Whose N real poles and NI complex poles are smple, is deter-amed by substituting eqs. 15 and 17 In eq. L. This gives

S*s Ao

112) %7 1 a -VtT 2-1

+ K~AI 2 + %iI[8

I - u-aIT co0T -1 + 2Ta-2

UNCLASSIFIED .

4.*0 LA14P-INVARIANT TRANSFORMATION

The z-transforn of this technique can be derived from the Laplace

transform by applying the same operations that yield the z-transformation

of the step invariance, the only difference being that the analog filter

at time tn is excited by a ramp such as:

1(t) - x[ (n - )T] + T~T - [n )T1 {[t -(n - )T] tg

p a(t -nT)I

If me apply the following procedure to a real-partial fraction

(eq. 9) of the expansion given by eq. 1:

1) Transformation of eq. 9 to eq. 10: this operation allows

us to consider initial conditions of the output variable at

t ime tn

2) Substitution of X(s) In eq. 10 by

x[( - )T]+ x(nT) -xI(n - )T11 ((n - )Ts -is 10

Ts 2 a.[0

which Is the Laplace transformation of the ramp function

(eq. 19) at time t

3) Solve eq. 10 to derive a relation as a function of time and-

replace t by aT to get the value of the output at time tn.

A difference equation that gives y(nT) from x(nT), x[(n-l)T] and

y [(n-l)T] can be derived. Its z-transformstion is

UNCLASSIFIED12

4! ( e-PTz - 1 -P( X-1

YC:) p 2TX(z) -ePTz - 1

In comparing eqs. 21 and 9, we deduce that the ramp invariance

can be obtained from H(s) by using the mapping relation

1 -plr-1-. .1

1 p p 2T 28 + p 1 -PT - 1 "'

For complex poles, we substitute the mapping relation of eq. 22

in each term of eq. 16 to find:

-1 -2c+ z - + K z -2 '"KC KDZ + z

-aT -1 -2aT -2 [3(-(2e cos 0 T)z + e z[-

where K, (C/d) + (C*/d*)

K 3 -[C/d2 + (C/d2 )*]/T

d d

-- TK 5 -- e (K3 cos T + K4 sin 0 T)

K-K- - aT e) L .KC 1, K3 (1 - 2 coos0T) +K 5

-aT co -20TKD(- K - cos B T- -

-K (2 e- coo 0 T) -2 ..

IC -2aT ( t + K ) + K

(K+L)%L .-i-

';", . -".I;.., """. , .'""""""'. . ,"i ".",". ," ... q-' , ,"_° .' ". ". ..- ," .•-". .".". '

* • •s . "*-. *%* *% ** *.a. *..-:- -- - -p.-; - "".* " " -""" ". -. . -'"''"" """" ' . .. .*"";" """" ""*. . * ". :._..- -

13

for shmple poles, the srp Ciasrsa tramefmatow. of aIIIIsogfilters charactsriusd by U ral poles amd. a complem poles gives:

R(S) AO+Y ApI p l )1IIPIt -PIT 9-1

-6 1 -T -26 71 - (20 Cos s+ al

5.0 CMuuaIAZI wP-u AIxLTO

Ina this chapter,, we will first establish a procedure thatderives the stop- and ramp-iuvariant transforma tions directly from theimpulse-inariant transformation. Then, this procedure will be used todetermne the step and rap invariances of mulog filters containing

multiple poles.

The mapping relation between the a and a planes that gives thestep Invariance Is from eq. 15

I (I/P) (I

This relation can he rewritten a

- )(....-L.... I/PT- _7_s____

where Z Is defined as the operator corresponding to the Invariant-

impulse response of analog systaIIIIIs or standard i-transformation. Also,

the mapping relation of the ram invariance (eq. 22) can be expressed as

"M

EAs

2Tm 02 (a+ P)

ThNs, it toasbus. that the step and rump ingviesmes am bedewiv from tde Impulse isvsulsmse. ThIs oerastion cossts is

sultip&7ig do tweete hu intift of an nou filter WO) by the

LSIeee temtios of tde mit-step fumetiemt (lie) or the milt-emp

function (lie'), fishe the atsr $-tre f-wtu ft athi W eisoltio.

emi, fi35117. dividing the resut by the stemimid -te hi s N I

the milt-se function *Miob to 11(I - 31 Or the mit-wep functions

The invatiant-step response $as) sad the invuwiut-raqi reponse

I(s) of an asales system se transf er fuMetis, Is UEs) oM thes be

put is the fomv

$as) a [zI! U8I (I - a)[26]

a T

Woi procedure allow the determination of the step ad rom

invarisuces of aalog filters shoe trasfer funct ion coutains mult iple

poles. In Appendix A, this transfer function is expressed ast

15

(+ I (. + d,)k + k [21I)- o (a + Pi) Ia I(+ i *L

hereN ad aI in the double sintion indicates the multiplicity

.of the real and complex poles.

The standard a-transform of these fuactioas is given in 3sf. 6

as:

I T e" l t(ai)z 1

-11+1 [291-

where D(a i ) Is defined by -

- 1 10 .0 0

21 1 1

-- 2)1

where ai e p IT and Do(ai) 1. For complex poles, P1 and al are0 -d LT

replaced by dI and bL respectively (bi - d

The derivation of the step invariance can be found by

substituting eq. 28 in eq. 26. This yields

UNCLASSIFIED16

SN-1 A A t aI D(a I)Sk , - Ik X-1 + P.p1 pi (I 1 )

I... +.11 b _ Dk z(b. )."1k;I k k -k k-9 bt -I I-:.d d d - (I , -)

I b'• *1 *

+Lk b 1 D,(b ) :"ii I N 1 -) + i ll ::d (I b I tx

[30]

The derivation of the ramp invariance consists In substituting

eq. 28 in eq. 17. This leads to

N -1N T i Ai Ai k(1-z )

A Lkik k Ck -1

Pi1 ,. T.

S+ - - [d-4"] + d-1[

I- A- dk1 T a D (a z-

+i i -I I

~ Lz-12 I 0 [ Clk b Di(b1 )

TT. z1 rO L1 k1.-i - I

*°.*° ;

1Ci C biDbi

.. .-

.'"S

Lk I I

. . . . . . . . • .. . -Q . . . .. - . -o • . o. o " .

5. • .%' % • ' , % . . ,5** . .% % .".% % % ' % " *% •5 ,% .5 *' ,• % ,%

*~~ [31]%.. .. * - . . . . ~ ~5~ ~5

17

Cbetters 3.0 to 5.0 derived the step- and ramp-invariant. digitalfilters train the partial fraction expansion of the ostanamet iestranse r functions. Wei does procedure leads directly toe& realizes-ties Is a parallel form. If the cascade or direct form is desired, it

a be obtained by substituting the mapping relations of eqs. 15 and 22Into the transfer function of analog filters expressed as a cascadeform of first- and seododr blocks*. The results are presented in

Apedix D.

6 .0 COWARIS 0F TOR NRODS

Use invariat-impulse, watched s- and bilinear transformationsor either of the too, methods developed In this report load to differentdiscrete-tim transfer functions and, In scm cases, to different real-isations. At sufficiently high fe/fp, their time, gain and Phaseresponses are nearly the se, and the step- and ram-Invariant tres-formations will be unnecessarily sophisticated. In this chapter, Itwill he abownn that, at Isa values of f 8Ifp, the sophistication inhor-out In those math- ode io necessary for transposing the character-istics of analog filters In the discrete-tims domain.

First, the approximation of the first- and second-order digitalfilters with the impulsem-, stop- and ramp-invariant transformationshave, been considered. The frequency responses In usgnitude and inphase of the transfer functions of the filters obtained by each sothodhave been produced for the second-order band-pass f ilter, the f irst-and second-order low-pass filters and the f irst- and second-order high-Pass f ilters. The frequency responses of the analog f ilters are givenas a reference while these of the digital filters are given for a setof three values of fe/f P: 100, 10 and 4. In all cases, the polef requenY f9 e nd the hain of the analog filter at f1 wre net toI Wkb and 0 4 respectively.

UNCLASSIFID18

The exercise was repeated for higher-order filters such as the

Butterworth and elliptic low-pass, high-pass, band-pass and band-reject

filters. In this case, only the frequency responses In magnitude were

produced. However, the impulse-invariant, bilinear, step-invariant and

ramp-invariant transformations were compared.

6.1 Comparison of the Methods for Deriving First- and Second-Order

Filters

This comparison shows the differences between these types of

digital filters in the tims, magnitude and phase responses when mall

f/f p ratios are used. It leads to the determination of the best

procedure for designing high-order filters since they are merely a

combination of first- and second-order terms.

First, let us consider a second-order band-pass filter with the

following specifications: quality factor Q 1 10, resonant frequency

fo M I kHz and gain at resonant frequency - 1. This Is the special

case of eq. A.5 where

(wolQ)s -'

H(s)- 2 /~a + (QQ)s + w 2

0 0

Figure 1 shows the performances of the three types of digital -"-,

filters for various sampling frequencies (fa 20 f0, 10 f0, 4 f and

2.5 f ) when the input Is excited by a step function of amplitude equal [--

to 1. Each time the ratio f /f is changed, the coefficients of the

filters are recalculated to maintain the resonant frequency constant.

From Fig. 1, we deduce that the step- and ramp-invariant procedures

simulate exactly the time response of the continuous filter. TheImpulse-invariant method generates an offset at the output of the

filter that increases when the sampling frequency decreases.

'P

p...4

p-,'.'

4, .4,%

UNCLASSIFIED19

Iupulse-invariant filter step- (or ramp-) invariant filter

.M1

smI'.02 a .02s

0 lkHz, f 20kHz

f 0l1k~z, fl10kHz

fl0kHz, f a .5kz

FTC= Cosparlo bewe h5tp o r )Ivratmto

FIGUR and th omrionpus-nan meetoeen the stp-(r m- iaresne toastep-function Input of a second-order band-pass filter

UNCLASSIFID20

The behaviour of the time response is explained by examning the

msnitude and phase responses of the three digital filters plotted in

Fig. 2 for a set of three sampling frequencies (fa - 100, 10 and 4 f ).

The responses of the corresponding analog filters are also plotted in

this figure. This provides a reference for determining the performance

of the digital filters. The ramp-invariant method produces a digital

filter with a frequency response in amplitude and in phase that corre-

sponds to the response of the continuous filter for a frequency band of

0 to f /2. In this region, its frequency response does not depend on

the sampling frequency. The step-invariant filter has a magnitude

response similar to that of the ramq-invariant filter. However, its

phase response does not match the desired response around the resonant

frequency. The difference between the analog and digital responses

does not exceed 30 as the sampling frequency is set to 4 fre The

response of the impulse-invariant filter is identical to that of the

analog filter only in the frequency band located around the resonant

frequency. Depending on the sampling frequency, the attenuation of the

filter in the frequency band mller than f0 becomes limited to a

fixed value rather than following the slope of 40 dB/decade. For

example, at fo/10, the digital filter gives an error In amplitude

that varies from 12 to 18 dB as the sampling frequency falls between

10 fo and 4 fe" A difference between the digital filter and its

analog counterpart Is also observed in the phase response. It can

attain 90 within the full bandwidth (f /2) when f is set to 4 f ea 0

Legend for Figures 2 through 6

............ outliuu-t lee filter

Digital filter uhen f• 100 kH

Digital filter when f - 10 kfz

-- Digital filter when f - 4 kHs

IL4

UNCLASSIFIED21

.6 31e0

---- -. - - - - F - ,>* - - f

-0.31 0-

200

22e

13e

0I23 4 2 4to 10 to 10 10 1 1 10 I0 10 10

*e 1 Y'... 6. IIS

3W

315P

a**

46.6.

IC 101 102 t 1 0 101 102 10

FIGURE 2 - Magnitude and phase responses of a second-order band-passdigital filter obtained by the impulse-invariant, step-invariant and ramp-invariant methods for various samplingfrequencies

e~ %P

UNCLASSIFIED22

I .o .... ......... ......... ........... .

-7 .5 g e° .. '

ft.'.

S.2. 3115

- t. 0 o ..010 10 10, 1

10

-32.!.........................................h. ..'0

S.8

270€

13 e

1.0 -20

-I. .. .. I . . . .. 1. . . . . 3 . . .'.. . . .. aa. . .. . . . .. . . .

*0 202OI l 0°

I0 O'1 a

00 .04e

-25 .. J

120 2 3 4

10 0 0 l0 0 10 a 10 10 10 10tO 10 1 4.

af. 300r~an I 4

-1.0 IF IN*

.\ % 13

0

4-35.0 1' :3-20.0

-25.0 e~4

10 I2 In1 o t ot10 1 0 0o 0 a 1 to 3 o

FIGURE 3 -Magnitude and phase responses of a first-order low-passdigital filter obtained by the Impulse-, step- and ramp-Invariant methods for various sampling frequencies

%4 %

;.

p* %f %

--% ... .... ... ... %h.. . .. .f~a %,* 4 -. -, .. - %,. --... . "- "-., - -4 . --." ' -- f-t.. *' . - .a '- "..f. . . .." ", . ". tf,

. ... , , , : , . .. .... . .. .:....:,: .:. ..:.:....-.f., .-t. :. .....-t.: :..:::. _. :-.a

UNCASIFIED.23

a..~~ AAH

-10.0 Iwo

-40.6 4e

00 2t o oto 10 ID 101 ~ 10 1 10 t

36.0 327e.

-19.0

-30. N go3

10.0 2e

0.0 2250.

0

-50.c1 0 I2340 234t0 10 ]a 10 10 to to 10 10 )a

FIGURE 4 - Magnitude and phase responses of a second-order low-passdigital filter obtained by the impulse-, step- and ramp-invariant methods for various sampling frequencies

.. . . .

- . . . .. . . . . . . . . . . * * . ~ - * * ..%

UNCLASSIFIED

4-.4

-20.

.- 3.* a- 2

i.,. .o... ... " -.

3A6

270

.28.9 / ; 'A, -0., ./,- -"

.,- U.. .. . .. . ... . :" •.

-30.0

•r' . . . . .. ,..." I,-.'aIA-.,

.3e

32

10 10 tO0 to to 1O0 t10 t10 1O" ".

-11.62700""-3,.0 me" 7° ':

-40.611P

""-N-,e ,..'-..

-.. .

0. 2 3 4 0 334

F.AA.,..Of No....'"qq-....

- 10.o 0 0too 1

-ia0 3le o0 b the Impuls- s0ep and ramp-

-na-7 0.o ' for .s ,"qun

100 101 102 Io 0o 2-'2

"'I-DI / 5 ID'- I I7 4VA A iPA4AAq A 4

FIGUR 5 antd n paersossof is-re ihps

digtalfilerobtine b th."p-e- te-an pinvaiantmethds fr vaioussamping requncie

A .A - A ' EAA~~A * ~ ~ *~ A-:

A,*.-A., . . A .,,.;,,s ~ A ~ * A .

.~~**A ~ % ' ~ A % AAAAA. % A A-- , A -[ A A--AA ::-''A - -

UNCLASSIFIED25

2.10 31A

10 .0 , 3 0

IS*S I~J270*A0 +

22e.q . .. E is

-60 0-• II00

-. -O-. -/, I-6 l -'.-'* I _ _ _ _ __o_ _ _ _ _ _ _ _

t0 10 10 10 1 1 10 10 10 to

V..

"Iij lieI 3l

400

10

to 10 10 30 I0 I0 toa 1 I0 I0a 3 I0 -

l".4

21.0 3"

• " .0 -2"70 ",

".o • "~~41.0 I 0€ '"'

1900-6 .0le

.0 0 0

-100.0 €o .

-120.O 5i; 0 4a°* 1 : -4 02

10 10 10 10 10 10 10 10 to 10

FIGUREI 6 - Magnitude and phase responses of a second-or~der high-pass -:

digital filter obtained by the Impuse-, tep- and ramp-.:Invariant methods for various spling fequencies

_ __ % % N.N

40.0.

0::3,.5

A.,

".-..' . .''' " ..- ". .". . ".".",- . ..", "•","'-•• - ".- * '-.-.' . '.' ' .....- ,. ""'" - '"..0. """" , 27.00 - .-. . q".'

. . o .= . % % % . % % . % . . . . •. •. •. •. •. . o . • .. •. . . . . .. . .. . . . . .q . . . . . ,,

'-.;- ;.- -. -, -.. -..' .0-..... *'.-.,. .-.,;: .. ' .'- 3.,;. 0..-,.. , ,.. ., . ... , .. :. - .,-.-.','.,',,,..'.,

UNCLASSIFID26

The discrepancies In the magnitude and phase responses also

exist for the low- and high-pas filters. Figures 3 and 4 Illustrate

these responses of the first- and second-order low-pass filters for the

step-, romp- and Impulse-invariant procedures when the sampling frequen-

cies are set to 100, 10 and 4 f p. The pole frequency and the gain of

the filters are maintained constant at I ka and I respectively. The

quality factor Q of the second-order filters Is set to 10. The Impulse-

invariant filter generates an error in magnitude that varies with the

sampling rate. When f is set at 4 f , the error is limited to 6 d for

the first-order filter and to 3 dB for the second-order filter In the

frequency band of 0 to f /2. On the other hand, the low-pass digital

filters from the step-invariant and rmp-invarlant mothods stimulate

exactly the characteristics of their analog counterparts for a frequency

band lower than fm/2, except that the phase of the second-order step-

invariant filter Is slightly disturbed around the resonant frequency.

Finally, the magnitude and phase response of first- and second-

order high-pass filters are illustrated in Figs. S and 6 for the ome

operating conditions as for the low-pass ones. For the first-order

model, the preservation of the analog filter's characteristics by the

ipulse-invariant method requires a high sampling rate ( 100 fp ). For

the second-order model, the analog filter's characteristics are pre-

served on a narrow frequency band. The attenuation of the amplitude

becomes limited to a constant value in the frequency band lower than

f /2 rather than following a constant slope of 40 dB/decade. The atten-

uation value and the starting frequency of this phenomenon vary with the

sampling rate. The step invariance preserves the characteristics of the

first-order high-pass filter within the frequency band of 0 to f /2 but,a V

at sall fa/fp ratios, a constant shift in the magnitude response Is Uri

observed. This shift varies in function of the sampling rate and, for

example, it Is about 8 dB at fe 4 f p In addition, an error in the

phase response exists around the pole frequency. For the second-order

filter, the constant amplitude attenuation encountered in the Impulse-

, - ml . AW"9 '4VmrI

~~ ---

UNCLASSI1I327

Inwariant filter is replaced by an attenuation reduced to a aloe of

10 dz/octave. The f ilter also generates a phase error of 90* in theuseful freqnucy band for sapling rates maller than 10 f p Theagaitude and phase responses of the first- sad mecond-order highpass

digital filters from the ramp invariance correspond to the responses ofthe highpas aalog filter, provided that the frequency band is lin-

ited to f /2.

Thus, step- and ramp- invariant filters give more accurate fre-

queacy response In magnitude and ia phase than the impulse-invariant

filter for all the cases of first- and second-order filters whens mall

I f Pratios are used. In this condit ion, the iMpulse invariance canbe performed only on the second-order low-pass filter. The step In-variance gives accurate frequency responses for low-pass and band-passfilters with small phase errors around the comlex poles. Finally, the

ramp invariance produces a close match between the frequency responsesof digital and analog filters.

6.2 Comparison of the Methods for Deriving Higher-Order Filters

The next set of figures (Figs. 7 to 15) shows the frequencymagnitude characteristics of digital filters obtained by the step-,

ramp- aad impulse-Invariant and the bilinear transformations of high-

order Butterworth and elliptic filters. For further details about

Butterworth and elliptic filter synthesis, see Ref. 7. In all exam-pies, the response of digital filters is presented for various sampling

frequencies (1000, 100, 10 ad 4 k~s) and It Is accompanied by the

j response of the corresponding analog filter. Figures 7 to 10 Illus-

trate fifth-order low-pass ad highpass Butterworth and elliptic

filters. The cut-off frequency and the gain of these filters are

maintained at I Hz and 1 respectively. In addition, for the elliptic

filters, the stop-band attenuation Is at least 40 dB and the pass-bandripple Is limited to 3 dB. The band-pass and band-reject Butterworth

if .'#

28

and elliptic filters of Fio. 11 to 14 wre derived frem a transform-

tioa performed eo the corresponding low-pass filters (Rsf. 7). Se

lmr- and upper-cutoff frequencies of these filters are located at 700

ki and 1.4 Wk. S poles and soros of these analog filters are given

In Appendix C. lh. cutoff frequency of the high- and low-pass filters

and the center frequency of the bad-pass and bend-reject ones are

noted by f p In addition, the step sad ranp invariances of analog

filters presented In this group of figures were performed on the

partial-fraction expansion which yielded a parallel realiztion.

This set of figures show, that the iupulse-invariant transfor-

mation can be applied only to Buttervorth low-pass and hand-pass

filters. In the first case, a close match betueen the analog and

digital filter is obtained. In the second case, the agnitude response

in the low-frequency band does not correspond to that of its analog

counterpart when the ratio f /f becomes smaller than 10. In all thespother examples, the Impulse-invariant mothod is unacceptable. For

elliptic low-pass and band-pass filters, the equiripple character of

the stop-band response ms destroyed by the aliasing effect. Finally,

the allasing renders this transformation entirely useless as a digital

band-reject or high-pass filter (Butterworth or elliptic).

The billnearly transformed filters are essentially identical to

the original analog filter when the ratio fs/fp Is greater than 10.

lovever, the poles of the digital filters are moved in relation to the

sampling frequency. This effect, called nonlinear frequency warping,

Is usually observed uhen the sampling frequency Is realler than 10 fp

Thus, this method does not correctly preserve the magnitude response of

analog filters at omall values of fs/fp,.

The step and ramp invarance of low-pass and band-pass filters

produce digital filters whose magnitude response does not depend on the

sampling frequency in the frequency band of 0 to f /2. For high-pass

5E

en beed-vejeet filters the asalmid gespomee of the eter-iavariastdigital fultue is ueetb at snall values of the ratio f/pJo rm-Lmcataa mathd can be used to dgitise high-pass and band-

reject filters but tde magnitude responses of analog ad digital

systos ame sot Identical when the ratio f/f 1 , become smailer themn

10. to this came, the equipp~e character of elliptic highpee. sa

band-reject filters Is not correctly simulated. Soe magnitude response

of the Butterworth hih-pass filter In the lw-frequency bad Is not

preserved by the romp-invriast filter. 11rsover, the attenuation of

the bad-reject, Vttervorth filter In the stop-band region Is limited

to a value that varies as a function of the sampling frequency.

If the realization In cascade Is used for the ram inrance of

high-pass end bad-reject filters, better approximation of the analog

filters Is obtained. Figure 15 reveals that the digital Butterworth

filters preserve the agnitude response of their analog counterparts.

lowever, a small shift In the meantude response of the digital high

pass filter can be observed when the ratio f aIf pbecome smeller than

10. For the elliptic highpass and band-reject filters, a close match

between analog and digital systems exists If the ratio f aIf pis main-

tamned above 10. Cascade realization can also be performed on low-pass

and band-pass step-invariant filters but the magnitude response of

these filters Is loe accurate than that obtained by the ramp-Invariant

filters realized in a parallel form.

Legend for Fisuree 7 through 15

____________Cntinuous-tim filter

* .. . .Digital filter when f - 1000 h~m

Digital filter Ame f 9- 10 Hz

Digital filter when f5am 4 kRm

UNCLASSIFIED30

-aon.o *oo

-44.0 -40.0

-W6. a 2

-C50CC' 10. -50.6

10-C 1 -0. .3 -20.60

FIGURE 7 -Magnitude response of a Butterworth low-pass digitalfilter obtained by the impulse-invariant, the bilinear,the step- and the ramp-invariant transformations forvarious sampling frequencies

r.

UNCLASSIFIED31

-3.

IOO . . ... . IO, . . .. .. 4

3 1~-0.0 0*

5-

2 3 2 2

.10 10 0 10 1

- o - o j- ....

.3 -O. .3 -40.0u

-.o 0 " -30.0

400 -.C .-.--S.--.-...-40.0

FIGURE 8 -Magnitude response of an elliptic low-pass digital filterobtained by the Impulse-invariant, the bilinear, the step-and the ramp-invariant transformations for various .sampling frequencies

% %

,'I'.

- HI.II -3.

"40.0" f -' 40.0 i, *.

• F'ee vemcq In I't ._ . Jl

'"'

.- 5

~ ~ " 5 d.4 f.9.5*4 ~V..*.'.'' ~ . . ~ .. - .,.

.32

-dLI

'AS *-M.0

- -. - .43.

.93 3 6 4 43.!10 10 I0 0D 10 10

1B.8 -M.

10 a / - 1033 0 031

*m~ a. IIq" IO

JO 9a. - Vmda epneo uttrot *.s iiafilter obtaned by the ~ulsem-Inveian h iier

.t.3 stp an th *p1vrattrnfrain o

varou sapln f43.3nle

19.

-08

33. -3.63

to t 10 ~ ine01$-qseqS.M F,"w,. i.

as M

-MIA-a 3'

10r 10101]a1

1 0 1 10 [ 3 .

.8o" ' .6F.o~ m6

FIGUU I' I X\epns faBtewrh adps iiafilte obaie byte *.lsw6 vra ,teblier h

stp and* th'apivr ttasfrain o aiu

ampln frequncie

7 ZN,-

UNCLASSI*IED35

I -., -- 0.0

-40.0

a ,

-300 -20.0I

-30.0 . . -4-0.0 C A

10 10 10 10 103 1

a4 3k 1k 4.

I .., . ,o~

l0 2. 0 IQ

. .e mKI . .-M.a0

S - 2. Mnu -20.0 , ta

-30.J -30.0 o C,. ,

.,,0tO lO 3 0 0 tO! I0 ,

IGURE 12 - Magnitude response of an elliptic band-pas digital filter [

obtained by the Iapulse-invariant, the bilinear, the step-and the ramp-invarlant transformations for various samplingfrequencies

'NI%

-'-C'

, . 1_ * , -, * .. : . ., , ... -. . ,- ,. "' ; . % % - . , ,'.' - -

UNCLASSIFIED36

3 -20.5 J -20.0

U.0 -30.0 I

-40.0 -40.

102 10 1102 10

-0.0 -30.0

-40.0 -40.0

2 -3. 10 -20.

400 -400 j01ID~ Ma~SAS IA-- -5.-2 ~ Ha I

FIGURE 13 -Magnitude response of a Butterworth band-reject digitalfilter obtained by the impulse-invariant, the bilinear, thestep- and the ramp-invariant transformations for various *

ampling frequencies

V!.

f ...

-act

to 10 0 is0t

Flom ~ ~ -1a..tIffller~btat"-y d Upzeoluart~t m .t3u..

tM Sopad o LrA*ftfrj~ 'tOxfrmaion.fovaiu .4*.gif 4tuO I

UNCLASSIFIED38

3 3

a0. 3 -23.0

b a n -r e e c B u t r o th a d e l i t c f l t r -

12;:

DUCLASSIPI39

7.0 CO LUSUOI

To methods for approzimtin digital recursive filters from

analog filters have been described. The invariant-step response is

maintained in one case while the invariant-ramp response is used in the

other case. The discrete-time transfer functions of digital filters

obtained by these methods were derived for parallel and cascade

realizations.

The performance of the proposed methods ws first determined by

giving the magnitude and phase responses of the first- and second-order

filters. It is shown that, for low-pass and band-pass filters, the

step-invariant procedure gives accurate magnitude responses that do not

vary with the sampling frequency in a frequency band of 0 to f /2. The

ramp invariance gives magnitude responses as accurate as those produced

by the step-invariant method for low-pass and band-pass filters and it

can be used to reproduce perfectly high-pass filters for ratios f /fp

maller than 10. In addition, a close match In the phase response

between the ramp-invariant digital filter and its analog counterpart

can be observed for all cases of first- and second-order filters.

Finally, a comparison between the step-, ramp- and Impulse-

invariant and the bilinear transformations uas performed on high-order

Butterworth and elliptic filters. At small sampling rates, the

impulse-invariant procedure realizes a close match with the analog

filter in the magnitude response only for the Dutterworth low-pass

filter. In the billnear transformation, the poles of digital filters

are moved in relation to the sampling frequency. This distortion,

caused by nonlinear frequency warping, becoms significant when the

ratio fs/f becomes smaller than 10.

ep!

UNCLASSIFIE40

T he step aid rmp invariance of low-pass and bad-pass filters

In a parallel implamsntatiou produces digital filters whose magnitude

aepos does not depend on the smepling frequency in the frequency

bad of 0 to f£ /2. TMe mantude response of the step-invariant high

pass mid bad-reject filters realized In a parallel or a cascade form

is =macceptable for small ratios f 6 f p(410). The ramp invariance

realised In a cascade form can be used to digitize highpass and band-

reject filters. The ratio f aIf can be decreased to 10 for Butterworth

filters but the equiripple character In the stop-band response of

elliptic filters Is preserved by~ a ratio f./f p of at least 10.

The step- and ramp-invariant methods are thus less sensitive to

frequency folding in comparison with the impulse-invariant. mthod.

Furthermore, step invariance can be applied to transfer functions in

which the degree of the denominator must exceed that of the nuserator

by at least one. In the rso invariance, the nuserator degree can be

-'.1 ashigh as the denominator degree. in addition, when the rasp-

invariant transformation io mpantdIn a parallel form for the

low-pass and band-pass filters and in a cascade form for the high-pass

and band-reject filters, the resulting match becomes better then the

one with the bilinear transformation for ratios f I f mlagler than 10.

UNCLASSIFIED p...41

8.0 REFERENCES

1 Rabiner, L.R. and Gold, B., "Theory and Application of DigitalSignal Processing", Prentice-Hall, Englewood Cliffs, New Jersey,1975. .

2. Oppenheim, A.V. and Schafer, R.W., "Digital Signal Processing",Prentice-Hall, Englewood Cliffs, New Jersey, 1975.

3. Golden, R.N., "Digital Filter Synthesis by Sampled-DataTransformation", IEEE Transactions on Audio and Electroacoustics,Vol. AU-16, No. 3, September 1968.

4. Temes, G.C. and Mitra, S.K., "Modern Filter Theory and Design", Ok-John Wiley & Sons, New York, 1973.

5. Antoniou, A., "Digital Filters: Analysis and Design", McGraw-Hill,New York, 1979.

6. Solodovnikov, V.V., "Statistical Dynamics of Linear AutomaticControl Systems", translated by W. Chrzczonowicz, D. Van NostrandCompany Ltd., London, 1965.

7. Temes, G.C. and LaPatra, J.W., "Circuit Synthesis and Design",McGraw-Hill, New York, 1977.

%.% ..-

• .. "'.

r. ....

. . . . . .... . , ,.- , ..... ... - , .. . . . .... .'. " '. ._'.... ...... _ '.%,

UNCLASSIFIED

42.-

£fPI I A

Rapresentation of the Continuous-Tims

Transfer Function of nalo Filters

Let us consider that

b.(s) . iFM [A.1]" UA(S) "X(s) a a Is

iJo

is the transfer function of a realizable analog filter if coefficients

ai and bi are all real, and the degree of the numerator is miler

than or equal to the degree of the denominator.

Equation A.1 can be expanded into partial fractions and, after

separating the real from the complex poles, it yields: A"

A k~ L IL 3210a+ BIikp)k I 1 2 + 0 2)kL"

+- (sp +i. - (22sa 2B)I I I I

H(s). - + N N Aik + _ + [A.3]I, X (s + p )k iXI (s + d )k (s + d*)k

where Cik - (B2ik12) + j 1(B2i10i- ILik)1 2 111

i [--adi " ai-J """""

The * indicates the complex conjugate. The uultiplicity MI and 11 J 6V

of real poles pi and complex poles (di. d*) are related to the degree

of the denominator of H(s) by

".'--:•

• o* .

-,

* a',

UNCLASSIFIED43

N NI11 - ,N + 2, I [A-4]

If we assum that all real and complex poles are simple, eq. A.3

*: becomes simply

A iNI CI

u(s)-A 0 + A1 + [A-d]0 -+pi + il + di + d[

Equation A.5 covers low-pass, high-pass, band-pass and band-

reject filters mong the ell-known filter classes such as the

Butterworth, Sessel, Chebyshev and elliptic filters.

Equations A.2 and A.3 are suitable for representing parallel

realization of analog filters. For serial or cascade realization,

eq. A.1 must be expressed as

A (a [A.6]: uH(s) - Ks i-l S+Us )

i I (s + pi)

If we separate the first- and second-order terms, eq. A.6 becomes2' + ... ...

H(s)-K [NA A + 1 1 [ i'A+ +'-i.7]

51=1 + P, 1=1 +~ 2 2[A7a- s + 2ale + al + 0 2 : - -

where N' = NA + 2M3.

Equation A.7 can also be written as

NA AliA i N3uijPTsie CA.8

N(.)- .[ ( + K(A [±+ + [A8

,. '.,

.4 21 - pa7- L

% .

.. . % P....- ..- .. % > W.. . - -. ' -

.., ,. . 4'. .. , %, -. * % * ",. ". ", . ".~. *;, .,', * . *. . -,. ":.. .- ". .. ' -. - .* .. **_.-,...- * ~ * .'.', . *.'.. .. ' ."V '.,' ..' , .'. ,..... .'. ,.,

.-- 3 ..4. .... Ct " t 'c..............

fli' t . -

.1" 777-

st IL

UNCLASSIFIED45 .

APPENDIX BS-.-

Realization of the Step- and Ramp-Invariant

Dgital Filters in a Cascade Form,.7 ..' "

If the cascade or direct form is desired, it can be obtained by

substituting the mapping relations of eqs. 15 and 22 into the transfer

functions of analog filters expressed In cascade form (eq. A.8).

For such a realization, the step-invariant transforeationbecomes

'''

-pTNA i(Aii/pi) -A 2 1 (1- e z

s(A) R 1 A21 + -p .T2--

NB K lIz +K zH [• z B + K:iz..i-I 51 1-2e - Kcoi T T] [+.1 ""-

Where BL ad . are directly obtained by substituting C by C1 I

eq. 17 and the ramp Invariance becomes

1(z)- N A A1+ .. .. - 1P1 [1 - ePiTz-1 (Ic - - I )(1 - -)1 &2 1 - • PiTz -1 T i p2T

-A -2 A--B KcIi + -liz + K 1 " (I21I + T -2Tp [22

~ i- -2

I -2e cos iTz + e z-

I.eq. 23.~

% *

UNCLASSIFIED

46

a.AFPEMDU C a

Continuous-Time Transfer Functions of

the Butterwrth and Elliptic Filters -

This appendix gives the continuous-tine transfer function of the

Butterworth and elliptic filters used in Chapter 8.0.

The transfer function of these filters is written as

Nt'., s N'A (I + pi) -;'"

i-I:

1) Fifth-order Butterworth low-pass filter:

- cutoff frequency- 1000 Hz

- M' - number of zeros - 0

- N' - number of poles - 5

- K. - 9.79 x 1018

- Pl,2 - 1941 ± j 5975

P3 , 4 = 5083 ± J 3693

P5 - 6283

2) Butteruorth high-pass filter derived from the low-pass

filter:

- cutoff frequency = 1000 Rz

M' -3,1' 5, Ks - 1

- poles are identical to those of the low-pass filter

144

-1' r eros ; areallloctedat-

%4 T wL = 6 -.. , .,,... .. ,--%•.,., .. - . ,....-. -

o--' -r---- - -.

--. "" " o

' '7; ", .. u

•

UNCLASSIFIED47

.4'

3) Butterworth band-pass filter derived frou the low-pass

filter:

- lower-cutoff frequency - 700 Rz

- upper-cutoff frequency - 1400 Hz- M, = 5, N, = 10, K. - 1.731 x 1018

- zeros are all located to 0

- p1 2221 * J .5877

P3,4 -1416 ± j 4866

p5 ,6 -2177 ± j 7478

7,8 466.6 J 4484

p9,10 -906.3 ± J 8709

4) Buttervorth band-reject filter derived from the low-pass

filter:

- lower-cutoff frequency - 700 Rz

- upper-cutoff frequency - 1400 Hz

- M4'- 10, N' 1 10, Ke - 1

- poles are identical to those of the band-pass filter

- 10 zeros are located at 0 ± j 6283

5)Fifth-order elliptic low-asitr

- cutoff frequency - 1000 Hz

- M' - 4, N' - 5, Ks - 240-a 1 ,2 - 0 ±J 9309, '3,4 ' 0:t j 6865 K

- 'l,2" 775.2± j 4367, p3,4 " 155.5 1 j 5802 p..

p 5 - 1479 "

-..

rr--,

UNCLASSIFIED48

6) Elliptic high-pass filter derived from the low-pass filter:

- cutoff frequency - 1000 Hz

- M, - 5. N. - 5, Ks- -

l,2 0 J4240, 0 t j 5750

a 5 - 0

- P2- 182.3 ± j 6799, p3 4 " 1555 t j 8763

p5 - 26689

7) Elliptic band-pass filter derived from the low-pass filter:

- lower-cutoff frequency - 700 Hz

- upper-cutoff frequency - 1400 Hz

- M' - 9, N' - 10, Ka - 169.72

- 2 -0 ± j 10324, '3, 4 = 0 ± j 3801

"5,6 - 0 ± j 9163, a7,8- 0 1 j 4308U9 m 0

- P1 , 2 379 j 4557, P3 , 4 - 72.08 ± j 8660

P5,6 - 208.6 ± j 4920, P7 , 8 " 339.5 ± j 8008

P9,10 = 5229 ± j 6261

8) Elliptic band-reject filter derived from the low-pass

filter:

- lower-cutoff frequency - 700 Hz

- upper-cutoff frequency - 1,400 HzM' -10, N' - 10, Ka -1

- 2 0 ± J 6283,,,, 0 J7959

-5,6 0 ± j 4960, m7,8 - 0 + j 8636

-9,10 - 0 1 j 4570

p1 - 16,476, p2 - 2396

P3, 4 - 41.4 ± j 4323, p5,6 - 87.4 ± J 9131

. 306.1 ± j 3889, V9,10 793.7 10086

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