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Full text of "A computer aided design of digital filters."

A COMPUTER AIDED DESIGN 
OF 
DIGITAL FILTERS 



Sal ih Kayhan El itas 



DUWITWWXUBRMY 

HAVAL POSTGRADUATE SCHOd. 






NAVAL POSTGRADUATE SCHOOL 

Monterey, California 




THESIS 



A COMPUTER AIDED DESIGN 
OF 
DIGITAL FILTERS 

by 

Salih Kayhan Elitas 

June 1977 



Thesis Advisor: 



S. G. Chan 



Approved for public release; distribution unlimited 



U7S651 



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READ INSTRUCTIONS 
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1. REPORT NUMBER 


2. GOVT ACCESSION NO. 


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4. TITLE (and Subtttlm) 

A Computer Aided Design of Digital 
Filters 


5. TYPE OF REPORT a PERIOO COVERED 

Engineer's degree thesis; 
June 1977 


6. PERFORMING ORG. REPORT NUMBER 


7. AuTHORfaj 

Salih Kayhan Elitas 


e. CONTRACT OR GRANT NL-MSERfaj 


9. PERFORMING ORGANIZATION NAME AND AOORESS 

Naval Postgraduate School 
Monterey, CA 93940 


10. PROGRAM ELEMENT. PROJECT, TASK 
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12. REPORT DATE 

June 1977 


13. NUMBER OF PAGES 
104 


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Unclassified 


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16. DISTRIBUTION STATEMENT ol :hi, Report) 

Approved for public release; distribution unlimited. 


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18 SUPPLEMENTARY NOTES 


It. KEY WORDS (Continue on rererae nam II neceeeery and Identity by block number) 

Digital Filters 


20. ABSTRACT (Continue on reveree tide II neceeamty and Identity by block number) 

Expressions for a generalized Modified Transitional 
Butterwor th-Chebyshev (MTBC) filter are derived. The 
characteristics of this filter as applied to digital filter 
design are investigated. It is shown that by adjusting 
location and order of the inserted zeros, the cut-off slope 
rate of the filter can be traded for maximum attenuation in 
the stop band. 



DD 



FORM 
I JAN 73 



1473 EDITION OF 1 NOV 6* IS OBSOLETE 

S/N 0102-014- 6601 | 



SECURITY CLASSIFICATION OF THIS PAOE (When Data Entered) 



ittiumTv CLASSIFICATION Q> Twi5 P»Gtrw.,« r>»>« tmiwj 



The performance of this MTBC filter is compared to 
that of Butterworth, Chebyshev, transitional Butterworth- 
Chebyshev filters together with those suggested by other 
investigators [l]-[3]. It is shown that the stop-band 
attenuation can be significantly increased without great 
sacrifice of cut-off slope rate. 

Step response of this MTBC filter is also obtained 
and compared with other filters. Various tabulations as 
well as graphs of this filter are given for design purposes 
A computer program is developed for the design of this 
filter . 



DD Form 1473 

1 Jan 73 

5/ N 0102-014-6601 sccu«itv classification or this PAoer**** o.*« En<»~<i> 



Approved for public release; distribution unlimited. 
A COMPUTER AIDED DESIGN OF DIGITAL FILTERS 

by 



Salih Kayhan ELITAS 
Lieutenant, Turkish Navy 
E.S.E.E., Naval Postgraduate Scholl r 1975 
M.S.E.E., Naval Postgraduate School / 1976 



Submitted in partial fulfillment of the 
requirements for the degree cf 



ELECTRICAL ENGINEER 

from the 
NAVAL POSTGRADUATE SCHOOL 
June 1977 



DUDLEY KNOX LIBRARr 
NAVAL POSTGRADUATE SCHOOl 



A3STRACT 



Expressions for a generalized Modified 
Transitional 3utter worth-Chebyshev (MTBC) filter are 
derived. The characteristics of this filter as applied 
to digital filter design are investigated. It is shown 
that by adjusting location and order of the inserted 
zeros, the cut-off slope rate of the filter can be 
traded for maximum attenuation in the stop-band. 

The performance of this MTBC filter is compared to 
that of Butterworth, Chebyshev, transitional 
Butterworth-Chebyshev filters together with those 
suggested by other investigators [1]-[3]. It is shown 
that the step-band attenuation can be significantly 
increased without great sacrifice of cut-off slope 
rate. 

Step response of this MTBC filter is also obtained 
and compared with other filters. Various tabulations 
as well as graphs of this filter are given for design 
purposes. A computer program is developed for the 
design of this filter. 



TABLE OF CONTENTS 



LIST OP FIGURES 7 

LIST OF TABLES 10 

ACKNOWLEDGEMENT 11 

I. INTRODUCTION 12 

II. DERIVATION OF MODIFIED TRANSITIONAL BUTTERWOETH - 

CHEBYSHEV FIITERS 16 

A. INTRODUCTION 16 

B. MODIFICATION OF THE TBC FILTERS WITH COINCIDENT 
TRANSMISSION ZEROS 17 

1. Slope at cut-off frequency 18 

2. Stop-band characteristics: 19 

C. MODIFICATION OF TBC FILTERS WITH DISTINCT 
TRANSMISSION ZEROS 22 

1. Stop-band and cut-off characteristics.... 25 

D. SUMMARY 26 

III. COMPARISON OF MIBC FUNCTION WITH B,MB,C,MC AND TBC 

FUNCTIONS 41 

A. INTRODUCTION 41 

B. MIBC FUNCTION V.S. B FUNCTION : 41 

C. MIBC V.S. M3 FUNCTION ; 44 

D. MTBC FUNCTION V.S. C FUNCTION : 45 

E. MTBC FUNCTION V.S. MC FUNCTION : 45 

F. MTBC V.S. TBC FUNCTIONS: 46 

G. SUMMARY 47 

IV. COMPUTER PROGRAM 64 

A. INTRODUCTION 64 

B. COMPUTER PROGRAM 64 

C. REQUIRED DATA CARDS 66 

D. REQUIRED SUBROUTINES/FUNCTIONS 68 

E. DESIGN EXAMPLE 68 



V. TIME DOMAIN RESPONSE OF DIGITAL FILTERS 72 

A. INTRODUCTION 72 

B. TRANSFER FUNCTIONS' POLES AND TRANSIENT 
RESPONSE 72 

C. SUMMARY 80 

Appendix A: COMPUTER PROGRAM LISTING 85 

Appendix E: COMPUTER PROGRAM LISTING 98 

REFERENCES - 102 

INITIAL DISTRIBUTION LIST 104 



LIST OF FIGURES 



1. general stop-band characteristics of the finite zero 
filters 20 

2. stop-tand attenuation of MTBC filter with two 
coincident zeros 23 

3. cut-off slope of MTBC filter with two coincident 
zeros 24 

4. cut-off slope of MTBC filter with two distinct zeros 
(w=1.06 , n=5) ,. 27 

5. cut-off slope of MTBC filter with two distinct zeros 
(w=1.06,n=6) 28 

6. cut-cff slope of MTBC filter with two distinct zeros 
(w=1.06,n=7) 29 

7. cut-off slope of MTBC filter with two distinct zeros 
(w =1.06 , n = 8) 30 

8. cut-off slope of MTBC filter with two distinct 
zeros (*=1.06,n=9) 31 

9. cut-off slope of MTBC filter with distinct transmission 
zeros (w =1.06 , n = 10) 32 

10. cut-cff slope of MTBC filter with distinct transmission 
zeros (w =1.06 , n = 11) 33 

11. stop-tand attenuation of MTBC filters with two distinct 
transmission zeros (w =1.06 , n=5) 34 

12. stop-band attenuation of MTBC filters with two distinct 
transmission zeros (w =1.06 , n=6) 35 



7 



13. stop-band attenuation of MTBC filters with two distinct 
transmission zeros (w =1.06 , n=7) 36 

14. stop-band attenuation of MTBC filters with two distinct 
transmission zeros (w =1-06 r n=8) 37 

15. stop-band attenuation of MTBC filters with two distinct 
transmission zeros (w = 1.'06 , n=9) 38 

16. stop-band attenuation of MTBC filters with two distinct 
transmission zeros (w =1.06 , n=10) 39 

17. stop-band attenuation of MTBC filters with two distinct 
transmission zeros (w =1.06 , n=11) 40 

18. ratio of cut-off slopes of MTBC and B functions v.s. w 
( m=1 ) 48 

19. ratio of cut-off slopes of MTBC and B functions v.s. w 
( ai=1 ) r 49 

20. the difference between the stop-band attenuations of 
MTEC and B functions v.s. w ( m=1 ) 50 

21. the difference between the stop-band attenuations of 
MTEC and B functions v.s. w ( m=2 ) 51 

22. ratio of cut-off slopes of MTBC and MB functions v.s. w 
( m = 1 ) 52 

23. ratio of cut-off slopes of MTBC and MB functions v.s. w 
( m=2 ) 53 

24. ratio of the cut-off slopes of MTBC and C functions 
v.s. w ( 1=1 ) 54 

25. ratio of the cut-off slopes of MTBC and C functions 
v.s. w ( oi=2 ) 55 

26. the difference between the stop-band attenuations of 
MTEC and C functions v.s. w ( m=1 ) 56 

27. the difference between the stop-band attenuations of 



8 



MTEC and C functions v.s. w ( m=2 ) 57 

28. ratio of the cut-off slopes of the MTBC and MC 
functions v.s. w ( m=1 ) 58 

29. ratio of the cut-off slopes of the MTBC and MC 
functions v.s. w { m=2 ) 59 

30. ratio of the cut-off slopes of the MTBC and TBC 
functions v.s. w ( m = 1 ) 60 

31. ratio of the cut-off slopes of the MTBC and TBC 
functions v.s. w ( m=2 ) 61 

32. the difference between the stop-band attenuations of 
MTEC and TBC functions v.s. w (m=1) 62 

33. the difference between the stop-band attenuations of 
MTEC and TBC functions v.s. w (m=2) 63 

34. cut-off slope of bilinearly transformed butterworth 
filter 67 

35. magnitude responses of (a) MB filter n=8, m=1 / w =1.36 
(b) MC filter n=7, m=1, w =1.36 (c) MB filter n=7, m=2, 
w=1.26 71 

36. step response of MTBC filter ( n=7, w =1.46 ) 81 

37. step response of MTBC filter ( n=5, m=1, k=1 ) 82 

38. step response of MTBC filter ( m=1 / w =1.36 ) 83 

39. step responses of B, MB, MTBC filters ( n=5, m=1, w 
= 1.36 ) 84 



LIST OF TABLES 



1. Formulas for cut-off slopes and stop-band 
attenuations (4 2) 

2. Possible solutions of example design (69) 

3. Pcles of HT3C filter (75) 



10 



ACKNOWLEDGEMENT 



The author expresses his greatest and sincerest 
appreciation to Prof. S. G. Chan and Prof. T. F. Tao for 
their guidance, tolerance, and assistance during the time of 
this study. 

The author would also like to extend his appreciation 
to the Turkish and United States Navies for providing the 
opcrtunity tc pursue this research. 

Last, tut not least, the author wishes to thank his 
wife Guzin, for her patience and understanding. 



11 



I. INTRODUCTION 



The digital filter is, as defined by Rabiner et al [10], 
" a computational process or algorithm by which a digital 
signal or sequence of numbers (acting as input) is 
transformed into a second sequence of numbers termed the 
output digital signal". 

The area of digital filtering can be divided into two 
major subdivisions as Finite Impulse Response (FIR) filters 
and Infinite Impulse Response (IIR) filters. 

During the development of digital signal processing, the 
interest cf the investigators in IIR and FIR filters varied. 
Before the introduction of the FFT algorithm by Cooley and 
Tukey (1965) IIR filters were much more efficient than FIR 
filters. Stccham's work [13] on the FFT method of 
performing convolution indicated that implementation of 
high-order FIR filters could be made extremely 
computationally efficient ; thus, comparison between FIR and 
IIR filters are no lcnger strongly biased toward the latter 
[5]. Because FIR filters require very high orders to 
produce a sharp attenuation shape, they are not often used 
for real-time filtering of waveforms. Recently, due to the 
increase in computing capabilities in digital signal 
processing and the availibity of long charge transfer device 
(CTD) tapped delay lines (TDL) , FIR filters are favored over 
IIR filters. However, in applications like design of 
digital ccmb filters IIR filters are the unique alternative. 

There are three basic design techniques of IIR digital 
filters [5]. 



12 



First method is the direct design, which is, 
appropriately placing poles and zeros to approximate 
required frequency response. 

A second method is to use an optimization procedure to 
place the foles and zeros to match arbitrary frequency 
response specifications. 

Finally the third technique makes use of highly advanced 
art of continuous filter design. This technique of 
designing digital filters from continuous filters by means 
of mathematical transformations is the most popular IIR 
digital filter design technique. 

Standard Z-transform, Bilinear Z-transform and the 
matched Z-transform make possible direct transformation from 
S-domain to Z-domain, preservinq essential characteristics 
of analog frequency response. 

Existence of frequency transformations reduces the 
problem to design a frequency normalized prototype low-pass 
filter. Then using appropriate frequency transformation, this 
prototype may be converted into desired band-pass, 
band-reject cr high-pass filter. Popular prototype filters 
are Butterworth, Chebyshev, elliptic and hybrid transitional 
filters. Frequency transformations for digital filters are 
discussed in various literatures ( [6] and [7] ). 

The problem of designing low-pass prototype filters, 
which possesses better stop-band attenuation and cut-off 
slope characteristics than existing prototypes has always 
attracted the researchers in the signal processing area. 

3udak and Aronhime suggested [1] modification of 
maximally flat rational functions by introducing a pair of 
finite transmission zeros such that the maximally flat 



13 



characteristic is maintained but the cut-off slope can be 
made steeper without great sacrifice of stop-band 
attenuation. 

Dutto Roy [2] investigated a more general case allowing 
insertion of multiple pairs of transmission zeros, either 
coincident or distinct. 

Introducing multiple pairs of jw-axis zeros in all pole 
Chebyshev transfer functions are investigated by Agarwal and 
Sedra [ 3 ]. 

The most attractive feature of these finite zero filters 
is that they offer the filter designer a great degree of 
freedom in choosing the location and order of the zeros to 
trade cut-off slope for stop-band attenuation. 

In this thesis, a modified Transitional 
Butterworth-Chebyshev filter is developed, which is a more 
general case, introducing finite coincident or distinct 
multiple pairs of transmission zeros in transitional 
Butterworth-Chebyshev filter. 

Trade-off's between tne order of the filter, the order 
of transmission zeros, stop-band attenuation and cut-off 
slope are pointed out. Graphs helpful in the design of such 
filters are obtained. 

Performances of Butterworth, Chebyshev, Transitional 
Butterworth-Chebyshev filters and the designs suggested in 
references [2] and [3] are compared with those represented 
in this thesis for the orders of three through eleven. A 
computer program is developed to implement the filters 
mentioned above. 

In addition, the time-domain response of digital filters 



14 



is studied. There are many applications, such as digital 
HTI filters, for which one is interested in the transient 
responses cf filters that are specified in the frequency 
domain. Step responses of the filters that are discussed in 
this report are plotted and compared. 



15 



II. DERIVATION OF MODIFIED TRANSITIONAL 
BOTTERWORTH-CHEBYSHEV FILTERS 



A. INTRODUCTION 



The most popular technique for designing IIR digital 
filters is tc digitize an analog filter that satisfies the 
design specifications [5]. There are many techniques for 
designing analog low-pass prototype filters. Among the well 
known analog filter classes are the maximally flat 
(Butterworth) and equal ripple (Chebyshev) filters. 

Butterworth filters are simple, excellent in the 
pass-band and monotonic in both pass-band and stop-band. The 
Chebyshev filters are superior at and near cut-off frequency 
and at stcp-cand. 

The transitional Butterworth-Chebyshev (TBC) filters 
combine the desirable attributes of these two filters in a 
single approximation that is given by 

where 

k= Weighting factor 

n= Order of filter 

C . (w) = (n-k)— order chebyshev polynomial. 
When k = n, |F(jw)| 2 is identical with the Butterworth 
function. When k=0, JF(jw)|2 is identical with the Chetyshev 



16 



function. With a varying value of k, a TBC filter possesses 
seme characteristics of each. As k approaches to n, a TBC 
filter behaves more like a Butterworth filter, as k 
approaches to zero, it behaves more like a Chebyshev filter. 

In this chapter, modification of TBC filters by 
introducing finite coincident or distinct multiple pairs of 
transmission zeros will be discussed. It will be shown 
that, using a weighting factor k, and the location and order 
of inserted zeros as parameters, attenuation in the 
stop-band may be traded for sharpness of the cut-off 
characteristics. 

Expressions for the cut-off slope and minimum 

attenuation in the stop-band are derived in terms of order 

of the filter, weighting factor, location, and order of 
inserted zeros. 



B. MODIFICATION OF THE TBC FILTERS WITH COINCIDENT 
TRANSMISSION ZEROS 



Introducing m identical pairs of transmission zeros at 
±j« to eg. (2-1) , we have 



In order to normalize |F(jw)|*, i.e. to force |F(jw)|* to be 
egual to 1/2 at m= 1 , the constant K should be 



17 



Then eq. (2-1) becomes 



F<j«) 



2m. 



C*S-Mi x ) + <W-t) 2, « ^ c n \ k M 



w 



where n>2m, because of the low-pass characteristics of the 
function, and the Chebyshev polynomial C^ (w) is defined by 



Cos (n Cos"' to) 



, toi I 



c>) = 



(«) 



2r\-l In. 



eos^Cn&slr'w)^ ***«?* , w )i (J-6) 



1 • Slo pe at cut-off fre quen cy 



Substitutinq eq. (2-5) into eq. (2-4), we obtain 



FCjw) 



(Wo*- U3 r ) 



2m 



Let 






h(co) = u^ 2t ^[("-It) Cos'io] 

Usinq these values, eq . (2-7) becomes 






■ffa) 



F( > )l " {(«,) + (w,M)* m w(«) 



(a-.o) 



18 



Taking the derivative of eg. (2-10) with respect to w, we 
get 






M 



At the cut-off frequency, w=1, we have 

f(w) = (*.*- i) 2 "" 

kfo)- 1 



( w 



Substituting these values into eg. (2-11) and simplif iying, 
we obtain 



f(y*)\ - 



m 



JT (uo^-O 



+ 



fe.4 (n-lp 



H 



For k=n , the result agrees with the cut-off slope of MB 
function, which is derived in [2]. For k=0, the result 
agrees with the cut-off slope of MC function as derived in 
[3]. 

2 • St o^^band char acte ristics 



In general, stop-band characteristics of finite zero 
filters will be of the form shown in FIG.1*. 



* Eecause of the inherent limitations in the plotting 
subroutine utilized to provide the graphs in this thesis, it 
was necessary to add supplementary axes to show proper 
scaling for seme of the graphs. 



19 



10 o Oad/sec) *■ 




T23 



. tf = l r( rvM 



5 -fop- bancL 



Figure 1 - 



GENERAL STOP-BAND CHARACTERISTICS OF THE FINITE 
ZERO FILTERS 



20 



The zeros introduced will cause a peak in the stop-*band, at 
a freguency vy>w . The minumum attenuation in the stop-band 
may be defined as 



c<*T8c - JO log Tp 



(hi) 



where 


i 


i 




F f - 


FCju) f ) 






In th 


e stop-t 


and eg. 


(2-4) becomes 



H 



t^) 



(u)o l -u)/) 



2»n 



(u3 *.^) 2 ^^ (Oo 1 --!) 2 " S^- 1 ^ 



(2-20) 



Let 



f l*r) = 

Vi (u) f ) = 



2n\ 



(uJo'-U). 1 ) 



(«f) 

(2-22) 



Taking derivatives of these values and substituting them 
into eq. (2-11) results in 



F(j^) 



f = V n-^ 



Thus 



1 60^ ) 2m ^ iU " kVz ^ (^^ [- w-^w) *» ] 

(^-23) 

(i-2v) 



Combining eg. (2-19) and eg. (2-23), we obtain 



u), 



f,= 



1+ (u0o-<; o2 



a' 



, „ Nf\-2m 2.rr\ 



U) 



l(r\-2m) 



H 



21 



And minunium stop-hand attenuation will be given by 

CX^tsc - 10 n log ( H \ + i0ry\ I09 (u) ) 



+,20 m io 



3 



^ A-2rr\ > 


I -p BW V ., 


n-£m 


10<M 


n 


U3o z 



f 6(n-*v\-te->t) 6-^ 



Plots of stop-band attenuation and cut-off slope of 
MTEC filters with two coincident transmission zeros, for 
orders 3 thrcugh 1 1 are given in figures 2 and 3. 



C. MODIFICATION OF T3C FILTEHS WITH DISTINCT TRANSMISSION 
ZZBCS 



Consider the n — order TBC function with in pairs of 
finite distinct zeros at±jw« , where 1=1,2,... ,1a and w- >1. 

The rragnitude squared function of TBC function with 
distinct transmission zeros will be given by 



m 



Kji) 



1 (^-taO 



»> 



* .Ik n' 1 



7f ( u>-. 1 - u>*) l + J c «*- o ^ 2 * c . k w 



i -i 



• = i 



Let 



•vi 



IS I 

Putting these values into eg. (2-27) results in 

i 



F( jw ) 



i+ 



gup uj^e^(^) 



(*«) 



H 



22 



Cfl 

-o 

d 

o 

*-r 

<H 

a 
c 

Qi 
-+» 
-f 

(!) 

c 
<a 



a. 

T- • ' 
VI 



/ 

/ / 
/ / / 

/ 

f / / / 

/ / 





A-sr.Ri 



r L -j ; z 3 •-)-!: 






-(.0 



t,r 



2-r ^ 7.o 

uJo L.c*<L/zec j >. 



?'5" 



^•o 



Figure 2 - 



ST0P-3AND ATTENUATION OF MTBC FILTER WITH TWO 
COINCIDENT ZEROS 



23 



! 

I \ 



\ 



f 
-J 

a. 





o 

I 

3 



\ 



\ \ 



\ 



V 



fii: 



n 



n i 



■ \ 







n*U 



r»- io 



n = <? 



n-sf 



A = > 



n^(* 



n = 5" 



44 



■f.r 



2.o 



£«*- 



ujo (fAcL/sec) 



30 



3<5" 



Figure 3 - COT-OFF SLOPS OF MTBC FILTER WITH TWO COINCIDENT 

ZEROS 



24 



1 • Sto£-band and cut-off characteristics 



The frequency w p at which the stop-band peak will 
occur, is given by 



FCj*) 



= 



H 



Ui=rU), 



Using eq. (2-6) and taking derivative of eg. 
respect tc w, we obtain 



(2-28) with 



KjuO 






JL\H$>)\ Wio)[u 560) a 



K(ta) 



1 



(«.) 



Combining eg. (2-29) and eg. (2-30) the equation to be 
solved fcr w_ may be found to be 



a 10 



f 



m 




i 



(i0; z -u) p z ) 



n = 



(«i) 



which agrees with eg. (2-24) when w ; »s are identical. Using 
real solutions of eg. (2-31), stop-band attenuation peaks 
may be fcund to be 



£<-. - JO iog tjo 



ft: 



where 



Fp ; = |F(jio Pl -) 



and minimum stop-band atttenuation is given by 



c< 



mm. = RtaT* 



H 



(is*) 
(4*3) 



(a.*y) 



25 



Using eg. (2-5) and differentiating eg. (2-27) with respect 
to w, the cut-off slope becomes 



FQ«) — 



I 



ZU? 



k + Ck-^) 2 + £ 



m 



«=i 



10;*-! 



H 



which agrees with eg. (2-17) when the w-, »s are identical. 

Plots of cut-off slope and stop-band attenuation of 
MTBC filters with two distinct transmission zeros, for 
orders 3 through 11 are given in figures 3-17. 



D. SUMMARY 



The Transitional 3utterworth-Chebyshev filter combine 
the best features of the Butterworth and Chebyshev filters. 
A Modified Transitional Butterworth-Chebyshev filter, 
obtained by introducing finite transmission zeros to a 
Transitional Butterworth-Chebyshev filter, possesses cut-off 
slope and stop-band attenuation, which are dependent on the 
modification parameters w and m. The closer the w a is to 
unity, the steeper the cut-off slope. However, this 
improvement in the cut-off region results in degradation of 
stop-band attenuation. Thus, using w and m as parameters, 
an advantageous trade between attenuation in stop-band and 
sharpness of the cut-off characteristics can be made. 
Graphs are given to serve as guides in trading cut-off slope 
for stop-band attenuation. 



26 



o 



V 



(0 







<4 



«4^ 

o • 

t 



o 
S 



K-5C,PlF- D^GGr-Oi UNITS Ihur 
■^-jHRLr- 1. QCEL+QC uKITS INCH. 




to,= /.5"6 



UJ. 



^r^cL/i£c) — p- 



to 



i.-r 



i.o 



J-5- 



■Sf.O 



Figure a - CUT-OFF SLOPE OF MTBC FILTER WITH TWO DISTINCT 

ZEROS <w=1«06, n=5> 



27 




• li it" -J L Un i ii 1 



irsOH 



nnr , H'" 1 , '■ !• : T T n I k! ; " v M 



U) t •=. J.Ot 



w.= M6 



U),«!.2.fc 



tO,= 4.3k 



u>,= <4£ 



tO. - (-5-6 



2-r 



H-o 



Figure 5 - COT-OFF SLOPE OF MTBC FILTER WITH TWO DISTINCT 

ZEROS <w =1«06, n=6> 



28 



,_erp L Fr-.,OCF-Qi UK ITS INCH 




vo 2 .(r2cL/sec i 



{ o 



2.o 



2-° 



2-r 



Figure 6 - 



CUT-OFF SLOPE OF 3T3C FILTER WITH TWO DISTINCT 
ZEROS < w =1 • 06 r n=7) 



29 














l * • 




«1 






- 


o 




3" . 




«» 






»^> 






, 


O 






f» * 






"N 






Q) £ 




-t- " 




a 




%m 




Qj 


o 


Q=» 


^ ' 


o 


*• 


- — 




«o : 




^u 




H- 




o 




-+-> 





o 


5 


: 1 


o 




o- 




^ ! 




1 


R > 




C 


a 




>i 


':.- 






3<3 ' 








:, 



K-SCRLE^OOF-Oi UNIT 



Mi i 




uj. si-rt 



U3 Z Cra-dL/sec.) 



<-o 



i-r 



30 



3-r 



Figure 7 - CUT-OFF SLOPE OF MTB-C FILTER WITH TWO DISTINCT 

ZEROS <w =1«06>, n=8> 



30 



ft 



Q 

i i 



51 



a* 



Hi 

I! 



I ! 

HI 



v 



- 



X-5CRL~ 5,0GF-Q1 



T 1 



rwru 

i- 1 1 v J 




W. = i.Ot 



U),= Ub 



u>,«i.z& 






WJ 



2. ( fza/sec) 



1.0 



t<r 



3-« 



Figure 8 - CUT-OFF SLOPS OF MTBC FILTER WITH TWO DISTINCT 

ZEROS <w =1«06, n=9> 



31 










O 



o 



QIS 








"tf - - 






^V 



3 
I. 



O 



<-3r;RLc---5«Gur.-0i UNITS INCH. 



jG UNITS IN^.h 




LO a (VadL/jsc) 



J.0 



i.5" 



;.a 



2o~ 



J.o 



-2-r 



Figure 9 - CUT-OFF SLOPE OF HTBC FILTER WITH DISTINCT 
TRANSMISSION ZEROS (tf =1«06 ,n=10> 



32 










ri- 



^ j 



UJ 



z (ra^//ec) 



*.o 



?.r 



Figure 10 - CUT-OFF SLOPE OF *TBC FILTER WITH DISTINCT 
TRANSMISSION ZEROS <w =1*06 ,n-11> 



33 



CO 

. o 

-r 

a 

C 

a) 
d 

c 
i 






u)i CraoL/^ec; 



V I -3CRLF= L.OOE+Oi 



UMIT5 INCH, 
JNIT5 INCH, 



\\\ S N 



u) t si.o6 



u>,»l- it 




Figure 11 - STOP-BAND ATTENUATION OF MTBC FILTERS WITH TWO 
DISTINCT TRANSMISSION ZEROS <w =1*06, n=5> 



34 



1.3 



1.0 



i-a 



k-o 



r.o 



U> 



6 



cO 

d 

o 

-r 



-5 

c 

1 : 
Q 



C] 



U3 a (rad/s<2cj - 

K-SHRLF. 1- CV r -0L UN 



-3C D|_p-„ i„ GGE-'-Oi 



!ki!Ts 



r i ; r- i_ 



! . s_i! • » 



\ 

. \ \ 

' \ 

\N N 

\\\ \ 

\ \ 
\\ \ 

■\\\\ \ 

\\V\ s 



%\\ 




Figure 12 - STOP-BAND ATTENUATION OF MTBC FILTERS WITH TWO 
DISTINCT TRANSMISSION ZEROS <w =1*06, n=6 > 



35 



1.0 



1-0 



Ut 






6-0 






•3- 
I 






1 - 



-3 



J ! 



r*- ., 



i 



C 
7 



i 



_ — * 



^ fradl/s^o) 



::■".-:: _•-: rs 



:: ;_ 



x 



- 



s~- 















. 




\ 



- 









' 



Figure 15 - ST0P-3ASD ATTENUATION 0? ilTBC FILTE33 WITH TIC 
DISTINCT TRANSMISSION ZZ50S C« = 1 •06 # n=7> 



36 



1.0 



2.0 



t.O 



S.o 



£.0 



: v 






o 





-T 
d 

Hi 

\ 

O 

-r 



U)2. CradL/s-ecJ 




- • o - • 



.n 



Figure 14 - STOP-BAND ATTENUATION OF I1TBC FILTERS WITH TWO 
DISTINCT TRANSMISSION ZEROS <v =1«06, n=8> 



37 



*o 



1.0 



3-0 



H»o 



r.o 



Co 

i 



5 

i 






I I 



C0 

o 

e 

5 ' 

c 
<i 

J> .. 

I ? 

4 

<0 



0)i (racL jse.c) 




U), = <.o6 



rigure 15 - STOP-BAND ATTENUATION OF MTBC FILTERS WITH TWO 
DISTINCT TRANSMISSION ZEROS <v =1«06, n=9> 



38 



1.0 



2.0 



2.0 



4-.o 

i 



f.o 



4,-0 



3- 

i 



<§1 



>, u 



d 

.o 



1 

c 
0) 



(3 

i 

O. 
o c? 

"+" S3 
1 



CH 




10 a (r^cl/secj »~ 






uJ,- i.ot 



u),= 1-34 




V^"N 



X 



^v-x X 



<- -r Pj 



i ; 







Figure 16 - STOP-BAND ATTENUATION OF MTBC FILTERS WITH TWO 
DISTINCT TRANSMISSION ZEROS <w =1*06, n=10> 



39 



1.0 



i-0 





3.0 



4.o 



f,° 



I 



u3i (/acL/sec) 



04 



c0 
1 

d 



-r 

9 
-V " 

a 
i 



T 



\ 

\\\ 

vV\\ ■ 



. . ~-l i ■ 1 1_- •— ju ■ - U/ i . ~ '_' !_. 



UNITS INCH. 






wo \ 






s 




%<i<: 



u, 


=s 


<.oG 


ttj, 


■= 


l.lk 


vo, 


= 


1.2.6 



ui -I ?i /' l*\v *> v - 



10, = |,f6 



/ 









■> V V > 



N 



^>> X 
XXX 



X v X x 



S;v 



Figure 17 - STOP-BAND ATTENUATION OF MTBC FILTERS WITH TWO 
DISTINCT TRANSMISSION ZEROS <w =1*06, n= 1 1 > 



40 



III. COMPARISON OF MrBC FUNCTION WITH B, t MB x C J .MC AND TBC 

FUNCTIONS 



A. INTRODUCTION 



In chapter 2, derivation of MTBC filter_is given. It is 
shown that, stop-band attenuation and cut-off slope rate of 
this filter depend on modification parameters m, k, and w . 
So the filter designer have the flexibility to trade cut-off 
slope for stop-band attenuation by changing these 
parameters, without changing the order of the filter. 

In this chapter, performance of MTBC filter is compared 
with B, MB, C, MC, and TBC filters. 

Formulas for stop-band attenuations and cut-off slopes 
of all these filters are given in Table I. 



B. MTBC FUNCTION V.S. B FUNCTION 



Cut-off slope of MTBC function is given by 






41 



o 

K 



Q 



i 



o 



9* 
x 

O 






l 

C 

+ 
3 

o 



O 



« 



~4 
I 

C 

CD 



l 
_£* 

l 

c 

+• 

3 

o 

d 
O 

II 

CO 

J- 

75 



I 

-at 
i 

II 



I 

u 






N 
CO 



^ 

Vi 



^ 



>o 



^ 



II 

o 



x 



c 
I 

t 






I 



^ 






u 



I 

+ 

-a? 
+ 
II 



^ 



O 



? 

^ o 



o 
I- 

n 



UJ 

Uj 

X 
O 



^1 

o 






I- 

o 

3 
v. 

Uj 
H 

C0 



w 

X 

> 

QQ 

111 

X 

u 



CO 



42 



Q 
W 

Z 



O 

u 



w 

CO 



^1 



<5 






3 

£ 
i 



£ 
i 

II 



g* 



o 



I 









rr~i 

3 









o 

r 



- «! 

i i 

E 



en 



NO £ 

li 

CO 



-ail 

I 

u 



i 1 

3 



s 

I 

C 




I 



KS1 

O 

— I 

o 

+ 



NO 
I 


h 

li 



^r 



07 



O 



N 



3 



r 
11 

Z 






+ 



X) 









3 






o 



+ 



+ 
o 

li 









X 








I 




H 


Q 




UJ 


o 


*» 


_> 


u. 


^ 


Q 


UJ 





»— 


£ 


1- 

— i 



cO 





> 


Q 


at 


U4 


_i_ 


|j 


<s) 




> 


u 


CO 


o 


In 


-L 






^' 



Q 
Lu 

ul 

a 

o 



i 

« I- \ 

2. s£ > 

2 §£ 

H; qt t/i 

c/i iu 5r 

■2 t tf 

c I" W 

H cQ U 



43 



From eg. (3-1) ratio of cut-off slopes of these two 
functions may be found as 



'MT6C./8 - "? 
06 



=■ fl-V- 



J2 



m 



n(\0oM) 






(••0 



r Mi8c/& v.s. w for n=3 through 11 and ra= 1,2 are given in 
Figures 18 and 19. Stop-band attenuation of MTBC may be 
written as 



^mtgc = °^q + 6(n-m-fe-i) ■+ o20 mr\ Loo 



n-im u3 x -4 



n 



lOo 2 " 



(3-0 



Then the difference between stop-band attenuations is given 
by 



d M T8c/s= <**t«c - °^g - £Gi-m-k-l)+AOmloq 



A-2yy\ lQo a --l 

n too 1 



H 



Plots of d 



HTtC 



/ Q v.s. w for n=3 through 11 and m=1 f 2 



are given in FIG 20 and 21. 

Figures 19-21 indicate that, MTBC filter can be made 10 
times steeper than B filter still having 40 dB more 
attenuation at the stop-band. 



C. MTBC V.S. MB FUNCTION 



Using Table I, cut- off slope ratios of MTBC and MB 
filters may be found as 



/ 5m +n (lOoM; 



(«) 



Plcts of ^ HliC J M a v.s. w for n = 3 through 11 and m=1,2 are 

given in figures 22 and 23. 

The difference between the stop-band attenuations of MTBC 



44 



and MB is given by 



Equation (3-7) indicates that, stop-band attenuation 
difference between these filters doesn't depend on w . For 
n=10, i=1, k=1, with 48 dB more attenuation, cut-off slope 
ratio may be changed from 2.5 to 8.0 by changing w© from 
1.06 to 2.4. 



D. MTBC FUNCTION V.S. C FUNCTION 



The ratio of cut-off slope of MTBC function to cut-off 
slope of C function is given by 



MTBC 



:/c =■ i + 



im 



k(l*+l-i*) 



A 2 (ul» l -0 



rV 



Plot of r MT g<i/c v.s. w„ for n=3 through 11 and m=1 ,2 are 
given in Figures 24 and 25. The difference between 
step-band attenuations of MTBC and C functions is given by 

- 

Plots of &mtcJc v * s w ° ^ or n= ~* "through 11 and m=1,2 are 
given in Figures 26 and 27. The Chebyshev filter is known 
to provide much steeper cut-off slope than the corresponding 
B and TBC filters. Figure 25 shows that the cut-off slope 
of the MTBC filter can be made 1.4 times steeper than that 
of the Chebyshev filter, with n=5, m=2, and w„=1.Q6. 



E. MTBC FUNCTION V.S. MC FUNCTION 



45 



The ratio of cut-off slope of MTBC filter to cut-off 
slope of MC filter is given by 






(3..0) 



Plots of Zhiqc/mc v,s * w « for n=3 through 11 and m = 1,2 are 
given in Figures 28 and 29. 

The difference between stop-band attenuations of aTBC and MC 
functions is given by 

For a given k, ^UTar/MC "*" s a ^-^ wa y s constant. 



F. MTBC V.S. TBC FUNCTIONS 



The ratio of the cut-off slopes of MTBC and TBC 
functions is given by 



^T6c/tbc ■! + 



m 



H 



0o. l -J) [n 2 -t k(k-v l-4<0] 

Plots of ^tcncljzc v « s » w » f° r n= 3 through 11 and m = 1 ,2 are 
given in Figures 30 and 31. 



The difference between the stop-band attenuations of 
these two functions is given by 



^M-rac/rac = - 6m -*■ 3.0 w Ua 



n-£rvi lo*M 



r\ 



Wo 1 



(«-b) 



Plots of 3 MTe £/Tec v ' s « w o for n=3 through 11 and m=1,2 
are given in Figures 32 and 33. 



46 



G. SUMMARY 



Using the location and the order of the inserted zeros 
and the weighting factor as parameters, the characteristic 
curve of the MTBC filter can be made steeper than that of 
the conventional all-pole filters without greatly 
sacrificing either stop-band attenuation or flatness in the 
pass band. A Modified Chebyshev filter provides slightly 
better performance than the MTBC filter, at the expense of 
sutstantial degradation of the pass-band flatness. Graphs 
are given to help in the comparison and in determining the 
numerical advantages gained by increased complexity. 



47 







V. 

a. 




I 





I 

! , 
! . 
o- • 



■ 

J- 



i-0 



K-5CRLE-i,00E-Gi UNITS INCH, 
^-SCRLE-- 2,0GF+00 UNITS INCH. 




n~3 



u 



•M 



<<>'i 



Jo (faci/secj 



LH 



a:: 



^.r 



Figure 18 - 



RATIO OF CUT-OFF SLOPES OF MTBC AND B FUNCTIONS 



48 



o 


E 




» 


1, 

O 


^i 


r 

(a 




y 

O p- 


- 


VI 






1 

o 


'<* 




= 





-. 





.; 



1 

\ 



\ 



V 



\ \ 



*s 



1' \ 

1 \ \ 



■, 1 \ 




v s 




\ \ 


v, 


\ \ 


'- 






\ \ 




\ \ 




\ 


\ 




> 


\ 


% 



X 



-> , 1 - 









nrrof! i iki T TS 



- • L i_ : . ■ •_' w 



Mj 



k 



\ 



A^N 



rwio 



n = 9 



n=.? 



n=^ 



ft-5" 




u)o Oad/sec) 



<.z 



-(.^ 



r,6 



<•<? 



2Z 



Figure 19 - RATIO OF CUT-OFF SLOPES OF MTBC AND B FUNCTIONS 

V«S« W <M=2> 



49 



3 






o 

(9 

C 




rt.=\i 



i\-io 



fl=? 



fl-=£ 



*»* 



rt^k 



n=5* 



n=f 



1=3 



K-SCR: 



ncr -i 






j ,— i . -i ■ 



II " I T~ T I • P- I ' 

UN 1 i j iNuH. 
UK: ^5 INCH. 



Figure 20 - THE DIFFERENCE BETWEEN THE STOP-BAND 
ATTENUATIONS OF 21TBC AND B FUNCTIONS V«S« W <M=1> 



50 




u,uuc-ui UNITS INCH. 



*-5rqLE-5,00E-0i 
^-5rflLr=-2.00E+01 UNITS INCH. 



Figure 21 - THE DIFFERENCE BETWEEN THE STOP-BAND 
ATTENUATIONS OF MTBC AND B FUNCTIONS VS* tf C M=2 ) 



51 



•J 



O 



S~ 






**ll 



3. ?G? -01 UN ! r 5 IHC 1 - 1 

1 p.'T x f|'-» . .t : t t.; Ti:r u 







-r 



o 



o 

I 

V 
o 







,L 



i.' / 









n=3 



r-o 



<r 



.Z-O 



2'i 



uJo (<*cL /sec) 



3-z- 



v.o 



Figure 22 - 



RATIO OF CUT-OFF SLOPES OF MTBC AND MB 
FUNCTIONS V»S« H <M=1 > 



52 




i.o 



w-r 



2-r 



10 



lu 



(r*cl/sec ) 



Z'S~ 



1.0 



Figure 23 - RATIO OF CUT-OFF SLOPES OF MTBC AND MB 
FUNCTIONS V«S« W <M=2> 



53 



I 



< -5TPi 



GGE-G.' 



rc\ ?: 



JNITS INCH. 

?GF.-OI LJNITS INCH, 




Figure 24 - 



RATIO OF THE C'JT-OFF SLOPES OF MTBC AND C 
FUNCTIONS V«S« W <M=1> 



54 



q| 

o 



o 

I 



01 

n < 
U '- 



■ II 



1 1 

i Ki . 



J-\ 




n=T 



/.o 



J-T 



U) (fad/sec) 



1.0 



s-r 



h-t 



Figure 25 - RATIO OF THE CUT-OFF SLOPES OF MTBC AND C 

FUNCTIONS V«S« W <M=2> 



55 



■i-O 



2 1 



uJo (racc/so-c) — 

3.t 3.2 



h>r 



£L 




Figure 26 - THE DIFFERENCE BETWEEN THE STOP-BAND 
A1TENQATICNS OF MTBC AND C FUNCTIONS V«S* W CM=1> 



56 



1,9 



V.? 



I 







\^f a 

n 

QJ ' 

o 

<z 
<D 

\. 

U 

S- o 

.0 

-r 

3 
C 

I) 

T 



l 



o 

is 

I 



u) o(fzcL/zec J 



i.2 



2.4 



*.< 



r.r 




^-5CRL~-9^ DG-.-G1 JN1T5 INCH. 



Figure 27 - THE DIFFERENCE BETWEEN THE STOP-BAND 
ATTENUATIONS OF MTBC AND C FUNCTIONS V»S» tf <M=2> 



57 




\.o 



fo 



1.0 , 2-r n 



3.0 



3-r 



Figure 28 - RATIO OF THE CUT-OFF SLOPES OF THE MTBC AND MC 

FUNCTIONS V«S« W <M=1> 



58 



4-0 



*-r 



K--SCRlF.---5.0GE -01 UNITS thcH- 
V , -SCRLF=--3.G0F--02 UNIT5 IHGH. 




;.0 ?'5- 

uj (rai/secj 



J.o 



Figure 29 - 



RATIO OF THE COT-OFF SLOPES OF THE MTBC AND MC 
FUNCTIONS V«S« W <M=2> 



59 



K - C ! i ~.p! 



inF-n i 



s 'N T 



i. : 1 •- V : I 

INC 1 -! 




f.O 



*.1 



1-H 



u3o Cr^d./%e.Cj 



f-r 



5-3 



Figure 30 - RATIO OF THE CUT-OFF SLOPES OF THE MTBC AND TBC 

FUNCTIONS 7«S« W <M=1> 



60 



J ! 

i 

i ! 

. ! 






U „ w' 






N3 



■ 



I i ! 



u. 



II : 





■V 

o •' 



i 

+ 

u 



! I 



' 




>S 



O J 



/•r 



J..O 



u) (VaoL/ s^cj 



j.r 



^.« 



Figure 31 - 



HATIO 0? THE C0T-0FF SLOPES OF THE MTBC AND TBC 
FUNCTIONS V»S« W <tf=2> 



61 




P» ~ _• v ■ ■ L- . . ' " tl 3 U -' ■- ■ <-* — U> . 1 - . -' - ■ • - 

On ' i I ' 7 ~T ~ F l :p[_ 
U uhii ! -J LhL-h » 



Figure 32 - THE DIFFERENCE BETWEEN THE STOP-BAND 
ATTENUATIONS OF MTBC AND TBC FUNCTIONS V»S« H <M = 1> 



62 



10 



11 






j.g 



2.o 



2.2. 

l__ 






Q> 



"Si 
o 



\ 



- - 

I 



Sj 

I 




Figure 33 - THE DIFFERENCE BETWEEN THE STOP-BAND 
ATTENUATIONS OF MTBC AND TBC FUNCTIONS V«S« H <M=2> 



63 



IV. COMPUTER PROGRAM 



A. INTRODUCTION 



in the previous chapters, expressions for MTBC filter 
are derived and the performance of this filter is compared 
with various filters. 

In this chapter, A computer program is developed to 
implement MTEC filter. It is pointed out that all five 
filters, which are used to compare with the MTBC filter, are 
the special cases of the MTBC filter. Thus the program 
developed in this chapter for the implementation of the MTBC 
filter may also be used to implement any one of these five 
filters. 

A sample problem is worked out to illustrate the use of 
the plots presented in earlier chapters, the use of the 
computer prcgram, and to point out the flexibility offered 
by the MTBC filter to the filter designer. 



B. COMPUTER PROGRAM 



In the most general form the transfer function of MT3C 
filter is given by eg. (2-27) , which is repeated here for 
convenience . 

1=1 



F(ju>) 



f Cw. 1 - 10*) z •* ICwM) u lW C n \vO) 

1 = 1 i=i 



H 



64 



where : 



4lt_ 
w = Location of l— inserted zero 



m = Order of inserted zeros 

n = Order of filter 

k = Weighting factor of Transitional Butterworth 
Chebyshev filters 

When k-n and m=0, |F(jw)| 2 is identical to the 
Butterworth function. With k=n and m*0, |F(jw)l 2 is 
identical to the modified Butterworth function which is 
discussed by BUDAK and ROY [1] - [2]. When k=0 and m=0, 
|F(jw) | 2 is identical to the Chebyshev function. With k=0 
and m*0 Modified Chebyshev function may be implemented which 
is discussed by AGARVAL and SEDRA [3]. 

When m=0 and k#0, |F(jw) | 2 is identical to the 
Transitional Eutterworth-Chebyshev filter. Finally allowing 
both k and m to vary, Modified Transitional 
Butter worxh-Chebyshev filters may be implemented. 

The program consists of two main parts. In the first 
part the analog filter, which is given by eg. (2-27), is 
implemented and the frequency response is plotted. 

In the second part, the analog transfer function F(jw) 
is first predistorted then transformed into z-domain by 
algebraic substitution method (using Bilinear 
z-transf ormation) to obtain H (z) . Digital transfer function 
[ H (z) ] is then factored into second order cascaded stages. 



65 



Finally, frequency response curves ( |F(jw)| v.s. w, 20 
log |F(jw)| v.s. v. and F ( jw) v.s. w e ) are drawn. 

3ilinear Transformation is preferred over the other 
available algebraic substitution metnods {i.e. Impulse 
invariant, Marched z-transf ormation) in obtaining H (z) , 
mainly for the following reasons [4] : 

(1) it has the property that realizable stable 
continuous systems are mapped to realizable stable digital 
filters. 

(2) iiidehand sharp cut-off continuous filters can be 
mapped to wideband sharp cut-off digital filters without the 
aliasing in the frequency response. 

(3) After Bilinear Transformation, the relation between 
the analog and digital frequencies is given by 

3y choosing cut-off frequencies approaching to f»/2 where f& 
stands for sampling frequency, extremely sharp cut-off 
slopes may be obtained. As an example plot of cut-off slope 
of bilinearly transformed Butterworth filter v.s. cut-off 
frequency is given in PIG. 34. 



C. 3ZQUIRED DATA CAHDS 



The data cards required to use the program are given 
below. 



Card 1 : Values of n,m,k in 312 format 



66 



"> L 



1 



q ' h 










tfitt'Off frsffocy 



Figure 34 - 



CUT-OFF SLOPE OF BILINEARLY TRANSFORMED 
BUTTERWOETH FILTER 



67 



Card 2 : Location of inserted zeros, w; where i=1,2,...,m 
in 8F10. 5 format 

Card 3 : Initial and final values of the frequency to be 
used for the frequency response plot in 2F10.5 format 

Card 4 : Number of solutions required in 13 format 



D. REQUIRED SUBROUTINES/FUNCTIONS 



In addition to the built-in subroutines, the following 
IBM source library subroutines are used. 

1 . POLRT 

2. PLOTP 

3. PSUB 

4. PMPY 

5. PADDM 



E. DESIGN EXAMPLE 



Suppose we want to design a digital filter with a cut 
off slope of 15 and minumum attenuation in the stop band of 
60 Db. 

Possible solutions for various types of filters are 
given in table II. 



68 



TABLE !!• POSSIBLE SOLUTIONS OF DESIGN EXAMPLE 



Or 


n. 


m 


at. 




. 

£T0P-&#klD 
ATTENUATION 


B 


43 


— 


— 


/SI 


Ci reader -than 


M6 


c*jO 


2 


Hi 


/o~.0 


\8o±& 


3 


-M4 


^tf.05" 


|20 dS 


c 


7 






/W5" 




MC 


7 


1 


122 


/f.o 


^</£ 


2 


J.2Z 


2(3.5" 


Je^s 


13C 


2 




/7"£T 


63 J8 


H1SC 


7 


2 




/f.4 


6o</8 


2 


1 


1 




60 J£ 


i 

i 


2 /.// 


£// 


60 d'S 



69 



The following input data specifies a HTBC filter of 
order with 111=2, w =1.36, k=1, sampling period of 1 sec, 
100 solution points, and frequency response plot from to 3 
Hz. 



I 


2,3,4,5 


s 


1 1 S 1 9 | 10 


II 1 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 


21 |22|23|24|2S|26|27|2S|29| 30 




?, .4 


I 


1 1 1 


1 1 1 1 1 1 I 1 1 


' 




l I I 




h.xZ-.t 


11.11 ill. .3,6 






I I I 




\O\.\o 


.I.... i3i.iO 




I 


0<O< i 




1 1 1 


1 1 I 1 1 1 1 1 1 






I I I 




1 1 1 


1 I 1 1 1 1 1 1 . 





Frequency responses of three of the possible solutions 
of the design example are given in Fig. 35. 



70 



udo (fad/ Sec J 




Figure 35 - MAGNITUDE RESPONSES OF (a) MB FILTER N=8, M=1 f 
(b) MC FILTER N=7, M=1, (c) MB FILTER N = 7, M=2 



71 



v - Hill DOMAIN RESPONSE OF DIGITAL FILTERS 



A. INTRODUCTION 



In previous chapters we have investigated the problem of 
desining digital filters in the frequency domain, i.e. to 
meet given frequency domain specifications. A filter 
designer should always consider the transient response 
characteristics of its filter. There are many applications, 
such as digital MTI filter, for which one is interested in 
the transient responses of filters that are specified in the 
frequency-domain. Time-domain and frequency^ domain 
characteristics of a filter will work against each other. 
Filters close to the ideal frequency characteristic can be 
designed. Filters whose time characteristic is close to the 
ideal can also be designed, but filters close to both cannot 
[12]. 

In this chapter time-domain response of digital filters 
will be discussed and it will be shown that the location of 
the transfer functions poles has a profound effect on the 
transient response of the filter. 



B. TRANSFER FUNCTIONS' POLES AND TRANSIENT RESPONSE 



Given a transfer function of a digital filter in 
factored form 

72 



where z, *s and p f ' s are the zeros and poles, respectively, 
of the filter. In general, the input signal's Z-transform 
is of the form 



X( 4 > = !lfe) 



H 



where g; 's are the poles of the input function. The 
response of the system may be obtained from the transfer 
function relationship 



Y(*)= Hfr) ■)<{*)- K. 



fe.p) . . . U-?n) 



Hit) 



C*-q,)---(l-lO 



(r-3) 

For simplicity, we assume that all of the poles of the Y (z) 
are distinct, and making partial fraction expansion of Y (z) , 
we obtain 



iM = k, 



teifc 



_] — + 



K-i 



+ 



C, * 



+ ... + 



C** 



The response of the system may be decomposed into [5] two 
parts called the input signal mode (ISM) and the system mode 
(SM) as 






Y«W- 



a-p, 






+ 



+ ••■ + 






Combining eg. (5-4) with eg. (5-5) and eg. (5-6) 






Eg. (5-7) indicates that the response of any linear system 
to any input will contain modes generated by the input 



73 



signals poles and the system transfer function poles. The 
time domain response of the filter may then be found by 
taking the inverse Z-transform of eg. (5-7) 

y(k) = k sk + ^, SM a) * ^ c\l) (s-t) 

where 

Equations (5-9) and (5-10) indicate that a fast responding 
system is one in which all of the system transfer function 
poles, p; , are sufficiently smaller than unity in magnitude, 
in order that the system mode will decay to zero rapidly. 
On the other hand, a slowly responding system is one in 
which the system mode decays to zero very slowly (i.e. at 
least one of rhe p is close to unity in magnitude) . 

Those poles closest to the unit circle will be called 
the system's dominant poles because they tend to dominate 
the characteristic of the resultant transient response. 

The dominant poles' response property is, as stated by 
Cadzow [14] "The response time of a linear discrete system 
is directly dependent on the locations of the system 
transfer function's dominant poles. Depending on the 
particular response- time requirement for a given 
application, we then have to correspondingly locate the 
dominant. poles of the transfer function. A fast responding 
system necessitates dominant poles of magnitude much less 
than one ". 

The poles of the MTBC filter for the orders 3-11 and for 
various values of m and w are given in Table III. 



74 



TABLE III* POLES OF HTBC FILTER 



1 

n 


i 
rn 


^■oca-ft'on. c -f /'i se^recC j evaj C^o) 




4.0(3 


4-oS 


-f.il 


<- 12. 





j -IK*j.O 


- ./to +j.o 


— /SO-t-J-O 

-•Sootj.iSl 


- • m +j . 

_.4£6±j.2Vo 


H 


i 

A ! 

J. i 
i 
| 




.HfiJ .Lo\ 


.3^ ±j.;^ 


.%Hj .i?r 

.2?i±j. #1 


ST 




♦26J ± j.o 


• W ±j.O 

-300 ±J. 92| 
»</0$-±.j .JIM 


.323lj.?24 
.^2^>±j.322 


. oS1 +j.o 
$ . 36! t\. o 


-•1/4 -t-j.c? 
•3o3^J-!36 


•3?i +j*0 
— iiV ±J-2o5" 


•3>2S^-r2t 


6 


4 .3?3±j-43o 

. J2c» ij • 765" 


•500 +j. 134 


-.5I>±J.|3> 

.Hn±J.¥tf 


.5or±j-<fk> 






•tlt±j .09/ 
.02? ±j. UZ 


•2ntj.3IO 
.(#?±. j.6^2. 


7 

i 
i 


4 


•534+j-c 




.570 + j.o 
.5^ + j-l62. 

• 422i j.T-fa 


.ff5"tj.29-V 
'5215. j-6~62 


4 


• 2J0l 1-J6O 


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. j I T + j • r53 


.^o+j-0 

•l"H±j .263 




3 
» 


•07J> tj-o 
- • 3 r^ t j • 

-•W-tj.o 
•«f±j OSS 
-221 iJ-!36 


-.102 +J o 


— -Oil f r )-0 

— IH3 ti. Ifl 

— ■ 2 5" ± j . 221 

( 





75 



TABLE III* CONTINUED 



a. 



rr> 



i_CKL3 -fioo. o^f \h.s.€.r -V«.cL z^ns (u)o) 



-f.ofe 



-T6? ±j.U4 
•5-1T ±J.3f? 



-/. 0$ 



.4te±j'£or 



4.10 



.60) ±j-H7- 
-49otj.6io 



Ml 



-£22-tj.624 



'4So rj. OS I 
'11* ±y If fl 



•111! ySQ) 



.49}±j.o*3 

•32?±j.f24 
• 26^ i j. «?o2. 



•3*0 ± J. $02 



.3#±j-ofl 
• 32.KJ.J46 
. 1^ ij .2*7 



• 2H ±j. o62 
•Z4>±J. )U 

• oS>±j.;zik 
,325^.4% 



. /o k ±\ . 0}3 
,06Zij.243 



--o4S~± j.oft 
-.26<f±j.34l 

-.64?±j.^3 



135 +j.o 
-106 ij.3lk 



?/ 






.20 1 "M .0 

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-*w±j.?2r ! 

— 06f ±j-ft3 I 



.tf3*j.33? 

—432±j-$J3 

- 05 9 tj.^Of 



2*2? -tj .O 
.166 ±j-343- 
.•2?9±j\?*o 



J2 



•3*2 ±j .3J- 



■5/4 ±j.o 
•3oolj«/2l 



0^4 +J.0 

../si + j-vsi 



•351 ij- 601 
5"|? +j.?32 



.611 +j.o 
H43±j-*tt 

.2/^ij^3r 



..o>6 t i»o 

-461 tj.30l ' 

•2^2 ij. 214 



_ .o^tj .0 

_.*3?*i'.3tf 

_.tf2tj W| 



■of* + 



J 



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o4?± j. \>l 
4»3+j.423 
£41 i\.5-63 



003 -h-o 
l9o ij.34l 
045"+j.}£7- 

3% ij-^2 



76 



TABLE III* CONTINUED 






i 

n 


m 


i-ooa-HoA. of »rvse.rAedL 2<eros C^3 ^ 


4.o6 


Jt.Oi 


J-i 


MO. 


to 


i 


.iL%±'yS2.o 

.^3o±j-l^o 
.6cT±joo5~ 
.$T/f ±j.S"iO 


— 3U>±J-??3 

./o£±j-Wf 

— -JI*+ ±j-6>o 

— ^34 ±j.£42 


— .10? tj. L2(> 


•264 ±j.|fc£ 
-.o?$±j.m 
-.Solij-W 
.|32±j.W 


1 


o/4 ± i.oSO 

- -Sil -Lj.62f 

— 3^f ±j-«2 
-.3W±j. 535" 


, 262±j.o% 
-•f24± j.^i 

-■ //2 £ j. W 


•lo2±j.U5" 

— oo 1 * -Li. 35*3 
-•5oi t j. ?62 


_.2T6±j- til j 
-•H3lrj.ro2 


3 


-.055 ±1.0*1 
-•2?3 ±*J.30fc 

-•^Jo+j.V/4 


-• 054 tj. 10 2 
— 661 *j W 


--04-9 ±jN& 
-.2^1 ±j.3o? 
-•/ri -tj ■ 2 2o 
-• 5/6 tj- ^f2^ 


— l»o±. j-^2^, j 


il 


i 


.201 j j -2 3 7 

J 

--33I *j -??? 


. 2?0 + J >0 
.226 ±j -20? 

-.326 ±i.?7* 
u 

,o5£±j -5?2 


- 3lo tj- 


.211 ± j- 32] 

j 


X 


•Otftj-o 
_.5J2tj -^ 


-/la i \. 

<Wi f.2S2 

— .$oo i \ . i&o 

-.f>53*j.?33 


./or ij.AVS" 
..20? ij.lol 

— •-/ <JO — ) '7 77 

— .nil + I'.^ff" 


• i*i+jf.o 

-intj.4ir 


3 


— •oft +y o 

-•052- ;./^ 

- • ill ± i.*Ss 


• C52 t V ^ 
-. 131 ij'-^H 


.051 -ri -0 
.0/0 t ,-. ig if 

264 i \ -^^° 

--427ij.-' 


| -0^6 t-j-o - 
L.oS} t j-2o| 

|-.1lt±.j.4 l H 



77 



Although this partial fraction expansion method helps us 
to understand the importance of the poles of the systems 
transfer function in transient response analysis of digital 
filters, evaluation of residues of corresponding poles of 
partial fraction expansion is not a trivial problem. 

We believe that the so called ■ transfer matrix ' 
method, which we are about to discuss, is more suitable for 
digital computer simulation. 

Given a digital filter weighting seguence h (n) and the 
input seguence x(n), the response of the digital filter may 
be obtained by convolution summation 



rt. 






H 



Eg. (5-11) may be written in matrix form as 

r KM o o • • • © i ^m 

kto) h(o) o • • • o 



gfc) 



3^ j 



WW WGi-0 kU) • • • W«) 



Y<0 



H 



Or 

i}- = fi. . X (s--ft) 

where G is the systems transfer matrix, which is defined as 



-ww o o ... o- 

KUj WW o • • ■ o 



G = 



. kit) wM M*-0 



WO 



H 



To find the systems response using eg. (5-13) , systems 
transfer matrix must be available. To obtain the weigting 



78 



sequence (impulse response) of the system to form the system 
transfer matrix in terms of the transfer function's 
coefficients, the difference equation of the system is 
solved for the impulse input, i.e., 



m 
ku) = O.o - J_ hi MCr\-L) , \)o (sws) 

where a- and b { are the numerator and denominator 
coefficients, respectively, of the transfer function of the 
digital filter. 

A computer program ( FORTRAN ) is developed to 
investigate the transient response of the digital filters, 
using equations (5-13) and (5-15). Program listing is given 
in Appendix 3. 

Step responses of MTBC filter for various values of n, 
m, and w are given in figures 36-39. Figure 36 and 37 
indicate that increasing the values of m and w also 
increases rhe overshoot and settling time, but doesnt have 
any significant effect on the rise time. The rise time tends 
to increase with increasing order of the filter. 



79 



C. S0HH1BI 



Transient responses of digital filters depend en the 
position of the poles of its transfer function. 1 fast 
responding filter has poles of magnitude much less than one. 
Modification cf ail pole filters increase settling time, 
decrease peak overshoot, and doesnt significantly affect the 
rise time of the filter's step response. 






30 



// ; 







ft 



nni~_.fi: ■ 'K! T i ^ 



fiT- 



Figure 36 - STEP RESPONSE OF MTBC FILTEB(N=7, W =1«U6) 



81 




,r I 



iLp--5^ QL 



Ou UNITS INCH, 



-t-g: c _ 



-'-['.£ 



^ -• U u r 



01 



UK 1 15 



T ki r M 



n 



Figure 37 - STEP RESPONSE OF THBC FILTER ( N = 5, M= 1 , K=1) 



82 







Irlf ±-~0 


■ ! ki T r % 


INCH 


Mr , -i « 
' . " — j I I 


• ILITTIT 


1 MS, ,rH 



rLT 



Figure 38 - STEP RESPONSE OF MTBC FILTER ( M=1, W =1«36) 



83 



- 
5 



- i 



/tf$N 




H? 



i 

/ / 



/ // 

/ / 



/ 



/ / 



/ 
/ , 

/ / / 

1 / f 






nT 



Figure 39 - 



STEP RESPONSES OF B, MB, MTBC FILTEfiS ( N=5, 
H=1, W =1*36) 



84 



APPENDIX A 



COMPUTER PROGRAM LISTING 



C 
C 

C 
C 



C 

c 
c 



c 

c 
c 
c 



c 

c 
c 



: 
c 
c 



CIMENSICN RNUMR(25 ) , RNUM I ( 25 ) , RDNUMR ( 25 > ,RCMMI(25) . 
*C1(25) tC (25 ) ,R1(25) ,F (25) 
CIMENSICN X(50)»Z(50),W(50),Y(50) 
CI MENS I C Is FCCTR(50) T RCOTI(5C),COF(5C ) 
CIMENSICN CM50) ,CY(5C) ,CZ(50) 
CIMENSICN RCOTDR(50 >,RGOTCI(50> 
CIMENSICN CYH50) 

CIMENSICN FX(100),PY(1CC),RC(1C0),RC1<100) 
CIMENSICN FN ( 25 I ,RC(25) 
OIMENSICN XT (25) ,YT(25),AX(25),AY<25) 
CIMENSICN CUMMY( 100),CLMMX(100)tITB(20)tRTB(30) 
CIMEISSICN CLMKAY(ICO) 
CCHPLE) Fl.PiCtCl 

FiAC THE CFCEF CF FILTER, CRCER CF INSERTED ZERCS AND THE 
WEIGHTING FACTCR 

RSAC (5 ,5C1 ) N,M,K 
5C1 FORMAT<2I2) 

PEAC LCCA7ICN CF ZERCS 

FEAC(5,5CC> (W(I),I=1,M) 
5CC FCRMAT (EPIC. 5) 

RE/O INITIAL ANC FINAL VALUES CF THE FREQUENCY 
(10 BE LSEC IN FREQUENCY RESFCNSE CALCULAT ICNS ) 

PEAC (5 ,5C2 I W8EGIN,V<LAST 
5C2 FCPMAT (2F1C5) 

PEAC THE NLNE EP CF SOLUTION POINTS 

PEAD(5,5CS) NPCINT 
5C< FCRMATU2) 



PEAC SAMPLING PERIOD CF THE DIGITAL FILTER 

7 



REA0(5 .7C ) T 
FORMAT (F1C.5) 



XPCINT=NPCINT 
WCELTA=('*LAST-WBEGIN >/XPOINT 

CAlCLLATICN CF NUMERATOR FCLYNCMAL 

ICIMY=1 

Y( 1) = 1.C 

IF(M.EC .C ) GC TO 401 

CC 2 I=lt> 

X( l)=Vi (I )**4.0 

X {3)--2 ,C*V (I )**2.C 

X(2) =C.C 

X(5)=1.C 

X(4)=C«C 

CALL FfrFY (2,IDIMZ,Y,ICIMY,Xt5) 

CC 3 J=1,ICIMZ 

Y( J) =Z (J) 
2 CCMIME 

ICIMY=1CINZ 
2 CCNTIME 

CCNVERSICN CF INDEPENCENT VARIABLE FROM fc TO S 

CO 26 1 = 2 ,ICIMY,4 
Yd >=-l.C*Y(I ) 



85 



26 CONTINUE 
C 

C FINC THE F-CCTS OF NUMERATOR SQUARED FUNCTION 
C 

PMJM*XCII'>-1 

CALL PCLR7 ( Y ,C0P , MNU N ,ROCTR ,RCOTI , I ER ) 
C 
C 

C SELECT THE RIGHT HALF PLANE RCOTS OF NUMERATOR SQUARED 
C FUNCTION 



C 



L = l 

CO 2 I=1,MNUM 

IFCRCCTMI ).GT.O.O) GO TO 20 

RNUMR(L)=FCC7R(I ) 

RNUMKL l = PCCTI(I > 

L = L+1 
2C CONTINUE 

MN=MNUN/2 

LK = L 

LZ=L-1 

IF(LZ.EC.NN ) GO TO 2 2 

LL=1.C 

CO 35 I=1,NNUM 

LX=L+1 

LV=I + 1 

IF (RNUM KLD.EQ.ROOTK I ) .OR .RNUM I ( LL > .EG. (-1 .0*ROGTI 
MI))) C C T C 3 5 

IF(LL.EC. 1) GO TO 26 

LJ=LL-2 

IF(RNLMI (LJ).EC.RCCTKI ) .OR .RNUM I < L J ) . EG .( - 1 .C*R0G7I 
MI ) ) ) GC TC 25 

IF (LL - E C .2 ) GO TO 26 

LI =LL-4 

IF(RNUMI(UI).EQ.RCCTI (I ) .0 F .RNUM I ( LI ).EG .(-1 .0*R0CTI 
MI )) ) GC TC 25 
It RNUMR( L) =FCCTR(I ) 

RNUMKL > = RCCTIU ) 

RKUMBCLXJ =FCCTR(UY) 

RNUMI(L>)=FCCTI(LY) 

IF{LX.Ed v N) GO TO 2 2 

LL=LL+2 
25 CONTINUE 
2 2 CC 2 2 JM,MN 

R€2 )=C>FL> (FNUMRU ) ,BMJNI (I) ) 
22 CONTINUE 

CAUL MKFCL (MNtRfC) 

^1 = ^+1 

C(MN1)=CMFUX(1. 0,0.0) 

CC 31 1 = 1, CM 

RCCI ) =P E AL ( C( I ) ) 
21 CONTINUE 

GC TC 5 
4C1 RC(1)=1.C 

MMM 

NN = C 

RNUMR( 1)=C.C 

RNUMH 1 )=0 ,0 

C^lCULATICN CF DENOMINATCR POLYNOMIAL 

5 NDNUM=2*N 

IF(M .EC .C .4NC.K.EC.N ) GO TC 402 

CCNST=1.C 

IFCH.EC.C ) GO TO 7 

CC 11 1 = 1, M 

CM L7MV»(I)**2. 0-1.0) **2.0 

C0NS7=CML1*C0NS7 
11 CONTINUE 
7 IF(K.EC.C) GC 70 52 

KK=2*K 

KKK = KM1 



86 



c 
c 
c 
c 



c 

c 
c 

c 
c 
c 



c 
c 

c 



c 
c 

: 
c 



c 

c 
c 



c 
: 
c 



CM(KKK)=CCNST 
CG 12 1 = 1, KK 
CMI )=C.C 
12 CONTINUE 

GO TC 52 
52 CM( 1)=1.C 

KK=0.C 

KKK=1.C 
52 NK=N-K 

NKKl=NK4l 

DETAIN THE CFEEYSHEV FCLYNCM I A L , SQUARE IT AND NULTIPLY 
THE RESULT EY EUTTERkQRTF SQUARED FUNCTION 

CALL OeSV(CYlfNK) 

CALL FNPY (CY,NKK,CY1,NKK1,CY1,NKK1) 
CALL FNF> <CZ,IDIMCZ,CY,NKK,CM,KKK) 

CETAIN CENCMINATOR SGUAPEO FUNCTION 

CALL PiBCCf CZtIDIMZ,Y,IDIMY,1.0,CZ,IDIMCZ) 

CCNVERSICN CF INOEPENCENT VARIABLE FROM W TC S 

t CG 17 I*3,ICIMZ,4 
Z(I)=-1.C*Z(I) 
17 CCNTIME 
6 NONUMICIN2-1 

FINC THE FCCTS CF DENOMINATOR SQUARED FUNCTION 

CALL PCLRT ( Z, COF , NDNUM , ROOTCR ,ROOTD I , I ER ) 
GC TC 4C2 
4C2 CALL RCCT ( NCNUM, RCCTCP ,RGQTCI ) 



SELECT THE FIGHT 
FLNCTION 



HALF FLANE RCOTS OF DENOMINATOR SQUARED 



AC2 J = l 

CO 21 I=1,NCNUM 

IF(RCCTCF ( I ) •GT.O.O) GO TO 21 

RDNUMF (J )=FCCTCR(I) 

RDNUMI < J)=RCCTDI < I ) 

J = Jf 1 

21 CCNTIME 
NC=NCMN/2 
CC 22 1 = 1, NC 

fild ) = CNFLXRDNUMR(I ) ,RDNUM (I ) ) 

22 CONTINLE 

CALL NXKFCL (MC,R1,C1) 

NCl=MC4l 

CKMC1 l=CVFLX(l,C,C.C) 
47 FCPMAT (2> ,25F11.3//) 

CO 32 1 = 1, NCI 

RCK I )=PEAL(C1(I ) ) 
32 CONTINLE 

NCRMALIZATICN CF NUMERATOR POLYNOMIAL 

FACTOR=RCl(l)/RC( 1) 
CC ICC 1=1, CM 
RC(I )=PC (I )*FACTGR 
IOC CONTINLE 

WRI T E < 6 ,57C) 

C l'f PUT SECT ICN 

57C FCRMAT ( ' 1'////) 

U R T T C f £ C 6 C ) 

5tQ FQRMATC2X,' LOW-PASS PROTOTYPE (CONT I NUOUS ) FILTER'//) 

bRITEU ,571) 
571 F0RM*T(2X,' CRDER OF FILTER •//) 



87 



c 
c 
c 



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57 
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51 
52 

51 



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CC) N 
, 12//) 
72) 

, ■ CRQER CF INSERTED ZEROS •//) 

CC) M 

72) 

, » WEIGHTING FACTOR CF TBC FILTER '//) 

CC) K 

74) 

t ■ LOCATIONS CF ZEROS '// ) 

CI) (w(I ) ,1=1, M 

1C.5//) 

76 ) 

,' COEFFICIENTS CF NUMERATOR POLYNOMIAL 

ING CROER ) •//) 

7) (RC( I ) ,I = l,yNl) 

75) 

, • COEFFICIENTS OF DENOMINATOR POLYNOMIAL 

If^G CRDER ) '//) 

7) (RCKI ) ,1 = 1, MCI) 

11) 

,'RCOTS CF NLVERATCR'// ) 

12) 

,'REAL PART • ,4-X, • IMAGINARY PART '//) 

1 ,¥h 

12 ) RNUMRd ),RNUVI( I ) 

,F10.5,4X ,F1C5/ ) 

14 ) 

,'RCCTS CF CENCMINATCR'// ) 

12 ) 

It PC 

12) RDNUMR(I) tRCNLMI(I) 

76) 

, • INITIAL ^ALLE CF FREQUENCY '//) 

1C) WBEGIN 

,F1C.5//1 

77) 

,« FINAL VALUE CF FREQUENCY •//) 
1C) «LAST 



CUTTING SECTICN 

WRITE(6 ,5<C) 
55 C FORMAT ('l 1 ) 

CALL PPLCT TC FIND FRECLENCY RESPONSE VALUES CF THE 
FILTER 

CALL PFLCT(RC,RCl,W5EGIN,wCELTA,NPOINT,PX,PY,fDl,MNl) 
CALL FLCTF ( FX , PY ,NFC INT , C ) 

PScCISTCRTICN FOR BILINEAR TRANSFORMATION 

SCALE=7A^ (T/2. ) 
CO 777 LK=1,MN1 
EXF=LK-1 

RN(LO=RC(LK)/(SCALE**EXP) 
777 CCNTINLE 

CO 776 LK=1,MD1 

EXP=LK-1 

R0(LK)=BC1(LK)/(SCALE**EXP] 

77c CCNTINLE 

n t = m fv 

CALL ZCMN (RN ,RD ,NK ,f«T,WBEGIN,WLAST, kCELTA ,NFCINT,7) 

CALL XFCRMRC,RC1,1,XT,YT,*N,MC,1CCC.C,:2.C) 

CALL PFLCT(XT,YT,5C0-C,10.C,1CC,AX,AY,NC1,VM) 

CALL FLCTF (AX, AY, ICO, C) 
STOP 



88 



c 

c 

: 
: 
: 
c 



c 
c 

r 



C> 



ENC 



SL3R0LTINE OBSV <CY,NC) 

5t?59JrIlt? J C FIND Tl ~ 6 CCEFFICIEM5 OF CFEEYSFEV 

FClYNC*IAL CF GIVEN CPCER 

NC : CRCEF CF CFEEYSFEV FCLYNOIAL 

: CALCILATEC COEFFICIENTS (CESCENOING C^CER) 

CIVENSICN CX(50) ,CY(5C),YY(50),ZZ(50 ),Z(50) 

N N = N C 4 1 

CX(1) = 1 ,c 

IF(NC.EC.C) GG TC 2 

CY(1)=C.C 

CY(2 1*1. C 

I c ( NC . EC . 1 ) GG TC 19 

YY (1 ) = C .C 

YY(2)=2.C 

CO 5 I = 2 t NC 

11=1+1 

11=1-1 

CALL ?NFY(Z,IDIMZ,YY ,2.CY,I ) 

I C I M Z Z = I C I fi Z 



c o 



1 • ICINZ 

) 



ZZ(J)=Z( 
CONTINLE 

CALL FSia(Z,I3IMZ,ZZ,i:iVZZ.CX,U) 



6 
LFLATE FCLYNCMALS 



CO 7 J = 1 , 1 
CX(J ) = CY ( j I 

7 CONTINLE 

CG 8 J=l , ICI vz 
CYU> = Z<JI 

6 CCNTIME 

5 CONTINLE 
l< RETURN 

i CY(1)=C> (1 ) 
GC TC IS 
ENC 



SLi 
FIL 

PCI 
PC 
N : 
V : 
Fl 
F2 
T : 



SLSPCL7INE 
TC 



ZDMN(RCtRCl,N,M,Fl t F2,WDELTA,NP , x ) 

FIND ANC FLCT FREQUENCY -RESPONSE CF DIGITAL 



ROUTINE 
TEPS 

: COEFFICIENTS OF DENCPINATGR OF CONTINUOUS FUNCTION 
COEFFIC I6NTS OF NUVERATGR CF CONTINUOUS FUNCTION 

CPCEP CF CEN0 W IN4TCR 

CPCER CF NUMERATOR 
: INITIAL FREQUENCY 
: = INAL FPECLENCY 

SAVFLINC- PERIOD 



CI 
CI 

PC 
X° 

ex 

CI 

CI 
FF 

:i 
,p 
ci 

:: 

CO 



VENSICN ECC(25) 

MENS ION FC(25),PCH25>tZX(25),ZXl(25) , XJ(25),YJ(25>, 

CTPC (25) tPCCTPN (25 ) ,RCCTIC(25) ,RGOTIN ( 25 ) , CY(25) , 

AY(25 ) ,>FA>DA( 25) , 

(25 » ,YFJY(2 ) ,YP£YDM2 ) 

KENSICN ACC(IO) ,Aii(10),*22( 10 ),B11 (iC) , 522(10) 

MENSICN >AXIS(l5C0),YAXISH150C)tYAXIS2(l5C0), 

(15CC ) ,CEL( 150C) 

MENSICN >CCF (3) , X2CCF(3) , COF ( 3 ) , RCCT .< ( 2 ) , FCCT I ( 2 ) 

C0T1 (2 ) ,;CCT212 ) 

VENSICN CLfKAY(lCC) 

UBLE PRECISION W,Y 

M*GN/FAW/W(100) ,Y( 10C>»ICALL 

AL K/Gh , MGNH,NL.*i ,NUN2 



39 



CX ,C>,XJ , VJ 

NLN,CENQV; 



9C 



COMPLEX 

CCNPLEX 

MJ=M+1 

NJ=N+1 

NNN=N-> 

call ez>ffm,rc,m,nvm ,zx,1) 
call s2xfpmrc1,n,nmn, zx1,c> 

sfact = zx<n,j)/zxi(nj) 

CC 9C 1 = 1, NJ 

ZXH I )=ZX1(I)*SFACT 

CCNTINLE 



WRITEU ,6C) 
C 

C COEFFICIENTS CF DENOMINATOR PCLYNOPIAL 
C 



i: 

6C 

7C 



61 

62 

62 
64 



WRIT 
WRIT 

CALL 
CALL 
CALL 
CALL 
CALL 
F = F1 
CO 1 
X = F* 
VAGh 
FF3 = 
CG 6 
AC = A 
A1 = A 
A2=A 
S 1 = 3 

E^ = e 

NLMi 
NUM2 
NLV = 
CeNO 
CENO 
CENC 
VAGh 
FH1 = 
PH2 = 
F!-2 = 
CC.NT 
VAGN 
PF(I 
FK = P 
XAXI 
YAXI 
>AXI 
MI) 
Y( I ) 
F = F + 
CCNT 
WRIT 
FORM 
FGRM 
hRIT 
FORM 
(PRC 
WP-IT 
FORM 
WFIT 
*RIT 
FORM 
WRIT 
WRIT 
FOR* 
WRIT 
FGRM 



£(6 ,45 ) (ZXH I » , I = 1 ,NJ ) 
EU t6C) 

PCLRT(ZX,CCF,N, FCOTRN ,RGOTIN,IER) 
FCL=T(ZX1,CCF,N ,RCCTRC,RCCTIO, I ER ) 
CCE^EN (N,MX) 

F^CTCR (ROGTRN,RCCTIN,N,VX,AGO,A11 ,A22 > 
F4CTCR ( ROGTR C, P CCTI C, N,MX, 3 22, Bli, ECO) 

C 1 = 1 , NF 

T 

*1.C 

c.c 

c c T 

n ( 

22 ( 
11 ( 
22 ( 
= (A 
= U 
CVP 
Ml = 
N2 = 
N=C 
= AE 
(AT 
(AT 
FF2 
INL 
F = 2 
) = F 
h(l 
S( I 
Sl( 
S2( 
= XA 
= W 
WCE 
INL 
EU 
ATI 
AT ( 
E U 
AT ( 
TCT 
E (6 
AT( 
E (6 
EU 
AT ( 
E( t 
EU 
AT ( 
EU 
AT ( 



1,N 

J ) 

J ) 

J) 

J ) 

J) 

2 + 1 

C-A 

LX ( 

(E2 

( 1, 

NFL 

S(N 

if\ ( 

AN ( 

+ ( F 

E 

C.« 

F2 

) 

)=F 

I ) = 

I ) = 

X IS 

XIS 

LT A 

E 

,6C 

■ 1 ' 

2X, 

,7C 

2> 

YF 

,6 

2X 

,4 

t 62 

2X, 

,45 

,62 

2X , 

U« 

IX, 



n*c:s( xj+ai 

2)*SIN(XI 

NLM ,NUN2 ) 

♦i.0)*ccs(x)+ei 

-E2 >*SIN(X ) 

X(CENGV1,CENCV2) 
AGh*CA3S(NL^/DENGV) ) 
NLN2/NUM1))*57.2957 7951 
CENCX2/CENCM) )* 57 .29 5 7795 1 
F1-PH2 ) 

ALCGIO(NAGF) 



NAGh 
f-AGNH 
(I ) 
1(1) 



) 
) 

1CF12.3/// > 
) 



3ILINEARLY TRANSFORMED LCW-PASS 
DIGITAL FILTER'//) 



COEFFICIENTS OF NUMERATOR 
(ZX(I ) ,I = i,NJ) 



PCLVNC W IAL »//) 



COEFFICIENTS OF DENOMINATOR POLYNOMIAL '//) 
(ZXK I ) ,1=1, NJ) 

NUMBER CF CASCADED STAGES'//) 
NX 
2) 



90 









r 

c 
c 
c 



WRITEU ,61) 
81 PgRMAT <2X, 'COEFFICIENTS OF CASCADED STAGES 1 //) 

.- ^|l|^2e2)JG0(I),All(I),A22(I),Bll(I) t e22<I) 

o: rG« M n (i)< t ;pl2.5// ) 

62 CCMIME 
WRITEU ,63 ) 

63 FORMAT (2X , 'ROOTS CF NUMERATOR '// ) 
CC 66 1=1 ,N 

WRITE<6,43) POGTRNd ),R0GTIN(I J 

66 CONTINUE 
WRITEU ,67) 

67 FORMAT <2Xf' ROOTS OF DENOMINATOR '//) 

WRIT E ( 6 t 5 5 ) ROOTROd ) ,RCCTIC(I ) 

68 CCNTINLE 
WRITE (6 ,S£ ) 

96 FCRMAT(2>,' SAMPLING FRECUENCY '//) 

WRITE(6,cc )T 
9<E FORMAT (2X ,F1C. 5//) 

N=NP 

WRITE(6,12 I 
12 FORMAT ( • 1* J 

CALu FLCTF OAXIS ,YAXIS1,\,C) 

WRITE(6,31 ) 

31 F0RMAT(//4CX, • ABS. GAIN V.S. FRECUENCY') 
*RITE< 6,16) 

16 FORMAT ( i i« ) 

CALL FLCTF OAXIS, YAXIS2,N,C) 
"RITE (6 ,3 2 ) 

32 FCRMAT<//4CX,» GAIN (C3 ) V.S. FREQUENCY •) 
*RITE(6 ,14) 

14 FORMAT ( ' 1 • ) 

CALL PLCTP (XAXIS ,FI-,N,C) 
WRITE( c,32 ) 
52 FORMAT (//4CX,» PHASE V.S. FRECUENCY •) 
RETLRN 
ENC 



SUERCUTINE CCEVEN(N,M ) 

SUBROUTINE TO FIND THE NUMBER CF CASCACEC S' 
N: Zmtz CF NUMERATOR FCLYNGMIAL 
M: CEGREE CF CENQMINATCP FCLYNCMIAL 



'AGES 



NN=(N/2)*2 
I F ( N . N c . V N ) 
M=N/2 

GO TC 2 
*=<N+1 )/2 

RETURN 
ENC 



10 TC 1 



SU 

CI 

* ,Y 

3 ,X 

SL3R3 

FClYN 

* : FC 

» : CR 

N f M = N 

AND C 

ZX: R 

I=mCT 

i fact 



BRCL 

MENS 

:jm 
fact 

UTIN 
CM I A 
LYNC 

CER 

-M ( 

ENCM 
ESUL 
= 1 
= C 



TINE 

ION 
) ,2 
23 ) 



EZXFRM (X, N ,NMM,ZX, I FACT ) 
>(25),Y(25),ZX(23),XCUM(2) 
1125),Z2(23),X1( 25>,X2(25> 
,ZXi(23) 

FIND ThE BILINEAR Z-TRANS FORM 



OF A GIVEN 



IAL 

CF F 

TM 

IN A 7 
TANT 

FCR 
FOR 



CLYNOHIAL 

:i c =ERENCE EET^EEN 
CR 

FCLYNOMIAL 
NLVERATCP 
CE NOMINA TCP 



ThE CRCERS OF NUMERATOR 



M=V + NM+1 



91 



N2=M*1 

CO 1 1=1 .M> 

ZX(I )=C.C 

CONTIME 

XDLN(1)=-1.C 



XDUMU 
YDIM1 
YOLM(2 
IZX = 2 

CO 2 
CALL 
CALL 
CALL 



) = 1 
) = 1 
) = 1 



,11) 

12) 

tI2) 



11 = 1 ,N2 

PCLE>F(XDUM,2,Hf,Xl 

FCLEXF(YCUM,2,N ,X2, 

FNPXZltlZi XI, II, X2 
FACT = X (II ) 
CALL FADCN ( 22 ,1 Z2 , ZX , I ZX , F JCT , Zl , I Z ) 
CC 4 K= 1 , 1 22 
ZX(K) = 22(K ) 
C3NTINIE 
NM = MM1 
N=P-1 
IZX=IZ2 
CCNTIME 

IF(IFACT.EC.C) GC 
CALL FCLEXF(YDUM, 
CALL FNFY<2X1,MZ, 
CC 6 I = 1 , N 2 
ZX( I )=ZX1< I ) 
CONTIME 
V=M2-1 
RETURN 
ENC 



TC 5 
2,NPM,XFACT,NXFACT) 
XFACT,NXFACT,ZX,IZ2) 



,PIN 
(25) 



SLERCL7INE FACTOR(RRN 
CIMENSICN RRN( 25) ,RIN 
* C(25),C1(25),AOO(25),A11(25),A22(25) 
CCNPLEX C,C1 

IMTIALIZE CCLNTERS 

IM = C 

IY = 1 

K = 

M=0 

IX = 1 

NXR = C 

NXI = C 

IFtV.NE.C ) C-C TO 18 

CC 17 1 = 1 ,>> 

AJO< I ) = 1.C 

All( I)=C .C 

A22(I )=C.C 
17 CCNTINLE 

GC TC 2C 
IE IF(M.NE.l) 

AOO( 1> = 1 .C 

A1K 1) =-1 

A22( 1)=C. 

K = 2 

NCIF=M> 

NXK=1 

GC TC 1C 
U CO 1 1 = 1 , 

IX1 = IX-»1 

IFdX.GT. 

IF(RRN(IX).EC.RRN(IX1).AND.RIN(IX).EC 
* (-l.C*F IN ( 1X1) ) ) 

IMI»+1 

XR(IN)=-1.C*RRN( IX) 

IX=IX+1 



K,MX,A22,A11,A00) 
, XR<25) ,XI(25),YR(25),YI(25), 



GC TO 16 
C*RRN< 1) 



ICC 

M GC TC 7 

) .EC.RRN(IXl) .AND.RIN(IX) 

GC TC 2 



92 



11 

1C 



NXR=NX 
GC TC 

iyi=iy 

YR(IY) 

YKIY) 

>R(IY1 

YKIY1 

IX=IX1 

M=NI + 

NXI=NX 

IY=I Y 1 

CONTIN 

NT=NXF 

IF(NT. 

IF(MX. 

NCI(= = N 

IC = 1 

CO 4- I 

IC1=IC 

ACG( I ) 

411 ( I ) 

A22( I) 

IC=IC+ 

CCNTIN 

N0IF1= 

NCCF=N 

NLAST= 

CC 5 I 

NCGF=N 

ACC(NC 

A1KNC 

A22(NC 

CCNTIN 

N6EGIN 

NCOMF= 

CC 6 I 

NC0MF1 

C(l)-C 

C(2) =C 

CALL V 

ACCtNe 

All( NB 

422(NE 

NCGMF= 

CCMIN 

GC TC 

NCIF=f 

IF(NXR 

CC 9 I 

ACCl I ) 

Alien 

422( I ) 

K = K+1 

CCNTIN 

IF(NXI 

NXK=NX 

CC 11 

NXK=NX 

IK=I+1 

C(1)=C 

D(2)*C 

CAuL M 

ACC(NX 

A1K NX 

A22(NX 

K = K + 1 

CCNTIN 

CG 12 

NXK=NX 

ACCtNX 

A1HNX 

A22(NX 



P + l 

1 

+ 1 

=RRN(IX> 

= FIN (IX) 

)=RRN(IX1) 

)=R1N(IX1) 

jl 

2 

1+1 

+ 1 

LE 

■»NXI 

EC. NX) GG 

CT.NT ) GO 

T-NX 



NCIF 



TC 
TO 



F (IC l+XR ( IC1 ) 
R (ICJ*XR(XC1) 



IF-f 1 

F*2 

-NXI 

CIF1,NLAST 

F4 1 

i ) = 1.0 

1 )=XR(NCCF) 
1 ) = C.O 

LAST+1 

BEGIN ,MX 

CCNF+1 

LX(YR(NCOMP ) ,YI(NCCMP) ) 

LX(YR(NCOMPl) ,YI(NCCMP1) ) 

PCL(2tD,Dll 

in )=i.o 

IN) = FEAL(C1(2) ) 
IN )=REAL(C1( 1 
CNF1+1 



I ) 



= 1 

41 
= 1 

=x 
= x 
1 

LE 

NC 

CI 

NX 

-N 

CC 

IF 

IF 

IF 

LE 

= N 

1 

= N 

= N 

NF 

NF 

*K 

EG 

EG 

EG 

NC 

LE 

2C 

X- 

.E 

= 1 

= 1 

= X 

= C 

LE 

.EC .C) GO TC IC 

r 

1 = 1, NXI 
K + l 

NPLX (YR(I),VI(II) 

NPLX (YF(IK) ,YI (IK) ) 

JKPCL (2,0, CI) 

K > = 1 .C 

K) =FEAL(D1 (2) ) 

K )=FEAL(01( 1) ) 

LE 
I=K,NCIF 

i< + ; 

K)=1.C 
K l=C .C 
K )=C.C 



NT 
CO 

,NXP 
.C 

R ( I) 
.C 



GO TC 15 



93 



12 CCNTIME 
QC TC 2C 

2 IX = 1 

LZ 13 1 = 1 ,NXP 
ACG( 11*1. C 
All! I ) = XR CI ) 
422(1 J=C.C 

13 CCNTIME 
NXX=NXF-H 

CC 14 I=NXX,vx 

I>l=IX4l 

C (1) =C*f LX(YR( IX ) ,YI (IX) ) 
C(2) =C*PL> (YFCIX1 ) ,YI (1X1) ) 

CALL M4KFCL (2,0, CD 
4CC(NX> 1=1 .C 
A1K NX> > =FEAL(C1 (2) ) 
A22(NX> )=FEAKCi( 1) ) 
IX = IXH1 

njx=nx>+i 
14 continue 

2C RETURN 

END 



C 

c 
c 

r 
C 

: 

L 

c 



SUBROUTINE PQLEXP( XX ,IOIFXX,H, YY ,1 DI I»YY1 

SLSR.CLTINE TC FIND THE C C*=RS CF GIVEN POLYNOMIAL 

XX : PCLYhCPIAL 

ICi V XX : CIMNSICN CF XX 

V: POKES TC EE SAISEC 

YY: RESULTANT POLYNOMIAL 

ICiNYY: CIPEhSIQN CF RESULTANT PCLYNOMIAL 

CIVENS1CN XX(25) ,YY(25 ) ,ZZ(25) 

IF IN .EC. 11 C-C T 4 

ICIMYY-1 

YY (1 ) = 1 .C 

IF (M. EC. CI C-C TC 2 

CC 1 1*1 f P 

CALL FNFY (ZZ,ICIfZZ,YY, ICINYY, XX, IDT-'XX) 
CO 2 J = 1 , 1 C INZZ 
YY( J I = Z 2 IJ ) 

2 CCNTIME 
ICI^YY^ICINZZ 

1 CCMIME 

GC TC 3 
A CO 5 1 = 1 ,ICI^XX 

YY ( I ) = XX ( I ) 

i ccntime 
ici. m yy = i::*xx 

3 RETURN 
END 



SUBROUTINE c c LOT(RC,RCl,weEGIN,«CELTA,NPCINT,?X,PY, 
*VC1,NN 1 » 
CI^ENSICN FC(IOO) tRCKlCC) 
CINENSICN FX(lJG),PY(i:C) 
COPLEX XMMtCNUM 
fc=*6EGIN 



CC 666 I=1,NPCINT 
b»h+fcCELTJ 

INITIALIZATION CF REAL ANC COMPLEX PARTS Jr NUMERATOR 
A NO DENCHNATCR 

XNUMA=C.C 

x n l y e = c . c 

XNUMC=C.C 



94 



_ 


>NLMD=C.C 


c 


XCNUNA=C.C 




XDMJNB=C.C 




XDNUMOC.C 




XCNUNC=C.C 


c 




c 
c 


INITIALIZATION GF CCLMERS 


NNLMC=1 




PNUM1=2 




NNUM2=2 




MNLM3=« 


c 






*»cnunc = i 




NCNUM1=2 




NCNUP2=2 




NDNUN3=4 


c 






XC=MNLNC-1 




X1 = MNLM-1 




X2 = MMN2-1 




X3=M NLN2-1 



XC0 = NCMNC-1 
XC1^CMM-1 
X02=*DM*2-1 

XC3=*CNLN2-1 

OG 665 J=1,1C0 

IF(,VNLf-3.LE.NNl) 

IF(MMJf2.LE.MNl) 
IF(MNUM .LE .MN1) 
IF(MNLNC .LE.NN1) 



I F ( I* C N L I* 3 
IFCMDMN2 
IF(VCNLM1 
IFiVCMVC 



LE.MD1 ) 
LE.^Oi) 
LE.M01) 
LE.MD1) 



XNUNC=XNUNC+RC(MNUM3)*W**X3 
>NLNC = XNLNC + RC(MNUN»2)*W**X2 
XNUNE=XNUV8+RC(MNUM 1I*W**X1 
XNUNA=XMJNA+RC(MNUMC)*W**XC 

XDNUMC=XCNUMD+RC1(M0NLM3 )*fc**XD3 
XCMNC = XCMJMC + RC1(PCNUM2 )*W**X02 
XCMjNB=XCNUMe + RCl(^CMJNl )*U**XD1 
XONUMA-XCNUMA+RCK NCNIVO )*U**XD0 



INCREMENT CCLMERS 

nncmc^mk+4 

NMJMl = NNLNl+4 

MNUM2=^NL*2+4 

fCNUNG = »-CMN0+4 
yDNUMl = MCMM + A 
VCMIN2 = NCMV2+4 
►0NU»'3»»'CNLI»3+4 

XC = XC*4 ,C 
X1=X1+4.C 
X2=X2*i .C 
X3=X3+4.C 

XC0-XCC*4,C 

>C1=XC1+4.C 
>C2=XC2+4.C 
XC2=XC2+4 .C 



C 
C 
C 



66 



IF( ( NDM^C.GT.MDU .ANC. (MNLMO .GT .MNi ) ) GG TC 668 
CGNTIME 



CAuCLLATE NAGMTUDE CF THE TRANSFER FUNCTION 

666 XMJM10MN4-XNUMC 
>NLM2=>NLNE-XNUM0 
XNUM=CFFLX(XNUM1,XNUM2I 

CNLN1 = >DMNA-XDNUNC 



95 



ttt 



CNUM2=>CNLME-XDNUMD 

CNUM = ^FLX (CNUM1 ,CNUN2) 

XMAGH=CAES(XNUM/DNLM) 

PX(I \ = V 

PY(I )=2C.C*AL0G10(XMAGH) 

CONTINLE 

RETURN 

END 



C 
C 

C 

c 

c 
c 

r 

c 

c 
c 
c 
c 
c 
c 



SIB 

MY 
N7Y 
N1Y 
MY 
N1Y 
X 

\ 

w 

N 

XT, 



SL3RGL7INE XFGRM ( X , Y , MYPE ,XT , YT , M , N , Fl , F2 ) 
RGUTINE FCF FREQUENCY TRANSFORMATIONS 



c 

11 



2C 



FE = 3 

PE = i 

PE = 2 

PE = 3 

PE = 4 

COE 

CCE 

CRD 

GRD 

YT : 

CIME 

NN=N 

N1=N 

M = M 

IF(N 

CO 1 

XM=N 

>T(! 

CCNT 

CC 2 

>N=N 

YT(I 

CCNT 

GO T 

IF(N 

XF=M 

YF=N 

FACT 

K = M1 

r O 3 

XN=M 

K = K- 

XT(I 

CCNT 

K = N1 

CC 4 

>N=N 

K = K- 

YT(I 

CCNT 

PETU 

END 



NC TRANSFORMATIONS REQUIRED 
LCfc-FASS TO LCW-PASS TRANSFORMATION 
LCU-PASS TC HIGH-PASS TRANSFORMATION 
LCfc-FASS TO 8ANC-PASS TRANSFORMATION 
LCV.-FASS TC EANC-STCP TRANSFORMATION 

FFICIENTS OF NUMERATOR POLYNOMIAL 

FFICIENTS OF CENCMINATCP POLYNOMIAL 

ER OF NLMERATCR PCLYNCMAL 

ER CF CENOMINATOR POLYNOMIAL 
TRANSFORMED NUMERATOR AND DENOMINATOR POLYNOMIALS 

NS1CN X(25>,Y(25),XT(25),YT(25> 

-M 
+ 1 
+ 1 
TYFE.NE.l) GC TC 11 

1 = 1, M 
1-1 

) = > (I )=MF1**XM) 
INLE 

1 = 1, M 
1-1 

) = t (I M(F1**XN ) 
INLE 
2C 
TYPE .NE.2) GO TO 2C 



-i F1**YF)/(F1**XF) 
+ 1 

1 = 1, M 
1-1 

)= (X (K |*(F1**XM ) )*FACT 

INLE 

+ 1 

1 = 1, M 
1-1 
1 

> = Y(K )*(F1**XN ) 
INLE 
RN 



C 
C 
C 

r> 

C 
C 

c 



SUSRGUTINE RCCT(N,RR,RI > 



SL3R0LTINE 
X**N+1=C 
N : ORDER 
RF :ARRAY 
RI :ARRAY 



TC FIND THE PCCTS CF THE EQUATION 

CF FCLYNOM IAL 

CONTAINING REAL PARTS OF CALCLLATED PCCTS 

CCf^TAINING IMAGINARY PARTS OF CALCULATED ROOTS 



96 



CINENSICh FF(25),RI(25) 

PI=3. 141592 

XN = N 

TETA=FI/XN 

DC 1 1 = 1, rv 

XI =1-1 

ARG=((2.C*XI ) + l)*TET4 

RR(I } = <IN UFG) 

RI (I )=CCS(4RG> 

CCMIME 

RETURN 

ENC 



97 



APPENDIX B 



COMPUTER PROGRAM LISTING 



COMMCN h( ICG) ,HM(1C0 ,100) ,X<100) ,V ( 1 00 ) , N, P ,NPULS E 
C 

C SLBRGUTINE tc find the time dcmain resfcnse 

C A DIGITAL FILTER 

C 

C N : FILTER'S CRCER ♦ 1 

C A : NUMERATOR COEFFICIENTS GF THE DIGITAL FILTER 

C e : CENCNINATCR COEFFICIENTS ( WHERE FIRST COEFFICIENT 

C WIuL BE NCFNALIZED ANC WILL NCT EE ENTERED TC THE 

C PROGRAM 

C 

P»l. 

N = 24 

NFULSE=2C 
WRITEU ,1C) 
1C FORMAT < '1 • ) 
CALL If»FlLS 
CALL FMR> 
CALL INFLT 

call cermet 

WRITEU ,1C ) 
CALL FLCT 
WRITE U ,1C) 
STOP 
ENC 



SLERCLTINE HMTRX 
C 

C SL8R0UTINE TC FIND SYSTEMS TRANSFER MATRIX 
C 

COMMCN h(100) t HM( 100,100 ) , X< 100) ,V (100) ,N,P,NPULSE 

NN=N+1 

NV=NFLLSE+1 

CC 1 1 = 1, NN 

CC 2 J = 1 »NN 

H N ( I , J ) = C . C 

2 CCNTINLE 
1 CCNTINLE 

CO 2 1*1, Nh 
CG 4 J = l ,NN 
IFU.GT.I ) GG TO 4 
L-I-J+l 
HM(I,L ) = H(J I 
4 CCNTINLE 

3 CCNTINLE 
WRITEU.2C ) 

2C FGRMAT(2X,' HNATRIX •///) 
3C FORMAT (2X,26F5.3// ) 

CC 5 1*1, NN 

WRITEU ,2C) (HM(I ,J) ,J=1,NN) 
3 CCNTINLE 

RETURN 

END 



SL8RCL1INE INPUT 
C 
C SLBROUTINE TC FORM THE INFLT FULSE SEQUENCE 



C 



COMMCN H(1CQ),HM(100,100>,X(10C),V(100) ,N,P,NPULSE 
NN=N+1 



98 



NM=NFUISE+1 

CC 1 I=1,NFLLSE 

>(I)=F 

CCNTIME 

CC 2 I»M» ,NN 

X(I }=0.C 

CCNTIME 

kRXTEUvlC) 

FCRMAT (2X, • INPUT VECTOR 

WRITE U,2C) (xm,I = l,NN) 

FORMAT(2> ,26F5.3//) 

RETURN 

END 



//) 



C 

C 

c 



SLBRCLTINE CCNVGL 

SL3R0UTINE TC PERFORM CONVOLUTION SUMMATION 

COMMCN M10J),HM(100,130),X(100),V(100) ,N,P,NPULSE 
NN=N*1 

co l i*i,m 

V(I )=0.C 
CC 2 J = l,Nf* 

V( I ) = V(I )+hMI ,J)*X(J) 
2 CCNTIME 



1C 
2C 



1 CCNTIME 
WRITE U ,1C) 
FORMAT (2*,' RESULT VECTOR 



WPITEU ,2C ) <V(I) ,1 = 1, NN) 
FORMAT (2> ,26F5.3//) 
RETURN 
ENC 



//) 



C 

c 

c 



SLERCL7INE FLOT 

SL8RGUTINE TC PLOT THE CUTPUT 

COMMGN H(10O),HM(lOO,100),X(100),V(L0O),N,P ,N PULSE 

CIFEhSKh T (250) ,Y(25C) ,Z( 25C ) 

NN=N+1 

NCELTA=1C 

NFCINT=(NFLLSE+3)*NCELTA 

CO 1 I=1,NFCINT 

XI = 1 

T ( I > =X I 

Yd )=C.C 

za >=cc 

1 CCNTINLE 
K = C 

CO 2 I=1,NFCINT,NCELTA 
IF(K.G1.NN 1 GC TG 5 
K = K+1 

CC 3 J«l,4 
L=I+J-1 
Y (L)=V (K ) 
3 CCNTINLE 

2 CCNTIME 

I CALL FLCTF(T,Y,NPCINT,1) 



FLCT INFLT FLLSES 

K = 

NM=NFLLSE-1 

CO 10 I=1,NPCINT,NCELTA 

[F(K.G1 .hf> ) GC TC 7 

K = K + 1 

CC 8 J=l,4 

L=I+J-1 



99 



Z(U=X (K I 

6 COM I ME 

1C CCNTINLE 

7 CALL FLCTF(T,Z,NPCINT,3) 
RETURN 

ENC 



C 

: 
c 
c 



SL8RCLTINE IMPULS 

SIBROLTINE 7C FIND IMPULSE RESPGNSE 
( SOLVING CIFFERENCE ECIATIONS ) 



OF THE FILTER. 



B(l4c 

00) 



c 
c 
c 



c 
c 
c 



CGMMCN H( ICC) .n 

cimensicn a(iog> 
cimensicn clmmy< 
cimensicn c(10c) 

REAC<5 ,3C ) N 
NN=N+1 

NNN=NN*2C 



INITIALIZATION CF VECTGFS 

CC 35 1 = 1, NNN 
C(I)=C. 
MX)*G« 
MX)*0. 

E(I )=C. 
35 CGNTINLE 
C(20)=i. 

SUFT ORIGIN TC 20 



HM(ICO.IOO) , X(100) ,V(1C0),N,P,NPULS 



3C 
2C 



C " 



6C 
4C 



U 



NQRDER 

PEAC(5 

REA0(5 

FORMAT 

FORMAT 

WRITEi 

FORMAT 

WRITE ( 

FORMAT 

WRITE( 

wRITE< 

FORMAT 

WRITE ( 

CO 4C 

FACT1= 

NK=II+ 

CO 5C 

NK=NK- 

FACT1= 

CGNTIN 

FACT2= 

NK = II 

CO 6C 

M<=NK- 

FACT2= 

CCNTIN 

H( 1 1 ) = 

CONTIN 

h B I T E ( 

FORMAT 

WRITE* 

FCRMAT 

PQRMAT 

KKK = 1 

CO 55 

A(KKK ) 

E(KKK ) 

KKK = KK 



= M4^ 

,2C ) 

,2C) 

(12) 

(SF1 

6,1) 

<2X, 

t ,2) 

(2>, 

6,3 ) 

6,4) 

(2X, 

6,3) 

11 = 2 

C . 

1 

IL = 2 

1 

FACT 

LE 

C. 

IL = 2 

1 

FACT 

LE 

FACT 

LE 

6,21 

(2X , 

6, 1C 

(2X , 

(2X , 

KK = 2 
= A (K 
= E(K 

K + l 



2C 



(Ad l,I=20,NCRCER) 
(E(I) ,I=2C,I^CRCEF) 

C.5) 

• ORDER OF THE FILTER IS s •*I2///I 

•NLMERATCR CCEFFICIENTS •///) 
(A(I >,i=2C,NCRCEP) 

'CENOMINATCR COEFFICIENTS •///) 

(Ed), I=2C,NCRCEP) 

CNNN 



CNCRDER 
1+A<IL)*C(NK) 

1,N0RDER 

2*8( IL)*H(NK) 

1-FACT2 

) 

'WEIGHTING SECUENCE'///) 
KKIlt I = 2C,NNN ) 
12F10.5//) 
12F10.3) 

C ,NORDEP 
K ) 
K ) 



100 



55 CONTINUE 
K K = 1 

CO~65 I=2C,NNN 
CUHMY(KK)=MII 

KK=KK+1 
65 CCNTINIE 

CO 75 l=ltNN 
h(IJ=CLNN\ (I) 

75 cqntinle 
xcum-eo ) 

B(3) =E(1 > 
E(1)=XCIN 
PEAC(5,1CC I F1,F2 
P5A0(5,1CC) T 
IOC FGPMAT<2F1C.5> 
NF=1CC 
>NP=1CC. 

FCELTA=(F2-F1)/XNP 

CALL ZCMN(AfB,M,M,Fl » F2 , FDELTA ,NP ,T ) 
PETURN 
ENC 



101 



REFERENCES 



1. Aram 30DAK and Peter ARONHIME, » Maximally flat 
low-pass filters with steeper slopes at cut-off", IEEE 
Trans- on Audi o Electro acou stic s, v. AU-18, p. 63-66, 

MAR 1970. 

2. S.C.Dutto ROY, " On maximally flat sharp cut-off 
low-pass filters", IEEE Trans. on Audio 
Electro acoustics, v. AU-19, p. 58-6 3, MAR 1971. 

3. M.C.AGARWAL and A.S.5SDRA, " On designing sharp cut-off 
low-pass filters", IEEE Tra ns. on Audio 
Electroacoustics, v. AU-20, p. 138-141, JUNE 1972. 

4. Lawrance R.RA3INER and Charles M.RADER, Digital Signal 
Processing, p. 7-19, IEEE PRESS, 1972. 

5. Lawrence R.RA3INER and Bernard GOLD, Theory and 
Applic ation s of Digital Signal Pro ces sing, p. 20 5-226, 

prentice hall, 1975. 

6. A.G .COSTANTINIDES , " Frequency transformations for 
digital filters", Electronics Letters , v. 3, p. 
487-489, NOV 1967. 

7. A.G .COSTANTINIDES, " Frequency transformations for 
digital filters", Electronics Letters, v. 4, p. 
115-116, APR 1963. 

3. M.T.JCNG, " Notes on Discrete-System Transfer 
Functions", I . E. E.E. Trans, on Circuits a nd Systems , 
v. CAS- , p. 634-635, October 1976. 



'• 



F.BROPHY and A.C.SALAZAR, " Synthesis of Spectrum 



102 



Shaping Digital Filters of Recursive Design", I. £. £ . E. 
Trans., on Circuits and Syst ems, v. CAS- 22, p. 
197-204, March 1975. 

10. L.R.EABINER and others, " Terminology in Digital 
Signal Processing", I5EE Trans. on Audio and 
Electro acoustics, v. AU-20, p. 322-337, DEC-1972. 

11. R.W.DANIELS, Approximation methods for Electronic 
filter Design, p. 282-292, Mc Graw-Hill, 1974. 

12. Stacy V. HOLMES, Theory of Operation and Applications 
of sampled analog devices in Re curs ive Comb Fi lt ers, p. 

263-307, Ph.D. Thesis, Naval Post Graduate School, 
1976. 

13. I.G.STOCHAM, " High-speed Convolution and 
Correlation", J966 S£ring. Joint Computer Conf . ,AFI PS 
Conf^ Proc^, v. 28, p. 229-233, 1966. 

14. J.A.CADZOW, Discrete Time Systems, p. 250-261, 
Prentice Hall, 1973. 



103 



INITIAL DISTRIBUTION LIST 



No. Copies 

1. Defense Documentation Center 2 
Cameron Station 

Alexandria, Virginia 22314 

2. Library, Code 0212 2 
Naval Postgraduate School 

Monterey, California 93940 

3. Department Chairman, Code 62ki 1 
Department of Electrical Engineering 

Naval Postgraduate School 
Monterey, California 93940 

4. Professor S.G.Chan, Code 62cd 4 
Department of Electrical Engineering 

Naval Postgraduate School 
Monterey, California 93940 

5. Lt. S. K. Elitas 2 
Cif tehavuzlar ,Cavit Pasa mah. 

Engin Apt. No. 14/14 
Istanbul, TURKEY 

6. Professor T.F.Tao, Code62Tv 2 
Department of Electrical Engineering 

Naval Postgraduate School 
Monterey, California 93940 



104 



otq; 



3 



■ 




Thes is . ; - : o O 

E324 Elitas 

c.l A computer aided 

design of digi tal f i 1- 

ters. 



; i it 



15 OCT C 




' ,3398 

El itas 

A computer aided 
design of diqital fil- 
ters. 



thesE324 

A compouter aided design of digital filt 




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