74D-A125 967 FREQUENCY DOMAIN DESIGN OF MULTIBAND FINITE IMPULSE- 1/2 'RESPONSE DIGITAL FIT..(U) ILLINOIS UNIV AT URBANACOORDINATED SCIENCE LAB G CORTELAZZO DEC 80 R-909

UNLSSIFIED, RFOSR-79-1629 F/G 9/5 NucRs-hhhhhhhhiIlsmhhhhhhhhhhhEhhhhhhhhhhhhIEhhhhhhhhhhhhEEhhhhhhhhhhhhIEhhhhhhhhmmhhhEmhhhhhhhhhmhhE

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11111 I 20lIIII25.

MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANOARDS-1963-A

REPORT R-906 DECEMBER, 1980 UILU-ENG 81- 2237

T1 c~ CoL er tHA SCIENICE LABORATORY

qttFREQUENCY DOMAIN DESIGN1 OF MULTIBAND FINITE

IMPULSE RESPONSE DIGITAL, I FILTERS BASED ON THE

MINIMAX CRITERION

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FREOUENCY DOMAIN DESIGN OF MULTIBAND FINITE Technical ReportIMPULSE RESPONSE DIGITAL FILTERS BASED_______________ON THE MINIMAX CRITERION 6. PERFORMING ORG. REPORT NUMBER

_________________________________R-906; UILU-Eng 81-2237

7. AUTI4OR(a) S. CONTRACT OR GRANTr NUMUER(oo)

NSF ECS 8007075

*Guido Cortelazzo &AFS-9029. PERFORMING ORGANIZATION MAMIE AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

AREA & WORK UNIT NUMBERS

Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaianUrbana, Illinois 61801

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National Science FoundationDembr10Air Force Office of Scientific Research 106NME O AE

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19. KEY wORDS (Continue i reverse aide it necueary and identity by block number)

Dicital Filters - FIR multibandMinirnax desian

20. A816TRAC7 'Contnue on reverse ,od* if necesayan d Identfy by block number)7:t is well known that in applying minimax criteria for the frequency do-maindesian of finite impulse response digital filters, the use of I dont't care'recions can :ead to the occurrence of undesirable resonances in the transiticihands of the multiband desians. Rabiner et. al. showed hcw zhis effect can 1:eeliminated by modifying the choice of design parazneters, althouah theirapcroach did not attempt to reformulate the design technicue. The nresent workaresents a modified desian technicue in which the des'-red function is sineci-fiec:also over the transition recions an,! the minimax octirnizatic.

DD ~ 1473

S6C;JRITY CL.ASSIPICATION OF THIS PAGIE(Whan Data Entg.El)

is carried out over the entire band [0,7]. Four example filters-of Rabiner et. al. are used as exanmles to illustrate thismodified technique. Theoretical results relatina to the desiantechnique are also presented.

Acoession For _

FNTIS GRA&I

SDTIC TAB oUnannounced []

Justificatio

SDistribtion/

Availability CodestAvail and/or

Dibt Special

I'D

S C'.j""f : .- AssI JcA rlON W '49S OAGE'*7 e .t ee .,..

FREQUENCY DOMAIN DESIGN OF MULTIBAND FINITE IMPULSE RESPONSE

DIGITAL FILTERS BASED ON THE MINIMAX CRITERION

BY

GUIDO CORTELAZZO

.aur., Universita degli Studi di Padova, 1976

THESIS

Submitted in partial fulfillment of the requirements

:or zhe degree of Master of Science in Electrical Engineering

in the Graduate College of the

University of Illinois at Urbana-Champaign, 1980

Urbana, illinois

ACKNOWLEDGEMENTS

I would like to acknowledge my advisors Professor Jenkins

and Professor Lightner for their supporting encouragement without

which this work would not have been possible.

hi iv

TABLE OF CONTENTS

IChapter Page

1. INTRODUCTION 1

2.LIMITS OF PARKS AND McCLELLAN's FORMULATION 8

3.NEW FORMULATION OF THE LINEAR-PHASE FIR DIGITAL

FILTERS DESIGN PROBLEM 27

4. IMPLEMENTATION AND PERFORMANCE OF THE NEW FOR-

MULATION 37

5.COMMENTS ABOUT CONRIP 62

6.SUMMARY 67

APPENDIX 1 70

APPENDIX 2 75

APPENDIX 3 82

BIBLIOGRAPHY 102I

UCHAPTER 1

INTRODUCTION

In the early 70's minimax criteria for frequency domain

design of digital filters received a vast amount of attention

([1], [21). The most popular among these formulations of the

problem is the one due to Parks and McClellan [3], probably

because it allows a very large class of design specifications

and it is available in an efficient implementation [4].

Parks and McClellan characterized the problem of the

design of linear-phase finite impulse response (FIR) digital

filters as the following Chebyshev minimization problem:

min n- max 4.W(e )(e H(eLh(i)}i. 0 We.i

Ewhere

W'(e j ) is a weight function (real and positive),

D(ej ) is the desired objective function,

"T-I)(e 4 h(l)e -

, where , is the order of nhe fitter,1=0

S. where B . . <'.u 4,01Izs = fsi fi' si

2

an Bi i~i 1,2 ....

yB. .1 i-..

with L den6ting the number of passbands and stopbands.

AjW

Fig. 1 gives a pictorial illustration of the criterion. We)

-weights the error I D(e) -W 1{'e j) I on ~.Parks and McClellan

mainly conceive piece-wise constant behavior for Wi(e jW), although

they leave open the possibility of other (strictly positive)

behaviors. The reason why Q@eJ ) cannot take on the value 0

AjW

will be explained in Chapter 2. D(e ) is a piece-wise constant-

function for the case of multiple passband-stopband filters.

U 9 is taken so that it doesn't contain any discontinuity points

of D(e ).The phase linearity of the frequency response of the

filter H(e ),implies that

jW jW Wjj.J

where 3 is constant and both Q(e )W and H(e )W are real functions

defined (41 in the following way:\

:n 12,-or real constant, the phase term is actually t;we neziect to take into account phase Jumps of s-ize 2- as i: iscustomnarv in the literature.

3

- Case 1: N oddQ(e j ) = 1

jW MH(e w) Z E a(K) cos(Kw)

K=O

N-Iwith M - - , a(O) - h(M) and a(K) - 2h(M-K), K- 1,2,...,M.

(

0

Fig. Chebychev criterion in frequency domain.for a low-pass filter

4

Case 2: N even

Q(ej W ) - Cos

M-lH(e ) = E a(K) cos(Kx)

K=O

where M - -and a(K)'s are defined by

h(M-1) = a(O) + -L a(1)

h(M-K) = - a(K-1) + T a(K) K=2,3,...,M-1

h(O) = - - a(M-1)

Substitucin3 (1.2) into (1.1) resalts iato the expression giveri

a (1. 3)

min imax W'eJ"))-: ) - H.'eJ)j (1.3)

ta "K)j K=O '

where

*jW j jWW(e ) =W(e ) Q(e )

D(e j D(ej W

) andQ(e O

The sets B. , i = 1,2, *L defined above are called pass-

bands if D(ej ) 1, ICB or stopbands if D(e = 0, X 3. The

sets Lti (0', 7 ] . < <, < , , with i = 1,2,... L-1 constitute

the so called "don't care" bands.

3 Equations (1.1) and (1.3) have the structure of a minimax

(or Chebyshev) approximation problem for generalized (trigonometric)

polynomials. The solution to this class of problems is character-

ized by the Alternation Theorem (51, pp. 75' E. Ya Remez [6]

provided algorithms for the computational s( tion of such Chebyshev

problems. J. McClellan et al. published a c uter program [4]

that uses the 2nd Remez Exchange Algorithm ..nd the solution

of a discretization of (1.3). Specifically, the problem solved

in McClellan's program is:

min max (W(eJ')jD(ej') - H(eJ')I} (1.4)M Unta(K)Ji.° d

i d a IU 6L and . -M , K positive integerJ

C is a parameter (positive integers) that can be controlled by

the user. It can be shown ([51, pp. 89-95) that the solution to

(1.4) approaches the solution to (1.3) as L tends to 0. McClellan

suggests that values of L - 16 produce satisfactory results.

'he choice of solving (1.4) instead of (1.3) is dictated for

reasons of computational efficiency.

McClellan's program works very well for designing low-pass

or high-pass digital filters. However, the multiband filters,

i.e., _he filters with a number of pass and stop-bands grea.er

:han 2. designed with McClellan's program ozten exhibit non

monotonic Iscmet-imes even resonanc) behavior in the "don't

care" bands. Figures 7 11 15, 19 illustrate se'eral cases

6

of this phenomenon.

Since McClellan's program became the tool most universally

used for the design of finite impulse response filters, Rabiner,UShafer and Kaiser (71 addressed the multiband design problem in

particular. The technique they propose is the following: if

McClellan's program returns a multiband design with resonances,

the filter specifications passed to the program are modified

according to empirical strategies until a filter without resonances

is obtained. Their strategies take into consideration the modi-

fication of the size of the stopbands (i.e., basically changes

of j) as well as the modification of W(e ). The nunmber of tries,

* with the McClellan's program, rkecessary to obtain an acceptable

filter varies from case to case. The implementation of

multiband filters with this procedure might easily become

cumbersome.

The present work reconsiders the problem of the minimax

design in the frequency domain of linear phase finite impulse

response (FIR) digital filters. The objective is to provide a

sacisfactory theoretical solution for the design of multiband

filters as well as a convenient technique for the implementation

such a solution.

:t was decided co _,se McClellan's program fcr the imple-

men-aticn of Ihe new soLution. This was natural since the new

a!-orizhm is an extension cf the Parks McCleLan technique and

7

because the new algorithm would be of interest to the great number

of filter designers currently using McClellan's program. Chapter 2

shows that the inadequacy of McClellan's program for the design of

multiband filters lies in the formulation of the problem. Chapter 3

presents a new formulation capable of handling these difficulties.

Chapter 4 introduces an implementation of the new solution and

examines the results. Chapter 5 points out the relationship

between the new program and the program CONRIP [8].

,I

i.

8

CHAPTER 2

LIMITS OF PARKS AND McCLELLAN'S FORMULATION

Two standard results of Approximation Theory are now

3 introduced: the Alternation Theorem and the 2nd Remez Algorithm.

They respectively constitute the theoretical and the computational

tools to solve Chebyshev problems.

Alternation Theorem: Let 3 be any closed subset

of [O,n] and let D(e j ) be any continuous real valued function

defined on 3. In order that H*(ej ) be the unique

best approximant on 3 to D(e jX), among the class of the

trigonometric polynomials of order M, it is necessary and sufficient

* that E(ej ), defined as

E(e ) = W(ej')ID(e j ") - H*(eJW)l (2.1)

exhibits on J at least M + 2 "alternations". Thusj -Ai lMax 'jw

EVe i)=-E(e )-+ E +I e- _+ , E(e j )I

with v " 2 -" < M+I and cx.

Proof: The proof can be found in [51, pp. 75.

Remark: The notation of the following chapters is

consistent with the one of the previous sections. j, H, D, W

therefore are as defined in (1.3).

Notice that the Alternation Theorem is an existance

theorem, it does not describe how to find the best approximant

I.

'.7

9

H*(eJW). Remez algorithms are iterative procedures for generating

H*(e ,-). The second one applies to classes of approximating

functions like the trigonometric polynomials.

2nd Remez Algorithm: Each step of the algorithm works with aM+l

set of M + 2 frequencies {w I} The frequencies are arbitrarilyK K-"0

chosen at the first step and updated at successive iterations

according to the particular algorithm ((5], pp. 97). The frequenciesr , M+l- K'K=0 are used to determine the following system of M + 2 equations

in the M + I a(K)'s and in p (M + 2 unknowns):

E(eJ )=W(ej') I D(e J"J) - H(eJ WK) -(-1)K p (2.2)

for K = 0,1,2,..., 1 + 1

The assumption W(ej ) > 0, 'i allows (2.2)to be written as:d

SCos 0 cos2, 0 .... cosMw0 W(ejL) a(O) D(eJ 0)

1 cosw 1 cos2 1 .... cosM -1 a(l) D(ej * )W(eJ U1)

(2.3)

SI a (M) D(e

i Cos .[ i cos 'z-, . . COsMCJ~ o4 e DM -. +I)I. .. M~W~e M+1+m+

- L

10

For reasons of efficiendy Parks and McClellan avoid the

matrix inversion in the solution of (2.3). This is done by first

calculating:

SM+Il WE bi D(e )

i-0 (2.4)

%1+l (-l)i b

i-0 W(eiwi)

where b (- )K M'l,

i-o i XKi#K

Xi cos,%

Then the Lagrange interpolation formula in the baricentric

form is used to interpolate H(e ) on the M + I points [uKK=K K=O

to obtain the values

C - D(ej u'K) _ (_,)K PK W (ej k) (2.5)

with K - 0,1,...,M.

This type of interpolation gives an equiripple fit to the m + 2

data points. This form of interpolation is required by the 2nd

Remez alg3orithm.

11

After the system is solved, the algorithm calls for the

Kcheck:

E(ejjd) 5 E(e j ' K) - p u Ej (2.6)d d(26

if there is some frequency of Jd not satisfying inequality (2.6),

a further iteration that begins with the update of the frequenciesr[4-g+lMKJK0 is necessary. The algorithm continues until inequality

(2.6) is satisfied over the whole ad* The H(e j W) obtained at

this step is the best approximant H*(e ) characterized by the

Alternation Theorem.

The Parks and McClellan's FIR design technique corresponds

to the application of the Alternation Theorem and of the 2nd Remez

algorithm to the solution of (1.3).

It must be noticed that the "don't care" bands constitute

a mathematically ambiguous feature of their formulation that can

be very dangerous. In fact, one actually does "care" about the

behavior of H(e' ) over the whole band [0,7 ] and, in particular,

one wants H(e yx) to be monotonic over the "don't care" bands.

Further, it is not clear that ignoring the behavior of the

filter over the "don't care" bands will necessarily result in the

desired monotonic response.

Before showing in detail the dangers connected with the

"don't care" bands, let's review a few athematical concepts

necessary to understand them. Recall that the extrema of a funct'.in

over a comoac: and bounded set cart only occur at the boundary points

S

12

of the set or at the interior points where the first derivative

of the function is 0. H(e jW) is a trigonometric polynomial ofdH(eiW

order M defined on (0,T]. Its derivative dw can have at

most M - 1 distinct O's in (0,") since it is a polynomial of

U order M - 1. The boundary points of M(e j ) are the set [On}.

Therefore H(e j ) can have at most M + 1 distinct extreme on [O,n].

The function:

jUW jW jWiE(e H(e ) D(e Wei (2.7)

where D(e ) and 3 are defined in formulation (1.3),is a trigonometric

polynomial of order at most M on :. Notice that E(e ) over CO, 7r]

is not a polynomial of order M, because of the discontinuities of

D(eJW ).

The cases of the low and high-pass filters will be con-

sidered first because of their special characteristics. Then the

multiband filters will be discussed.

For low or high-pass filters 3"= [0, 'J [ws, ], thes

interior of 7 is J - (Owp)-(cs,.r) and the boundary of j isinero ofiti (0, W )PS

j, to, W 0 7rJ. E(e ) can have at most M - I distinct extrema

on 5 (since E(e ) is a polynomial of degree at most x in J) and

at most four extrema on the boundary (O,WPs" 7j. Therefore

E(e can have at mosc N + 3 extrema on Zi. Furthermore (2.7)' iL

implies that all the extremal points of E(e ), except possibly

the extrema at p and w are extremal pcints of H(e J:), namelyp s

I

13

E*(ejW H* (a i - D(e ) (,, (2.8)

will have at least M + 2 distinct extrema alternating in 3 by

the Alternation Theorem. Specifically, E*(e ) can have at

*least M + 2 distinct extrema on 3 either having M - 1 distinct0

extrema in ; and at least 3 extrema in [0,w ,p wIr) or having

M - 2 distinct extrema in 3 and four extrema in[O, ,ps, W .

More specifically E*(e ) can have:

i)M-1 extrema in and extrema at (0,W 31s, ~0ii) M-1 extreme in J and extreme at £0 w 70

iii) M-1 extrema in it and extrema at 10,w 71pii) M-1 extrema in and extrema at tO,WpT s}

iv)sv) M-I extrema in 3 and eltreme at tw'os'

vi) M-2 extrema in d and extrema at (0,W p,"s,7.

Some of the cases of E*(e j ) listed above leave open the possi-

!)bility of a corresponding H*(ej ) that would be unacceptable

as a low or high-pass filter. Case ii), for instance could

correspond to and H*(e j ) having an extremum at w (see Fig. 2c).p

Case iv) could bring an H*(e ) with a saddle point in (x, S)

(see Fig. 2f) and case vi) an H*(e ) with a local maximum in

(wP. S) (see Fig. 2e). The reasons such possibilities don't

occur are explained in the following theorem.

I"

14

Theorem: The low and high-pass digital filters designed via

the Remez algorithm, according to formulation (1.3) where

B " Nw < W < u; ,w w 6[0,11 is the don't care band, havep S p5

the following properties::jW

a) w and w are always extremal points of E*(e j),D S

(i.e., the cases ii) and iii) above cannot occur)

" b) ,; and w are not extremal points of H*(eJ)

p sjWc) H*(e is strictly monotonic on B, namely:

dH*(e j )

" < 0 , uCB for the low-pass filters

dH*(e w)and

and dw >0 , CB for the high-pass filters

Proof: the Proof takes into consideration each of the

cases of E*(e j) listed above and shows the necessity of

properties a), b), c) for the cases whose occurrance is

unot a contradiction (i.e., all but case ii) and iii)).

Case i) satisfies properties a), b), c)

Property a) is satisfied by definition.

E*(e' ) has now M + 3 extrema on '. Therefore H*(e )

has M + 1 distinct extrema in j: namely M - 1 in a

and 2 at 0 and ,, respectively. The points '- and £jp

therefore cannot be extrema of H*(e j ) (property b)) and

further there cannot be any extremal point in B (property c)).

15

Case ii) and iii) contradict the Alternation Theorem

Consider first case ii), i.e. E*(e j ) has M-1 extrema

on 7 plus 3 extreme at {0, ,1T }. Notice that 0 and iT are

also extrema of H*(eiW), therefore 0H*(eJ) - 0 cannot occurdw

at any w C B (property c)). Also, the point w cannot be anp

extremal point of H*(ej ) otherwise H*(ejw) also has M + 2

extrema on j . Since , by assumption w is not an extremal

point of E*(eJ ), E*(ejW) does not alternate (see fig. 2a,b)

and therefore the contradiction is achieved.

A completely analogous argument proves that case iii) cannot

occur.

Case iv) and v) satisfy properties a), b), c)

Consider case iv) first, i.e. E*(e ) has M-1 extrema in 1

plus 3 extrema- at 10, us w }. Notice that 0 is extremal point' p

also of H*(eJ ).therefore H*(e ) has at least M extrema in 7.

It is worthy to distinguish 3 subcases:

Subcase I: and w are both also extrema of H*(eJ). This is ap s

contradiction because H*(e J ) would have M+2 extzema.

Subcase II: p is an extremum of H*(e ) and w is not (the caseps

s is an extremum of H*(eJ) and w is notis compietely analo-

gous and leads to the same conclusions ),4

By hypothesis H*(e) I has M-. extrema on- since H*(e' "

has at most M-I ::+rema on (0, -) it is dh* (ej )d # 0 B

(recall that the roots of derivatives correspond to the interior

extremal points).

S

1.6

a) H(ijW)

Ua

FP-6924

b; E Jci')

Fig. a~.Case ii): E *(e1j) with M -i e-xtrena iand three extrerna at o,,7r- The frequenc,.-

*L is not extrem~al ooint'ol. E (elJ

17

In this subcase the contradiction is achieved because

U E*(e j ) violates the Alternation Theorem (see Fig. 2-c,d).

Therefore subcase II cannot occur.

Subcase III: Neither w nor w are extrema of H*(eJW). InU p sthis case property a) and b) are satisfied by assumption.

4W 0H*(eJ ) by assumption has M-1 extrema in a )therefore dH*(e) 0dw =

can not occur in B otherwise H*(e j ) would have M extrema

in (0, 7). This proves property c).

The same argument shows also that the properties a), b),

c) are satisfied for case v) (it is enough to interchange the

roles of 0 and ir as extremal and non-extremal points of E*(ej")

respectively).

Case vi) satisfies properties a), b), c)

E*(ej ) is assumed to have four extrema at( 0, w ,s' 7

i and M-2 extrema in C . This implies that H*(eJ ) has 2 extremaa

at 0 and 7 plus M-2 extrema inJ3, i.e. M extrema in 7.

It is again convenient to distinguish three subcases.

Subcase I: and w are extrema of H*(e J). This cannot occurs p

because H*(e J ) would have M+2 extrema in 7.

Subcase IH: is an extremum of H*(ejW) and s is not (equiv-ps

alent to the case - is an extremum of H*(eji ) and - is not).s p

H*(e j-) has by assumption M - 2 extrema in :, two extrema at

.0, - and one extremum at . H*(e2 ) therefore has by assump-

tion M I extrema in . This excludes the possibility o:

U(

c) E *(eiw)

FU.., -cd aei) *e()wt 1eteai

Fi.,. Cas is) assue~ tit be- extrema in o *ej-

&- 19

dH*(e 0 at some w E B (property c), otherwise H*(eij )

would have M + 2 extrema in 3 . In this subcase contradiction

is achieved because E*(e j ) violates the Alternation Theorem

(the situation is analogous to the one of Fig. 2-c,d).

Subcase III: w and w are not extrema of H*(eJW). This casep s

specializes into four situations:

(a) Maximum (or minimum) in B. I. e.:

dHei)d 2H~eW

d0, 2 0 , w e B (2.8)du d2

Since H*(eJ WP) # H*(e jws) as long as w w S and neither w norp sp

w are extrema of H*(eJW), the presence of a maximum (or a

minimum) in(w p, w s) implies also the presence of a minimum

(or a maximum) in (wp, Ws) (see Fig. 2e). The contradiction

is achieved since this requires H*(eJW) to have M + 2 extrema

on Tr

(b) Saddle point in B (see Fig. 2f)

dH*(e) 0 d2H*(eJ) 0 W£ B (2.9)d d2

Recall that a saddle point of a polynomial H(e j ) corresponds

to a zero of the derivative of multiplicity at least 2. By

assumption H*(e ) has M - 2 extrema in ) this means that

dH*(e )/dw has M - 2 zeros in . Condition (2.9) implies that

dH*(e )/d has a zero at least of multiplicity 2 in B. This

is a contradiction because it implies that dH*(ej;)/dw has M

zeros in (0,7) and dH*(eJ,)/dw is a polynomial of order M - 1.

1 20

ZOOO

0Fig. 2e. Case vi): E*(eiw) with M -2 extrema in

and four extrema at {o,w wsq} H*(ejw)isassumed to have a maximum in (w gws

cip ~ ~ p FSW 7

Fig. 2f. Case vi): E*(ejw) with M -2 extrema in Z7and four extrema at {o~upw 5 ' N*(e2j) isassumed to have a saddle-point in ( 4p,ws).

21

jW ie.i wM-2

(C) Further extremum of H*(eJW) in t, i.e. if 150

denote the set of the extremal frequencies of 5 assumed by

hypothesis, this circumstance is expressed as

dH*(e iW) tw M-2

This situation satisfies properties a), b), c) by assumption.

- It should be noticed that, since w is not accounted among

--2the extremal frequencies {w V the Alternation Theorem

imposes

IE*(ejW )I < p - max IE*(ejW )I JE(e iWi)IW L;

(d) No further extremum on (0, ). I.e., if {w} denotes0 i=

the set of the extremal frequencies of;Y, this circumstance is

expressed as

dH* (eW) 0 WEWE 0-r ( M2 W W1.dw s p

This situation satisfies properties a), b), c) by assumption.

Remark: For simplicity E(ejW) has been used in the theorem according

to definition (2.8) instead of definition (2.1). The theorem can be

jWproved also for a weighted error as in (2.1) with W(e~ 0 and real

Also in this case E(e ) turns out to have all its extreal points,

except possibly pand , in conon with H(e) and E*(e j) has

extrema as in the six cases previously considered.

22

U

1 0wsp W / 7r

0

Fig. 2g. Case vi): E (ejtO) with M -2 extrema in aand four extrema at {o ,Wp, Ws Tr} H*(eiw).,sassumed to have a further extremum in U

I

23

The preceding theorem shows that the mathematical structure of

the approximation problem (1.3) corresponding to the design of high

and low-pass filters guarantees strictly monotonic behavior in the

"don't care" band. It is legitimate to ask if formulation (1.3)

also guarantees strict monotonicity in the "don't care" bands for

multiband filters. The answer is "no". The reasons behind it can

be illustrated by an example. Consider a 3-band filter like the one

shown in Figure 3-a,b. In this case : 0, usL J Wf2 ,s 2] 'Wf3'S

the interior of is - (0, Ws1 W) f 2( ' Ws2 ) U (Wf3' 7) and the

boundary of 7 . { -0, W sl' wf2' ws2' wf3' I} "

E*(e j ) given by the 2nd Remez algorithm has at least M + 2

extrema in T. Therefore E* can have the structure of any of the

cases below:

o(1) M - 4 extrema in and 6 extrema in I3:1 subcase

0 M(3) M - 3 extrema in a and 6 extrema in I:6 subcase

() M - 3 extrema in j and 6 extrema in 15 subcases

(4) M - 2 extrema in 3 and 4 extrema in U": 15 subcases

0 0

(5) M - 2 extrema in 3 and 5 extrema in :,. 6 subcases

( 0

(6) M - 2 extrema in j and 6 extrema in 15 1 subcase

(7) M - 1 extrema in 3 and 3 extrema in ,3. 20 subcaseso 0

(3) M - 1 extrema in 3 and 4 extrema in : 3 15 subcases

(9) N4 - 1 extrema in 7 and 5 extrema in 7 6 subeases

o m - I subcase.(10) M - 1 extrena in 3 and 6 extrerna in v 3 ubae

1. 24

* T ,

a) 0*(e

b) E*(e.jw)

Fig,,. 3a,b. Example of 3-band filter with an extramunaof q *(eiW) in (w,. w,a:)

25

Each one of these cases can specialize in many subcases dependingIfrom the points of U5 that are assumed to be extrema (similarly to

what was done for the low or high-pass filters). The number of the

subcases is 6!/(K!(6-K)!), where K is the number of extrema in U .

It is not difficult in this forest of cases to find a situation

unable to guarantee monotonic behavior in the"don't care bands without

violating the Alternation Theorem or the relationships between the

order of a polynomial and the number of its extrema. Select, for exam-

Sple, the subcase of case 9) corresponding to M-1 extrema in ,and

extrema at {0, Wsl' wf2 ' wfl} " If Wsl' wf2 ' ws2' 'f3 are not extrema

of H*(eJ ), H*(eJ ) has M extrema in 3 one at 0 and M-1 in a.

H*(e j ) can have a further extremum in (w slf 2), as shown in Fig.3 a,b

without causing any contradiction.

In general for an L-band filter

IJ L j.1 fj' sjp

and E(e j ) can have up to M-l + 2L distinct extrema in n The 2nd

Remez algorithm guarantees only that E*(e j ) has at least M + 2

alternating extrema in o. The possibility that the Remez algorithm

takes into account extrema of E*(ejW) not corresponding to extremal

point of H*(e j ) is very high. Therefore for an L-band filter (L > 2)

some of the M + I extrema of H(e ) can occur an vhere ohiO, - '. If

they occur in , the second Remez algorithm constrains them to 2ive

deviations from D(e4 ) within the final minimax error p = max r

26

If they occur in the"don't care"bands they are unconstrained and canEnot only spoil the monotonicity but also become resonances for the

filter. Unbounded extrema of H*(ejW) in the"don't care"bands are

reported [7] to occur experimentally in 9 out of 10 multiband filters

designed with formulation given in (1.3).

In the next section a new formulation of the filter design

problem is presented. This new formulation is immune from the draw-

backs of the McClellan formulation and capable of the straightforward

design of the multiband filters.

N

w

27

CHAPTER 3

KNEW FORMULATION OF THE LINEAR-PHASE FIR DIGITAL FILTERS DESIGN PROBLEM

The elimination of the "don't care" bands in the formulation of

the linear phase FIR digital filter design problem calls for the

following type of formulation:

min max W(e j ) ID(ej ) - H(eJW)1 (3.1)

{a(k) k-O WE

where the symbols are as defined in (1.3).

The presence of D(eJW), as defined for (1.3), i.e., as a piece-

wise constant discontinuous function, brings two major difficulties

to the formulation (3.1). The first difficulty is that the discon-U

tinuities of D(eiW) will seriously limit the convergence of the

approximation (3.1). For instance it is easy to see that if W(eJW)=I

the minimax error will not become smaller than

max ID(eJ) (3.2)E discontinuity 2

points of D(e]')I

The second difficulty is that the Alternation Theorem cannot be

used to characterize the solution H*(e jW) of (3.1), since it requires

the continuity of the function to be approximated. It will be noticed

that since the approximating functions in (3.1) are trigonometric

polynomials it does not even make sense to require a discontinuous

IL behavior from them. These reasons motivate the use of a continuous

D(e ") in (3.1). Figure -'a shows an example of a Diecewise

28

ab

a)j

Fi2. 4a -Example of D(ejW) for the McClellan's formulation

b- Example of continuous D(eJi) for the new formulation(technique L)

c -Example of continuous DVejw) for the new formulation(technique P)

29

discontinuous D(e j ) and Fig. 4b and c two continuous versions of

D(e W) for use in the new formulation (3.1). A method for choosing

such a continuous D(ejW) is another major point of the new formvla-

tion of the filter design problem, whose detailed discussion will be

given later in the section.

The relationship between the Parks and McClellan's formulation

(1.3) and the new formulation will now be discussed. Such a compar-

- ison turns out to be rather informative and gives the motivation for

the choice of W(ejW) and D(ejw) in the new formulation (3.1)

The Parks and McClellan formulation (1.3) can be obtained as a

particular case of the new formulation (3.1). This equivalence occurs

when

W(eJW) = 0 wD {WIE[[0, ,r3,w€ } (3.3)

is used in (3.1).

It will now be shown why the formulation of (1.3) is mathemat-

3 ically preferable to the formulation (3.1) with condition (3.3),

although the two are absolutely equivalent in meaning.

Formulation (3.1) for W(ejw) > 0 constitutes a minimization

problem with respect to a weighted minimax norm defined for real

functions supported by [0, 7-*) Condition (3.3) turns (3.1) into

(*) A norm is a functional f over a vector space X, with the following

properties:

(i) f(x) Z 0 for all x E X

(ii) f(ax) = x tor all scalars a and x X

(iii) f(x ,) f(x) + f(y) for each xy £ X

.'.') f(x) = 0 i: an,! onl" if x .

30

a minimization problem with respect to a weighted seminorm definedfor the real functions supported by[o;n ]. Such a seminorm defines

a norm for the 10, it] real functions supoorted by 3(**). To write

(3.1) with condition (3.3) as formulation (1.3) corresponds to con-

sidering the seminorm for function of [0, iT] as a norm for functions

of 1. This point of view is of theoretical as well as computational

utility. The theoretical utility comes from the fact that formula-

tion (1.3) entitles us to apply to a seminorm problem the results given

for the norm problems (such as the Alternation Theorem and the second

- emez algorithm). In order to understand the computational utility

notice that W(ej ) - 0 for w e Bi and Bi considered as a"don't care"

band will produce the same effect, namely eliminate the occurrence of

* extremal frequencies on B . In fact, W(e j ) 0 will force E(e j ) - 0

on W E Bi, therefore the second Remez Algorithm will not find any

extremum of E(eJW) for w F B . If Bi is a "don't care" band, it is just

not taken into account during the operation of the second Remez Algo-

rithm, therefore it cannot deliver any extrema of E(e w). Therefore

the total effect of the two techniques is identical, but the use of

- the "don't care" band is more efficient. The efficiency lies in saving

the actual evaluation of E(e jW) over E B. as well as in allowing

the efficient non-conventional solution of system (2.3) via (2.4) and

(2.5) (made possible by the fact that the weight W(e j ) is never 0 in a).

(**) A seminorm is a functional g over a vector space X, satisfying

the properties (i), (ii), (iii) abcve.Sb

31

A computational check of the above claimed equivalence between

"don't care" bands and regions supporting 0 weight was tried.

McClellan's program was used because minor modifications can

oU convert it into a tool implementing formulation (3.1) with condition

(3.3). A direct verification of the equivalence was found not to be

possible since the McClellan's program does not allow 0-weights. This

limitation derives from the calculation of p via (2.4). Also an indirect

verification by means of small weights, ideally tending to 0, was not

easy to obtain. Specifically, filters of relatively high order (above

60) can call for weights of order 10- 7 or less in order not to exhibit

any extremum over the "don't care" bands. McClellan's program starts

to lose its nurerical accuracy for weights of this order(as reported

*in C7] ) and the results may not be reliable. However for filters of

lower order the indirect verification is possible, as the examples

of Fig 5 and 6 show.

gDuring these experiments it was noticed that a complete removal

of the extremal frequencies from the transition regions gives a better

performance in terms of reducing ripple over the "care" bands than a

partial removal of extremal frequencies. It was noticed, in practice,

that a fewer number of extremal frequencies in the transition regions

results in a smaller ripple.

The idea behind the choice of the values of W(e') and D(e )

over the transition regions for use in (3.1) is to have as few extremal

frequencies as possible over these regions (possibly none). The choice

of the values of W(ej ) and D(e J ) ove: the pass-bands and the stop-

bands follows the usual criteria taken for the Parks and McClellan's

32

_ _ _ I _ _

U.2 I i '

M i

g3. 5a " ,

.2 2._ 2___S9.

- -* r?-i

NORMALIZED FREQUENCY

Fig. 5a McClellan's program: exampese

of low-pass with N = 30

° I I i 1.4 -z II_ _-_ _ _

" -J .

I _ ____3- I * *

NCPRMALIZED FPXUENCY

Fig. 5b ExremaL frequencies of the 1ow-jassof Figure 4a

33

~2.79-

z S _ _ _ _ _ _ _ _

2. 2-

2.2.1 93.2 a. 3 .4 .NORMALIZED FRQUENCY

Fig. 6a The low-pass of Figure 4 designed withthe new program

I "I,. t f_ _ _ _ _ _

E -f

2..S

NORMAL:zD FREUENCY

Fi=. 6b Extrea frequencies of :he low-passci Figure 5a

34

formulation.

LI The W(e ) to be assigned to the transition regions comes from

a trade-off between two conflicting requirements. W(ejW) should be

snall in order to keep E(e j ) small, so that no extrem-i frequency

is detected by the Remez algorithm. But W(ejW) cannot be too small,

since the smaller W(ejW) is, the larger the term JD(ejW) - H(e J)I

becomes for a given minimax error E(eJW). As a matter of fact, the

difficulty with W(eJ ) = 0 (or "don't care" bands) is that resonances

of H(e> ) (i.e. points where the term ID(ej ) - H(eJ )l is very large)

can occur without being detected as extrema of E(e J). Therefore the

weight over the transition regions has to be taken small, but non-zero,

in order to prevent resonances.

The following two observations are useful for the choice of

D(e ) over the transition regions. The first observation is that

every multiband filter can be thought to be the compositi.), of several

low-pass or high-pass filters [ 7 1 Such low and high-pass filters will

be called "prototypes" in the present work. The second observation is

that the prototypes, by the theorem of Chapter 2 have a monotonic

transition region. Thus, they can be safely implemented with the

McClellan's proaram. This suggests that the transition reqions of the

prototypes be taken as a model for the transition regions of D(e>).

Two techniques for obtaining the transition regions of D(e ) have been

tried- a multisegment piece-wise linear approximation of tlie trans::on

regions of the prototypes, and t:.e direct use or the transition re2 ons

of the prototypes in D(e- ). For ,-onvenince these "ill e roferred

on as technicue L and ? , rosnectiverv. .,etaz , fiscussion e

L

35

implementation and performance will be given in the next section.

r The new formulation together-with technique P prevents an extre-

mal frequency from occurring in the transition region for the design

of low or high-pass filters, since the error E(ejW) over the trans-

ition region is made 0 by the term ID(eJ W) - H(eJW)1. This result is

independent of the value of W(ejw) over the transition region.

In the design of multiband filters, the complete elimination of

the extremal frequencies from the transition regions is not to be

expected, even with technique P, since every prototype induces extremal

frequencies into the transition regions of the other prototypes.

Finally, notice that the new formulation (3.1) allows a general

control over the transition regions of the filter. This characteristic

can be used to obtain monotonic behavior as well as any other desired

behavior in the transition regions. The new formulation is therefore

very suited to the design of filters for which the shape of the trans-

ition regions is important. Thus if transition band performance is

important in a high or low-.pass filter formulation, (3.1) is to be

preferred to the McClellan's formulation. Figure 7 shows an example

of a low-pass filter that could not be obtained with McClellan's form-

ulation.

,U

*6

36

2S i

, _ I I

- I \ I

c. .5-iL~A I

4 .' I - " I j I J i

_ _ _ .1 U. 4_ 0.

NORMALIZED FREQUENCY

Fig. 7a Example of a filter that can be designedwith the new program and can not be designedwith McClellan's program.

7 1I I v

3.2. .2 2.S 2.4 a.SNORMALIZED FREQUENCY

Fib. 7b Extremal frequencies of the low-pass of

of Figure 7a

-I

L

37

CHAPTER 4

IPLEMENTATION AND PERFORMANCE OF THE NEW FORMULATION

The new formulation of the linear phase digital filter design

problem (3.1) has the structure of a minimization problem in minimax

norm for trigonometric polynomials, similar to the Parks and McClellan

formulation (1.3). The Alternation Theorem and the second Remez

algorithm are still applicable for computing the solution to problem

(3.1).

The program written by J. McClellan contains a general purpose

subroutine for performing the Remez algorithm. The program is designed

in three parts; an input section, a computational part, and an output

section. The first part is devoted to building W(eJ), D(eju) from the

input data and to setting up the approximation problem. The central

computational part is the implementation of the Remez algorithm. The

third part is devoted to the display of the results. Such modularity

facilitates possible modifications of the program.

It appeared convenient to incorporate the implementation of the

new formulation (3.1) into McClellan's program. This would make the

new formulation directly accessible t( l who currently use McClellan's

program.

The key idea of the implementation was to insert in parallel

with McClellan's program an alternate pattern in 3 parts devoted to

the implementation of (3.1). The first part can accept the input data

peculiar to the new formulation. The second part prompts the operaticn

Of the second Remez Alorithm over the full band [0, -1. The third

38

part is devoted to the display of the results of interest. The new

program so obtained can be thought of as a modified version of the

McClellan's programallowing the selection of two different "modes"

of operation. Namely the "regular mode" that prompts the new program

to implement formulation (1.3) (i.e. to behave exactly as the

McClellan's programiand the "custom mode" that prompts the implementa-

tion of the new formulation (3.1). A user-oriented description of this

program is presented in Appendix 1.

The rest of the section is devoted to the presentation of the

results obtained with the new program together with some practical

observations useful for the choice of W(ejW) and D(ejW) over the trans-

ition regions.

For comparison purposes the four filters reported in [7] as

typical cases of multiband filters with non-monotonic transition

regions have been designed with theMcClellan's program,the new program,

*and CONRIP (which is amother digital filter design program discussed

in Chapter 5). The filters examined are labeled Design 1, 2, 3, or

4 as in [7 3 The new program has been used with both the techniques

L and P, described in Chapter 3.

Figures 8-23 constitute by themselves the best comments on the

performance of the new program versus the other two programs. It

can be seen that the new program gives strictly monotonic behavior in

the transition regions also in cases where McClellan's program and

CONRI? do not.

Several comments are now presented about the choice of ()

39

I I I I

! A A

hi

9 9.1 ,2~ Q0.4 2.9

-NC1ALZ= PQUNC7

Fig. 8a - McClellan's program: H*(e ) ofDesign 1

__ _ _ __ _ _ I_ _ _" _ _ _ ____ __•

! .", r IqU 4

Ia 1*

Fig. 8b .cClelian's proaram: extreIl

frequencies of Design 1

ru - - - : o" " ": ' ° -- - ' " " " J * . . . - " - - "- -- - ; . -. - -;- - - " -.*

I 40

i.25 --

I- --CD

8.25-

8.8 8. 8.2 8.3 8.4 8.5

NORMALIZED FREQUENCY

Fig. 9a New program, technique L:

H*(e j ) of Design 1

0.5-

0.4

0 .3- --

3.3.

.< 8.2-,

9 °S

EXTR. FREQ.NUM ER

Fig 9b New program, technique L: extremalfrequencies of Design I

41

-23

Z -48-_ -- -

8.9 8 .1 .2 8.3 8.4

NORrALIZED FREQUENCY

Fig. 9c New program, technique L:

H*(e j ) of Design 1 in dB

r

U ~ II-..

I I" 1

rIMCIALIZED FREQUE:ICFig. 10 New program, technique P: desired

function D(ej ) for Design 1

* 42

Wi

p~g-

NORMtALIZED FREOUENCY

Fig. 10a New program, technique P:

H*(e j of Design 1

i

zCa

, .

I" 7

"'" -T i ; i . 1 i t 1 ' ; I

" I ,". ">.. "" 2. ...1NORMALIZED FREQUE ICY

Fig. lOb New program, cechnique P: extremalfrequencies of Design-J i . ,, _ . .. w .

43

U-

4-' j

1,-

mC

- 0 0. 0.2M .3 0 .4O .

NORMALIZED FREQUENCY

Fig. l0c New program, technique P:

H*(ej') of Design 1 in dB

1.( m ,I LU 0.0- 33

-0.157 a

O.ooco O.ccoO 0.2g00 a. coa a .. o o . oFREOUENCY

:iz. 1. CCNRIP: ll*(eJL) of Design iL .. (from [8], pp. 66)

44

N '4%^ . ,__ _ _ _ __ _ _ __ _

-Ai I

I • I

Fi. 2 Mcll a' prgrm

° I

3'1 2. Z,__

, I

Fig. 12a McClellan's program:

H*(e) of Design 2

1- - - -I__ _ _ _ _

.i . .. VIA, ,i

g: __ __ _ ± _

Fig. 2._ McClellan's program: ex~remalfrequencies ofi Desig n 2

b 45

!J ._ __ _ _ ---

- 9D-

0.3 2.1 9.2 0.1 2.4 A.S

NORMALIZED FREQUENCY

Fig. 13a New program, technique L: H*(eJw)of Design 2

U

9.4-

C,

N

-J

14 t

-I-

TrT-TlirT.

R 1L3 Is 2S

EXTR. FREQ. NUt"SE7

Fig. 13b New program, technique L: extremalfrequencies of Design 2

46

ma

.. 3 2.4 2.z

NORIALIZED FREQUENCY

Fig. 13c New program, technique L;

H*(e 2 ) of Design 2 in dB

II "°.- / --

8d -. - -- ....

3A I •H

8.2- - -. . . I _

NORMALIZED FREQUENCY

Fig. i New program, technique P: desired

function D(e2 ':or Desi.;n .

47

I .25- - _ _ _ _ _ - _ _ , _

lAJ 9.75-

I.-

Z: 9.58----

9.25- - _._ _

9.89- T1F i i i " i i- i --i i

8.1 0.: 0.2 81. 8.4

NORMALIZED FREQUENCY

Fig. 14a New program, technique P:

H*(e j ) of Design 2

S25-- _ _ _ -_ _

9_-1_ _ _ _ _ _ _ a - -s i ,

" Is ---- _ _ _- _ - _ _ _

I,-

J.o 0.1 8.2 0.. a.. o.

!NORMALIZED FREQUEfJCY

Fig. 14b New pro gram, technique ?: ex.-rernalfrequenci.es of Design '2

48

S-2-4 - __° /

- - -T I -r i l l

jJ

P.P 13;8. .3 0.4 3.5

NORM'ALIZED FREQUE.NCY

Fig. 14c New program, technique P:

I H*(e ) of Design 2 in d3

I °..ea

LU .3

z

0.167

O...0 C-000 0.2tCa %'ZCO 0FR~EQUENCY

-ig. 1.5 CONRP: H,(e' ) of Desin 2(Srom [l3, p. 69)

IL

49

Fijg. 16 Mclla'sporm

2-

2-

0:3 .1 a21 3.4

Fig,. 16a McGlellan's programa: xreaH*(reec s ) of Desin

1* 50

CD

08.1 0.2 9.3 03.4 .1

NORMALIZED FREQUENCY

Fig. 17a New program, technique L:

H*(e ) of Design 3

UU

CD

PL . .

NOMLZD RQEC

7bI __ Ne__gam ehiue ete?feuece o. -. sg 3 _ _ _ _ _ _

51

r

-

-29- -- -

Hz

40--

'C

-~ T~fl TFT-rrr- I rTT" ' IT

..t 9.2 .0.4 9.S

NORMALIZED FREQUENCY

Fig. 17c New program, technique L:H*(e J W) of Design 3 in dB

.-

_ _ _ _ I •

0.

NORtALI.ZED FREQUENCY

FL-'. 18 New program, technique P: desired

function D(ei ) for Design 3

52

!" -- *i

w!

U. 7S --

- n.2~ - *~\ /i

.gD - I I T- I I I I I ' | I I -T

•2. ,.., ".- .1 P-.2 n.- -1 .4 2.

NORMALIZED FREQUENCY

Fig. 18a New program, technique P:

H*(e j ) of Design 3

gw

Lu -

Lu

__]i i i I_ _i I

.i ,_ _ _ _ I [

NORIIALTZED FREQUENCY

Fig. l3b ,ew program, :ec"-'niue e: e:.:tremraLfrequencies o. Design 3

9

53

0-

Fi.18c New program, technique P:

H*(e j) of Design 3 in dB

Li0.833-

C

FREQUENCY

Fig. 19 CONRI?: '4[*(ej'L) of Desi-r. 3(fr.om [8], PP. 72)

54

IfI

- 4O4AL= rftW

Fig. 20Oa McClellan's program:

1{*(e j) of Design 4

I'8

. .. .. . . . . .

a.2 2.t 2.2 2.2 2.4

S20b McClellan's program: extrema.freauencies of Desi-n 4+

55

* -I

3.S-

0.1. 0.1 0.2 11.3 2.4 9.5

NORMALIZED FREQUENCY

Fig. 21a New program, technique L:

H*(ej ) of Design 4

I"

0-

-j

I i I I r 'I 1rT1 I T I IP.R O.t A . .4 O.S

EXTR. FREQ. rIUrIBER

Fig. 21b New program, technique L: extremalfrequencies of Design 4

56

$1.0 -. I 0.2 -. 3 0.4 B.5

NORMALIZED FREQUENCY

Fig. 21c New program, technique P:

H*(ej W) of Design 4 in dB

I I .")- - /

Fig 12•e rgatcnqeP eie

I-

- -- X

ni

NORMALIZEP FREQUENCY

Fig. 22_ New program, technique P: desiredJ function D(e ) for Design 4

57

0 8.75-

o. -- / -

0.2 .lt 9.2 0.3 S. 4 .

NORtIALIZED FREQUENCY

Fig. 22a New program, technique P:

H*(e ) of Design 4

i 3 -

I I,

NORMALIZED FREQUENCY

Fig. 22b New program, technique P: exremalfrequencies of Design 4

4

58

~1--r--7 - I0.2 9. 0.3 ~ j I,

0. 0 .

Fig 23CONH*(e ) of Design id(fo 81 p

59

and D(ej ) for the new program when technique L, i.e., the multisegment

U piecewise linear simulation of the transition regions, is used.

Technique L implies the subdivision of each transition region into

several bands over which D(e j ) has a prescribed slope. The values

of D(e j ) on each band are chosen to linearly approximate the filter

prototypes. A criterion for the choice of the weights on each band,

that seems very effective, is to assign the smallest weights to the

internal sub-bands of the transition region where D(e j ) is steepest.

Greater weights should be assigned to the other sub-regions where the

absolute value of the slope of D(eju) become smaller. For instance,

if the transition region between the band Bi with weight a and the band

B i 1 with weight 2 is divided into 5 sub-bands che preceding criterion

can be expressed by'the two following choices of the weights (it is

assumed .:t 2 3, without any loss of generality).

Choice 1 Choice 2

Band 1

Subregion 1 SC 0.1

Subregion 2 ssc 0.ct

Subregion 3 sss, 0.00a

Subregion 4 ss3 0.0'

Subregion 3 s: 0._

Band 2

Choice 2 is a particular case of Choice 1 and it has been extensively

used in the design examples presented in this section.

60

Choice 1 assumes a > sa > ssa > sssa and ss3 < sS < 8 and it is

recommended when Choice 1 gives poor results (for instance in Deisgn

4). A non-uniform (piecewise) constant choice of the weights in the

transition regions seems to significantly contribute to the elimina-I

tion of extremal frequencies on them. Technique P corresponds to

assigning the values of the transition regions of the prototypes to

the corresponding regions of D(e JW). The weight assigned to a trans-

ition region with this technique should be the smallest weight still

capable of giving monotonic behavior.

The order of the prototypes is a point that needs some comment.

If the multiband is thought of as a cascade of prototypes, their order

should be equal to the order of the multiband divided by the number

Lof the prototypes. If instead the multiband is thought of as a parallel

of prototypes their order should be equal to the order of the multi-

band. Actually the relationship between the multiband and the proto-

types is not clear. Therefore, the question of the order of the proto-

types doesn't have a definite answer. Fortunately, it appears that

the order of the prototypes doesn't influence the result very much,

probably because of the effect of the weight that helps keep E(e j )

small over the transition regions. In the design examples shown in

this section the prototypes were usually taken of the same order of

the multiband. In Design 4, though, the multiband is of order 73

and the prototypes were taken of order 41. Prototypes of different

orders, according to the empirical rules of Rabiner et al.F 7 1were

tried, but no improvement over the other choices cf -he order was

noticed.

61

The plots of this section show that technique P gives better

I monotonicity of the transition regions than technique L. However,

the two techniques have comparable performance in terms of elimina-

tion of extremal frequencies from the transition regions. This is

explained by the two following facts: every prototype induces extremal

frequencies into the transition regions of the other prototypes,

therefore, as mentioned earlier, extremal frequencies in the transi-

tion regions should be expected. Furthermore, technique L benefits

from the non-uniform weights in the transition regions, while tech-

nique P, in the present implementation, doesn't have this feature.

(This further modification has not been implemented because the results

of technique P were already satisfactory).

To obtain a multiband filter with monotonic transition regionspwith the new program is very simple. Some common sense is needed to

choose the weights of the transition regions, which is the only euristic

part of the procedure. The rules are the following: if the initial

weights give resonances, then increase their value; if the initial

weights give monotonic behavior then try a smaller weight that might

reduce the ripple. The criteria for the choice of the weights in the

transition regions should be used in these changes of weight.

All the filter examples shown in this section were obtained on

the first try, with the exception of the filter of Design which

required two attempts (it is not excluded that prototypes of order 73.

instead of l, -culd have zi'ven a satisfactory result on th& firs r

62

CHAPTER 5

COMENTS ABOUT CONRIP

This section presents the relationships between the new program

discussed in Section 4 and the program CONRIP, written by M. T.

McCallig [8]. CONRIP implements the design of FIR filters according

to a "constrained ripple" formulation, due to B. J. Leon and M. T.

McCallig [8]. Their formulation is the following: given continuous

functions U(ejW) and L(eju) on [ 0, iT such that U(ejW) > L(e j ) find

the polynomial

W MP(ej ) Z a(K) cos(Kw) such that:

K=0

(i) L(e j W) -- P (ei : U(ej ) W E [ 0, 7]

(ii) P(e ) is monotone on specified subintervals of [0, 7]

(iii) P(ej2) is the minimal order polynomial meeting conditions (i)

and (ii).

It is worthwhile to note that the preceding formulation is not a

traditional approximation problem (like (1.3) and (3.1)) and it is a

full band formulation (like (3.1) and unlike (1.3)).

CONRI? allows the design of multiband filters that do not exhibit

strictly monotonic behavior in the transition regions but which dc

not nave resonances like the ones obLained with McClellan's program.

The results of using CONRP for the filters o4 Designs 1, 2, 3, 2re

jhcnM in figures 11, 1;, 19 and 3.

The similarit: between he cor.strained ri ppe ro!)e nd ma e

63

Chebychev approximation problems (1.3) and (3.1) is greater than it

might appear at first sight. In fact the computational solution of

the constrained ripple problem is obtained with an algorithm (reminis-

cent of the second Remez algorithm) which searches for the polynomialsiw jtangent to U(e jW ) or L(eJw) at their extrema. Two theorems of

'McCallig state precisely the similarity of the two methods.

Theorem 5.1 Let H*(ejw) be the solution to the Chebychev approx-

imation problem (3.1). Let

p = max W(e j ) IH*(ejW) - O(eJ W) (5.1)

[O, .7]

The constrained ripple problem having

L(e) -_ D(ejW) -p

U(ejw) = D(ejw) + p (5.2)

admits unique solution P(ej ) = H*(eJW).

Proof: 13], pp. 42. The converse of Theorem 5.1 stated as

follows.

Theorem 5.2: Let the boundary curves U(ej and L(e>) be given

such that the unique solution to the constrained ripple problem

is ?(eJW). The Chebychev approximation problem (3.1), having

D(e j ) = l(u(eJ) + L(eJ ))

W(e) -D(e

U(e j -) - D(e J )

admits unique solution H*(e' ) = P(eJ).

64

Note: The assumption U(ej ) > L(ej ) guarantees the denominator

U(ej )- D(ej ) - (U(eJ )- L(ej ) > 0.

Proof: [8], pp. 42.

RIt is important to notice that Theorem 5.1 says that the approx-

imation problem (3.1) needs to be solved in order to obtain the equiv-

alent constrained ripple problem. (The two problems are said to be

equivalent if they have the same solution). In fact the final devi-

ation p (5.1) is not available before the solution of (3.1).

Theorem 5.2 similarly says that the constrained ripple problem

needs to be solved in order to obtain the equivalent approximation

problem. The order of the polynomial to use in (3.1) is not avail-

able without solving the constrained ripple problem. Some cosequences

I of the above two theorems having practical interest are now _,:-<:ussed.

The following equalities

L(e j ' ) = D(e j ) 1 i.)

We (5.4)U (e l) - D (ej ) [ + Wej )I

W(ejW)]

can be used to solve via CONRIP the following Chebychev approximation

problem:

rain rain max W(e D) D (ej ) - H(eJ) (5.5)

M:Z -a(K) o,K=0

This last aspec: of the equivalence between constrained ripple prob-

Lems and aoproximation problems needs a comment.

65

Let

! p(a) min max ~ W(ejw) D (ej() H H(ej w) (5.6)

a(K I 6 e O

with real and positive. The best approximant H*(e jW of (5.6)

is independent of S, but the minimax error p($) is dependent on ~

The constrained ripple formulation, as intuitively understood and as

precisely stated in Theorem 5.1, is sensitive to the minimax error

p(a). Therefore the filters found via formulation (3.1) and the new

program are independent of the weights. However, scaling the weights

in formulation (5.5) implemented via CONRIP by (5.4), give different

filters (even differences in the order of the filter must be expected).

The new formulation (3.1), on the converse, suggests different

i ways of using CONRIP. Perfect monotonicity of the transition regions

should be obtained by picking L(e ) and U(e ) via (5.3) where the

D(e ) is chosen with the help of the prototypes and the W(ej ) is

chosen with the criterion of Section 4. The new program presented

in Section 4 can be used to implement the following problem which is

related to a constrained ripple problem:

Given continuous functions U(ej -) and L(ej ) on[ 0,.] such that

U(e j ) > L(ejW), and a given order M, find the polynomial

M

P(ei" ) = -a(K) cos (K-) such that:K=0

(i) L(e>) - (M) . P(e) U (e ) -- 1) for some

(M) > 0 [0, -

(ii) P(e3-) is monotone on specified subintervals ofO,-.

66

It should be noted that the 6(M) of point (i) depends on M,

and there exists an M such that VM > M 6(M) 0.

The experimental verification of the use of CONRIP to solve

approximation problem (5.5) and of the use of the new program to solve

Iproblems similar to the constrained ripple problem was beyond the

objective of this work. Nevertheless, such verification would be

an interesting project.

I

67

CHAPTER 6

SUMMARY

The problem addressed in this work was finding a technique for

if the straightforward design in the frequency domain, using a minimax

criterion of multiband linear phase FIR digital filters.

To the author's knowledge the only similar attempt has been the

work of Rabiner, Kaiser and Schaffer [7]. Their work has the merit

of having brought to general attention the problems arising in multi-

band FIR filter design with McClellan's program. However, their

solution doesn't address the essence of the problem which lies in the

inadequacy of Parks and McClellan's theoretical formulation of the

filter design problem for the multiband filter case. The "strategies"

U proposed in [7 ]to design multiband filters are empirical and their

application is generally not straightforward.

Chapter 2 presents an original analysis of Parks and McClellan's

filter design formulation. It is clearly stated that its mathematical

meaning over the band [0,7 ]is that of an approximation problem not

according to the Chebychev norm but according to a seminorm. The Parks

and McClellan formulation is shown to be ideal from the filter design

point of view for the high and low-pass filter cases. It is similarly

shown that it is mathematically inadequate from the filter desizn

point of view where there is more than one "don't care" band as in the

multiband filter case.

Chapter 3 proposes a new formulation of the filter desizn

problem. The new formulation corresponds to a minimization 9roblem

- -

68

in Chebychev norm over the full band [0, Tr]. It requires the use of

p continuous desired functions D(eJw). The Alternation Theorem and the

second Remez algorithm still apply. The new formulation is immune

from the drawbacks of the McClellan formulation and it is theoretically

adequate for the multiband filter design problem.

Chapter 4 discusses an implementation of the new formulation

based on McClellan's program. Considerations about the choice of

W(e j ) and D(ejw ) having practical interest are also introduced. The per-

formance of the new program is compared to that of McClellan's program and

McCallig's program CONRIP for the multiband filter case. It is empha-

sized that the new program seems to be the only one capable of giving

strictly monotonic transition regions in a straightforward way without

changing filter specifications.

Chapter 5 presents the relationships with McCallig's program

CONRIP. The possibility of the use of CONRIP to implement a design

criterion very close to the one presented in Chapter 3 and the possi-

bility of the use of the new program to implement a design criterion

similar to a constrained ripple problem are both introduced and dis-

cussed.

From the author's point of view McClellan's filter design formu-

lation (1.3) is a particular case of the new formulation (3.1). This

made it natural to incorporate McClellan's program as the core of a

program implementing formulation (3.1). The practical result obtained

in this thesis therefore is a kind of "extended McClellan's pro-ram"

that is identical to the original one when appropriate, such as for

69

the low and high-pass filters, or can be used for its new features

13 (formulation (3.1)) when McClellan's program is not adequate, as in

the design of multiband filters.

U

I

70

APPENDIX 1

Appendix 1 provides a user-oriented description of the new

program discussed in Section 4. The program operates interactively

and asks the user a sequence of questions part of which are derived

from McClellan's program and part are original. The meaning of the

questions in terms of McClellan's program will not be reviewed here

and the word "standard" will be used in place of the actual answer

for questions from McClellan's program. All the comments to the answers

will be on the modifications for the new program. The questions will

be numbered for convenience of reference.

1. TYPE FILTER ORDER

Standard

2. ENTER FILTER TYPE

Standard

3. ENTER NLMBER OF BANDSIEnter the number of bands where D(e j ) changes slope if the transi-

tion regions will be piecewise linearly simulated. Enter the total

number of passband, stopband, and transition bands if the transition

regions of the prototypes will be used.

4. TYPE GRID DENSITY

Standard

5. ENTER - T- -PLCT

Standard

6. ENTER LINE PRINT lFG (0 - DO NOT PRINT, 1 - PRINT

Sta-ndard

71

7. STANDARD WEIGHT AND TARGET, TYPE 0, CUSTOM WEIGHT AND TARGET, TYPE 1

0 selects the operation of the program as regular program of

McClellan. 1 selects the "custom mode" operation, implementing

formulation (3.1) (all the comments to the questions assume 1 is

entered here)

8. TO CHANGE BAND EDGES TYPE 1, OTHERWISE TYPE 0

The band edges previously entered will be kept if 0 is entered

- (this happens because the program can cyclically call itself).

New band edges must be provided if 1 is entered.

9. ENTER THE BAND EDGES - - NO. = NBANDS

Enter the sequence of the edges of the bands. Since the new pro-

gram assumes full-band operation the edges corresponding to two

contiguous bands have to be 1 sample apart. In order to facilitate

this feature, the program just uses the edges 0, 0.5 and the even

ones (the second edge, the fourth, and so on) to calculate the odd

xones via an increment of 1 sample.

Since the important information for this answer is the edge 0,

the edge 0.5 and the even edges, the odd edges are usually assigned

to dummy integers like 0 or l(because they are convenient to type!).

10. FILTERS PROTOTYPE USED? NO = 0, YES = I

Enter 1 for technique P. Enter 0 if no filter prototype will be

used.

11. ENTER SrMULATION FLAGS

Question 11 appears only if I has been entered at question 10.

Enter a vector associating a number with each band. If the band

is a transition region the number associated must be the number cf

. . . ... . . .. . . . . . . . . . . . .. . . .. . . .. . . ..

72

the file containing the impulse response of the prototype for that

region. The number 0 has to be given for the pass and stop-bands.

12. ENTER ORDERS OF THE PROTOTYPES

gThis question appears only if 1 has been entered at question 10.

Enter a vector associating a number with each band. If the band is

a transition region the associated number must be the order of the

prototypes used for it. Any number can be associated with the other

bands. This feature allows the use of prototypes of different order.

13. PIECEWISE LINEAR DESIRED FUNCTION? YES - 1, NO - 0

Enter 1 for technique L. Enter 0 if technique L is not wanted.

It should be pointed out that the current implementation allows

also the use of technique L and technique P together, that is some

transition regions can be taken from prototypes and some others

can be piecewise linearly modeled.

14. ENTER DESIRED FUNCTION AT THE EDGES

This question appears only if 1 has been entered at question 13.

Enter the values assumed by D(e j ) at the band ed es.

15. ENTER THE CONSTANT VALUES FOR EACH BAND

This question appears only if 0 has been entered at question 13.

Enter 1 corresponding to the pass-bands, 0 corresponding to the stop-

bands, and any number corresponding to the transition regions.

' ENTER WIGHT FACTORS FOR EACH BAND

Enter the weights corresponding to each band. :t should be empha-

sized that the current implementation calls -or unif:rm weizhts

over each band. A piecewise-constant weight could -e 21so )tnirii,

over a region b' means of its subdivision int2 se:eral band .

- - .

73

Figure 24 shows an example of conversational terminal using the new

* program with technique P.

I

m

m

I. 74

X: ..oaieq

7TFE 1 '0 CV 7irfa -3 7:

EfIS FIM. TME (!.2,3)

STWOMITAS NF 20TM.T7,

TO CiJE ? 0 7.7, 1, j-,E mmI ,YE a

a 8. '43-7i3a a

3 3.41744

8 a ! a

V 1S a 3!TP ,

-1. ;ES. . 1C.

82!a =!

aAS

La d

75

APPENDIX 2

IThis appendix offers numerical examples of filters designed with

techniques L and P. The examples correspond to the filters denoted

Design 1, 2, 3, 4.UBoth techniques require one to design filter prototypes. Tech-

nique L needs them as modils for piecewise linearly simulating the

transition regions. Technique P needs them to use the actual values

of their transition regions into D(e J).

The prototypes are designed with McClellan's program. Their

, specifications are directly obtained from those of the multiband filter

corresponding to them.

For every design example the information relative to the proto-

w types will be presented first, and then the information relative to

the multiband filter.

(a) Technique L

Design 1

Prototype 1.1 (low-pass); Filter order = 75

Sand no. Firt Edae Second Edae ) (e

C .. 6 =_ 2 9.

76

IPrototype 1.2 (high-pass) Filter order = 75

Band no. First Edge Second Edge D(e j ) W(e j ,)

1 0 0.41679744 0 0.03853275

2 0.37032451 0.5 1 1

The transition region of Prototype 1.1 was modeled by means of

2 line segments. The one of Prototype 1.2 was modeled by means of

3 line segments.

Multiple passband-stopband filter 1

aBand No. 1st Edge 2nd Edge D(eJ) D(ej )(e

(ist Edge) (2nd Edge)

1 0 0.14375973 1 1 0.04588809I

0 0.159 1 0.07 0.004598809

3 0 0.165 0.07 0 0.04588809

4 0 0.37032451 0 0 1

3 0 0.385 0 0.09 0.0383833275

6 0 0.021199 0.09 0.91 0.003853275

7 ,) 0.'1679744. 0.91 I r.03853275

3 0 0.o i 1 0.03532,3

77

IDesign 3

Prototype 3.1 (Low-pass); Filter order = 57

Band No. 1st Edge 2nd Edge D(ejW) W(ejW)

1 0 0.11018835 1 0.19386528

2 0.00820222 0.5 0 0.17459027

Prototype 3.2 (High-pass); Filter order = 57

Band No. ist Edge 2nd Edge D(ej ) W(e>)

1 0 0.26931373 0 0.17459027

2 b.2C585967 0.5 1 1

?rototype 3.3 (Low-pass); Filter order = 57

Band No. !st Edge 2nd Edge D(ej )

1 0 0.37673351 1 1

2 0.31138715 0.5 0 0.13180259

I S nInlI ' I

78

Prototype 3.4 (High-pass); Filter order = 57

* Band No. 1st Edge 2nd Edge D(e> ) W(e j')

1 0 0.46299174 0 0.18180259

2 0.3892995 0.5 1 0.21319649

IThe transition regions of Prototypes 3.1, 3.2, 3.3, 3.4 were

simulated by 3,3,5, and 3 line segments respectively.

Multiple passband-stopband filter 2

Band No. 1st Edge 2nd Edge D(e j> ) D(e3' ) W(e j -)

(1st Edge) (2nd Edge)

1 0 0.00820222 1 1 0.19386528

2 0 0.03439052 1 0.96 0.0193

3 0 0.04139057 0.96 0.91 0.0193

4 0 0.077 0.91 0.09 0.00174

3 0 0.084 0.09 0.04 0.0174

6 0 0.11018835 0.04 0 0.0174

7 0 0.20585967 0 0 0.17459027

8 0 0.216 0 0.065 0.017459027

9 0 0.2591734 0.065 0.935 0.0017 9027'

10 0 0.26931373 0.935 1 0.01749027

1 0 0.3130715 1 1 1.

0.313,2266 i 0.953 0.01

13 0 0.3699 0.953 0.9)7 0.00i

q 6..37673531 0 .07 0 ?.O1S1,9.258

0.239:995 C 0 .1 3 80_5

79

Band No. 1st Edge 2nd Edge D(ejW) D(e j ) W(e j )

(1st Edge) (2nd Edge)

16 0 0.4045 0 0.006 0.018180258

17 0 0.44779124 0.006 0.994 0.0018180258

18 0 0.46299174 0.994 1 0.02139649

19 0 0.5 1 1 0.21319649

(b) Technique P

Design 2

Prototype 2.1 (High-pass) ; Filter order = 27

Band No. 1st Edge 2nd Edge D(e j ) W(ejW)

1 0 0.22754845 0 1

2 0.0728033 0.5 1 0.1616032

Prototype 2.2 (Low-pass) , Filter order = 43

Band No. ist Edge 2nd Edge D(e) W(e

1 0 0.3445328 1 0.1616032

2 0.29030124 0.5 0 0.0034068

The prototypes used in this example are of different arders (their

orders correspond to those suggested by Rabiner et al. [7]).

McClellan's program stored the coefficients of Prototypes 2.1

and 2.2 in disk-files number 21 and 22 respectivelv.

80

Multiple Dassband-stopband filter 2

Band Simulation Prototype 1st 2nd D(ejW) W(eiW)

No. Flags Orders Edge Edge

1 0 0 0 0.07280333 0 1

2 21 27 0 0.22754845 1000 0.01616032

3 0 0 0 0.29030124 1 0.1616032

4 22 43 0 0.3445328 1000 0.00340608

5 0 .0 0 0.5 0 0.00340608

jW

The dummy value 1000 in the transition regions of D(e )has been

used to indicate the use of Prototypes.

Design 4

Prototype 4.1 (High-pass) ;Filter order =41

Band No. 1st Edge 2nd Edge D(ejw) We~)

1 00.13199438 0 0.0959953

2 0.08886197 0.5 1 0.11187421

Prototvpe 4.2 (Low-pass) ;Filter order = 41

Band No. 1st Edge 2nd Edge D(e>') e)

1 0 0.27193968 0 O.137, 21

2 0.18550831 0.5 1 1

Protocv: e 4.3 (High-pass) ;Filter order =-

"and No. 1st Edg-e 2nd Edge D(e- ") 'Ve-

1 0 0.33373202

2 0.2331105 0_7 i Q177'!-Q

81

Prototype 4.4 (Low-pass).; Filter order - 41

Band No. 1st Edge 2nd Edge D(e j ) W(ej )

1 0 0.45732656 1 0.11177379

2 0.43737502 0.5 0 0.05401694

The prototypes in this case were taken of orders different from

the order of the multiple passband-stopband filter. McClellan's

- program wrote the coefficients of Prototypes 4.1, 4.2, 4.3, 4.4 in

disk-files number 21, 22, 23, 24 respectively.

Multiple passband-stopband filter 4

Band Simulation Prototypes 1st 2nd D(e j ) W(e j )

No. Flags . Order Edge Edge

1 0 0 0 0.08886197 0 0.0959953

2 21 41 0 0.13199438 1000 0.09

* 3 0 0 0 0.18550831 1 0.11187421

4 22 41 0 0.27193968 1000 0.11

5 0 0 0 0.28819105 0 1

6 23 41 0 0.35373202 1000 0.11

7 0 0 0 0.43737502 1 0.11177379

8 24 41 0 0.45732656 1000 0.05

9 0 0 0 0.5 0 0.05401695

82

APPENDIX 3CDThis Appendix contains the computer printouts with all the

numerical informacion referring to the filters shown in the plots

u of this work.

In order to facilitate the association of the plots with the

information tables corresponding to them, the tables of this Appendix

-are labeled with the same number as the figures containing the plot

to which they refer. Therefore, Figure 4 corresponds to Table 4,

Fig. 13 corresponds to Table 13, and so on.

4 It should finally be noticed that the printouts of the new

program used with technique L under the voice "DESIRED VALUE" report

the slopes of D(e>) over each band.

I-

h 83

TABLE5

FTPlIl IMPULSE RESPONSE (FTNI

LINO-AR PWASE O1GZAL FfLYER IESIGNRE MEZ EXCHA-409 ALGONlZo

VANDPA33 F!LTEW

FILTERt LINGTm a 33

110 .1121qIa a pi 3aM a l'u *.3vU3S41.32 a 4(9 ae)

Mc 4) .4.4459b4366-4 a 4( 2714C 114 e88,9ila(S.i a MC 12

M'C 718 1,13543364c.at a 4( j42

4( a)* ..17ta4E.at At " 3)

mC 102' 0.341619E.41 "( 21)a"t ilia 6,I~31.1a 4c 44114C 1270 *4.3733878G(..81 a wt 19)Pit M3e -.,8472316C.al a "C to)KC 1410 4,146772%colle a "t 1714C 15)6 3,44L9444379024 a 00(1

dANO I dAN 2 ANOL.:iWEm SANO !061 ,I99~ E qS~6OVUPPER dANO EDGE a,agee~eg e0931RED VAL.UE 1,3g030300 aagoi?"llOCZG~tTING 1.3190~0 1J.81oeDEIAIIIN 300a264722 .Z9J3£ oVIA?:o'4 IN 00 .53,1196lI395

E3EMAL FOEQUENC!ESa.3IIeaeI 8,3373690 1,3756062 0.1243333 3.4661.17a8333 0.191bb8? 3,2320@60 a.3j@0049 h.34633333.32%lb.7 3.336524 1337118 0,Q071819 l,'1!mlmfQ

a,4'83 3333

84

TABLE 6

L1411AR PwAsE oCl1I'AL FIV'ER 1ES!nkQE"t? CxCANGE ALZG OM

9ANOPA34 FILT ER

..... IMPUL39 ESgPON31 ea..

- "C 123 a.11b8 J a "t 31)"( Us. 0.21,b62f1.42 Wt 29)Mc 316 u*36a5896e..da -4 is)"4C 4)v -8.631534221.a2 a 4( 21MC 9)8 .. l9gEd2 14~( 26)"C~ b~. 9,69014a9e.di a oc 23)

nC ?in2. g,44a23E.J1 a KC 242"C a' .a.lb43331bE-d1 a "t 232

9)4~2 .0.24leatil~E-t a 4( 222IC -alg g.,2*4qgqf.a l (C 212

pit 13)a .487324-a NCM 162

mC 14la . II'54734(.*2 * 4( t?)

SAND I SAND a 9440 30010~ 3AP40 EDGE 1. lu3' a11I33 .382033OPBER qANO EQGE 0,luagiifl a3raegg .dJMs1OES:"qU V*L.JE 1.3UU9ve9 tjIeolg3 233,gq

et-cc'AL U(;-jf.4jE

3,433633333~~'S~ ~ a~a~3

TABLE 7

FINIT'E IMPULSE RESPONSE ( FIR)L:"E~AA 2WASE o'1Z!AL FILTER JESZGN

F!ITER LENGT4 a 40

"C 12' 3.1185a3ec.42 ac 185~i2(1m' M,84947q531E.J3 *C 3t )

4~32' J.114a3889E.a2 a At 38)Ma -4) a2 -a.1l~eqt.sJE-43 a P4( 373

3C52' .-).4e55QgiI2E-a3 a MC 36)PC b2' .. ?94b3a3IE~a2 a 4f 311'4C ?20 3.349lS021EJ3 a '4C 341KC 53 O.!*e1Ca a 4C 332"tM 92'1161671d a 04 313

ct 1o2a 1.7q2G61qIE.13 a "it 31)P1( Ill* *.I1479632E.Jl a 4 t 362Pq( 1212 493231761eE.42 a ot 29)4MC 1320 -a.314139lE.aa a Mt 162"t 142'1a 41g1~1 a 04C 172NC Is2a -0a6A31d 1 M( 1424( 14' b 11e~31fJ a OC 212m e 173318 4511.l'~ 4

Mt 192' ?.1346839e*1aa MC 2114 20'1.433b8320!.daw .t c 12

aANl I 9AO O 3

DESIRED I'&LA a,z3a061'0 .3.333333333 2,3zogeseesi~m:4.I.ouoo 1.2d000000V 0I~6l9

!V1QE-AL '2E~iJECZE3J'AZO0040 3.a39aft2, 3.3671375 9.389ata 2.~~fe

a,2324115 2.,25q3750 3.2819371 a1,3125a~v 8,337!060a.34.O6.l5 2,aS .401562! 0.41413621 a,4.ae07-i

86

TABLE 8

*........... ... 0...

WT!LwrO LEN'GIr 8 73

I...:PULSF 0F5iP 1dE S..

(~ t~ ,l. ?935TE-01 b4)73

**05 1' 3 8 7

121* -)A .72635AE-al mf b-1 )

,(~~~~~ 7. *"aE.: ( 621

10) 4a 12 1 A t7 t C -4 1 ( 681)C a1 206304 3d1E-6' I a4 " 4-c ~l 1.!2!11!57-a't a 4C W

C 22)v a . 4,132 541

at 1)* .476031!E-~1 a 4( Sal271Ma) -2.114737-pl '4( 5

-'C 4 ?A. 14424'546..4% 41355

3-1 a2) .43015125IE-1 a -4 5161

3'26) 0I4197C4 -C 441

"C 33). i.361 UVUSIS-2 C 431

'C 39.A) Ia' -e34493a51J q. ISE?0"C3.J1 --I.35547. 9E-e'1 a -e'46

"C36)a j.32b1b?94EvAI a -C )"C3712a .2*a4bQT27E..II -4( 341~381 2a 2 e 1315 E. -0Z t 34)

SANfl 4A4 4 I"L:-C4 5AN'O !O~ 3.:1321000"3 3.1065339433 3.41679744aAE ts'10 E- G 1.437S9731 2.373324,510 25440a

5I8899 .~7234g 3.1345327307.7~1'e ;.!,3~24,t ;4.3855220145

J2 ~-~.t75633179 -.Q.0416542ij -J..3584343466

~d.~13L7Q ~.~'13I a.67715' ~j~5UO57 a.;6t4.4?q2.,7d3z5q2 a..,972335 1. I 3~ . :15dea 3. :37333!

Z.1-37547 .3,.13S33,44 .1*.b6214 1.17"S4322 21.19514.3

1 . qO-54Q4 -.. lai1172 1.22:!14!j t P.23 :2513 ".3~2-adabJ.250223 ;'.2bAQ!7* .~15 J.241batq Z.33 Q64t1j,31.p3pjvl .2'1'. 73'6 J.3sam"1. .b3'I.3b7tIA;I d.'2~~4171 .4];44242 .454a

3.4d7a1 3.~21~ Z.48V64 a . _9a a dA 63

87

TABLE 9LI *~ewe~esg~eu e..tg~f........................ *********

MANOPASS FILTEO

if F:LTE LENGM a 73

***** IMPULSE RESPOINSE e.MC 139 3,11877904[oai x "C 73)AC a]* P.207a6496s(.d a At 7%]of 332 0.21593887e.al a ' 712"t 4)m -. 37731763C.32 a Nf72)

"c512 of'aha~a*" 711

"C728 .8.927223341.02 A ( 69)At UC 3, a12117133C.82 At "C6)MC 912 -0.4d3134S3f--12 AC 6( eA C Ills 3.1151P732E.I1 mf" bdh)"C Ills n.,2q1a82iE-a2 a At 65)14( 122' *I.1218I'a27f-a2 a of 64)14C O3)e -0.23264496E.I2 a A( 63)of 14t .41ls491i a Of 62)HC 15)' 1222222244E-42 4 C 613"C 16)v3 .115qbqaaE.at A( baof 17123 .qQ2?776.c.3 4(N 54)of~ 1630 3.1237209aC.J1 4 C 56)04C 19)' aI693643E-4t a-4( 57)MC ail) aJ.16MA113Eal1 a A C 56)oC 212' 9.131'14329.82 a of 51)NC 2212 i'6S'E a mC 54)Al( 2310 233373331.al a A( 53)4C 2')' a.9l2sa.4.a a of 5algPIC a1,. .,34588102C.at a mc 51)H( We) -a.224389u1E-32 a of SRI4C 27)8 -4.49l43657f.J1 of" 4q)A C 20)9 p.t1637873e.al A (C 48)"C 292' 3.2674a696i-8l a -of 471'4C 306 3.13065313E-.' a "( 46)AC 3128 2.1qP24254E.01 8 c 45)AC 321m -a.?3872283E-d1 m3 aC4MC 3373 .2.4S5saqT2F.8I of 4C.31of Ws - .2'A54 I542E .,a "C -421"C 35)' A~.W722634E-al 4 C 4tjA(C 36)8 1.36122E.1 v At 46)

oC 371s9 .591i~Ed a AtC 39)of 386 2.32162215E*44 " 38,

SAND I 4*"O 2 5*'.0 3 SAND' 4t.DWER 3&AAO ID0E U,~960Umgp2 a.134562996 3.15982236a a.1658223b60P"R SAND0 EDGE a,I1a3739730 3.13"G66062 X.161220400 3.3'Z32431A3131RE VALUE ~ 0*kst9*A.J22143939 1.664,

Is.Grh ,aaSS88aq@ 3.J1458d19 Z.AI4548J te4

SAND 5 9140 ft SAma 7 840L~.'1R SAND !OGE 0.37:14687S 8.18!822366 3.442q422ta a.Whiq298OVER SAND EDG 0.365ep,!aev a.'A2211440 a.4t67Q74a 4.7aa2131RE0 V&L6 6,132',0W!' 47.497476031 b.t3t31?73a41I;?*TI'G 2,238527106 3.30335327! 3.J3&532'!2 j.j3532'!1

!KrsE'hI. Fqe~uer:ES39 3 13 2.1263158 1.3411163 a.a29 3.36;0739g~a832592 23q?33q A.:Iidlql 2.:asapeg 3,1371355I,1437197 Z.lb!822' 4.1t41116 3.:765132 21.136391bJ.9772 3,226S855 1'.22sq2t: 3.2332566 3.24550293,2s579276 3,2732b32 4.2425447 1.2q49344 a.327254,0,3106053 2,33104a a.334139 0.341447 1.3bJi3sla.3?13241 *1.*46a24 1.4tolq'z J.42ql330 1,4.393!6

88

TA.BLE 10

LINEAR Pw43E DISZ1&A4 rXI..EP 7EsS54aE"SZ !XCI4ANGE AL50OM4

3AN00433 FZLT!9Q

F:6rEN LINGTK 8 75

am*** IP'UI.SE Psolsle eee..AtC 1)* 3,107I1ab6aE0d1 a At 75),4c a]@ 3.328saybfE.2 a c 74)Ac 330 3.t9413180C-S1 a O( 733of 43 -41.16231182f-02 0 P4 ('2)"f 523 a.~4o1* AtM 71)MC *Is -4.169931639-@t a Af 74)"t 733 -. 07g1L30C-la a "( fig]At $12 *0.24549,5agiaa a At balKc 9)v 2.171304421.03 v At 673AtC UP! avd0043ll~jwa2 a mc 60)AtM II)s 3,362630448-02 a wt 053At 126 .6.sa7~q24t9.aa a "( 087m( 131s -4,719311 2,e-aa s A( b3)A( 1430 ea,98243421Ca32 mt nC01A~c 153' .jqqq4932jf.32 A( MC 1)MC 160 4,121565669-61 a -I( WlM( 1770 3.'91?%laq.aa a M( so)

" ( 1839.9,640779819.42 a 14C 18MC 19)z *0.39611jiSlf.42 a M( 57314( 22)8 -e.2164Q24ag-a1 v M( 56)mc 20 1) 911I947W4.12 a At 55)A( 2130 *e,3043Gq3te-a2a v 4A( 232' 33)~'8L11a~14f a4)4 .IIqI.1a~ S2)At 2930 -4.18'sl19931-al 3 ( So)"t a0)9 .4,2742S10gei.a2 4f go)AC 273. q4.4tq96.99eL.41 s Af 49)At 28)9 *.5321500g(.83 s Ac as)A( 293' 8.37abs7419-al a -6c 473AtCW 36 . .53aSjasGE.Ja a -o b

308 313157329,a29-al a A( 453S3216 -. *1Jb0346LvJ1 a 4C 44)M33)s *8.51722297t.11 AtM 43)-34)e *a.96a47' lC-42 4( &a]£M35)1 .am.b94917219C. Af M 1,~30)' 1a30eltlwfav 4f M '), 3733 4.679'3a3~lf.J "(a 39)

4( 38 a,S39l117oEf* "( 361

OANO I JANO a BANC 3 BANC 460WER SANO toGE a'aafees 4g.I418eqa~ a,:6650311 a.371168OjPP9 SAMO 90GE g*.&37519?3g a,1653339a3 3.37a324112 3,11o774Aa

0E99!I aA.E L186066eI 1gefa41ge 1JIsI66 12.9fl69

SANO I @ANDLOWER BANCo I GE a.417boeaUPPER sAho !OGE 3.a.5662g69MIRED V&464 1.3iglgoggu

melso"N8. 2361 327!0

EX191M8L V.g:U94C:E33.8123355 S,aa71332 3.3411134 8.85!987 3*3 0 qg7 lq8,2832S;2 a,2973395 3,lL1131q, 4,5234 a.1373353,1'3719V 3.166150 8.16418s 3.!768471 3,L5671530.074863 *3Q 3.22844 .2!!a Z,23314J5 1.2459201a,2sai01. a.278537% a~asaqsao a.2qboe6 a,308aaba3,321lhIt a.33227,47 3.3446123 1.35!32:a 1.3651sq!3.37131. 3.3913179 3,413awt a.3.263126 1,a431323,a576q3j 2.471896L 3.8.~5&7,4 3.5Z23602

89

TABLE 1

-7- 1!' - Z

- . Iq w I.m)-

;I it 1 :1 1

.4~~ ~ ~ F lN"r.- T

*~: -r IJ*

.~~~~~ ~ .z :~ .* . . . . . . . . . . . . . .

- e,

J ~***~* *=I**It. N '. *':. r - 1 :

-A -- c -50%~I~ 0 . . . -

LL . . . . . p 'n-

90

TABLE 12

I L:%'!AR Pw4,SE 'I!GfTAL FILTER OE31GNJ

*wee 1' -LSC 4E1 PO3E 2 0* ' 3"C 1) -4.467142a3E-.'2 KCM 421

339 41 .1531 3E-41 a C41)AC 51 ~3m 14549P20E.a2 a MIC 3914f bs -0.244666'3.01 8 H( 361"C ?1* -. 631961E.Jt a 'IC 37')4f us0. '2,113364 a C-01 u NC 36)

"c lals a)177!~.1 " '( 34)4C U1113 .394171ab(-al a 4f 33)"~14( .a89el2Cd a OC 323

C 1316 -4. q#03 ?4eo.4d a 14 31)"C1412 .1.1177334CE3a7 a 04 36)3"Cs1)' *j80, 1 a 'Oc 2q)

mf to)* O.2?713146Eoeo a Wf 26)14C 17)v: ~. I34Z 1AA7!. * a "C 271HC a6 Atb4110~3 u " 1

"C V' ~3,7q8334.44a f 25)1C 2010 .I.4325!ob'?!*0O a -C 241

?1C t1 .2*q.d7O2E.J' -0 9 32 11' 3.bq9qAtbS3E.J0 a q ( 221

.A l1 14NO 2 SANO 3

J~! ~~4f ~ a*MF283311 .4 3.54a089pop

UEV!Aj2:4 3.123?tq9. a.R60475698 3,P2S78718:Cvrta4 J 3.llqaig 6 *4q1 6 3.2qee

EXraE'*L q.I(PE

I AD-i2S 987 FREQUENCY DONIAIN DESIGN OF NULTIBAND FNT MUS- 2-RESPONSE DIGITAL FILT.. (U) ILLINOIS UNIV AT URBANACOORDINATED SCIENCE LAB G CORTELAZZO DEC 88 R-986

UNCLRSSIFIED AFOSR-79-8929 F/G 9/5 NL

EL

111 1.0 t:' 28 12.5La 1 1..2

1.45 1.6

MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDARDS-I963-A

91

K TABLE 13

p.. ~FIXNR9IMZPULSE RESPONSES CP?97LINEAR PWASI OWGTAL FILTER 01314P*gwgj EXCHANGEI ALGONIbOM

EAMOPA3S FILTER

FILTER Lgwg?'* a 63

,sets Z'4PUL3E RESPONSE ..

4'C Jim .*.71T46097f a 14C 42)14C 33a 6.976716911-s? a 14C 41)14 41 *lts *a398131.f6 a 14t 4S)Mc 5)e *,1331.13 a N4C 39)H( 6) 4:62498089£.42 a WC 31"t ?1: . 324?94171.81 a It( 3?)of( 43a .6.129794369.6t a WC 36)Ht( 9)0 *,1663o9?loIt a MC 353)MC 1838 foa17aal51-4 a 4C 34)At &1)* 6,8310919-41 a 14C 337

WI HC tala 4,26763t4taftel a NC Ii)PIC 131s o.d09?10E41i a HC 3t)"C 14)0 oI.463t3g1?g.81 a N(C 301N4C IS)* I'istl0??1-st "a NC 1NC W 101 ,94749ME1.81a mc allNC 1710 *.161614691* a N a?)ot ci I&* .117aO3011.66 a of( 10)HC 1910 .6,399573399.11 a NC 2S)"t im .e.Ial4aqeaqa~v9 a NC 141hC at), *,W10 6181 a "t 13)NC 2ls *.3478IS1*14.6 a NC 2al

$AN Ioc SANG I06 S2?A405 3 RANG45LOWER SANG 106£ 0.3090060e I9000I6I *.112330P66 lo.29351?l018191 SAD DG 5,37936 26164996162 O'lalamoves 13970130.C61N6 VLU 4,84I6*I664 1,86897213 4.a1t666 t.66610416a

491 ATNG tuo:olos evloes I'aANGo 24M64324AN

SANG 9 SANG II SANG 7SN

uP'39~AN 9AN S06G g*a4311 *.sgsuggUPPER SNG EDGE 634021* losaosl06660131910 vAt.Uf -16,538659670 g,alglraeelg

*.8161036 3,'d164659 4,9369314 8.1539173 6.1007%14u,58hu 3 *,1092289 2,19277li 0.2283972 4.23469471.1488153s 0,1614413 4.27&64& IsISIOIIO 1.3mollg6,369933 1,335594 1.3772133 4.34q933% 1,4114917

1. 6449006 8,472146 3.909666

92

TABLE 14

FINITE IM9PULSE RESPONSE (FIX)

LINEAR PHASI OISITAL FILTER 09113

FILTER LENGTH 9 43

:"6'IIPULSE RESPONSE see**liC 13. 0.14361431198a I Nt 43)"tN 2)6 06695943497909a a Nc 42)"t 3)6 0.81*6*719.82 a lit 41)N(t 436 *9,4?51536E.0 a Ht 49)"i f 99 =66.19377349.82 a "t 39)it 030 0.487?61146 9 Nt 36)HCt 7)' w661)41w62 a N( 371

N 8 30 m4.r*36?4?fE3 a N( 36)NtIla 9). s,4g556U 1 a "it 39)"it Isla .0,08464743904a p "t 34)

- HC H)* a6ii7116 "tN 33)lit $a,: 26*30641-91 a "t 313"it 13) w.821!64Ew61 a HC 31)M( 14)' .6.161143180.1 a i 36)lit IS). *.5T3*E.19 t*)"t 1*)' 29661SuS t1)HC III* 9,*5132633twI*2 "t 1?)HC( 1430 *1W803991099 a HC W*lit 96 s6.36319111-l 0 Off 25)HC 2636 .a6.99136106 6 H( 14)Nt II)a *,23933077E.I1 a Ht 13)Mc Ba)e 33e9139bE.SS a Ht 22)

@ AND I SAND a -SB AND 4LOWER SANS DGEOO ggggg994989 2.1?4I3769 4,28896I90S 1.191711*95UPPER SAND EDGE 17g2893330 0,22T54456 5,1940391849 1,3453860DESIRED VALUE possesses1 1694849499909 1*S**I6660 I68,09616696aWEIMYINS 1,80I6966 6,11*16321 U,16163294 6,89346440

LOWER SANS D E S00 *34993agUPPER SAND forge 9.50is0geggDE3IRED VALUE uIV9I66@OfeWEIGHTING s,@@360*469

EXT"CNAL FREOUENCtEIG..Ouraess 8,M136*1 566 094136 4.2~6t99 4,87284330.111S7*1 6143416t *.1?2a331 6.268961 1.1119*69

8,44964$4 6,4752146 S.!SU0fl66

93

TABLE 15

. : :. . o .-M CO ,o z .C. #z CD ._ o a,.. N q. .,w ,Z f.

AC R - V V*

:e. N V,==========

cu V--4 0 .7. -'- 0--N

I aa "- . i *; . :. ; ! I I I I I i

~~LC M - -

17 .1

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I- f,' . .. ' , *. s

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, U . - -.- " . , -m

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94

TABLE 1

WtpL?gI PIPULI(45'E S P~LZ'!A -~As*0 ,)?JTAL FTL'FFR OCS?144

PILTEI .CMG?14 a S?

114013LSI VVe3PrNSE sa*ao

1)*a3 .0.4*03'1(16 *aItr4 4(54

2:6 73 .Aj33q3?91664A a It C I

MC 419'IdJaj-a . 6Mf gig~~3554Q a 04 c~

4c 13)' -.. 1739649641d a #q( gal- ,. 1)* 24,47?l'441sd a "( 91)

418'a.9aI,~G a 14c 446)hC1,318 '133bE4 a P1 45)

UPS gC tel. 0,477646t$1.41 0 FIC 47

4C 13)a O.M3"341*43 aC 4( 33tC 1)8 0.4776U596.at * A( 41

M( ibis .j6li~ a "( 413" c Iris . 1 mal3ote-ill a "'c 61)"t 19)z n,1~I?(O "t~ 393H'c 121 .0.a~tl?9ssI.40 a 4( 361A( ails -4.2aaaeigSI*44 * "c 3?)PC ( a1' .0,993826811.dI a 04 36)?It a])* .0.1124'371evia a "C 35)mc .24)0 ,L9.4ee a "t 34)-I( is' 49S'a6IE.4U a 331"( able P.19642?349.49 a lMt 32)mC I?)* J.,72394SI6e.O a 4( 31)fit Ila .Si1362 1 at *si a wMc 33

a" l M.7 6,e8I tE'e a MC 2q)

441.0 1 UANO a sA%0 3 &ANDo A

p4 01 06.3431t14" -. 113183341 *?.86aa313730 *&4.514a111

t..JW1 S* S LGE X2.aSM 24342991S

!S~~VALLE214900 Isltpe 'RO40

EVIATION VjOq-&. zaalat 440314 J42qbfflVaibp 0 5.374833t4 -413311-93jj3q-454M

t)1436 , )11'421 4.322064S Is135659? 2,26431372.2?14ea' 8.2790ta T42847ta3 *,90461 13334t@6

2,1 571 *37073,59 2*3?449g7 1,36S316a1,*3619q6~'.G1i~O.4bnleS 4,4739224 0,487776a a,94egese

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C. - -. . 77.7 7 - ... - -. C -

95

TABLE 17

F~kIT1 V'PULSI 13PONSE CPT*)p LINEA" PM&09 DIGITAL. v?Li,1' '3135VIpf"EZ 1zCN4104OE ALPt?

FILM1 LEMG?M a IT

ge*00 IMPULSE *1sD'4E sees*

"C lie *."2163!1-ea "cM ,77"t a)$ a..0.'E. * p* 56)

14 ' 318 0,49460331 0 M 53)pot 61' 10032099s11-Ja a "t 543M(71 53 .a**49a00331..g a "t 13

W( 10 -. 3343au 63iuE.a Nt 50IN(c ai. .e.100ullI7E!.ao MC 50)Nt Me. .e.383e5&a16.ea mt &6 emf Isla .j7163'oI&.1.a "t we 4Nt 141U a55ia8~ ~ *M ad)

%6 1 -4.13141m11v a "t as)078 ii. 4i0t11 a29eoEat a mc 457

M151' Xea!17921-.e a "( 4314C 2161 -:.381348799.42 a WC 141MC 209 - ,,t3'8ua4E-t a "t 37)MC 2191 a01**7l1 * f 36)let 8312 81548478e'99-41 a 0C 311Mt 1616 *..10b61Ie1e a Mtt 341MC 833' .141084t.5EIfe a Ht 33Mt 831a ,'g~'g. a M( 38)N( 241' .. 9t'9911.At a MC 31)

Ill 163 .6*1M71.di a Mf 383

IV) 891 *443129t7coof a Mt Eel

SANo 1 44140 3 *Ari 3 *&NO 4LOWER 4&NO Mot 1,0696"eggg 2aG~9879Gfe R.239468196 aee85UPPER IAND tcOS 9,16898ls 2.834399516 P,341340170 ~ ?Pu'mDESIRE0 VALUE a'satesuflas -1.327390644 -7.14606113 -93.i87648136

W~G0,tG .9mless0 ReStl3seepa X.at93a0060 a.3617402aq

salvo 5 IAMO 6 *&NeO 7 @ANDaULOWER 5a4" coal 1,87816 89UC9g?7386 wet1I2e593& 3*263756UPPER blvo furyE *,8,a.rea 8*112181356 P.iSqS96?2 avilbaees0E31RED YAi~uc *'.14al7la3 *telol?3967aa 2,3US400l 6AfQ?9

'EZMY6 .i74OV40g *.Il7aeeevu 1.174390l 0121?a8000@

&AND 9 SAND to SAND 11 @AMC 1aLOWER SAM0 EDGE 4,2171774366 2*26425996 0.2723'1316 a.31a6.&736UPPER same C06l 84259171490 fea0913?3 3.1158Y151 3.316.81~ba2951010 VAUE 11.1313,06461 6.4111a?77' I*23094geg -*b.IWasas64

lAND 13 BANC 14 SANd tS %&NO tbLOWER SANO 1061 0,31 0324446 9.37g4??5ao 3,37faI319e 2.3903?726UPPER lAND toot 6.3644444fg. 003T6715112 2.33aga9Ss 3.4046129;DESIRED vh4.uc -1*q9ig *.1794149e1 0,36020894f 0,394723a94

IANfl t7 BAMC Is SINO 19 SAND1.0169 *&pV EDGE 4113577166 30446668436 2,364obsa.UPPER SameNDG 061 6,'77'ilel 8#46a441742 a.Seeeveeee0EIZRUC VALUE 1aaI.Isa1 1.344723a8 2.160'40pofl0112NZma I'Sai3t9644 80.113196d96 0,213t96494

2.3ag8000 180118 30741114 2.1840%G6 8171077b

*.geII949d6 3,51o .484 .4883143 a.59116004

F-

96

TABLE 18

LOW* PHASE ONVAL MYER DESIGN

FILT14 LENGTH a 37

We 0(~ : eg:ibu3924iiui3 a e 7

"t 31098,e334791801083 0 We 53)He 4): of,3466S346 9903 o .it 91)He 3)3 4,1686111c.6 * me 53)He 6):0 *,191 ?481E002 a He 533pit 7). .@,35S?964E.Il *aM 13

MeOl 9) , 4.7095341106a a We 49)He Is)@ U.40431994948a 0 We 462

We III: ;96,163163013.aI v 4C 46)He 13)0 0.67911 offC 41Me Idle o@,1917'@6g.3 a We $4Me 15)0 e4,27?l 11 a We 43)He 163w .4.14966434994t v HC 4a)He Ing0 I'Salagiscoa a we all

. ma H sale .6,.v677a4g9ea * He 46)He 19)09,6.4a4697124.ea a Olt 39)HeM e 203 ,2i161433.I1 a Me 34)He We) -O.1946?33161.11 a Pit 37)"t fle 1,736,134151.11 a MC 36)He a3). 8.!1a94776(.61 a Ot 31)me 2419 -0.9903366.1 a He 34),4( Me) 3,281911443.8 9 We 33)MC 2630 9.1134656619040 a We 3a)me 27)* 4.5SI8894to81 a We 31)4C 147' oo&6*45366I.644 14C 31

LOWER BANC 1061 od o 6.3264746 2.377636 4e3963726

UPPER1 BACu EDE66.66aal 3113130338 8,1S4670.6 GoI6.1373666DI1IXIt M6c 1,319694164 14*3609916 893964615606 9*096

$AND I $&NO BAC7AN S

UPOER $ANO 1061 9,3154go 06311 gq9gss 06a1065hR0 VALUE 1.39294 seamloe 334196

r MIZGHING 1,4111849 836412 216bas elodo

EXTRI44L FREQUENCIES1,636.666 8,w66111 4,334o643 13,16166 031649916.111ab59 f,1111696 ::131741 ,a93 ,6966,1786763 6,191 4 a ,1743 a 2891397 81834491,271146! 2,26.%ago 4.2C476?q 1,3663637 3,341261.3113471 3.341794% a.3746? 6,3491916 2,464341

2,45931 144366 1,69G73 3,434699 316166

97

TABLE 19

oz - - M c .,.... eo i. O .!.. - j co .- . . P.- M

0 r i J ' 17 0A . ^. 0r V .1* . - 0 A .6;j V ,..t

21.1 1*11. 1i s

-,P

Z - I I I i t H -1 It of'I *' I I t I t H 1 of :1 11 9 1 I 9 1 It I t I 1 ;1

.u in of 1. .tw z ~ Q .. I 01e U ) I='. N '00 tZ A -M L.

42%*7 -Z% g.M -Z-Ma- 0 2- - Z -Z.:

B =2

-r .. . .

p -~kj - *..J-

S-Z .- Z

-5- - *S - c. z. -Z

. .. .- ..

98

TABLE 20

FI..'!' LVOV"t a ?I

w**&~ tI3..t.S "FSPO'iSE ***go4( tie .2.306730431.g1 a PIC 713

it pit a 4.!blh39*34mA1 a nt ?a)I C 33 0,1431769tdcoat a 4Ic 711"t 438 3e.1174t1*r.tI a iq( 70)

Ile 3. tqI9.4?E.01 a bs 0"C 719 St 47?41*011* a q( thy

'ICila .4ikidSmSa'C e.)

dimc Ills Kc1~a7l.' a 3)taC 01'*199??447E*4g a Wt 641)

'C133 0.931S1I41ofd~ a pf C 612PIC 1410 .. b71aLj1C.3 a "C 64)IfC 1113 -4. 94a7,51afoas a 4c 5q2

9C M7$ bl.i1(G Ph C( 71-~ AC l1a 4*qe7gA a 4C 56)

nC III@ -A. 15'64400*41 a pI( C 5OfC 23w -e.ajaa6t36F*03a P IC 54)At ails A.172199*0.e a M( 5314C Vale *eIJe32§31I*a *( 52 lOf C 132. a~it~~gl 4( 'IC51A C l21 AN.191a641390at a PIC Solbt1' s) 413',92l3todt a "IC 4q)

"C dl] a *1376AA971#.dg a P4 ( 47)"Cf 261a *I.aV4139779.GI a -1( 463C 2qla -4.3ap3aaet.*4 a~ WC a1"C 341a 2.a92446431.I1 a "t 64)(I 3093a .153464495g* 1 a o ( 432

'I 11 Me q?15~d a Fit 421

"t 3320. 1? l7,17 54! *a1 a Pot 412'C3413 4. 119263511E0;1 a 4IC 631"C331a 1.3aRS9967C*41a 341

31j 3~a .,Oqql.g*'C34)K 4( 371a -JO"313464og.1 P# 'C 373

SA~ft 1 Sj~fl 6A4 3*. SAP40

*43 ~ *df ~'; '.deqaaI12 J.1'55d8313 1,i36l'1I50 1043737'Iaaq

.1;;-8t533 20111474211 1.204026I68 J6111773740

vZ?1',1' ~d 34.34J334416 *35.6719eedee .94.*9734414 v3!,meZ139957

SAFhC' 5 I&NQ

4ppem 94000 5ij 2.S~w'Io'gpdd

jeVIATIO e3340641 1639V?^?1O4 14 go a 4.499?

I,38555.3 2,3426#b1 p.32484 2.034814 3'aJtl03?.016334 0.173Ll3' 3.AO437375 3.348, 3.a13;44

a,.4133734

1.4i66878 J.!0400S30

77... . .. . .. . . .7. . .. ..71.

99

TABLE 21

FILT9@ LSMOTH a 73...o IMPULSE ,4eSPONS1 moos*'C1) .4 1@,g4tGI*I4E1t a "f 73).ca): *g4:l9e ?..1 a b4( 71)

MC 330 U,?a9332439mea a -of 711NC4)' -.. 331a19311.ai a iwc 76)

$( )a 8,133076791.01 a '4( 69361C -4 611116639o$4 a HC 68)

mc 71' .4 AR16&643EwdJ a MC61

m( %Is ag3~i3Ci l it 615)

4IC IIl' .g.9613343411.6l a me 6d)14 C 112' : *9*96431g11*1 a mt §3)4CM e i 6:j4638663EqOl a if( 68)me 13)2 *85811I a Of f 1)M( 1436 9.131i541[oft a WC 63

N( f)a .8135( a. 93)MC168'*2~5l( a 36 5)

NC tyle 0.2165690331.11 a 04f 57)"t Isla .O,~187873.waa a "tC 58)ht 197' *10810131*St at we 9)Mt 193a *I.31173461(.al aC N( 4)Nt ails I,13?337361.*1 1a 9i 3)Hc 22). *.16991aa9104l A ( la)4C ails S35118(.1al 0 " 51)NC 14)9 *of2aGS4799*41, a Mt 3g)NCa' *.t755Ea a off £9)

MC 27)' -6,19313691*1 a wt 47)tit Isla 9.34673871.61 a we 463Nfc av)a .@,315519(1at 0 M 45)N4C JIa 6124762490.1 a NC 443MC 31)s v.*931783ifaa1 a Nt 43)"( 32as 0,16447416109 a *( 4a)"t 3310 a..111S591*4e a Mf41MC 34). w.93931979aOg 4 Nt 44)',t 39)8 -. 4,6i6g1j(. a "t 39)M( 386 *6*83390a561.11 a "t 38)

**t40 I SAND a SAND 7 AN

ujpptR wo EM 10"861" g135fooffoo 0,12944,996 8.131q94380OlSIRrD VaAU @suee@* *glaSle3gg 2I921329 .9837314

lANa I @AND0 g SAND ii SAND a~

40W1 9*140 101 ,M639716 291?1399as 2,3161915 3,SIIOISIISUPPRSNO YA0UE 2,195614S 1,269S34190 o8,I**6101 3t238m5I9"4DESI Gt6 3,3@0992000 .4****6t wa*96611561 -3,717b26168

SAND 13 SANDG to ANO It SAND tzLOWmef 3*14 to0 239444399' 1.22853545q *3 52145788 3.4sq3I114

0951420 VALUE Sl,35866IMS 01,21s9586 611 0,890608.143492t9'u(GMTZM6 uaimi7737 ,17776 3111773906 311223016"a

SAND 13 SAND L4 IAN* IS SAND tLOW $APO EDGE 8:329664t7S 143011999 V341766i 15519t

UPPzEoa*gG aaL M7.~1191 .4333741 1,4217sas ,422O6

SENK7ING lzt.26 71 141177 3"a 2,111773606 aeatto1nale

LOWE SAD E a g3s3a46t178 3,45 3e~1 3.6581?tteQPU4 AN* t10 :11850642 3*3 31 812,1 A213990

g.,:0INGV4 **1558431 ae ~,43g9 2.lsoci 3,34094 8

,3%094000 P.11031 924381 56 341 314451%

8132774%1 3.3395454 P,3121ta 4.366a.0d9 1:,341's813134460 v,3vbqAj3 P.423413321A.322703 8.43137!0

100

TABLE 22

WiNItE IMPULS9 W93PONSE (7!6'L14N(*v p"Ast OtGITAL FT!p 0151014t'I ~E'ig 4~CMA1 64 LGORIT

8AMCPA38 F?6?EAX

**u ZVI4S 111901"Se ....."t 11' 3,164877499-a1 8 At 73)of I)m 1,433649:1394a a At 1)At $14 496Al6tm6l a At 71)Of 4)8.2,I3497'96.d3 0 Pit 76)pit Ile 064496869463 a At 89)At( Q0' 47363811004 a lot $41N( 7)v 4,1834841162 a lit 67)lit &a' .4,3311599494a3 a At 61)At 9)s 06,481flas964u4a M 8 A 1)

AtC tale .a,69.a8g.4a a of 3At Ills 06,714a534614 a "t 81)

Of 143m 6.124a&9961-al s At 62MC Isle *I.,337434841 a wt 59)

4( 171s m4,3366435'l-4 a At 97)4( 17)' v,2,b387669Zt.01 a NC 56)

h'c 19)3 06O6eaallt.1 a NC 55)H( 1209 ofI816739eila v Mt 94)pit ai)s fl17646471-St a O 13)At ii' 9,444900,J79663 9 lot 513OC Me2 4,31161723E4t 1*( 911s"t W111 064037sia14.61 a AC 3104C Isle .145851613104a q 14f 40)NC as)@ .48*4976649.61 0 At 46)M( a712 06.1513 1154.61S a Of 4?)A(M e 492 .3*110t340411 a A( 4)

P 1 4 2a36913-A t 4

HC M 34. -77274711leaI a NC 40)"t 35). *81.7309t711 a K( 39)N( 38). a47~6S1d wt" 36)

Me31 ,a4154.3*N 3?)

6*140 a 1410 51 3 SANa aLzdtm 6*10 Z01 *,aseese Gea&97658 9,1123aI9a7 3,86332903UPIR 6*140 1069 6.1Gs6188:0 9,131904380 6,18S346318 3.2719398ola1MoCD U vau ,364680002 gjeueoe 1,344666666 ioe auesd(:SNT1NG 6,31991366 3,296440666 8,111374218 3,11asevag6

S404O 5 BANC 8 50140 7 3*140 4L S'ER E*4 009 0,1794273 2,21649 6,354176615 a34110815*jP*! 310 1 004 gaa9tils 4,333732922 8,43737561 3.45731886a

f331ME 'eai.Ut 2,34uu6e2e 111,32686440 1,34eeass 1846326 0964N71G1,33668664 2091166666 861117737114 2,31260

62e 9N [a 1417114VLPLP6 3140 1064 8,30261'i1!uP2 3 1 40 1064Q 8 .200g 48440

01116TING6,3546186948

!X4"L FOOUCNCI3

6 , 6 6366 1 1 6 25 8 . V 1 6 1 8 3 . 31 1 2 t § 2 . 1 1 8 5 6 7 6

0,4a~6 31539139 2.18311 11,186861 111986461,81661 3.2243597 8,2387176 2,214751t 0,272t6a!8,73t vla17142' J.3042381 1.313411 3.34438

9;:6.1 331122 6.17367 3.3434256 .13V'si9~ a1737453 1,A4J54a 4,45732%6 4.31617112,4649a70 a,4641334 0,56306

101

TABLE 23

I* NrTr C 01 c:%, C~ c .C- *: . 0 IV U

C a. %1 05 -oQ . 0 .0 A A

ft ?I F. %. L ~ 4 W W?% ' 11,N4VV* 0 1 - 1

1 i ! 3 II t I t * i t II * Ii *, iI I isi ' I ; I a . T1-. ~j- 4 -u---~J u-------~--------(U*N-

W i li IA. W -; t .- ; J L

Il I a' t *il N i II I

- -7II1 H II IB N U 11T1 ' Ni "- It-N, "tyi 0 ) I "X Cit it I . j -7 .2.

1JO & A C .-) ,W. . . . .- - - .... . ..U M - .- r.- - 7-

- -. - - -c~ -

Q ***M ~ .oe.t- I cz -. ..

I~~~~~C Ca ." f ~ 'W . u 0,

Q1 eD Ct C. C .II z

= -t P- *a K .

- . u... C.

-Z C. -M C

M - c .c* c* O. Il .I

.26

p -

ak t.,

COPY* avial toDI oeU:

pemi ful leil epouto

102

BIBLIOGRAPHY

1. 0. Herrmann, "On the Design of Nonrecursive Digital Filters with

Linear Phase," Elec. Lett., Vol. 6, No. 11, 1970, pp. 328-329.

2. E. Hofstetter, A. V. Oppenheim and J. Siegel, "A New Technique for

the Design of Nonrecursive Digital Filters," Proc. Fifth Annual

Princeton Conf. Inform. Sci. Systems, 1971, pp. 64-72.

3. T. W. Parks and J. H. McClellan, "Chebychev Approximation for

Nonrecursive Digital Filters with Linear Phase," IEEE Trans. on

Circuit Theory, Vol. CT-19, No. 2, pp. 189-194, March 1972.

4. J. M. McClellan, T. M. Parks and L. R. Rabiner, "A Computer

*Program for Designing Optimum FIR Linear Phase Digital Filters,"

IEEE Trans. Audio Electroacoust., Vol. AU-21, pp. 506-526, Dec.

1973.

5. E. W. Cheney, Introduction to Approximation Theory, New York:

McGraw-Hill, 1966.

6. E. Y. Remez, General Computational Methods of Chebvshev Approxi-

mation. The Problems with Linear Real Parameters, United States

Atomic Energy Commission, AEC-t2-4491, Book 1, pp. 76-101.

7. L. R. Rabiner, J. F. Kaiser and R. W. Shaffer, "Some Considerations

in the Design of Multiband Finite-Impulse-Response Digital Filters,"

IEEE Trans. Acoustics, Speech and Signal Processing, Vol. ASSP-22,

No. 6, p. 262-472, Dec. 1974.

-Cr: . . - " j .. -. : . -: : : v i

103

8. M. T. McCallig and B. J. Leon, Design of Nonrecursive Digital

Filters to meet Maximum and Minimum Frequency Response Constraints,

School of Electrical Engineering, Purdue University, Report

TR-EE-75-43, Dec. 1975.

g

U

I