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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
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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
V* *v1
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4. TITLE (and Subtilel) S. TYPE or REPORT & PERIOD COVERED
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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IS. SUPPLEMENTARY NOTES
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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ar IM
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I 'o 0 0 c. 0,0 . 0 .L ~ O
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•- @ 0 ', -. ,:
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I- f,' . .. ' , *. s
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-. - - - - . - : -- - - - -
-. - - - 'I_ .
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Copy available to DTIC does not) Permit fully legible reproductioni
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$1.41 0 FIC 47
4C 13)a O.M3"341*43 aC 4( 33tC 1)8 0.4776U596.at * A( 41
M( ibis .j6li~ a "( 413" c Iris . 1 mal3ote-ill a "'c 61)"t 19)z n,1~I?(O "t~ 393H'c 121 .0.a~tl?9ssI.40 a 4( 361A( ails -4.2aaaeigSI*44 * "c 3?)PC ( a1' .0,993826811.dI a 04 36)?It a])* .0.1124'371evia a "C 35)mc .24)0 ,L9.4ee a "t 34)-I( is' 49S'a6IE.4U a 331"( able P.19642?349.49 a lMt 32)mC I?)* J.,72394SI6e.O a 4( 31)fit Ila .Si1362 1 at *si a wMc 33
a" l M.7 6,e8I tE'e a MC 2q)
441.0 1 UANO a sA%0 3 &ANDo A
p4 01 06.3431t14" -. 113183341 *?.86aa313730 *&4.514a111
t..JW1 S* S LGE X2.aSM 24342991S
!S~~VALLE214900 Isltpe 'RO40
EVIATION VjOq-&. zaalat 440314 J42qbfflVaibp 0 5.374833t4 -413311-93jj3q-454M
t)1436 , )11'421 4.322064S Is135659? 2,26431372.2?14ea' 8.2790ta T42847ta3 *,90461 13334t@6
2,1 571 *37073,59 2*3?449g7 1,36S316a1,*3619q6~'.G1i~O.4bnleS 4,4739224 0,487776a a,94egese
Copy civailabla to DTI C do-> notpermit fully legiblo r' pr~lUc4jt
C. - -. . 77.7 7 - ... - -. C -
95
TABLE 17
F~kIT1 V'PULSI 13PONSE CPT*)p LINEA" PM&09 DIGITAL. v?Li,1' '3135VIpf"EZ 1zCN4104OE ALPt?
FILM1 LEMG?M a IT
ge*00 IMPULSE *1sD'4E sees*
"C lie *."2163!1-ea "cM ,77"t a)$ 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 $ANO 1061 9,3154go 06311 gq9gss 06a1065hR0 VALUE 1.39294 seamloe 334196
r MIZGHING 1,4111849 836412 216bas elodo
EXTRI44L FREQUENCIES1,636.666 8,w66111 4,334o643 13,16166 031649916.111ab59 f,1111696 ::131741 ,a93 ,6966,1786763 6,191 4 a ,1743 a 2891397 81834491,271146! 2,26.%ago 4.2C476?q 1,3663637 3,341261.3113471 3.341794% a.3746? 6,3491916 2,464341
2,45931 144366 1,69G73 3,434699 316166
97
TABLE 19
oz - - M c .,.... eo i. O .!.. - j co .- . . P.- M
0 r i J ' 17 0A . ^. 0r V .1* . - 0 A .6;j V ,..t
21.1 1*11. 1i s
-,P
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B =2
-r .. . .
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98
TABLE 20
FI..'!' LVOV"t a ?I
w**&~ tI3..t.S "FSPO'iSE ***go4( tie .2.306730431.g1 a PIC 713
it pit a 4.!blh39*34mA1 a nt ?a)I C 33 0,1431769tdcoat a 4Ic 711"t 438 3e.1174t1*r.tI a iq( 70)
Ile 3. tqI9.4?E.01 a bs 0"C 719 St 47?41*011* a q( thy
'ICila .4ikidSmSa'C e.)
dimc Ills Kc1~a7l.' a 3)taC 01'*199??447E*4g a Wt 641)
'C133 0.931S1I41ofd~ a pf C 612PIC 1410 .. b71aLj1C.3 a "C 64)IfC 1113 -4. 94a7,51afoas a 4c 5q2
9C M7$ 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
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SA~ft 1 Sj~fl 6A4 3*. SAP40
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.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.
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.26
p -
ak t.,
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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
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