The discrete prolate spheriodall (DPS) filter is one of the glass of nonrecursive finite impulse response (FIR) filters. The DPS filter is superior to other filters in this class in that it has maximum energy concentration in the frequency passband and minimum ringing in the time domain. A mathematical development of the DPS filter properties is given, along with information required to construct the filter. The properties of this filter were compared with those of the more commonly used filters of the same class. Use of the DPS filter allows for particularly meaningful statements of data time/frequency resolution cell values. The filter forms an especially useful tool for digital signal processing

Breakall, J. K.

Karawas, G. K.

Mathews, J. D.

English

NASA Technical Reports Server

563 
8.8A THE DISCRETE PROLATE SPHERIODAL FILTER AS A DIGITAL 
SIGNAL PROCESSING TOOL 
J. D. Mathews*, J. K. Breakall**, and G. K. Karawas* 
*Department of E lec t r i ca l  Engineering and Applied Physics, 
Case I n s t i t u t e  of Technology , Case Western Reserve University, 
Cleveland, OH 44106 
*%awrence Livennore National Laboratories, P.O.B. 808, Livermore, CA 94550 
ABSTRACT 
The d i sc re t e  prolate  spheriodal (DPS) f i l t e r  i s  one of the class of non- 
recursive f i n i t e  impulse response (FIR) f i l t e r s .  The DPS f i l t e r ,  f i r s t  
introduced by TUFTS and FRANCIS (19701, is superior t o  other f i l t e r s  i n  t h i s  
c l a s s  i n  that  i t  has maximum energy concentration i n  the frequency passband and 
minimum ringing i n  the t i m e  domain. 
DPS f i l t e r  properties,  provide information required t o  construct the f i l t e r ,  and 
compare the propert ies  of t h i s  f i l t e r  with those of the more commonly used 
f i l t e r s  of the same class.  We note that  use of the DPS f i l t e r  allows fo r  
pa r t i cu la r ly  useful statements of data time/frequency resolut ion "cell" values 
and t h a t  overal l  it forms an especial ly  useful t oo l  for  d i g i t a l  s ignal  
processing. 
INTRODUCTION 
We give a mathematical development of the 
The reduction of d ig i t i zed  data to  f i n a l  form usually involves what i s  
broadly described as d i g i t a l  f i l t e r i n g .  
the domain i n  which sampling occurs (e.g., time or space) o r  i n  the 
corresponding transform domain (e.g. , frequency or s p a t i a l  frequency). 
r e s u l t  of d i g i t a l  f i l t e r i n g  i s  always t o  "smooth" o r  r e s t r i c t  the frequency 
content of the data.  However, it i s  o f t en  very important t h a t  t h i s  smoothing o r  
f i l t e r i n g  process be done with minimum and well-understood d i s to r t ion  t o  the 
r e s u l t s  i n  both domains. An example of an often misused f i l t e r  i s  the equal 
weight ("square") running average which "smooths" i n  the domain of appl icat ion 
but r e su l t s  i n  a ( s i n  x/x>' l i k e  passband shape i n  the transform domain. 
( s i n  x/x)' behavior may have unacceptable sidelobe l eve l s  and/or spacing. 
This process may be applied i n  e i the r  
The net 
The 
The d i f f i c u l t y  of minimizing d i s t o r t i o n  due to  f i l t e r i n g  starts with a 
statement of the c r i t e r i o n  fo r  judging when a f i l t e r  i s  most sat isfactory.  
w i l l  follow the approach of TUFTS and FRANCIS (19701, PAPOULIS and BERTRAN 
(1972), and SLEPIAN (1978) i n  which an optimum f i n i t e  (length) impulse response 
(FIR) non-recursive f i l t e r  i s  derived on the bas i s  of maximum energy concen- 
t r a t i o n  i n  the passband of the f i l t e r  r e l a t i v e  t o  energy i n  the t o t a l  bandpass 
up t o  the Nyquist frequency LO~(=ZIT/ZT with T the sampling in t e rva l ) .  
t h i s  energy concentration c r i t e r i o n  leads t o  f i l t e r s  described i n  terms of 
d i s c r e t e  prolate  spheriodal sequences and wave functions. These sequences and 
wave functions have many desirable  properties which were f i r s t  investigated i n  
the continuous case by SLEPIAN and POLLAK (19611, LANDAU and POLLAK (1961, 
19621, and SLEPIAN (1964). 
We 
Choice of 
This paper i s  basical ly  t u t o r i a l  i n  t h a t  our major goals a r e  to  demonstrate 
the usefulness of FIR f i l t e r s  based on maximum passband energy concentration and 
t o  supply the information necessary for the "construction" of these f i l t e r s .  We 
do however introduce several  apparently new r e s u l t s  which prove useful i n  
numerically solving the  matrix equations which describe the f i l t e r .  
following sect ion w e  develop the general equations for.  symmetric FIR d i g i t a l  
f i l t e r s  and then obtain solut ions based on the energy concentration c r i t e r ion .  
Next we demonstrate t ha t  these f i l t e r s  have f ea tu res  which are desirable  
I n  the 
https://ntrs.nasa.gov/search.jsp?R=19840019106 2020-03-20T21:58:04+00:00Z
564 
especially when compared with more t r ad i t i ona l  f i l t e r s ,  and f i n a l l y ,  we present 
a summary and conclusions. 
MATBEMATIGAL DERIVATIONS 
(a) A General Non-Recursive FIR F i l t e r  
Using the t i m e  and frequency domains, a tapped delay l i n e  implementation of 
a non-recursive FIR f i l t e r  act ing on analog s ignal  x ( t )  i s  shown schematically 
i n  Figure 1. The r e su l t an t  output s ignal  has the form 
K 
n=-K 
y ( t )  = 1 anx(t-nT) 
where the coeff ic ients  an are  r e a l  and where the t i m e  reference is, fo r  l a t e r  
convenience, a t  the center  of the delay l ine.  Assuming t h a t  the Fourier trans- 
form of x( t )  ++X(w)  exists w e  Fourier transform (1) and f ind  
i s  the "voltage" t r ans fe r  function and j = d-1. 
t r ans fe r  function i s  
The corresponding 
K -j (n-m)wT s ( w )  = H(U) ~ * ( w )  = 1 aname 
n ,m=-K 
K 
n=-K m=-K 
( 4 )  
Figure 1. Block diagram of the non-recursive FIR f i l t e r  
described by Equation 1. 
565 
while the inverse  Fourier transform of (3)  i s  
K 
n=-K 
h ( t )  = 1 a,b(t-nr) (5) 
the  f i l t e r  impulse response with 6 ( t ' )  the  un i t  impulse. 
volved with x ( t )  y ie lds  equation (1). 
Note t h a t  h ( t )  con- 
The f i l t e r  described by (3) o r  (5) becomes a d i g i t a l  f i l t e r  i f  we uniformly 
and instantaneously sample y ( t )  a t  i n t e r v a l s  of T t i m e .  Then (1) becomes 
where the index j r e f e r s  t o  consecutive members of the  set of sampled signals.  
I f  we r e s t r i c t  x ( t )  t o  be bandlimited t o  the  Nyquist frequency (2a/2r) or  less, 
then the  sampl ingi f i l t e r ing  process occurs without a l i a s ing .  
(b) Choice of Coeff ic ien ts  
Equations (3) o r  (5) dtdcr ibe  the e f f e c t s  of the f i l t e r  on an input s igna l  
given a pa r t i cu la r  set of coe f f i c i en t s  {a }. These coe f f i c i en t s  are often 
chosen such tha t  the d i g i t a l  f i l t e r  chara? te r i s t ics  a r e  s imi l a r  t o  one of 
var ious  common analog f i l t e r s  (e.g., the  "ideal" f i l t e r ) .  The process of 
eynthesizing these f i l t e r  cha rac t e r i s t i c s  of ten involves smoothing (windowing) 
of the r e su l t an t  coe f f i c i en t  sequence to  suppress r ing ing  (BLACKMAN and TUKEY, 
1958; HAMMING, 1977). 
As mentioned i n  the  introduction, we propose t o  choose the sequence (a } 
by maximizing t h e  energy concentration of S(w) (equation ( 4 ) )  i n  some in t e rva l  
[ - w ~ , w ~ ]  on I-w,,~,] where wC i s  the  f i l t e r  "cutoff" frequency. Thus l e t  
n 
w w 
a 5 1 S(w) dwll S(w) dw (7) 
-w -w 
C n 
and f i n d  {a,} such tha t  a i s  maximized. 
(7) and performing the  in t eg ra t ions  y i e ld  
Subs t i t u t ing  equation ( 4 )  fo r  S(w) i n  
s i n  (n-m)m K 
n ,m=-K 1 (n-m)r ( 8 )  a =  
K 2  
1 an n=-K 
where E = wc/wn. I f  we define {a,) as elements of vector then ( 8 )  becomes 
aT - * !K * ~ - a & a = O  
where each 3 has ( 2 K  + 1) elements and the  elements of matrix& are 
e = sin (n-m)m 
mn (n-m)T 
(9) 
(10) 
We are thus searching f o r  such t h a t  a i s  maximum. From Lagrange 
566 
mul t ip l i e r  theory, it i s  clear tha t  (9) i s  s a t i s f i e d  by the set of (2K + 1) 
orthonormal eigenvectors &(I) and the corresponding eigenvalues X i  of the 
system 
(I  - ai:) = o (11) 
w h e r e  1 i s  the iden t i ty  matrix. 
eigenviktor associated with X 
"optimum" f i l t e r .  
K and E (i.e., wc) values i n  question. 
d i sc re t e  prolate  spheriodal sequences and wave functions,  respectively.  He 
demonstrates t h a t  i f  we order the eigenvalues according t o  magnitude, 
We choose the elenents of the 
the largest eigenvalue, as coeff ic ients  of our 
T h a t  is, g?;) maximizes equation (8) with CL = A, for the 
SLEPIAN (1978) ref ers t o  these eigenvectors and corresponding B(w)'s as 
1 > 10 > > e. .  > A2K > 0 
and i f  a,(i) i s  the n ' t h  element of the eigenvector corresponding to Xi then 
Let t ing b (j) be the n ' t h  element of the eigenvector corresponding t o  eigen- 
value X. '  "calculated fo r  w: = wN - wc then (SLEPIAN, 1978; sect ion 2.2) J 
Slepien a l so  notes t h a t  the d i sc re t e  prolate  spheroidal wave functions U(w) 
s a t i s f y  a Sturm-Liouville equation of the form 
(13) d dU - [coswr - c o s r ~ ]  - + r2  [K(K+l)cosw-r - 91 U = 0 d w  dw 
where 9 i s  the eigenvalue (different  from A )  corresponding t o  a pa r t i cu la r  
wave function and a l l  other parameters are as before. 
from (3)  i n t o  (12) we f ind  
Subst i tut ing H ( w )  (=U) 
or  equivalently 
1 2 1 - (K+n) (K-n+l)an-l + (n ~ 0 8 7 ~ ~  - ei)a, + #K-n) (K+n+l)a 2 n+l = 0 
56 7 
The matrix e' i s  t r idiagonal  and thus equation (14) is  much easier t o  
solve numerically than equation (11). 
eigenvectors as (11) howwer the corresponding eigenvalues are d i f f e ren t  
although w e  s t i l l  seek the eigenvector corresponding t o  the  l a rges t  eigenvalue 
eo. 
essen t i a l ly  halvg the s i z e  of e' by invoking synunetry (see equation (12)) 
which yields  
Equation (14) yields  the same normalized 
Since w e  are in t e re s t ed  only i n  which is  symmetric w e  can 
- 
-ei a. + K(K+l)al = 0 
plus equation (16) with 1 S n 5 K. 
values corresponding t o  "even" eigenvectors and the  r e su l t an t  a's may be 
normalized such t h a t  aT*a = 1. 
The compact form of (14) yields  only eigen- 
From equation (14) we f i nd  that fo r  w = 0, eo  = K ( K + l ) ,  and an(lnl  < K) = 
(i.e., E = 1 )  w e  f i nd  fusing the  wc = 0 case and (12)) t h a t  1 while fo r  wc - 
eo = K 2  and K - (n-l)] 
E 
1 ( -n  (-K 
2 We observe experimentally t h a t  K(K+1) 2 Bo 2 K f o r  a l l  0 5 wc 5 wN and t h a t  
... > a io '  2 o (0) (0) (0) > a. > al > a2 
where a.(o)(wc) r e f e r s  t o  the i ' t h  component of the eigenvector corresponding 
t o  the i a rges t  eigenvalue fo r  0 2 w, < . 
of 8 0  may be used i n  f inding eo  from r13. 
COMPARISON OF FILTER PROPERTIES 
The range constraints  on the value 
In the p rwious  sect ion w e  showed t h a t  FIR f i l t e r s  with d i sc re t e  prolate  
spheriodal sequences a s  coeff ic ients  (see equation (1)) have maximum energy 
concentration i n  the passband. These f i l t e r s  are known as d i sc re t e  prolate  
epheriodal (DPS) f i l t e r s  and i n  t h i s  sect ion w e  compare t h e i r  propert ies  with 
those of more t r a d i t i o n a l  FIR f i l t e r s .  
These " t radi t ional"  f i l t e r s  are a l l  based on various windowing (weighting) 
functions applied t o  the coeff ic ients  of the i n f i n i t e  Fourier series representa- 
t i o n  of t h e  idea l  l a r p a a s  f i l t e r .  The coe f f i c i en t s  of t h i s  series are 
- s i n  n m  
n = 0,+1,+2, ... n nr 
and the windowing function W(n) i s  such t h a t  W(n) - 0 fo r  1. 2 K thus 
truncating the series so t h a t  the r e su l t an t  t r ans fe r  function becomes 
(17) 
568 
The W(n) (-K i n  LK) are chosen t o  minimize i n  some sense the ringing (Gibb's 
phenomena) which r e s u l t s  from truncat ion of the series representat ion of the 
idea l  lowpass f i l t e r  (HAMMING, 1977; Chap. 5 ,  o r  RABINER and GOLD, 1975; Chap. 
3). 
(18)) of same length using rectangular,  Kaiser, Von Hann, Dolph-Chebyshev, and 
Hamming windows. 
referenced i n  Table 1 and are discussed i n  more d e t a i l  by HANMING (1977) and 
RABINER et  al .  (1979). 
I n  t h i s  discussion, w e  compare the DPS f i l t e r  with the Hw(W) (equation 
The W(n) corresponding t o  these windows are given or 
Figures 2 and 3 compare the frequency response ( i n  dB) p lo t t ed  versus 
normalized frequency f (Nyquist frequency f N  = 0.5) and impulse response 
envelope plot ted versus t i m e  i n  samples f o r  a l l  s i x  f i l t e r s .  
son w e  choose K - 15 ( t o t a l  length 31) and adjust  f i l t e r  parameters such t h a t  
the -3 dB l eve l  of each f i l t e r  occurs near f = 0.05. 
order o f ,  i n  our opinion, acceptabi l i ty  with the rectangular windowed f i l t e r  
least acceptable and DPS f i l t e r  most acceptable. 
For t h i s  compari- 
The f i l t e r s  appear i n  
The frequency domain d i s t o r t i o n  of a s ignal  spectrum i s  j u s t  the f i l t e r  
frequency response as  shown i n  Figures 2 and 3. However, the t i m e  domain 
d i s t o r t i o n s  due t o  the impulse responses are not so obvious. 
show the e f f ec t s  of these f i l t e r s  i n  the t i m e  domain on a square wave plus 
Gaussian d i s t r ibu ted  noise. 
"ring". 
frequency domain behavior. 
I n  Figure 4,  we 
As expected, a l l  f i l t e r s  except the DPS f i l t e r  
The DPS f i l t e r  "smooths" the input s ignal  while re ta ining a "smooth" 
To a i d  i n  understanding the "evolution" of the DPS f i l t e r ,  we show i n  
Figure 5 the DPS f i l t e r  frequency response cha rac t e r i s t i c s  fo r  normalized cut- 
off  frequency f, = 0.2 as K i s  increased from 1 (3 coe f f i c i en t s )  t o  8 (17 
coeff ic ients) .  
(f i 0.2) i s  the maximum possible fo r  the number of coeff ic ients  used. 
SUMMARY AND CONCLUSION 
R e c a l l  t h a t  i n  each case energy concentration i n  the passband 
We have investigated the propert ies  of the d i sc re t e  prolate  spheroidal 
f i l t e r  r e l a t i v e  t o  other common FIR f i l t e r s .  
f i r s t  der ive the general propert ies  of FIR f i l t e r s  and then special ize  these 
To accomplish t h i s  comparison w e  
Table 1. Wei ht ing coeff ic ients  of various windows 
(W(Yn1 > K) = 0 for  a l l  windows 
RECTANGULAR WINDOW: W(n) = 1 
KAISER WINDOW: 
I 
Io(adl - (n/K)L) 
Io ( a )  W(n) = 
m 
and we use = 3.395 (40 dB minimum stopband at tenuat ion)  
1 + cos m / K  
VON HANN WINDOW: W(n> = -2- 
DOLPH-CHEBYSHEV WINDOW: See HELMS (1968) and RABINER st a l .  (1979) 
HAMMING WINDOW: W(n) = 0.54 + 0.46 cos (2an/(K-l)) 
569 
properties t o  the case of maximum energy concentration i n  the f i l t e r  passband. 
The condition of maximum energy concentration gives rise t o  an eigenvalue 
problem which has the d i sc re t e  prolate  spheroidal sequences (SLEPIAN, 1978) ' a s  
eigenvector solutions. 
i s  used a s  the coeff ic ient  sequence i n  the DPS filter. 
The eigenvector corresponding t o  the largest  eigenvalue 
SLEPIAW (1978) a l so  points out t ha t  the prolate  spheroidal wave functions 
sa t i s fy  a form of the Sturm-Liouville equation. 
eigenvalue/eigenvector problen which i s  much simpler t o  solve numerically than 
the f i r s t  problem. 
corresponding t o  the l a rges t  eigenvalue (9 
This f a c t  leads t o  a second 
We note that  the elements of the eigenvector (a(O)) 
0 0 or  X ) a re  symmetric around the center value (a,) and t h a t  
n-1 
a ('1 lan(')> o (1 5 n 5 9). 
n 
0 
-n 
IRI 
.n 
-100 
0 
-25 
1001 
-50 
-7s 
I '  
- 1 0 0 , .  , , , , . . , , , , ' , , , , , , , , , , ~, 
0.0 0.1 0.2 0.3 0.4 0.6 
Figure 2. Filter frequency response (dB) plotted versus normalized 
frequency (Nyquist frequency fN+).5) and corresponding impulse 
response envelope plotted versus time in samples for rectangular, 
Kaiser and von Hann windows of length 31 (see Equation 18 and Table 
1). In all cases the normalized 3 dB frequency is about 0.05. 
570 
When used, DPS f i l t e r s  have several  features  which w e  f e e l  make them 
superior t o  more standard FIR f i l t e r s .  Specif ical ly ,  t he  DPS f i l t e r  i s  "smooth" 
in both domains and thus d i s t o r t i o n  of the s ignal  i s  minimal and eas i ly  
described i n  both dornains. Thus f o r  example, a f t e r  removing high frequency 
noise, a time domain f ea tu re  may be more easi ly  "restored" by deconvolving the 
DPS impulse response than by deconvolving the other s i n  x f x  l i k e  impulse 
responses. Another useful  f ea tu re  of DPS f i l t e r s  i s  t h a t  descr ipt ions of time 
and frequency resolut ions of published data  can be more readi ly  appreciated and 
used by the reader. I n  f ac t ,  w e  suggest t h a t  the DPS f i l t e r  might form a bas i s  
fo r  standardization of information concerning the "resolution ce l l s "  of data. 
0:o F:l o:r 0:s 0:r 0:s 
Figure 3. Similar to Figure 2, but for Dolph-Chebysev and Hamming 
windows and the prolate spheroidal filter. The normalized cutoff 
frequency of the prolate spheroidal filter is 0.2. The filters 
in Figures 2 and 3 are ordered according to overall quality with, 
in our opinion, the rectangular window filter worst and prolate 
spheroidal filter best. 
571 
Figure 4 .  We compare the time domain effects of the various filters on a 
Note that all filters *ex- square wave plus Gaussian distributed noise. 
cept the prolate spheroidal filter "ring" and that the rectangular window 
filter rings most with the Kaiser window next while the Dolph-Chebysev, 
Hamming, and von Hann windows are similar in this case. 
0 
-2.5 
1DBl 
-a 
-n 
-100 
0:o 0:l 0:2 0:s o:r 0:s 
Normalized Frequency 
Figure 5. The evolution of the prolate 
spheroidal filter frequency response 
as K increases from 1 through 8 for 
a normalized cutoff frequency of 0.2. 
572 
ACIWOWLEKEMENTS 
This work has been supported by the National Science Foundation Atmospheric 
Sciences Division under grant ATM 79-18379 t o  the Case Western Reserve 
University , 
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The discrete prolate spheroidal filter as a digital signal processing tool

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