Based on the multistage approach, a design procedure is presented for finding a spectral factor of an mth-band filter and for designing multistage decimation filters. The proposed design method finds spectral factors of mth-band FIR (finite-impulse response) filters without direct computation, and yields filters with much higher attenuation than would be possible by conventional methods. Such mth-band filters are used in filter-bank designs, including perfect-reconstruction systems

Nguyen, T. Q.

Saramaki, T.

Vaidyanathan, P. P.

Caltech Authors - Main

EIGENFILTERS FOR THE DESIGN OF SPECIAL TRANSFER FUNCTIONS 
WITH APPLICATIONS IN MULTIRATE SIGNAL PROCESSING 
D3.8 T. Q. Nguyenl, T. Saramaki* and P. P. VaidyanathanS 
$Dept. EE, Caltech, Pasadena, CA 91125 
*Dept. EE, Tampere Univ. Tech., Finland 
Abstract: Based on the multistage eigenfilters approach, 
we present a design procedure for finding a spectral factor 
of an m-th band filter and for designing multistage deci- 
mation filters. The proposed design method finds spectral 
factors of m-th band FIR filters without direct computa- 
tion, and yields filters with much higher attenuation than 
would be possible by conventional methods. Such m-th 
band filters find applications in filter-bank designs, includ- 
ing perfect reconstruction systems. 
I. Introduction’ 
Let G ( z )  be a nonnegative valued m-th band FIR fil- 
ter. Its spectral factor e ( ~ )  where G(z)A&(z)  e (~-’)  
finds many applications in multirate signal processing [2- 
41, and in QMF banks 15-71. Designing the nonlinear phase 
FIR transfer function G ( z )  with high attenuation often en- 
counters problems if conventional spectral factorization [8] 
is used. In this paper, an approach is outlined to over- 
come this difficulty. We shall show that the design of G ( z )  
can be done directly using the eigenfilters approach [ l ] ,  in 
such a way that its stopband energy is minimized. There 
is no need to directly find high order spectral factors in 
the design process. The second application which will be 
emphasized is the design of multistage decimation filters 
[3] using eigenfilters. These multistage filters are used to  
implement decimation filters efficiently. 
Bold faced letters denote both vectors and matrices in 
this paper. Superscript (*) stands for complex conjugation 
and &(z)  denotes the spectral factor of G(z) .  
11. Eigenfilters - A review. 
Let Ho(z)  be the amplitude response of a linear phase 
lowpass FIR filter of order 1 - 1 and cutoff frequencies wp, 
ws.  We consider the case where e - 1 is even (the odd case 
can be easily handled). Define the stopband error of H ( z )  
to be E s - l: IHo(e’”)12dw. Similarly, the passband 
error is formulated as E,, = /rl IHo(eio) - H~(e’”)l’dw 
and the total error of H ( z )  is thus 
W P  
E = (1 - .)E,, + aEs. (1) 
‘Work supported in part by the National Science Foundation 
grants DCI 8552579 and MIP 8604456. 
Let us rewrite Ho(z) in a more convenient form using the 
sequence b(n), defined as 
Here, h(n)  is the filter coefficient of H(z ) .  The magnitude 
response of H o ( z )  is compactly represented as 
Ho(z) = bTc(w) (3) 
where b = [b(O) b(1) ... b ( K -  1) b(K)]’, C(W) = [I 
cosw cos2w ... cosKwjT and K = -. Writing in 
terms of b and c(w),  the total energy in (1) is E = bTPb 
where 
e - 1  
2 
W P  
P = (1 - a) (1 - c ) ( l -  C)Tdw + q; ccTdw (4) 
and 1 = [l 1 . . . 1IT. The minimization requirement in 
(1) is now equivalent to  finding a vector b such that E is 
minimized where P in (4) is a real, symmetric and positive 
definite matrix. By Rayleigh principle 191, the  eigenvector 
associated with the smallest eigenvalue of P minimizes the 
total error E .  In summary, given L -  I,w,,,wS and a, the 
eigenfilter coefficients can be obtained from the eigenvector 
corresponding to the smallest eigenvalue of P .  The eigen- 
filter approach can be extended to  incorporate weighting 
functions on the total error E .  Consider the following 
total weighted error E = bTPb where 
+ a l: Ws(eiw)ccTdw. (5) 
We still can apply Rayleigh principle and obtain the eigen- 
filter which minimizes the total weighted error as long as 
W,(eiw) and Ws(ei”) are nonnegative valued functions. 
As elaborated further in [I], eigenfilters design procedure 
can incorporate both time and frequency domain specifica- 
tions. Thus, Nyquist, m-th band, equiripple, maximally 
flat, and multiband linear phase FIR filters can all be de- 
signed using the weighted eigenfilters approach (abbrevi- 
ated as the WE approach). 
1467 
CH2561-9/88/0000-1467 $1.00 0 1988 IEEE 
111. Spectral Factors of m-th band FIR 
filters. 
Let G ( z )  be an  even order linear phase FIR m-th band 
filter with order L - 1 and real coefficients g(n). To avoid 
the case where g(0) = g(L  - 1) = 0, we assume that L - 1 
is not an integer multiple of 2m. Since G(z )  is a linear 
phase filter, its zeros on the unit circle occur in conjugate 
pairs whereas its zeros on the real axis occur in reciprocal 
pairs and its zeros neither on the unit circle nor on the real 
a x i s  occur in reciprocal conjugate quadruples. To be able 
to have a spectral factor, G(eiw) has to be a nonnegative 
valued function. Consequently, its zeros on the unit circle 
have to be double zeros. In other words, if zo is a zero 
of G(z )  on the unit circle, then both zo and are double 
zeros. On the other hand, if zo is a zero neither on the 
unit circle nor on the real axis, then zi, 20' and (20'). 
are also zeros of G(z ) .  Similarly, zo and 20' is a pair 
of zeros on the real axis. Let's group the zeros on the 
unit circle into G ' f ( z )  and the zeros not on the unit circle 
into Go(z) .  We then have G ( z )  = Go(z)G;(z),  so that  
the spectral factor e(,) is Go(z)Gl(z)  where Go(z) is the 
spectral factor of PO(Z).  The problem statement is thus 
as follows: Find G ( z )  such that 
1. G(z )  = & ( Z ) & ( Z - ' )  is an m-th band filter and 
2. E = l: (&(e jw)12dw is minimized. 
Minimizing E in step 2 above is the same as minimizing 
E = l: IG(ejw)ldw = 1: IGo(ejw)llGl(e")12dw. ( G )  
We now discuss an iterative procedure to design the m-th 
band filter G ( z )  and its spectral factor & ( z ) .  The iterative 
procedure works as follows: 
Find Gl(eiw) such 
IGl(eiY)12dw is minimized using the eigen- 
1. Initialize Go(eiw) to be unity. 
that  
filter method. 
2. Fix G l ( z )  and readjust Go(z )  by solving a set of 
linear equations, so that G ( z )  = Go(z)Gq(z) has m- 
th band property [2]. 
3. By fixing Go(z) to be the solution obtained in step 
2, find G l ( z )  using the WE approach such that 
E = 6: IGo(eiw)llGl(ejW)l2dw is minimized. 
4. If the resulting Go(z)Gq(z) is not satisfacting, go to 
Very few repetitions of these steps results in excellent 
design. Trivial solutions to the minimization steps can 
be avoided by constraining the total energy of G l ( z )  to be 
unity. Even though the passband error of G ( z )  does not 
enter the error function in ( G ) ,  the passband of G ( z )  comes 
out to be good because of the m-th band property of G ( z ) .  
The order of Go(z )  is typically much smaller than that of 
step 2. 
G ( z ) ,  and moreover i t  has no zeros on theunit  circle, so it is 
a simplematterto find a spectral factor Go(z) of Go(z) and 
obtain G ( z )  = Go(z)Gl(z) .  The readjustment of Go(z) in 
step 2 is elaborated here using m-th band property. 
Let 2&,, L1 and e -  1 be the orders of Go(z ) ,  G l ( z )  
and G ( z ) .  Thus, G ( z )  and & ( z )  have orders 2(& + (1) 
and (LO + &) respectively. Denote the filter coefficients 
of Go(z),  G l ( z ) ,  G:(z)(&G'f(z))  and G ( z )  by go(n), gl(n), 
g:(n) and g(n) respectively. As mentioned earlier, t o  + 
el  # km for any positive integer k. Since G ( z )  is required 
to  be an m-th band filter, every mth coefficient from the 
mid-point has to  be zero, i.e., 
( g ( L 0  + L 1 )  = L. m 
The number of known coefficients in G ( z )  is therefore 
In terms of filter coefficients, G(z )  = Go(z)G;(z)  = Go(z). 
G:(z)  is equivalent to 
2 f n  
(9) 
k=O 
For a given filter G:(z) ,  there are ( 2 & + 1 )  unknown coeffi- 
cients of Go(z) and L fixed values of g(n). Consequently, 
if L = 2& + 1, then we can uniquely solve for go(n) in 
terms of g:(n) such that (7) holds. In other words, gO(n) 
is uniquely determined in 
, 
Using (8) in L = 210 + 1, we have 
Equivalently, 
1468 
As long as the order of G i ( z )  (which is 2 f 1 )  satisfies (13), 
we can always solve equation (10) for go(n). (IO) can 
be further simplified by observing that g(n) ,  go(n) and 
L + l  gi (n)  are symmetric sequences. Thus, only the first -
2 
equations are sufficient to  describe ( l o ) ,  i.e., 
L - 1  
2 
where e '=  -. (14) can be compactly rewritten as 
Ago = d (15) 
where go = [ g o ( O )  . . . go(&)]', d = [0 . . . 0 $1' and A is 
as in (14). A, go and d have dimensions (Lo + 1) x (Lo + I ) ,  
(LO + 1 )  x 1 and ((0 + 1) x 1 respectively. By noting that 
all elements in d are zero except the last one, go can be 
obtained by 
where Ai,j and A are the cofactor of the ( i , j )  element and 
the determinant of A respectively. 
In summary, given 6 ,  wpl w s  and m, we pick el  us- 
ing (13). Having fixed the orders of Go(z) and G l ( z ) ,  
we design the m-th band filter by itefative procedure de- 
scribed above. The spectral factor G ( z )  of G(z )  is thus 
& ~ ( z ) G l ( z )  where & ( z )  is the spectral factor of G o ( z ) .  
Ezample 3.1: We design a 3'd band linear phase FIR 
filter of length 62, with cutoff frequencies a t  .2% and .4%. 
Here, fo = 10 and f 1  21. Fig. 1 shows the magnitude 
responses of G ( z )  and G ( z ) .  &(z)  has order 31 and has the 
largest possible number of zeros on the unit circle (under 
the constraint that & ( z ) e ( z - ' )  is an m-th band filter.) 
lV. Design of multistage decimation fil- 
t er 
We will present here an iterative procedure to design 
the multistage decimation filter [3] in fig. 2(a). Redrawing 
the structure as in fig. 2(b) for analysis purposes, we see 
that the effective transfer function to be designed has the 
form H ( z )  = H o ( z ) H l ( z 2 ) H ~ ( z ' ) .  The filters Ho(z) and 
H1 (2) are required to be lowpass with minimum stopband 
energies so that aliasing is minimized. The passband re- 
quirements of H o ( z )  and H l ( z )  are not severe; however, 
the final stage H 2 ( z )  is required to be such that the overall 
system has a good passband (maximally flat or equiripple 
for example.) Suppose that the cutoff frequencies of H ( z )  
are wp = t - E and ws = f + E ,  then the appropriate 
stopband edges of Ho(z) and H l ( z )  are I + E and 5 + 2 ~ .  
Similarly, the passband edge of Hz(z)  is A - 4 ~ .  We will 
design the overall filter to  have equiripple passband. We 
use an iterative method, where at  each iteration we design 
one of the three filters Ho(z),Hl(z),Hz(z) while the other 
two are held fixed at  the 'current solution'. The procedure 
works as follows: 
1. Initialize Hl(eJ") and H2(eiw) to  be unity. Design 
IHo(eiw)12dw is minimized Ho(eiw) such that 
using the eigenfilter approach. 
2. Fix &(eiw) (obtained from step 1) and H2(eJw),  
and using the WE approach, design H l ( e J w )  such 
that 1' IHo(eJg)121Hl(eiu)12dw is minimized. 
3.  Fix &(eiu) and Hl(e iw) ,  design H2(eJ") such that 
its passband 0 5 w 5 A - 46 is equiripple using 
Remez exchange algorithm [ lo] .  The weighting and 
desired functions for the Remez exchange algorithm 
are: 
W(eiw)  = IHo(eiF)llHl(eiy)l (17) 
I' 4+r 
4 +2r 
4. Use the current H l ( e j u )  and H2(eiw), and design 
Ho(eiw) using the WE appoach such that 
1 I H1 ( e j b  12 IH2 ( e j 4 w  12 IHO ( e j w  l2dw 
5 +€ 
is minimized. 
that 1 ' IHo (e' ) I IH2 ( e j Z w )  l2 IH1( eiw ) I2dw is min- 
imized using the WE approach. 
satisfied. 
5. Fix Ho(ej") and H2(ejw) ,  and design Hl(eJ")  such 
f + 2 €  
6. Go to step 3 and repeat until the resulting H ( z )  is 
The design procedure usually converges to a good solution 
after 5 iterations. By constraining the total energy of 
either Ho(z) or H l ( z )  at the corresponding minimization 
step, the trivial solution is avoided. 
Ezample 4.1 We design a multistage decimation filter 
with the orders of Ho(z) ,  H l ( z )  and H2(z)  to be 10,5 and 
10 respectively. Thus, the overall filter has length 61. 
Here, = 0 . 0 5 ~ .  Fig. 3 shows the frequency responses of 
Ho(z), H l ( z 2 )  and H2(z4) respectively while fig. 4 shows 
the frequency responses of the overall filter H ( z ) .  
References : 
[ l ]  P.P. Vaidyanathan and T.Q. Nguyen, "Eigenfilters: 
A new approach to least-squares FIR filter design 
and applications including Nyquist filters," IEEE 
Trans. on CAS, pp. 11-23, Jan. 1987. 
[2] F. Mintxer, =On half-band, third-band and N-th 
band FIR filters and their design,” IEEE Trans. on 
[3] R. E. Crochiere and L. R. Rabiner, Multirate Digital 
Signal Processing, Englewood Cliffs, NJ, Prentice- 
Hall, 1983. 
[4] M. Renfors and T. Saramaki, ”Recursive N-th band 
digital filters,” (parts I and 11), IEEE Trans. on 
CAS, pp. 24-51, Jan. 1987. 
[5] P. P. Vaidyanathan, “Theory and design of M chan- 
nel maximally-decimated quadrature mirror filters 
with arbitrary M, having perfect reconstruction prop- 
erty”, IEEE Trans. on ASSP, pp. 476492, April 
1987. 
[6] M. J. T. Smith and T. P. Barnwell 111, “Exact re- 
construction techniques for tree-structured subband 
coders”, IEEE Trans. on ASSP, pp. 434-441, June 
1986. 
[7] F. Mintzer, “Filters for distortion-free two-band mul- 
tirate filter banks”, IEEE Trans. on ASSP, pp. 626- 
630, June 1985. 
[8] G. A. Mian and A. P. Nainer, “A fast procedure to  
design equiripple minimum-phase FIR filters”, IEEE 
Trans. on CAS, pp. 327-331, May 1982. 
[9] B. Noble and J.W. Daniel, Applied Linear Algebra, 
Englewoods Cliffs, NJ., Prentice Hall, 1977. 
[lo] J.H. McClellan and T. Parks, “A unified approach 
to the design of optimum FIR linear phase digital 
filters,” IEEE Trans. on Circuit Theory, pp. 697- 
701, Nov. 1973. 
ASSP, pp. 734-738, Oct. 1982. 
0. 
-30 n 
z 
H 
W 
rn z 
0 
rn 
W 
pc; 
-60 
-90 
w -120 n a w 
U 
2 -150 
0. -1 02 .3 .4 .5 c3 
NORMALIZED FREQUENCY 2 
Fig. 1 Ex. 3.1 Magnitude responsss 
of G ( z )  and its spectral factor G ( z ) .  
3 (a) 
Fig. 2 The multistage decimation 
structure (a) The actual circuit 
(b)  The redrawn version. 
F4 O *  
Ei -20 
n 
W 
rn z 
0 -40 Pl cn w tz 
W -60 n 
E 
2 -100 
H -80 z 
c3 
0. .1 .2 .3 .4 .5 
Fig. 3 Ex. 3.1 Magnitude responses of 
H0(z) ,  H1(z2) and H2(z4). 
NORMALIZED FREQUENCY 
0; 
-20 
z 
H 
rn z 
0 
W 
!% 
$j -40 
-60 
a w 
U -80 
-100 
3 
I 
0. .1 .2 .3 .4 .5 
NORMALIZED FREQUENCY 
Fig. 4 Ex. 3.1 Magnitude response 
of H(z), 
1470 


Eigenfilters for the design of special transfer functions with applications in multirate signal processing

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Eigenfilters for the design of special transfer functions with applications in multirate signal processing

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