We consider the optimization of pre- and post filters surrounding a uniform quantizer such that the mean square error due to quantization is minimized. Unlike some previous work, the postfilter is not restricted to be the inverse of the prefilter. With no order constraint on the filters, we present closed form solutions for the optimum pre- and post filters. Using these optimum solutions, we obtain a coding gain expression for the system under study. We then repeat the same analysis with first order pre- and post filters in the form 1+αz^-1  and 1/(1+γz^-1) providing some examples where we compare coding gain performance with the case of α=γ

Tuqan, Jamal

Vaidyanathan, P. P.

English

Caltech Authors - Main

I S C A S ,  A t l a n t a ,  May 1996. 
OPTIMUM PRE- AND POSTFILTERS FOR QUANTIZATION 
Jamal Tuqan and P.P. Vaidyanathan 
Department of Electrical Engineering 136-93 
California Institute of Technology 
Pasadena, CA 91125, USA. 
E-mail : tuqan@systems .caltech .edu, ppvnat h@sys. caltech .edu 
Abstract. We consider the optimization of pre- 
and post filters surrounding a uniform quantizer 
such that the mean square error due to quantization 
is minimized. Unlike some previous work, the 
postfilter is not restricted to be the inverse of the 
prefilter. With no order constraint on the filters, we 
present closed form solutions for the optimum pre- 
and post filters. Using these optimum solutions, 
we obtain a coding gain expression for the system 
under study. We then repeat the same analysis with 
first order pre- and post filters in the form 1 + az-' 
and 1/(1 + yz-') providing some examples where 
we compare coding gain performance with the case 
of a = y. 
I. INTRODUCTION 
Consider the general scheme shown in Fig. 1 where 
the box labeled Q represents a uniform quantizer. 
The input sequence z (n )  is passed through a pre- 
filter G(ej") and produces an output y(n). The 
sequence y(n) is then quantized and filtered with 
a postfilter H(ej")  to reproduce an estimate of the 
input denoted by k(n) .  Assuming that the quanti- 
zation system is constrained to have a budget of b 
bits, the main theme in this paper is to jointly opti- 
mize the prefilter G(ej") and the postfilter H(ej")  
such that the mean square value E{e2(n)} of the 
reconstruction error where e(n) = k ( n )  - ~ ( n )  is 
minimized. 
The renewed interest in the above classic problem 
[l] was motivated by the growing activity in op- 
timizing perfect reconstruction filter banks in the 
presence of quantizers [2], [3], [4], [5]. When quan- 
tizers are present, the FB output k ( n )  is the original 
input z (n )  plus a filtered version of the quantization 
noise e (n ) .  Searching over the class of perfect re- 
construction filter banks (PRFB), the goal is to find 
a set of analysis filters to minimize the quantiza- 
tion noise e(.). Although this problem was solved 
for the class of orthonormal FB [ideal filter case] 
[6], [7], [8] [9], the M channel maximally decimated 
A 
0-7803-3073-0/96/$5.00 '1996 IEEE 4 61 
optimum biorthogonal FB remains for example an 
open problem. Only the solution of the one channel 
case is well established [lo]. Furthermore, it is well 
known [ll], [2] that, in the presence of quantizers, 
the synthesis polyphase matrix is not necessarily the 
inverse of the analysis polyphase matrix. Restrict- 
ing ourselves to the class of biorthogonal FB when 
quantizers are present is therefore a loss of gener- 
ality. The joint optimization of the analysis bank 
and the synthesis bank together with the allocation 
of subband bits is quite a challenging problem. In 
this paper, we will provide a joint optimum solu- 
tion of the pre- and post filters for the special case 
of M = 1. The system of Fig. 1 can indeed be seen 
as the one channel case of the more general and 
difficult M channel problem. It is also a general- 
ization of the so-called half-whitening scheme [lo] 
where the postfilter is assumed to be the inverse of 
the prefilter. 
11. THE OPTIMUM POST FILTER 
In this section, we assume that all random processes 
are zero mean, real and jointly wide sense station- 
ary. The input z(n)  and the quantization noise q(n)  
are uncorrelated, i.e., E{x(n)q(m)} = 0 V n,m. 
The quantization noise q ( n )  is white with variance 
where U: is the quantization noise variance, c is a 
constant that depends on the statistical distribution 
of y(n) and the overflow probability, and U; is the 
variance of the quantizer input. The filters H(ej")  
and G(ej") are not constrained to be rational func- 
tions, i.e., the optimum H(ej")  and G(ej") can be 
ideal filters. Furthermore, no causality constraint is 
imposed. To develop optimum closed form solutions 
for both filters, we first fix the prefilter G(ej") and 
optimize H(e j") .  The optimum post filter solution 
is given in the following theorem. The proof of all 
four theorems included in this paper can be found 
in [12]. 
Theorem 2.1. For a fixed prefilter G(ej"), the 
optimum postfilter H,,,(ej") is the well-known 
Wiener filter and is given by : 
Using the optimum post filter solution (2.2), 
the orthogonality principle, Parseval's relation and 
the constraint (2.1), we can derive the following 
expression for the mean square error € only in terms 
of the prefilter G(ejw): 
The problem now reduces to finding the prefilter 
G(ej") that minimizes E .  Since the mean square 
error expression (2.3) is a function of IG(ej")12 
only, we will be actually seeking an expression for 
the squared magnitude response of the prefilter 
rather than G(ej"). Instead of attacking the 
problem directly, the idea is to transform the above 
unconstrained integral (2.3) to a communication- 
like problem, i.e., an integral objective function 
with a power constraint on the prefilter output. 
The problem then becomes more mathematically 
tractable and a closed form expression for IG(ej")12 
can be obtained. The equivalence of the two 
problems is established by the following claim [12]. 
Theorem 2.2. The squared magnitude response 
IGOpt(ejw)l2 that minimizes E(IG1, b )  is also the 
solution of the following constrained optimization 
problem : 
subject to: 
SzZ(ej")lG(ej")l 2- dw = 1 (2.5) 27r 
111. THE OPTIMUM $REFILTER 
The goal now is to find IG(ej")l that minimizes the 
integral in (2.4) under the integral constraint (2.5). 
Since the magnitude squared response is always a 
non negative function of w ,  the optimum minimiz- 
ing solution we seek must be non negative. This 
implicit condition is incorporated in the optimiza- 
tion problem as a pointwise inequality constraint. 
The next theorem [12] gives an expression for the 
optimum magnitude squared response of the pre- 
filter. 
Theorem 3.1. The magnitude squared of the 
prefilter IGopt(ejW)12 that minimizes the integral 
in (2.4) under the constraint (2.5) must have the 
following form: 
C2-2b ) J z p J  . ( J:JzpJ& 1+c2-2b - 
(3.1) 
Jm 
for all w E [-7r,n] for which (3.1) is non negative. 
Otherwise, the value of the optimum magnitude 
squared response is set to zero. 
IV. THE CODING GAIN EXPRESSION 
Assume that we quantize x(n)  directly with b bits. 
We denote the corresponding mean square error 
(m.s.e) by &direct. We then use the optimum pre 
and post filters around the quantizer. With the 
rate of the quantizer k e d  to the same value b, we 
denote the m.s.e in this case by E,,,. The ratio 
Gopt = &direct/Enew is called the coding gain of the 
new system and, as the name suggests, is a measure 
of the benefits provided by the pre/post filtering 
operation. The coding gain of the scheme of Fig. 1 
is given in the following theorem [la]. 
Theorem 4.1. With the optimal choice of pre- 
and postfilters, the coding gain expression for the 
scheme of Fig. 1 is 
A 
as long as (3.1) is non-negative V w .  Here Ghw is 
the coding gain of the half whitening scheme and is 
given by 
It is quite clear from theorem 3.1 that the system 
of Fig. 1 will always outperform the half whitening 
scheme as long as equation (3.1) remains non 
negative for all frequencies. The difference in 
performance is essentially a function of the bit rate. 
The lower the bit rate the higher the coding gain 
will be. However, the problem is that as we quantize 
462 
at lower bit rates, the quantizer assumptions made 
at the beginning of this section fail and all the 
previous analysis is not valid anymore. On the other 
hand, as b becomes large, one can easily check that 
the prefilter (3.1) becomes 
and is positive V W .  Therefore, the coding gain 
expression derived in Theorem 4.1 can be used and 
in fact becomes equal to Ghw. Hence, at high bit 
rate, there is no loss of generality in using the half 
whitening scheme. The same observation can be 
also found in the work of Goodman and Drouilhet 
[13] although their approach is different than ours. 
V. FIRST ORDER FILTERS 
In this section, we will constrain H(ej“) and G(ej”) 
to be first order causal filters in the form 1 - 
QZ-’ and We again jointly optimize 
the first order pre- and post filters to minimize 
the m.s.e under the constraint (2.1) and the other 
assumptions of section 11. We consider two main 
cases: a) an FIR prefilter with an IIR postfilter and 
b) an IIR prefilter with an FIR postfilter. The mean 
square error and optimum coefficients expressions 
for each case are not included in this paper due to 
lack of space and can be found in [12]. We will 
instead directly provide some numerical results for 
two specific examples of input z(n) with comparison 
to the a = y case. 
Example 4.1.1 Case of a MA(1) process. Assume 
that the input z(n) is a zero mean MA(1) process 
with an autocorrelation sequence in the form 
1 
1 - yz-1. 
( 1  IC = 0. 
IC = 1, -1. 
0 otherwise. 
The optimization of the coefficients for the IIR 
prefilter-FIR postfilter case and the FIR prefilter- 
IIR postfilter with Q # y were all done numeri- 
cally using MATLAB’s optimization toolbox rou- 
tine “fmins.”’. The plots of the coding gain are 
illustrated in Fig. 2 (a) and (b) for the FIR/IIR 
case and in Fig. 2 (c) and (d) for the IIR/FIR case. 
The vertical axis represents the coding gain mag- 
nitude in db and the horizontal axis represents the 
parameter 6 defined above. From these figures, we 
can observe that as the bit rate increases, there is 
no loss of generality in assuming Q to be equal to 
y. We also note that the coding gain obtained in 
the FIR/IIR case is higher than the dual case for 
the same process and bit rate. This is primarily 
due to the fact that the optimum coefficients in the 
IIR/FIR case are numerically close to zero and the 
coding gain is therefore close to one. 
Example 4.1.2 Case of an  AR(5) process. Table 
1 summarizes our coding gain results in db for the 
different cases and bit rates. Again, as b increases, 
we observe that there is almost no loss in coding 
gain if we assume that CY = y. We also observe that, 
at low bit rate, e.g. b = 1, the coding gain of the 
more general system is very small. This suggests 
that the gain obtained from searching over a more 
general class than the biorthogonal class may not 
be worth the added complexity. Finally, as was the 
case for the previous example, the FIR/IIR scheme 
outperforms substantially the dual case. 
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x(n) -b G(e’”) L Q -  ~ ( e i ” )  +-4(n) 
-0 0.5 1 
(a) 
c -  
/ 
FIR/IIR 
a+ Y 
FIR/IIR 
a = y  
IIR/FIR 
a# Y 
IIR/FIR 
a=Y 
Table 1. The coding gain obtained from first 
order filters for the AR(5) case. 
0.8 
0.6 
0.4 
0.2 
(cl 
Fig.  2. (a) FIR prefilter, IIR postfilter , b = 1, (b) FIR prefilter, IIR postfilter, b = 2, 
(c) IIR prefilter, FIR postfilter, b = 1, (d) IIR prefilter, FIR postfilter, b = 2. For all 
plots , .... : equal coefficients and - - - : unequal coefficients. 
464 


Optimum pre- and postfilters for quantization

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Digital coding of waveforms,

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Optimal orthonormal filterbanks”, submitted,

Optimum analysis/synthesis filter bank structures with application to subband coding systems”,

Principal component filter banks for optimal multiresolution analysis”,

Signaladapted multiresolution transform for image coding”,

Statistically Optimal synthesis banks for subband coders reconstruction”,

Statistically optimum pre- and post filtering in quantization”, Submitted,

Subband coding systems incorporating quantizer Models”,

the optimality of ideal filters for pyramid and wavelet signal approximation”,

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Optimum pre- and postfilters for quantization

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