In this paper, we statistically optimize a well known class of IIR two channel orthonormal filter banks parameterized by a single coefficient when subband quantizers are present. The optimization procedure is extremely simple and very fast compared for example to the linear programming method used in the FIR case to achieve similar compaction (coding) gains. The special form of the filters assure the existence of a zero at π which can be important for some wavelet applications and eliminate some of the major concerns that arise in the FIR design case. Finally, the compaction gain obtained is high and numerically very close to two (ideal case) for low pass spectra, high pass spectra and certain cases of multiband spectra. For these cases, the use of higher order IIR filters does not increase the compaction (coding) gain

Tuqan, Jamal

Vaidyanathan, P. P.

English

Caltech Authors - Main

OPTIMUM LOW COST TWO CHANNEL IIR ORTHONORMAL FILTER BANK t 
Jamal Tuqan and P. P. Vaidyanathan 
Department of Electrical Engineering 136-93 
California Institute of Technology 
Pasadena, CA 91125 
tuqan@systems.caltech.edu, ppvnath@sys.caltech.edu 
ABSTRACT 
In this paper, we statistically optimize a well known 
class of IIR two channel orthonormal filter banks 
parameterized by a single coefficient when subband 
quantizers are present. The optimization procedure is 
extremely simple and very fast compared for example 
to the linear programming method used in the FIR 
case to achieve similar compaction (coding) gains. 
The special form of the filters assure the existence 
of a zero at T which can be important for some 
wavelet applications and eliminate some of the major 
concerns that arise in the FIR design case. Finally, the 
compaction gain obtained is high and numerically very 
close to two (ideal case) for low pass spectra, high pass 
spectra and certain cases of multiband spectrum. For 
these cases, the use of higher order IIR filters does not 
increase the compaction (coding) gain. 
1. INTRODUCTION 
There has been a considerable interest in designing 
filter banks while taking into account the effect of 
subband quantization [l], [2], [3]. Given a fixed budget 
of b bits for the subband quantizers, the goal is to 
simultaneously optimize the analysis and synthesis 
filters and to choose a subband bit allocation strategy 
such that the average variance of the error e(n) at the 
output of the subband coder is minimized. 
The energy compaction problem. Consider the 
scheme of Fig. 1.1 which shows a filter H(ej")  with 
input z(n) and output y(n). The input z(n) is assumed 
to be a widesense stationary process. With the input 
power spectral density S,, (ej") fixed, the compaction 
filter problem is to find H(ej")  such that the variance 
of the output, given by 
is maximized under the constraint that IH(ej")12 is 
t Work supported in parts by Office of Naval Research 
grant N0001493-1-0231, Tektronix, Inc., and Rockwell 
International. 
a Nyquist(M) filter. The design of such compaction fil- 
ters is important in its own merit because of the usual 
question of finding the best wavelets for a given appli- 
cation. For a two channel orthonormal filter bank, the 
two problems are ideintical : optimizing one of the sub- 
band filters for maximum energy compaction is equiv- 
alent to designing a two channel optimal orthonormal 
subband coder. To see this, recall that the coding gain 
[4] of an orthonormal filter bank under optimum bit al- 
location and with the high bit rate quantizer assump- 
tions is given by GsEtC(2) = where U: is the 
input signal variance and U: is the variance of the 
kth subband signal. Using 2u: = a;, + U:, , the above 
expression can be rewritten as 
(7; 
J z z  
(2) 
1 
GSBC(2) = - 
lJGco"(2 - Gcomp(2)) 
where GComp(2) is the so called compaction gain and 
is equal to U : ~ / U ~ .  The compaction gain therefore 
uniquely determines the coding gain of a 2-channel 
orthonormal subband coder. It is important to keep 
in mind that the maximum possible compaction gain 
Gcomp(2) is equal to two whereas the coding gain 
G S B C ( 2 )  can be arbitrarily large. 
2. TH€ PROBLEM SET UP 
The aim of this pa,per is to statistically optimize a 
two channel orthonormal filter bank when subband 
quantizers are present at low cost. Two channel 
orthonormal filter banks are of special interest because 
they form a basic budding block in the design of wavelet 
transforms. Low cost filters are quite attractive in 
image and audio coding applications. The requirement 
for a very efficient two channel system motivates the 
investigation of filter banks based on IIR filters rather 
than FIR ones. Moreover, previous work on finite 
order compaction filters and/or finite order two channel 
optimum orthonorrnal filter banks has been mainly 
dedicated to the FIR case (see for example [5], [6], and 
[7] to name afew). To meet the above requirements, we 
propose the optimization of a class of two channel IIR 
orthonormal filter banks based on the sum of two all 
0-8186-7919-0/97 $10.00 8 1997 IEEE 2425 
pass filters [8]. In specific, consider the single coefficient 
system: 
(1) 
where a is real and -1 < a < 1. The synthesis 
filters are given by Fo(ejw) = H;(eJW) and F1(ejw) = 
H;(ejw).  The two channel system is shown in Fig. 
2.1. Note that since the polyphase components of 
the analysis filters are stable causal all pass filters, 
their reciprocals will produce unstable synthesis filters. 
To overcome this difficulty, Ramstad [9] proposed to 
implement the inverse filters as anti-causal stable IIR 
filters. Although (1) is a seemingly restrictive case, 
the proposed form of the filter Ho(z )  is a special case 
of the more general structure introduced recently by 
Phoong and al. [lo]. It has been shown that this type 
of filter provides several excellent advantages [lo]. For 
example, the filter Ho(z) (and therefore H l ( z ) )  can 
have a very good frequency response. Furthermore, the 
special form of the filter assure the existence of a zero 
at T which can be important for wavelet applications. 
For the purpose of this paper, our results indicate that 
for the cases where we can obtain high compaction gain 
with the special filters in (I), using higher order filters 
does not increase the compaction gain. 
3. THE ANALYTICAL RESULTS 
Consider the set up shown in Fig. 2.1. where the input 
signal z(n) is a zero mean wide-sense stationary (WSS) 
random process with a power spectrum Szz(ejw). 
Each subband quantizer, labeled by &, represents a 
scalar uniform (PCM) quantizer and is modeled as 
an additive zero mean white noise source q(n) with 
variance 
0 2  Q = c2-2b02 X,  (1) 
where ai is the quantization noise variance, c is a con- 
stant that depends on the statistical distribution of the 
subband signal zi(n) and the overflow probability, and 
ai, is the variance of the ith subband signal. The sub- 
band coding problem reduces to finding the optimum 
coefficient aopt that maximizes the compaction gain 
(alternatively the subband variance) at the output of 
one of the subband. The specific form of the analysis 
filters given in (1) guarantee automatically the Nyquist 
property and transforms the constrained optimization 
problem into an unconstrained one. A closed form ex- 
pression for the compaction gain is given next. 
Proposition 1. Consider the scheme ofFig. 2.1 under 
all the previous filter and quantizer assumptions. The 
compaction gain at the output of one of the subband 
filters, say Ho(z) ,  can be expressed as follows : 
The proof can be found in [ 111. The infinite summation 
in (2) is the result of the IIR nature of the filter bank. 
The above equation was written specifically in the 
above form in order to emphasize the following points: 
First, when Q is equal to zero, the compaction gain is 
equal to 1 + RzZ(l)/RZx(0), which is simply the 2 x 2 
KLT compaction gain. This indeed makes sense since 
the structure of Fig. 2.1 reduces to the 2 x 2 universal 
KLT. Second, when the input signal is white noise, 
i.e. R,,(k) = 6(k), the compaction gain is equal to 
one. Finally, observe that the above equation involves 
only the odd samples of the autocorrelation sequence 
Rzz(k), due to the Nyquist constraint on ~ H o ( e ~ w ) ~ 2 .  
Therefore, if the input signal z(n) is such that its power 
spectrum Sxz(z) takes the form S ( z 2 ) ,  the compaction 
gain is equal to unity. 
The goal now is to find the optimum coefficient aopt 
that maximizes (2). In general, it is difficult to 
obtain analytical solutions due to the complexity of 
the expression in (2). We will therefore present 
analytical solutions for the optimum coefficient aopt, 
the compaction gain Gcomp(2) and the coding gain 
GSBC(2) only for specific examples of the input z(n) 
such as the MA(1) and AR(1) processes. For a general 
random process z(n), the optimum coefficient aopt is 
obtained numerically through a MATLAB program. 
Example 3.1. 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 
k = 0. 
k = 1, -1. e 
otherwise. 
where 8 is between -1 and 1. It can be shown [ll] 
that, for this case, 
aopt = -0.5 
Gcomp(2) = 1 + 5Rxz(1)/4Rzz(O) (3) 
GSBC(2) = 1/d1 - 25R&(1)/16R;,(O) 
It is interesting to note that the optimum coefficient 
aopt is independent of the signal statistics. 
Example 3.2. Case of an AR(1) process. Assume 
now that the input z(n) is a zero mean AR(1) process 
with an autocorrelation sequence in the form R,, ( I C )  = 
2426 
plkl where p is between -1 and 1. It can be shown [ll] 
that, for this case, 
We note that the optimum coefficient in this case is 
independent of the sign of p, is always negative and 
between 1 - f i  and -0.5 (the case where p = 0). 
Furthermore, one can show that there is a negligible 
loss of compaction gain even when aopt is implemented 
using very small number of binary shift and add 
operations [ll]. As p approaches unity, the scheme is 
asymptotically equivalent to the 2 x 2 universal KLT. 
4. EXAMPLES FOR MORE GENERAL INPUTS 
We give several examples where the optimum coeffi- 
cient aopt is computed numerically through a MAT- 
LAB program. The program uses the compaction gain 
expression derived previously with input R,, (k) and 
output aopt. Although written in MATLAB, the pro- 
gram converges in fractions of a second. This is an or- 
der of magnitude faster than linear programming tech- 
niques used to design high order FIR compaction filters 
to achieve similar compaction gain. We vary the input 
z(n) such that the power spectral density shape spans 
a variety of choices from “low pass” to multiband with 
energy concentrated in a specific region to multiband 
with more even energy distribution. The magnitude 
squared response of the optimum IIR compaction fil- 
ter together with the ideal optimum compaction fil- 
ter magnitude squared response and the input power 
spectral density are shown in Fig. 4.1 to Fig 4.4. We 
adopt the following convention for all the plots : the 
solid curve denotes the input power spectral density, 
the dash-dot curve denotes the magnitude squared re- 
sponse of the optimum IIR compaction filter and the 
dashed curve represents the magnitude response of 
the ideal optimum compaction filter. This last curve is 
obtained by optimizing an FIR filter with order equal 
to 65 using a linear programming approach. 
5. CONCLUDING REMARKS 
We have addressed in this paper the problem of 
optimizing a two channel orthonormal filter bank 
with a cost constraint in mind. To achieve this, 
we optimized a well known class of IIR two channel 
orthonormal filter banks parameterized by a single 
coefficient. The resulting optimum filter bank provides 
some good advantages that are not available in the 
FIR case. First, the Nyquist property is satisfied 
automatically because of the special form of the filters. 
Second, in FIR compaction filter design, one would 
traditionally find IH(ej”’)I2 and then perform a spectral 
factorization to obtain H(ej“). The positivity of 
the solution is not usually guaranteed and spectral 
factorization can be prioblematic in case of unit-circle 
zeros. In our scheme, these concerns do not exist 
since Ho(z) is directly found. Third, the form of the 
filters assure the existeince of a zero at T which can be 
important for some wavelet applications. Fourth, the 
filters have only one coefficient which can be quantized 
without a major sacrifice in compaction gain as we 
demonstrate in [ll]. Finally, the compaction gain 
obtained is high and very close to two (ideal case) for 
low pass spectrums, high pass spectrums and certain 
cases of multiband spectrum. The only weakness of the 
filter bank under consideration is its poor performance 
for the case of general multiband spectrums. This is 
mainly due to the monotone property of the phase of 
an all pass function. For such cases, aopt can be set to 
zero to obtain the 2 x 2 universal KLT. 
References 
[l] Kovacevic, J., “Sulbband coding systems incorpo- 
rating quantizer Models”, IEEE Trans. on Image Pro- 
cessing, pp. 543-553, May 1995. 
[2] Uzun, N. and Haddad, R.A. , “Cyclostationary 
Modeling, analysis and optimal compensation of quan- 
tization errors in subband codecs”, IEEE Trans. on 
SP, pp. 2109-2119, September 1995. 
[3] Gosse, K. and Duhamel, P., “Perfect reconstruction 
versus MMSE filter banks in source coding”, Submitted 
to the IEEE Tkans. on SP. 
[4] Jayant, N.S. and Noll, P. Digital coding of wave- 
forms, Prentice Hall, Inc., Englewood Cliffs, 1984. 
[5] Moulin, P.,“A new look at signal adapted QMF 
design”, Proc. ICASSP, pp. 1312-1315, May 1995. 
[6] Caglar, H., Liu, Y .  and Akansu, A.,“Statistically 
optimized PR-QMF”, lSPIE vol. 1605, pp. 86-94, 1991. 
[7] Kirac, A. and Vaidyanathan, P. P., “Theory and 
design of optimum FIR compaction filters”, submitted 
to the SP special issue on filters banks and wavelets. 
[8] Vaidyanathan, P.IP. Multirate systems and filter 
banks, Prentice Hall, Inc., Englewood Cliffs, 1993. 
[9] Ramstad, T.A., “IJR filterbank for subband coding 
of images”, Proc. ISCAS, pp. 827-830, June 1988. 
[lo] Phoong, S.M., Kim, C.W., Vaidyanathan, P.P. and 
Ansari, R., “A new class of two channel biorthogonal 
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[ll] Tuqan, J. and Vaidyanathan, P. P., “Low cost 
optimum IIR compaction filters and their application 
in image compression”, in preparation. 
pp. 649-665, March 1995. 
x(n) -4.;.;”;t++ y(n) 
Fig. 1.1. Schematic of the energy compaction problem. 
2427 
Quantizer 1 - 
Fig. 2.1. The class of two channel orthonormal filter bank under consideration. 
normalized frequency 
Fig. 4.1. Case of a low p a s  AR(5) 
process : IIR compaction gain = 1.95, 
theoretical compaction gain = 1.951 
and IIR FB coding gain = 5.00 db. 
U 
U 
v) 
0 0.5 1 
normalized frequency 
Fig. 4.3. Case of a multiband AR(10) 
process : IIR compaction gain = 1.52, 
theoretical compaction gain = 1.61 
and IIR FB coding gain = 0.68 db. 
-0 0.5 1 
normalized frequency 
Fig. 4.2. Case of a multiband AR(12) 
process : IIR compaction gain = 1.951, 
theoretical compaction gain = 1.97 and 
IIR FB coding gain = 5.14 db. 
0 0.5 1 
normalized frequency 
Fig. 4.4. Case of a multiband AR(5) 
process : IIR compaction gain = 1.387, 
theoretical compaction gain = 1.6 and 
IIR FB coding gain = 0.35 db. 
2428 


Optimum low cost two channel IIR orthonormal filter bank

A new class of two channel biorthogonal filter banks and wavelet bases”,

Akansu, A.,“Statistically optimized PR-QMF”,

Digital coding of waveforms,

IJR filterbank for subband coding of images”,

Low cost optimum IIR compaction filters and their application in image compression”, in preparation.

Modeling, analysis and optimal compensation of quantization errors in subband codecs”,

Multirate systems and filter banks,

new look at signal adapted QMF design”,

Perfect reconstruction versus MMSE filter banks in source coding”,

Sulbband coding systems incorporating quantizer Models”,

Theory and design of optimum FIR compaction filters”, submitted to the SP special issue on filters banks and wavelets.

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Optimum low cost two channel IIR orthonormal filter bank

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