We consider the efficient quantization of a class of non bandlimited signals, namely the class of discrete time signals that can be recovered from their decimated version. By definition, these signals are oversampled and it is reasonable to expect that we can reap the same benefits of well known efficient A/D conversion techniques. Indeed, by using appropriate multirate reconstruction schemes, we first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals. To further increase the effective quantizer resolution, noise shaping is introduced by optimizing linear time invariant (LTI) and linear periodically time varying (LPTV)M pre- and post-filters around the quantizer. Closed form expressions for the optimum filters and the minimum mean squared error are derived for each case

Tuqan, Jamal

Vaidyanathan, P. P.

English

Caltech Authors - Main

THE DESIGN OF OPTIMUM FILTERS FOR QUANTIZING A CLASS OF NON 
BANDLIMITED SIGNALS+ 
Jamal Tuqan and P .  P .  Vaidyanathan 
Department of Electrical Engineering 136-93 
California Institute of Technology, Pasadena, CA 91125, USA. 
t uqan@systems. caltech. edu, ppvnat h@sys. caltech. edu 
ABSTRACT 
We consider the efficient quantization of a class of non ban- 
dlimited signals, namely the class of discrete time signals 
that can be recovered from their decimated version. By def- 
inition, these signals are cwersampled and it is reasonable 
to expect that we can reap the same benefits of well known 
efficient A/D conversion techniques. Indeed, by using ap- 
propriate multirate reconstruction schemes, we first show 
that we can obtain a greilt reduction in the quantization 
noise variance due to the oversampled nature of the sig- 
nals. To further increase the effective quantizer resolution, 
noise shaping is introduced by optimizing linear time invari- 
ant (LTI)  and linear periodically time varying (LPTV)M 
pre- and post filters around the quantizer. Closed form ex- 
pressions for the optimum filters and the minimum mean 
squared error are derived for each case. 
1. INTIXODUCTION 
If a discrete time signal z(n) is bandlimited to [-n/M, 7r/M], 
it can be recovered from its decimated version z ( M n )  by 
low pass filtering. Consider now the class of discrete time 
signals that can be modeled as the as the sum of the outputs 
of L < M interpolation filters, where M is the interpola- 
tion factor, as shown in Fig. 1. If a discrete time signal 
z(n) is obtained using the model in Fig. 1, it is natural 
to expect that it can be Iecovered from its decimated ver- 
sion, z ( M n ) ,  even though it is in general not bandlimited. 
As a simple example, assume that z(n) satisfies the model 
of Fig. 1 with L = 1. If Fo(ej") is a Nyquist(M) filter 
[l], then, z ( M n )  is equal to y(n) and we have the rela- 
tion z(n) = x z ( k M ) f o ( n  - kM). In other words, z(n) 
is completely defined by the samples z ( M n )  even though 
the filter Fo(ej") is not necessarily ideal. In a similar way, 
we can talk about a decimation system for the case where 
Fo(z) is not Nyquist(M) and for the case of L # 1. The 
details can be found in [2, 31. 
The main issue in this paper is how to take advan- 
tage of the signal model (Fig. 1) in preparing a quantized 
or compressed version of' z(n). Our study is inspired by 
important concepts found in A/D conversion applications 
that exploit the bandlimited property of the signals to be 
quantized, such as, oversampling PCM techniques and noise 
t This work is supported in parts by the Office of Naval 
Research grant N00014-93-'1-0231, Tektronix, Inc., and Rockwell 
International. 
k 
0-7803-4455-3/98/%10.00 1998 TEEE 
shaping. To elaborate more, consider the schematic shown 
in Fig. 3 where the box labeled Q is a simple uniform 
roundoff (PCM) quantizer. Assuming that the signal z(n) 
is bandlimited, we can low pass filter the quantized sig- 
nal z(n) + e (n)  where e(n) is the quantization noise. The 
ideal low pass filter on the right removes the noise in the 
stopband but does not change the signal component. The 
signal power remains unchanged whereas the noise power 
decreases proportionally to the oversampling ratio. By ex- 
ploiting the oversampled nature of the signal z ( n ) ,  we can 
therefore trade off quantizer complexity for higher resolu- 
tion. This technique is usually termed oversampled PCM 
conversion [4]. Consider now the system of Fig. 4 where 
P(ej") is a linear time-invariant (LTI) filter. The input sig- 
nal z(n) is still assumed to be bandlimited. In addition to 
the benefits described above, it can be shown that a clever 
choice of the filter P(eJ") in Fig. 4 produces a further de- 
crease in the noise power. The filter P ( e j W )  affect the noise 
component e ( n )  but not the input signal z(n). The sys- 
tem of Fig. 4 introduces noise shaping in the signal band to 
allow higher resolution quantization of bandlimited signals. 
With these ideas in mind, observe now the output z(n) 
of Fig. 1. Even though z(n) is not bandlimited, it can be 
reconstructed from its downsampled version as explained 
previously. In this sense, it can be considered as an over- 
sampled signal. The following questions then arise: Can 
we obtain the same advantages of the oversampling PCM 
conversion technique for a non bandlimited signal satisfy- 
ing the model of Fig. 1 ? Furthermore, for a fixed set of 
filters Fk(ej"), k = 0,1,. . . , L -  1, what is the optimal filter 
P(eJw)  that minimizes the noise power at  the output? Since 
the signal z(n) is cyclo-widesense stationary of period M 
[ (CWSS)M] [5], restricting ourselves to linear time invari- 
ant noise shaping filters is a loss of generality. So, by using 
a linear periodically time varying (LPTV)M scheme, can 
we decrease the error further ? These are the questions we 
address and answer in this paper. 
2. FILTER AND QUANTIZER ASSUMPTIONS 
Filter assumptions. The FIR filters Fk(ej") of Fig. 1 
are assumed to be the synthesis filters corresponding to 
the first L channels of an M-channel principal component 
orthonormal filter bank. Although not necessary for the 
derivation of the results of this paper, the assumption is 
motivated by the following fact: Given a wide-sense sta- 
tionary (WSS) signal with energy concentrated mostly in 
certain subbands, the problem of finding the best signal 
v-33 
model in the mean squared sense reduces to that of finding 
the filter bank that produces the L most dominant sub- 
bands. If the filter bank is orthonormal, the modeling issue 
reduces to the design of a principal component filter bank 
( L  # 1) or the design of energy compaction filters ( L  = 1). 
The design of FIR PCFB and FIR globally optimal energy 
compaction filters is discussed in [6, 71. 
Quantizer assumption. The box labeled Q represents 
a scalar uniform (PCM) quantizer and is modeled as an 
additive zero mean white noise source q(n). Because the 
model filters are not ideal, the input x(n)  is a zero mean 
(CWSS)M process. Since the input to the quantizer x(n)  
is a (CWSS)M process, its variance ul(n)  is a periodic 
function of n with period M .  Define u: to be the average 
uZ(n). Then, choose variance of z(n), i.e., u; = - 
the fixed step size A in the uniform quantizer such that the 
quantization noise variance u: is directly proportional to 
the average variance of the quantizer input ~ ( n ) ,  that is 
M -  1 
1 
M 
n = O  
(1) 
2 c2-2b 2 
uq = 0, 
where up" is the quantization noise variance, c is a constant 
that depends on the statistical distribution of x(n)  and the 
overflow probability, and a$ is the average variance Qf the 
quantizer input. The above relation is justified for a PCM 
quantizer using 3 (or more) bits per sample (see chapter 4 
in [8]). 
3. INCREASING THE QUANTIZER 
RESOLUTION BY MULTIRATE FILTERING 
Consider the set up shown in Fig. 5. The t,ilde accent on a 
function Fk(ejW) is defined such that Fk(e3w) is the conju- 
gate of F'k(e jw) .  In the absence of quantization, the scheme 
is a perfect reconstruction system. In the presence of the 
quantizer, the output 2(n)  in Fig. 5 is equal to the original 
sequence x(n)  plus an error signal e ( n )  due to quantiza- 
tion. The following result shows that, by using the above 
scheme, a significant reduction in the average mean square 
error E = - E(e(n)}' can be obtained in comparison M 
with the direct quantization of ~ ( n )  shown in Fig. 2. 
M-1 
A 1  
n=O 
Theorem 1 Consider the scheme of Fag. 5 where the L 
filters Fh(ejw) are assumed to  be any L channels of an M -  
channel critically sampled orthonormal filter bank. Under 
the above quantization noise assumption, the average mean 
square error E is equal to -ai. 
The proof can be found in 191. The quantization noise vari- 
ance u," obtained by directly quantizing x(n)  as shown in 
Fig. 2 is now reduced by the oversampling factor M I L .  The 
signal variance 6: on the other hand did not change. We 
note the following: 
1. The signal $(T I ) ,  modeled as in Fig. 1, is oversampled and 
therefore, contains redundant information in the form of an 
excess of samples. We are therefore effectively quantizing 
with a higher number of bits per sample. This trade off, 
between the quantization noise variance (effective quantizer 
L 
M 
resolution) and the sampling rate is the underlying principle 
of oversampled AID converters. 
2. The parameter L, defined to be the number of chan- 
nels, alternates between two extremes : L = 1 and L = M .  
When L = 1, we get the best SNR improvement at the ex- 
pense of a more narrow class of inputs x(n). When L = M ,  
no noise variance reduction is achieved since the class of sig- 
nals is now unrestricted. We can also see this by noticing 
that the multirate interconnection in Fig. 5 becomes a per- 
fect reconstruction filter bank that is signal independent. 
The parameter L therefore determines the tradeoff between 
the generality of the class of signals, modeled as in Fig. 1, 
and the reduction in quantization noise variance. 
4. NOISE SHAPING BY LTI PRE- AND POST 
FILTERS 
Following the philosophy of sigm&delta modulators, we now 
perform noise shaping to achieve a further reduction in the 
average mean square error. To accomplish this, we propose 
using LTI pre- and post filters around the PCM quantizer 
as shown in Fig. 6. The goal is to optimize these filters such 
that the average m.s.e. a t  the output is minimized. The 
noise shaping filters to be optimized are not constrained 
to be rational functions (Le., of finite order). Non causal 
solutions, for example, are accepted. 
Although our quantizer design assumptions are the same as 
before, the quantizer input is not anymore the (CWSS)M 
process x(n),  but a filtered version of it, which we denote by 
z(n).  Following (l), the noise variance in this case is given 
by ui = ~ 2 - ~ ' u f  where uf is the average variance of the 
process z(n) .  It is then possible to express u: in terms of 
the prefilter P(ej'") and the so called average power spectral 
density of the process x(n),  denoted by SzX(e jw) ,  as follows: 
The average power spectral density is a familiar concept 
that arises when "stationarizing" a (CWSS)M process [lo] 
and satisfies the well known properties of the power spec- 
trum of a WSS process. It is defined to be the discrete 
time Fourier transform of the time averaged autocorrela- 
1 tion function k,,(k) given by - M 
M-1 
E[x(n)x*(n - k)]. 
n=O 
Theorem 2 [9] Consider the scheme of Fig. 6 under the 
assumptions of section 2. The optimum filter P(eiw) that 
minimizes the average mean square reconstruction error has 
the following magnitude squared response: 
For the case of L = 1, the expression becomes: 
(3) 
(4) 
and can be regarded as a multirate extension of the half 
whitening filter [8]. The coding gain of Fig. 6, defined as 
v-34 
the ratio &d;rect/&minr where &direct is the mean squared 
error obtained in the direct quantization case (Fig. 2) can 
be derived under the assumption that the JWSS processes 
yk (n ) ,  k = 0,1,. . . , L - 1, are uncorrelated: 
M J : ~  kzi s,, ( e jMw IFk (ej" ) l 2  $ 
Qopt = 
(J:~ d-:j~.)l~$ (ejw 112 IFn (ej" 112 
(5) 
The coding gain expression for the L = 1 case becomes 
where Bhw is the half whitening coding gain of the Wss  
process y(n) [8]. The factor M in (6) is due to the over- 
sampled nature of the signal z(n). It is interesting to note 
that the noise shaping contribution to Qopt in (6 ) ,  which we 
denote by Ghw, is exactly the coding gain we would obtain 
by  half whitening the W S S  process y ( n )  in the usual way. 
Due to space limitation, specific numerical examples are not 
included in this paper but can be found in [9, 111. 
5. NOISE SHAPING BY (LPTV)M PRE- AND 
posrr FILTERS 
In this section, we consider using ( L P T V ) M  pre- and post 
filters instead of LTI ones surrounding a periodically time 
varying ( (PTV)M) quantizer. Since the signal model z(n)  
is (CWSS)M, restricting ourselves to linear time iruvariant 
noise shaping filters and quantizers is a loss of generality. 
Any optimum configuration for such processes should con- 
sist of (LPTV)M filters surrounding a ( (PTV)M) quan- 
tizer. Using some well known multirate results, it can be 
shown that this new quantization configuration is equiva- 
lent to an M-channel maximally decimated filter bank with 
M subband quantizers [l]. Because the general ( L P T V ) M  
problem is difficult to track analytically, we will only study 
a special form of a perfect reconstruction filter bank. In spe- 
cific, we assume that the analysis polyphase matrix E(ej")  
is diagonal with diagonal elements equal to Vk(eJw).  The 
synthesis poIyphase matrix R(e j " )  is then also diagonal 
with diagonal elements elqua1 to l / V k ( e J w )  for each k.  
The quantization configiiration is shown in Fig. 7. The 
scalar quantizers labeled Q are modeled as additive noise 
sources Q k ( n )  and individually satisfy relation the (1). w e  
assume that the subband quantization noise sources qk (n) 
are white and pairwise iincorrelated, i.e., the noise power 
spectral density matrix iis given by 
The goal is then to joiiitly allocate 
under a fixed bit rate 
M - 1  
1 
b := ?i;i b k  
the subband bits bk 
(8) 
k=O 
and optimize Vk(ej") in order to minimize the average m.s.e. 
at the output of Fig. 7. 
Theorem 3 Consider the scheme of Fig. 7 under the above 
assumptions. The optimum filter Vopt,k(eJ") (for each k) 
that minimizes the average mean square reconstruction er- 
ror at the output has the following magnitude squared re- 
sponse: 
L-1 
where Sk(ej") = z S V i ( e j " ) l & k ( e j " ) l 2  is the power spec- 
trum of kth channel and k k ( e j w )  is the kth polyphase com- 
ponent of the i th  filter d ( e j " ) .  Using the above optimum 
filters, the coding gain of Fig. 7 is then given by: 
i=O 
The case of L = 1 yields again an interesting result. 
Theorem 4 Consider the scheme of Fag. 7 under the same 
above assumptions and with L = 1. The optimum filter 
Vopt(eJw) that minimizes the average mean square recon- 
struction error at the output is independent of k and has 
the following magnitude squared response: 
where S V V ( e J w )  i s the power spectrum of the WSS process 
y ( n ) .  With the above optimwnflter eqwessioon, the coding 
gain is then given by: 
where Rk(eJ")  is the kth polyphase component of F(eiw).  
The LTI case is indeed a loss of generality. Since 
the class of ( L P T V ) M  filters and ( P T V ) M  quantizers in- 
clude the LTI case, it is clear that the performance of this 
more general class of filters and quantizers is at least as 
good as the LTI one. We have already shown that the opti- 
mum (LPTV)M filter for the case of L = 1 reduces to a LTI 
one. The question then becomes: Is the (PTV)M quantizer 
providing any excess gain over the LTI case and if so, by 
how much ? It turns out that, even in this restricted form 
of ( L P T V ) M  filters, the coding gain of the above scheme is 
always greater than the LTI one except when the magnitude 
squared response of the polyphase components &(e'") of 
F(ej")  are equal for all k .  Indeed, it can be shown 191 that, 
the denominator of (6) is always 2 than the the denomi- 
nator of (12) with equality if and only if all l&(eJ")J2 are 
equal. Since the numerator is  the same in both cases, the 
claim is proved. For the more general case ( L  # l), we again 
expect the coding gain of the more general (LPTV)M case 
of Fig. 7 to be higher than the analogous LTI one of Fig. 6. 
However, the complexity of the expressions (5) and (10) in 
this case prevents a formal mathematical proof. 
v-35 
6. REFERENCES 
[l] P. P. Vaidyanathan, Multirate systems and filter banks. 
Englewood Cliffs, NJ: Prentice Hall, 1993. 
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[7] J. Tuqan and P. P. Vaidyanathan, “The role of 
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Quantizer 1 ,  1 
Figure 3: Schematic of the oversampling PCM technique 
Figure 4: The quantization scheme of Fig. 3 with noise 
shapers 
Quantizer 
x(n) 
Figure 5: Multirate quantization scheme for the multiband 
model 
L A  
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Figure 6: Noise shaping by LTI pre- and post filters for the 
multiband model 
Y 0 (n) z(n) 
Y , (n) 
0 
Figure 1: The multiband model 
Figure 2: Direct quantization of z(n) 
Figure 7: Noise shaping using (LPTV)M pie- and post 
filters for the multiband model 
V-36 


The design of optimum filters for quantizing a class of non  bandlimited signals

An overview of sigma-delta converters,”

Digital coding of waweforms.

Effects of multirate systems on the statistical properties of random signals,”

Globally optimal two-channel FIR orthonormal filter banks adapted to the input signal statistics,” ICASSP Proc.,

Multirate systems and filter banks.

Oversampling PCM techniques and optimum noise shapers for quantizing a class of nonbandlimited signals,”

Reconstruction of sequences from non uniform samples,”

The role of the discrete-time Kalman-Yakubovitch-Popov (KYP) lemma in designing statistically optimum FIR orthonormal filter banks,” ISCAS Proc.,

https://authors.library.caltech.edu/10786/1/TUQiscas98a.pdf

The design of optimum filters for quantizing a class of non bandlimited signals

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