A new source coding technique called MODULOâPCM. (MPCM) is ... linear predictive coding or transform coding. In contrast with ..... [6] R. Zelinski and P. Noil: Adaptive Transform. Coding of ... Acoustics, Speech and Signal Processing., Vol.
MODULO-PCM:
A
NEW SOURCE CODING SCHEME
Thomas Ericson and V. Ramamoorthy
The University of Linköping, Linkoping, Sweden
ABSTRACT
A
new source coding technique called MODULO—PCM (MPCM) is presented and it shown that this new scheme has essentially the same performance as linear predictive coding or transform coding. In contrast with the conventional schemes, MPCM employs a simple memoryless encoder and a moderately complex decoder incorporating the Viterbi algorithm. Bounds for distortion in MPCM systems fora first—order Gauss—Markov process are numerically
a)
ij
B1
B=
n
N-i
U
b)
i=O
A
B.=R 1
reconstruction unit is a mapping r:A —' IZ; i = O,M—l. The qunatizer is charac4 terized by the partitioning B-j}; the reconstruction unit is characterized by the reconstruction levels {yj}. Together they constitute a quantization scheme. This scheme is characterized by the quan— tization function f:R —* R; x —' where i -,
calculated.
r(i) yj;
f(x);
1. Introduction f(x) Most source coding schemes are such that the complexity is concentrated to the encoder while the decoder is usually very simple. In the present paper we propose a scheme where the opposite is true. The scheme is called modulo—PCM (MPCM). The fundamental feature can be described as a modulo N reduction of the quantized samplevalues. The purpose is to take advantage of possible correlation between successive samples in order to reduce the bit rate.
xEB;
i=O,M—l.
The rate of the quantization scheme is R
log M.
Let X be a random variable over R with probability density p(x). The distortion when quantizing X by means of the above quantization scheme is E[f(X) -
D
XJ
N-i =
i=O
J (y-x)
p(x) dx
B
This distortion is known as mean square error.
Our presentation is organized as follows. In chapter 2 some basic concepts and results regarding ordinary PCM are stated for reference purposes. In chapter 3 the modification defining MPCM is introduced. A heuristic discussion of the inherent possibilities of the proposed coding scheme is carried out in chapter 4. One of the basic problems in the reconstruction is to undo the modulo N reduction. This problem will be discussed in chapter 5. It will be shown that dynamic programming is a key to the solution of that problem. Finally, in chapter 6 some numerical results will be presented.
In ordinary PCM the sets B. are intervals. Let us introduce the notation B14xi,xl÷l), where x04-co; xp44c. We define
4 Jdx
=
(x)
xB.;
ii)
i=O,M-l.
The quantization is fine, i.e.
B
p(x')p(x')
Saturation is negligible, i.e. Pr{X
i=O,M-l.
i=O,P4—l
tions:
x',x"
Let R denote the set of real numbers and let A be the finite alphabet A = {O,l,...,M—l}. A quanti— zer is a mapping q: R -, A; x - q(x). N is called tIT number of quantization levels. We define
x1 — x;
Throughout we will make the following two assump-
1)
2. Basic Concepts and Results on Ordinary PCM
q(i);
A r[q(x)] =
B0
u BN1}
0.
Under these two conditions the following well known approximations can easily be derived:
The sets B1 are called quantization regions. They form a partitioning of R , the following two conditions are satisfied:
i.e.
DJ(x)2 p(x) dx 419
og2
r'.
r dx
As compared with q the quantizer q has a reduced number of quantization regions M, rather than KM. This implies a reduction in bit rate by logK. Still, as will be seen shortly, the performance might be essentially the same at the price, though, of a somewhat more complicated decoder. The problem is, of course, to reconstruct at the receiver the factors k..,€{O,l,...,K—l} for which q'(x\))=k.)F4÷q(x), as only q(x) ',=O,±l,... are available at the receiver. For a memoryless source, i.e. when the source variables x, v=O,±l,...,are statistically independent, this is clearly impossible for any K>i. For sources with memory, however, the correlation between successive source outputs may be successfully used to estimate k. However, before discussing that problem let us first investigate the inherent possibilities of our proposed coding scheme.
j
These approximate equations provide a parametric relation between rate R and distortion 0 in terms of the function (x). Of special interest is the case of a random variable X with uniform quantiz— ation, where xI< amax
(x) =
lxi> amax In that case we have
D'1a2max
2-2R
Often the overload value amax is chosen proportional to the standard deviation o of the source process. With such a design the distortion is
D=ccy2
2
4. Heuristic Evaluation of Performance Let us consider a Gaussian source and with spectral density
-2R (1)
where c is a constant.
(e)
An important characteristic of ordinary PCM is that the quantization as wefl as the reconstruction are performed by means of memoryless operations as described above. This is appropriate - although not optimal — when the source to be transmitted is mernoryless. An efficient coding ofasource with memory (i.e. with probabilistic dependence between successive samples) requires some procedure for making use of the inherent memory structure. Usually this is done by introducing memory in the encoder as well as in the decoder (c.f. DPCM). In MPCM, however, memory is introduced only in the decoder. MPCM is characterized by the fact that the sets B1 of the quantizer are not intervals. Instead we assume that each set is the union of some intervals Aij:
B
-1 U
A..;
=0
1
K-l
B.1
u
A., M
;
i=O,M—l.
k=O
where{A} defines a uniform quantizer, i.e.
all
the intervals Ai except A0 and AM_i have the same Notice, that the quantizer q corresponlength ding to {Bi} can be implemented as a uniform quan— tizer q' , where
.
q'(x)
=
i;
followed by q(x) =
q(x)
x€A1;
a
where p(k)Ex.+k x. We take mean square error as our measure of performance and we consider high bit rates R. Then by rate-distortion theory (see Gallager [1] or Berger [21) the mihimum distortion available with any coding scheme is O
= a2expJ log (0)dB
- 2R}
(2)
0
where
=
f(e)de 0
is the variance of x.,. As is well known several constructive practical coding schemes like transform coding and linear predictive coding meet this bound within a few dB. See, for example, Huang [4], Habibi [5], Zeiinski and Nell [6]. We will indicate that this is true also for MPCM. Without going into details regarding the procedure for estimating the sequence {k}— this problem will be discussed in the next chapter — we simply observe that what is needed is to be able to estimate q'(x.) with an error magnitude less then 1/2 in order to determine k., and once this quantity is known we simply use the formula q'(x)=kM+q(x,) to determine the exact value of q(x ). Hence, with such a procedure our MPCN system will give a mean square error which is the same as that obtained with an ordinary PCM system using the rate R=R+logK.
i=UMT.
An illustrative example is shown in fig. 1. In paticular,wewill be interested in the case when the sets B. have the form =
e20k
k
3. Moduio-PCM
B.1 =
p(k)
x with Ex=O
Suppose everything has worked well until time -l. This means that we have q(x._), q(x.)_2) available for estimating q(x). The process is Gaussian with spectral density (®). the quantization is fine - as we assume - the problem of estimating q(x) is nearly the same as that of estimating given x._l, x_2,... But this is the well known one step prediction problem. It is also known that the error variance of linear prediction is
If x
i=0,KM-l
x
reduction modulo N, i.e.
mod N.
This fact motivates our terminology, modulo-PCM.
420
n
expf
log
(O)
P[x'(l,n)] =
do)
0
Now with this error variance it is reasonable to design our system so that K=c/ci. With such a design the probability of estimating q'(x) with an error magnitude larger than M/2 will be of the same order of magnitude -as the probability of overloading the system, Pr{xEBQUBM_l}, which we, in accordance with well established design principles, have assumed to be very small. Hence, we have
and by (1)
Dc
2
=
2
2
f log (G) CO we obtain
log
1og(-)2
—2R'
c exp2{f 0
=co 2
log2
—2R
2
(O)
do
K
-
do
r(x)
{logP[x]÷
z
Eq[y(1,n)] =
max
logP[xIx1]}
logP[xIx1J+r_1(x,;_1)}.
Eq(y1) Hence, the maximization can be carried out recursively. In fact, the above structure is exactly the same as the one occuring in the recently much studied Viterbi algorithm, see for instance Forney [3].
The performance of a MPCM system has been numerically evaluated under the assumption that x, can be approximately modelled as a Gauss-Markov se-
of performance as obtained and with linear predictive
x = px1 +v'2'w;
{k}
Let be the output from the quantizer q' at time -v, i.e. =q'(x). We use x'(l,n) as a notation for the sequence xl, i.e. Now the problem is to x'(l,n)x1, estimate x'(l,n) given y(l,n)y, y2,...,yn; where y=x-kM; k€fO,l,...,K-l}; '=Ti. Let K{1),l,... ,K-l.} and let us define Eq(y) {.x': x'=kM+y;kEK}; Eq[y(l,n)]{x'(l,n):x=kM+y; kEK; =l,n}. Hence, Eq[y(l,n)]is exactly the set of x'-sequences from which y(l,n) can be generated. We now observe
x
P[x'(l,n)Iy(l,n)] =
x ( ,n) P[y(l,n)1 0
P[x'(l))
as
.
' X
r ii ,n fl1 LqLy
{w}
7. Conclusion
A
new source coding scheme called modulo-PCM, (MPCM), has been described. This coding scheme might be useful in situations where the source sequence is highly correlated. At high bit rates the performance is essentially the same as that of transform coding and linear predictive coding. The foremost difference between MPCM and most other source coding schemes is with respect to implementation. For a MPCM system implementation complexity will be concentrated at the decoder while the coder will be very simple. For most other source coding schemes the opposite is true.
; x'(l,n)Eq[y(l,n)]
a
p1
