We derive the rate-distortion region for the two-channel multiple description problem on stationary discrete ergodic and nonergodic sources with alphabets admitting an ergodic decomposition. The results do not provide a single-letter representation for the rate-distortion region on i.i.d. sources

Effros, Michelle

Fleming, Michael

English

Caltech Authors - Main

ISlT 2000, Sorrento, Italy, June 25-30,2000 
The Rate Distortion Region for the Multiple Description Problem' 
Michael Fleming 
Dept. of Electrical Eng. (136-93) 
Cdiornia Institute of Technology 
Pasadena, CA 91125, USA 
Abstract - We derive the rate-distortion region for 
the two-channel multiple description problem on sta- 
tionary discrete ergodic and nonergodic sources with 
alphabets admitting an ergodic decomposition. The 
results do not provide a single-letter representation 
for the rate-distortion region on i.i.d. sources. 
I .  INTRODUCTION 
In multiple description (MD) source coding with two channels, 
a source is described at  two different rates, and each descrip 
tion is sent over a separate channel to the receiver. Each 
channel has some probability of breaking down, in which case 
all of the data sent on that channel is lost. :only channel i 
is working, the receiver makes reproduction X; with average 
distortion D; using the rate-& description sent on channel i. 
When both channels work, the receiver makes reproduction 
X12 with average distortion D12 using the description pro- 
vided by combining the information on both channels with 
ap additional ratel312 description. The descriptions of both 
XI and X2 are a e a b l e  when decoding X12; the additional 
rate Ria spent on X12 can be treated as refinement and split 
arbitrarily between the two channels. 
Other authors have found upper and lower bounds on the 
achievable rate distortion region (the set of achievable vectors 
(R,D) = ( R I ,  R2, R12, D1, D2, D12)) for the two-channel mul- 
tiple description of an i.i.d. source. The bounds do not match 
for all sources. We present a new achievability theorem and 
matching converse giving the achievable rate-distortion region 
for both ergodic and nonergodic sources. On i.i.d. sources, our 
converse is similar to that of [l], and our achievability result 
uses an existing achievability result from [2]. We use a La- 
grangian approach and closely parallel [3]. 
Both the converse and achievability proofs make use of dis- 
tributions with a property that we call n-block conditional 
independence, defined in the following section. The use of 
distributions with this property arises from the obseryation 
that the bounds of [l] and [2] match when XI and X2 are 
conditionally independent given X. While such conditional 
independence is not observed on a symbol-by-symbol b+asis,it 
arises naturally when using n-dimensional codes since X,", X," 
and X Y 2  are all uniquely determined by X". 
11. N-BLOCK CONDITIONAL INDEPENDENCE 
Let the elements of a one-sided infinite sequence Y be 
denoted Yl,  Y2,. . . . We divide these elements into n- 
blocks as Y(i)  = yLl),,+'. We say that distribu- 
tion q(x12, kl, falx) has n-block conditional independence 
if q(x12, xl , x2 Ix) = nCp=, q2 ( x 1 2  (k), xi (k), x2 (k) Ix(k)), and 
q,"(i$,2;lzn) = q2(2;lz")q2(&;Izn). Define r(n) to be the 
'This work was supported by a Pickering Fellowship, NSF grant 
CCR-9909026, Caltech's Lee Center for Advanced Networking, an 
F.W.W. Rhodes Memorial Scholarship, and a Redshaw Award. 
Michelle Eflfos 
Dept. of Electrical Eng. (136-93) 
California Institute of Technology 
Pasadena, CA 91125, USA 
set of all distributions q(f12, kl ,  f2 Ix) with n-block condi- 
tional independence for a particular n, and let I' = U,"==, I'(n). 
111. RESULTS 
Let A be a discrete source alphabet, and define A" to be 
the set of one-sided sequences from A. Let p be a stationary 
source with alphabet Aw and marginal p" on A". Let A be 
a discrete reproduction alphabet, and let p : A x A -+ [O,oo) 
be a nonnegative distortion measure. We assume that there 
exists a reference letter y' E A such that Epp(z ,  y) = d' < 00. 
Define p(z",y") = Cyz1 p(zi,y;). 
We denote an MD quantiEer of blocklength n by Q" = 
(Q;,Q;,QY2). For each S, Qg maps 4" onto some finite 
or countable set of codewords C," from A". We assume that 
the descriptio? of X,"2Amade by QY2 is a refinement of the 
descriptions X," and X," made by Q; and Q; respectively, 
since these indivi$ual descriptions are available to the decoder 
when decoding Xr2. The codeword descriptions are assumed 
to be uniquely decodable. 
The set RCp) of asymptotically achievable rate-distortion 
vectors (R,D) is, by a timesharing argument, a convex set, 
and can be entirely characterized by its support functional 
The weighted rate-distortion function is delbed as 
J(a,P,p) = vJn(a,P,P), 
where 
Stationary Ergodic Sources: When p is stationary and 
ergodic, the following result holds. 
Theorem 1: j(u,@,p) = J ( a , P , p ) .  
Stationary Nonergodic Sources: When p is stationary 
and nonergodic, but an ergodic decomposition (px : x E A - }  
of p exists, then the following results hold. 
Theorem 2: J ( a , P , p )  = J(a,P,px)dp(x). 
Theorem 3: j(a,P,p) = J(a,p,p). 
REFERENCES 
[l] S. Sher and M. Feder, "A converse theorem for the multiple 
description problem," Eighteenth Convention of Electrical and 
Electronics Engineers an Israel, pp. 1.1.411-4, Mar. 1995. 
[2] A. El Gama1 and T.M. Cover, "Achievable rates for multiple 
descriptions," IEEE %ns. Info. Theory, vol. 28, pp. 851-857, 
Nov. 1982. 
[3] M. Effros, "Distortion-rate bounds for fixed- and variablerate 
multi-resolution source codes," IEEE %ns. Info. Theory, vol. 
45, pp. 1887-1910, Sep. 1999. 
208 
0-7803-5857-0/00/%10.00 02000 IEEE. 


The rate distortion region for the multiple description problem

A converse theorem for the multiple description problem,&quot;

Achievable rates for multiple descriptions,&quot;

Distortion-rate bounds for fixed- and variablerate multi-resolution source codes,&quot;

http://authors.library.caltech.edu/7411/1/FLEisit00.pdf

The rate distortion region for the multiple description problem

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