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[7] I. Mantin, “Predicting and distinguishing attacks on RC4 keystreamgenerator,” in Adv. Cryptol. — Eurocrypt 2005, LNCS 3494. NewYork: Springer-Verlag, 2005, pp. 491–506.

[8] I. Mantin, “A practical attack on the fixed RC4 in the wep mode,” inAdv. Cryptol. — Asiacrypt 2005, LNCS 3788. New York: Springer-Verlag, 2005, pp. 395–411.

[9] R. J. Jenkins, Jr., ISAAC Fast Software Encrypt., FSE 1996, LNCS1039. New York: Springer-Verlag, 1996, pp. 41–49.

[10] B. Zoltak, “VMPC one-way function and stream cipher,” in Fast Soft-ware Encrypt., FSE 2004, LNCS 3017. New York: Springer-Verlag,2004, pp. 210–225.

[11] A. Maximov, “Two linear distinguishing attacks on VMPC and RC4Aand weakness of the RC4 family of stream ciphers,” in Fast SoftwareEncrypt., FSE 2005, LNCS 3557. New York: Springer-Verlag, 2005,pp. 342–358.

[12] Y. Tsunoo, T. Saito, H. Kubo, M. Shigeri, T. Suzaki, and T. Kawa-bata, “VMPC and RC4A attack,” in SKEW 2005 — Symmetric KeyEncryption Workshop, 2005 [Online]. Available: http://www.ecrypt.eu.org/stream/papersdir/037.pdf

[13] E. Biham and J. Seberry, Py (Roo): A Fast and Secure Stream CipherUsing Rolling Arrays Report 2005/023, 2005 [Online]. Available:http://www.ecrypt.eu.org/stream/py.html, eSTREAM, the ECRYPTStream Cipher Project

[14] S. Paul, B. Preneel, and G. Sekar, Distinguishing Attacks on theStream Cipher Py Report 2005/081, 2005 [Online]. Available:http://www.ecrypt.eu.org/stream/py.html, eSTREAM, the ECRYPTStream Cipher Project

[15] P. Crowley, “Improved cryptanalysis of Py,” in SASC 2006 — StreamCiphers Revisited, Work. Rec., 2006, pp. 52–60 [Online]. Available:http://www.ecrypt.eu.org/stvl/sasc2006

[16] E. Biham, Pypy: Another Version of Py [Online]. Available:http:www.cs.technion.ac.il/biham FSE 2006 Rump Session

[17] Y. Nawaz, K. C. Gupta, and G. Gong, A 32-bit RC4-like KeystreamGenerator [Online]. Available: http://eprint.iacr.org/2005/175 Cryp-tology ePrint Archive 2005/175.

[18] H. Wu, Cryptanalysis of a 32-bit RC4-Like Stream Cipher [Online].Available: http://eprint.iacr.org/2005/219 Cryptology ePrint Archive,2005/219.

[19] I. Mantin and A. Shamir, “A practical attack on broadcast RC4,” inFast Software Encrypt., FSE 2001, LNCS 2355. New York: Springer-Verlag, 2001, pp. 152–164.

Multiterminal Source–Channel Communication Over anOrthogonal Multiple-Access Channel

Jin-Jun Xiao, Member, IEEE, and Zhi-Quan Luo, Fellow, IEEE

Abstract—We consider the problem of multiterminal source–channelcommunication where a number of distributed and possibly correlatedsources are transmitted through an orthogonal multiple access channelto a common destination. We provide a characterization of the optimaltradeoff between the transmission cost � and the distortion vector Das measured against individual sources. Our approach consists of twosteps: 1) a multiple-letter characterization of the rate-distortion regionof the multiterminal source coding and 2) a source–channel separationtheorem ensuring that all achievable pairs of (�; D) can be obtained bycombining the rate-distortion region and the orthogonal multiple accesschannel capacity region. As a corollary, we determine the optimal powerand distortion tradeoff in a quadratic Gaussian sensor network underorthogonal multiple access, and show that separate source and channelcoding strictly outperforms the uncoded (amplify-forward) transmission,and is in fact optimal in this case. This result is in sharp contrast to thecase of nonorthogonal multiple access for which separate source andchannel coding is not only suboptimal but also strictly inferior to uncodedtransmission.

Index Terms—Gaussian sensor networks, multiterminal source coding,rate-distortion region, source–channel separation theorem, uncoded trans-mission.

I. INTRODUCTION

Consider a multiterminal source–channel communication systemwhere L correlated sources are transmitted to a common destinationthrough a multiple access channel (see Fig. 1). The encoders aredistributed and cannot cooperate, and are subject to a transmissioncost constraint � (e.g., transmission power). The receiver wishes toreconstruct the L sources using the received signals to achieve certainfidelity levels signified by a distortion vector D 2 L. The multiter-minal source–channel communication problem is to characterize allcost–distortion pairs (�; D) achievable by any coding strategy in aninformation-theoretic sense regardless of delay and complexity.

The source coding part of this problem is to characterize the rate-dis-tortion region which consists of all rate-tuplesR = (R1; R2; . . . ; RL)that allow for the reconstruction of the L source signals within theirrespective fidelity levels (denoted by vector D), when the sources aredistributively encoded at rates not more than Ri respectively and themultiple access channel is noiseless. The multiterminal source codingof correlated sources was first studied by [22] in which the well-knownSlepian–Wolf region was obtained for the case of lossless coding ofdiscrete memoryless sources. The lossy coding of correlated sourceswas first studied in [30], and subsequently in [3], [17], [23], [27], and[29]. Despite these efforts, the lossy multiterminal source coding re-mains an open problem to this date. Various inner and outer bounds ofthe rate-distortion region have been proposed in the literature, includingthe Berger–Tung inner region which was derived using the random bin-ning technique [9], [22] and the generalized Markov lemma [14]. The

Manuscript received November 8, 2005; revised February 27, 2007. Thiswork was supported in part by the National Science Foundation under GrantDMS-0610037 and by the USDOD ARMY under Grant W911NF-05-1-0567.

The authors are with the Department of Electrical and Computer Engineering,University of Minnesota, Minneapolis, MN 55455 USA (e-mail: xiao@ece.umn.edu; luozq@ece.umn.edu).

Communicated by R. R. Müller, Associate Editor for Communications.Color versions of Figures 1–3, 5, and 6 in this correspondence are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2007.903142

0018-9448/$25.00 © 2007 IEEE

3256 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007

Fig. 1. A multiterminal source–channel communication system.

Berger-Tung inner region was first given in [3] and [23] for sourceswith finite alphabets, it was then extended to the Gaussian source casein [17]. For the special case of two correlated Gaussian sources undermean squared distortions, the tightness of the Berger–Tung achievableregion has been proved in [28].

In several important cases of the source–channel communicationproblem, source coding and channel coding can be separately pursuedwithout incurring any performance loss. For example, in his originalinformation theory paper [21, Theorem 21], Shannon proved that,in a point-to-point link, source coding and channel coding can beperformed separately without performance degradation if the sourceand channel are both discrete and memoryless. This source–channelseparation theorem is quite appealing from a practical standpointsince it implies that source coding can be performed without channelknowledge and similarly for the channel encoding. Unfortunately, thesource–channel separation theorem does not extend to general networkcommunication models [9, Ch. 14]. For instance, an interesting coun-terexample was provided in [8] for the case of lossless transmissionof correlated sources through an interfering (non-orthogonal) multipleaccess channel. In this case, the separate source and channel codingis not optimal and is strictly inferior to the joint source and channelcoding.

In this correspondence, we consider the problem of multiter-minal source–channel communication under orthogonal multipleaccess. Examples of such multiaccess schemes include the standardstrategies of time–frequency/code-division multiple-access (TDMA/FDMA/CDMA). We first give a simple multiple-letter characteriza-tion of the rate-distortion region for the lossy multiterminal sourcecoding part of this problem. The achievability of this region isderived using a generalized random binning technique, while theconverse is established via elementary information inequalities. Thismultiple-letter characterization of the rate-distortion region enablesus to prove a source–channel separation theorem for this class ofmultiterminal source–channel communication problems. Our resultshows that all achievable transmission cost and distortion pairs (�; D)can be characterized by the rate-distortion region for the multiterminalsource coding and the channel capacity region under input constraint�. A general separation theorem has been established in [15], andlater in [1] and [16] for the lossless transmission of correlated sourcesfrom finite alphabets through independent channels. Our work extendsthe separation results to the lossy transmission case for orthogonalmultiple access channels.

Our work is closely related to the so-called CEO problem [4]where L distributed nodes make noisy observations of a commoninformation source and transmit locally encoded messages througha multiple access channel to a fusion center for the final estimationof the source. The CEO problem naturally models some recently

emerging applications such as the joint estimation of a common sourceby a sensor network with a fusion center [31], [32]. We show that theaforementioned multiple-letter characterization of the rate-distortionregion and source–channel separation theorem can be extended to theCEO problem under orthogonal multiple access. For the special caseof a Gaussian source estimation under the mean squared distortionmeasure, our results show that the optimal tradeoff between totalsensor power and final distortion is achieved by separate source andchannel coding, and the resulting “digital” strategy strictly outper-forms the “analog” uncoded transmission strategy. This result shouldbe contrasted to the case of non-orthogonal multiple access for whichthe “analog" uncoded transmission strategy is known to significantlyoutperform the digital approach of separate source and channel coding[10]–[12].

Throughout this correspondence, we adopt the following notations.i) A capital letter (e.g., U ) denotes a random variable, and the cor-

responding lower case letter (e.g., u) denotes its realizations. Fora random variable U , its distribution is denoted by PU (u) (orP (u) for simplicity), and the set from which U takes value isdenoted by the calligraphic letter U .

ii) A random vector (U(1); U(2); . . . ; U(k)) is denoted by Uk .iii) Notation A ! B ! C means that fA;B;Cg forms a Markov

chain.iv) Let IL

def= f1; 2; . . . ; Lg. For any index set

A = fi1; i2; . . . ; ikg � IL

the notation XAdef= fXi ; Xi ; . . . ; Xi g.

v) For any vector D = (D1;D2; . . . ; DL) and scalar �, the sumD + �

def= (D1 + �; D2 + �; . . . ; DL + �). For any two vectors

D(j) = (D(j)1 ; D

(j)2 ; . . . ; D

(j)L ) where j = 1; 2, the notation

D(1) � D(2) means D(1)i � D(2)i for all i.

vi) For any set R and scalar c, the multiplication cRdef= fcR jR 2

Rg. The sum of two setsR1 andR2 is denoted byR1+R2def=

fr1 + r2 j r1 2 R1; r2 2 R2g. Also for any set S, conv(S)denotes its convex hull.

II. MULTITERMINAL SOURCE–CHANNEL COMMUNICATION

Consider the multiterminal source–channel communication systemshown in Fig. 2. Let fXi(t) : t 2 +; i = 1; 2; . . . ; Lg denoteL tem-porally memoryless (with time index t), but spatially correlated sources(with spatial index i). EachXi takes values from a set of discrete alpha-bets. The instantaneous joint probability of fXi(t) : i = 1; 2; . . . ; Lgis denoted by PX ;X ;...;X (x1; x2; . . . ; xL).

For each 1 � i � L, encoder i maps a block of k source symbolsXki to a block of n channel symbols U

ni . The corresponding encoder

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007 3257

Fig. 2. A multiterminal source–channel coding system with source–channelcoding rate k=n � �.

function is denoted by fi so that Uni = fi(Xk

i ). The orthogonal mul-tiple access channel is described by a probability function

PY jU (y j u) =

L

i=1

PY jU (yi jui)

where U = (U1; U2; . . . ; UL) are the channel inputs, andY = (Y1; Y2; . . . ; YL) are channel outputs. We define cost func-tions associated with the transmission of inputs Ui for all subchannels:

�i : Ui 7!+; for all 1 � i � L: (1)

The transmitted channel symbols fUni : 1 � i � Lg need to satisfycost constraints

E (�i(Un

i )) � �i; where �i (Un

i )def=

1

n

n

t=1

�i(Ui(t)) (2)

for some � = (�1;�2; . . . ;�L) > 0. The decoder generates esti-mates of the input symbols fX̂ki : 1 � i � Lg based on the re-ceived signals fY ni : 1 � i � Lg according to a decoding func-tion g = (g1; g2; . . . ; gL) so that X̂ki = gi(Y

n

1 ; Yn

2 ; . . . ; Yn

L ). Theresulting distortion levels achieved at the destination is denoted by� = (�1;�2; . . . ;�L) with

�i = E di Xk

i ; X̂k

i

where di Xki ; X̂k

i

def= 1

k

k

t=1 di(Xi(t); X̂i(t)) (3)

and each di(xi; x̂i) is a given nonnegative distortion function associ-ated with source Xi.

Suppose the source–channel coding rate is fixed to be �, or in otherwords, on average every � source symbols are mapped into one channelsymbol. For any cost-tuple � = (�1;�2; . . . ;�L) and distortion-tupleD = (D1;D2; . . . ; DL), we say (�;D) is admissible if there arecoding schemes which can achieve distortion infinitesimally close toD while the required transmission cost is not more than �. More pre-cisely, (�;D) is admissible if for any � > 0, there exist k; n � 1 withk=n � �, and coding scheme ff1; f2; . . . ; fL; gg satisfying the trans-mission cost constraints (2) such that �i � Di+ �, for all 1 � i � L.Our goal is to determine all admissible pairs of (�;D).

A. Multiterminal Source Coding: A Characterization of theRate–Distortion Region

The source coding part of the problem is to characterize the rate-dis-tortion region. For any distortion D = (D1;D2; . . . ; DL), we saythat a rate vector R = (R1; R2; . . . ; RL) is achievable if for any� > 0, there exist some integer k � 1, L distributed source encoders

ff(s)1 ; f

(s)2 ; . . . ; f

(s)Lg such that for each i; Xki is encoded into code-

words Ci = f(s)i

(Xki ) satisfying

1

kH(Ci) � Ri + �; for all 1 � i � L (4)

and a joint decoding function g(s) = (g(s)1 ; g(s)2 ; . . . ; g

(s)L

) such thatthe estimates X̂ki = g

(s)i

(C1; C2; . . . ; CL) achieve distortions

E di Xk

i ; X̂k

i � Di + �: (5)

For any distortion D, the rate-distortion region R(D) is simply theset of of all achievable rate tuples. We list some properties of the rate-distortion region R(D) in Appendix A.

Definition 2.1: Given any rate distortion pair (R;D) and k � 1.We say (R;D) is k-admissible if there exists a random vector(Z1; Z2; . . . ; ZL) such that

i) ZA ! XkA ! (Xk

A ; ZA ) for any A � IL.ii) There exists deterministic functions gi such that

E(di(Xk; X̂k)) � Di where X̂ki = gi(Z1; Z2; . . . ; ZL) for

all 1 � i � L.iii) For any A � IL, there holds

i2A

Ri �1

kI XkA;ZA jZA :

Also, we let QkX denote the convex hull of all k-admissible pairs of(R;D), that is

QkXdef= convf(R;D) j (R;D) is k � admissibleg: (6)

The convex hull of all admissible pairs of (R;D) is denoted by

QXdef= conv (R;D) 2

k�1

QkX :

For the multiterminal source coding problem, Berger and Tung [3],[23] derived the following achievable region.

Theorem 2.2 (Berger–Tung Inner Region [3], [23]): For anyD � 0,the following rate region:

RBTin (D)def= (R1; . . . ; RL) j (R;D) 2 Q

1X

is achievable. That is, RBTin (D) � R(D).

The above Berger-Tung inner region is a single-letter characteriza-tion (i.e., k = 1), while such an inner region can be directly extendedto a larger region using a multiple-letter representation. The followingresult shows that the Berger–Tung’s inner region extended to the mul-tiple-letter representation is exactly the rate-distortion region.

Lemma 2.3 (Multiple-Letter Characterization of the Rate-DistortionRegion): For any D � 0, let us define

R�(D)def= f(R1; . . . ; RL) j (R;D) 2 QXg: (7)

Then R�(D) = R(D).Proof: The achievability of any rate-tuple in R�(D) is a direct

application of Theorem 2.2 on block source fXki : 1 � i � Lg. Wethus obtain R�(D) � R(D). Next we prove the converse: R(D) �R�(D).

Assume that the rate tuple (R1; R2; . . . ; RL) is achievable at distor-tion D, i.e., (R1; R2; . . . ; RL) 2 R(D). Then for any � > 0, thereexist some integer k � 1, encoding functions (f (s)1 ; f

(s)2 ; . . . ; f

(s)L

)

3258 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007

and decoding functions (g(s)1 ; g(s)2 ; . . . ; g

(s)L ) such that (4) and (5) are

satisfied.We can see that for any A � IL, the mutual information

I(CA; (XkA ; CA ) jX

kA) = 0 since for each i; Ci = f

(s)i (X

ki )

where f (s)i is a deterministic function. This implies that CAand (XkA ; CA ) are conditionally independent given X

kA. So

CA ! XkA ! (X

kA ; CA ) is a Markov chain. As a result, we have

H(CA) � I XkA;CA � I X

kA;CA jCA : (8)

Therefore, we obtain

i2A

(Ri + �)(a)

�i2A

1

kH(Ci) �

1

kH(CA)

(b)

�1

kI XkA;CA jCA (9)

where (a) follows from (4), and (b) is due to (8). Thus, from (5), (9) andDefinition 2.1, we obtain (R1+�; R2+�; . . . ; RL+�) is k-admissiblefor distortion level D+ �. It follows from the definition of R�(D+ �)in (7) that

(R1 + �; R2 + �; . . . ; RL + �) 2 R�(D + �):

Since (R1; R2; . . . ; RL) is any rate-tuple in R(D), so

R(D) + � � R�(D + �):

Letting �! 0 and applying Lemma 4.2 in the Appendix, we obtain

R(D) = lim�!0

R(D) + � � lim�!0

R�(D + �)

= R�(D):

The proof is complete.

Lemma 2.3 provides a multiple-letter characterization of the rate-distortion region for multiterminal source coding. Unfortunately theregion is not efficiently computable due to the necessity of checkinginfinite block length k. In this sense, a single-letter characterizationof the rate-distortion region is desirable, and it still remains an openproblem. However, the multiple-letter representation of the rate-dis-tortion region will be used to determine all admissible cost–distortionpairs (�; D) for the multiterminal source–communication problem. Inwhat follows, we establish a separation theorem for the multiterminalsource–channel communication system.

B. A Separation Theorem

Let the orthogonal multiple access channel be describedby the conditional probability distribution PY jU (y ju) =

L

i=1 PY jU (yi jui), where U = (U1; U2; . . . ; UL) are channelinputs, and Y = (Y1; Y2; . . . ; YL) are channel outputs. For any givencost � = (�1;�2; . . . ;�L), introduce

Ci(�i) = maxE(� (U ))��

I(Ui;Yi): (10)

Then it is easy to see that capacity region C(�) of this orthogonal mul-tiple access channel under input cost constraint � is given by

C(�) = f(R1; . . . ; RL) j 0

� Ri � Ci(�i); for all 1 � i � Lg: (11)

We now present the following separation theorem which givesa characterization of all admissible pairs of (�;D) for the

source–channel communication problem using the rate-distortionregion R(D) given in Lemma 2.3 and the channel capacity region(11).

Theorem 2.4: Let � be the source–channel coding rate for the com-munication system depicted in Fig. 2. Then, the transmission cost anddistortion pair (�;D) is admissible if and only if C(�)\�R(D) 6= ;.

To prove Theorem 2.4, we need the following lemma.

Lemma 2.5: For any jointly distributed random variablesfX1; . . . ; XL; Y1; . . . ; YLg, if

PY jX (yI j xI ) =L

i=1

PY jX (yi j xi) (12)

then for any index set A � IL, there holds that

I(XA;YA) �i2A

I(Xi;Yi):

The equality holds if and only if PY (yI ) = i2I PY (yi), i.e.,Yis are independent.

Proof: By the definition of mutual information, we obtain

I(XA;YA) = h(YA)� h(YA jXA)

�i2A

h(Yi)� h(YA jXA)

(a)=

i2A

h(Yi)�i2A

h(Yi jXi)

=i2A

I(Xi;Yi)

where (a) follows from (12) which implies PY jX (yA j xA) =

i2A PY jX (yi j xi).

Proof of Theorem 2.4: When C(�) \ �R(D) 6= ;, the admis-sibility of (�;D) follows from the strategy of separate source andchannel coding and the operational meaning of rate distortion regionand channel capacity. The detailed argument is analogous to the proofof [21, Theorem 21]. Next we show the converse by proving that if(�;D) is admissible, then C(�) \ �R(D) 6= ;.

If (�;D) is admissible, then for any � > 0, there exist integers k andn with k=n � �, distributed encoders fi and joint decoder g whichare depicted in Fig. 2 with achieved distortion �i � Di + �. For any1 � i � L and 1 � j � n, we define

Ri(j)def= I(Ui(j);Yi(j))

and

�i(j)def= E(�i(Ui(j))): (13)

Then it follows from (10) that Ri(j) � Ci(�i(j)). The cost con-straint (2) implies that 1

n

n

j=1 �i(j) � �i. If we introduce R�i =

1n

n

j=1 Ri(j), then

R�i =1

n

n

j=1

Ri(j) �1

n

n

j=1

Ci(�(j))� Ci(�i) (14)

where the last step results from the convexity of capacity region ininput constraints, which can be proved by a time-sharing argument.Thus from the definition of channel capacity in (11), we obtain

(R�1; R�2 ; . . . ; R

�L) 2 C(�): (15)

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007 3259

On the other hand, we notice that

R�i(a)=

1

n

n

j=1

I(Ui(j);Yi(j))

=1

n

n

j=1

(h(Yi(j))� h(Yi(j) jUi(j)))

(b)=

1

n

n

j=1

h(Yi(j))�1

nh (Y ni jU

ni )

�1

nh (Y ni )�

1

nh (Y ni jU

ni ) =

1

nI (Uni ;Y

ni )

(c)

�1

nI Xki ; Y

ni (16)

where (a) follows from the definition of R�i ; (b) is due to the discretememoryless assumption of the channel; and (c) follows from the dataprocessing inequality. Therefore, there holds

i2A

R�i �1

ni2A

I Xki ;Yni

(a)

�1

nI XkA;Y

nA

(b)

�1

nI XkA;Y

nA j Y

nA : (17)

In (17), (a) is due to Lemma 2.5, while (b) follows from the propertyof Markov chain Y nA ! X

kA ! Y

nA . Thus, from the fact k=n � �

and (17), we obtain

��1

i2A

R�i �n

ki2A

R�i �1

kI XkA;Y

nA jY

nA ; for allA � IL:

Choosing fY ni gLi=1 as auxiliary random variables and using Definition

2.1, we conclude that (��1R�;D + �) is k-admissible, where R�def=

(R�1; R�2; . . . ; R

�L). Thus, we have (�

�1R�;D + �) 2 QkX � QX ,which by Lemma 2.3 further implies that

��1R� 2 R�(D + �) = R(D+ �)

or equivalently

R� 2 �R(D + �): (18)

Conditions (15) and (18) imply that C(�) \ �R(D + �) 6= ; for any� > 0. Since C(�) and �R(D) are both closed sets, it follows from theright-continuity ofR(D) (c.f. Lemma 4.1 in Appendix A) that C(�)\�R(D) 6= ;.

We remark that it is possible to extend the above multiterminal sepa-ration theorem to the case when sources and channels are stationary andergodic (rather than memoryless). A detailed discussion on the separa-tion theorem for a point-to-point communication system can be foundin [24] and [25].

III. APPLICATION TO THE CEO PROBLEM

The so-called CEO problem was first introduced in [4], and subse-quently studied in [4], [6], [18], [20], and [26]. In this section, we focuson the source–channel communication of the CEO problem under or-thogonal multiple access; see Fig. 3.

In such a system, there is a discrete memoryless sourcefS(t) : 1 � t < 1g which has instantaneous probability dis-tribution PS(s) and takes value from a discrete-alphabet set S . Forevery t, there are L observations fXi(t) : 1 � i � Lg of S(t), whichare also with discrete alphabets, and follow a conditional probabilityPX ;...;X S(x1; . . . ; xL j s). Thus, (S;X1; . . . ; XL) have a jointdistribution

PS;X ;...;X (S; x1; . . . ; xL) = PS(s)PX ;...;X jS(x1; . . . ; xL j s):

Fig. 3. The CEO problem with orthogonal multiple access.

As depicted in Fig. 3, the source–channel encoders ff1; f2; . . . ; fLgare distributed and can not cooperate, while the encoded signals aretransmitted through orthogonal subchannels to a common destinationwhere the final estimate Ŝk is generated using the received signals(Y n1 ; Y

n2 ; . . . ; Y

nL ). The decoding is performed through a deterministic

function g with Ŝk = g(Y n1 ; Yn2 ; . . . ; Y

nL ). The achieved distortion �

is defined as

� = E(d(Sk; Ŝk))

with

d(Sk; Ŝk)def=

1

k

k

t=1

d(S(t); Ŝ(t)) (19)

where d : S � Ŝ ! + is a given distortion function.Similar to the general source–channel communication problem, for

any given source–channel coding rate �, we say that the transmissioncost and distortion pair (�;D) is admissible if for any � > 0, thereexists k; n � 1 with k=n � � and coding schemes (f1; f2; . . . ; fL; g)satisfying the cost constraints (2) and achieving a distortion level � �D + �. Our goal is to determine all admissible pairs of (�;D) in aninformation-theoretic sense.

Remark 3.1 (Relation to the Multiterminal Source–Channel Commu-nication Problem): The CEO problem is a special case of the multiter-minal source–channel communication problem in the following sense:each CEO problem is equivalent to a multiterminal source–channelcommunication problem with higher dimension, but in which somecost and distortion values are trivially assigned. Specifically, for theCEO problem with source S, observations fXi : 1 � i � Lg, costconstraints f�i : 1 � i � Lg, and distortion D, it is equivalent tothe multiterminal source–channel communication problem with sourcefXi : 0 � i � Lg with X0

def= S, distortion tuple fDi : 0 � i � Lg

with D0 = D and Di = +1 for all i � 1, and cost constraintsf�0;�1; . . . ;�Lg with �0 = 0.

A. Rate-Distortion Region and Separation Theorem for the CEOProblem

For anyD > 0, the rate-distortion regionR(D) of the CEO problemconsists of rate-tuples (R1; . . . ; RL) such that for any � > 0, the sourcesignal S can be reconstructed at a distortion level not exceeding D+ �,when the sensors’ observations are encoded at a rate of no more thanRi + �.

Similar to the multiterminal source coding, there is a correspondingBerger-Tung [6] inner region for CEO problem. In fact, such a regioncan be obtained from that of the multiterminal source coding based onobservations similar to Remark 3.1. For the CEO problem with obser-vations fXi : 1 � i � Lg and distortion D, we define a multiterminalsource coding problem with source fXi : 0 � i � Lgwhere X0 = S,

3260 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007

distortion tuple fDi : 0 � i � Lg with D0 = D and Di = +1for all i � 1. We useRmul(D;1; . . . ;1) to denote its rate distortionregion. Then the rate-distortion region of the CEO problemR(D) cor-responds that part of Rmul(D;1; . . . ;1)g when the rate assigned toX0 (or S) is zero, i.e., R(D) = f(R1; . . . ; RL) j (0; R1; . . . ; RL) 2Rmul(D;1; . . . ;1)g, from which we can obtain the multiple-lettercharacterization of R(D).

Definition 3.2: Given any distortion D, rate-tuple R =(R1; R2; . . . ; RL), and k � 1, we say (R;D) is k-admissible ifthere exists a random vector (Z1; Z2; . . . ; ZL) such that

i) ZA ! XkA ! (Sk;XkA ; ZA ) for any A � IL.ii) There exists a deterministic function g with

Ŝk = g(Z1; Z2; . . . ; ZL) and E(d(Sk; Ŝk)) � D.iii) For any A � IL, there holds that

i2A

Ri � 1kI X

k

A;ZA jZA :

Lemma 3.3 (Multiple-Letter Characterization of R(D)): For anyD � 0, the rate distortion region of the CEO problem

R(D) = f(R1; R2 . . . ; RL) j (R;D)is k�admissible for some k � 1g: (20)

From Remark 3.1 and the separation result in Theorem 2.4, we obtainthe separation theorem for the source–channel communication of theCEO problem.1

Lemma 3.4: Let � be the source–channel coding rate in Fig. 3.Then a cost–distortion pair (�;D) is admissible if and only if C(�) \�R(D) 6= ;, where C(�) is given by (11).

We next specialize Lemma 3.4 to a Gaussian sensor network (underorthogonal multiple access) and show that separate source–channelcoding achieves optimal power-distortion tradeoff.

B. Gaussian Sensor Network

In a Gaussian sensor network (see Fig. 4), the source data sequencefS(t) : 1 � t < 1g is independent and indentically distributed(i.i.d.) Gaussian with zero mean and variance �2S , and is observed byLdistributed sensors. For each i, the observations fXi(t) : 1 � t

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007 3261

The rate-distortion region of the quadratic Gaussian CEO problem canbe described as follows [6], [18]–[20].

Theorem 3.5: The rate-distortion region of the quadratic GaussianCEO problem is

R(D) =(� ;...;� )2�(D)

R D; �21 ; . . . ; �2L

where

R D; �21 ; . . . ; �2L

= (R1; . . . ; RL)i2A

Ri�I(XA;ZA jZA ); for all A�IL

(25)

in which fZi : 1 � i � Lg are given in (22), and �(D) is given in(23).

Notice that the AWGN channel capacity region when there is powerconstraint Pi imposed on sensor i is

C(P ) = (R1; . . . ; RL) 0 � Ri � 12log(1 + giPi) : (26)

Combining this with the rate distortion region from Theorem 3.5, weobtain from Theorem 2.4 the following optimal power-distortion re-gion for the Gaussian CEO problem under orthogonal multiple access.The proof involves simple calculation of the mutual information termsI(XA;ZA jZA ) and is thus omitted; also see [6].

Theorem 3.6: Assuming the source–channel coding rate is �, thenfor any distortion D � 0, the power distortion region P(D) can bedescribed as

P(D) =(� ;...;� )2�(D)

P D; �21 ; . . . ; �2L

where �(D) is given in (23) and

P D; �21 ; . . . ; �2L

= (P1; . . . ; PL)i2A

(1 + giPi)

� 1D

i2A

1+�2i�2i

11�

+i2A

1� +�

�

for all A�IL :

It can be easily seen that the power-distortion region P(D)is convex. To minimize the total power consumption Ptotal =P1 + P2 + � � � + PL, we obtain the following problem:

min

L

i=1

Pi

s.t. (P1; P2; . . . ; PL) 2 P(D):Applying Lemma 3.4 and (3.4), we obtain that the above problem isequivalent to

min

L

i=1

Pi

s.t. Pi � (22R � 1)=gi; 1 � i � L(R1; R2; . . . ; RL) 2 �R(D):

It is easy to see that the solution for optimal power allocation is uniquesince (22R � 1)=gi is strictly convex in terms of Ri, and R(D) is

Fig. 5. Uncoded (amplify-forward) transmission in a Gaussian sensor network.

convex. This is different from the optimal rate allocation (which mini-mizes the total rateRtotal = R1+R2+ � � �+RL in the rate-distortionregionR(D) for any given D) discussed in [6], where the optimal rateallocation region is a polymatroid with L! vertices.

2) Uncoded Transmisson: In this section, we assume thesource–channel coding rate � = 1. Consider an uncoded (am-plify-forward) transmission system in a Gaussian CEO sensor network(see Fig. 5). For any t, encoder i simply amplifies its input signal sothat the total transmission power is Pi:

Ui(t) = �iXi(t); where �i =Pi

�2S + �2i

:

The received signal Yi(t) at the central node is

Yi(t) =pgiUi(t) +N

ci (t)

=giPi

�2S + �2i

S(t) +giPi

�2S + �2i

Ni(t) +Nci (t)

where N ci (t) are channel noises that are i.i.d. over time withunit variance. Due to the memoryless property of S(t) andYi(t) over t, the mmse estimator of S(t) from the receivedsignals fY1(t); Y2(t); . . . ; YL(t) : 1 � t < 1g is simplyŜ(t) = E(S(t) j Y1(t); Y2(t); . . . ; YL(t)). The resulting mse Dsatisfies

1

D=

1

�2S+

L

i=1

1

�2i +� +�

g P

:

Therefore, the achieved power-distortion region Pa(D) by the un-coded transmission can be described as follows.

Theorem 3.7: For any D � 0, the uncoded transmission achievesthe following power-distortion region

Pa(D) = (P1; . . . ; PL) 1D� 1

�2S+

L

i=1

1

�2i +� +�

g P

and Pi � 0; for all 1 � i � L :

The optimal power allocation for an uncoded transmission that min-imizes the total transmit power Ptotal = P1 + P1 + � � � + PL whileachieving a given distortion D can be formulated as

min

L

i=1

Pi

s.t. (P1; P2; . . . ; PL) 2 Pa(D): (27)

3262 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007

The optimal solution can be explicitly computed as (see Appendix B)

P opti =ai�i

(�0 � ai)+ ; 1 � i � L (28)

where ai = (�2S + �2i )=gi, and �0 is a constant which depends on

D; f�2i : 1 � i � Lg and fgi : 1 � i � Lg.3) Comparison of the Uncoded Transmission and the Separate

Source–Channel Coding: Lemma 3.4 implies that the power-dis-tortion region achieved by the separate source and channel coding(i.e., the digital approach) is optimal. Therefore, there must holdPa(D) � P(D).

Alternatively, we can analytically verify thatPa(D) is a strict subsetof P(D) by converting the uncoded transmission to an equivalent dig-ital system with separate source and channel coding. Specifically, for afixed distortion level D, suppose (P1; P2; . . . ; PL) 2 Pa(D). Define

�2idef=

�2S + �2i

giPi; for all 1 � i � L: (29)

Then from Theorem 3.7, the achieved distortion satisfies 1D� 1

�+

L

i=11

� +�, or equivalently, (�21 ; . . . ; �

2L) 2 �(D) (see (23)). The

uncoded transmission of the observations from sensor i to the cen-tral node is equivalent to a digital system with source coding that is aGaussian quantization of Xi(t), followed by capacity achieving codeswith power constraint Pi. The achieved information rate in the equiv-alent digital system is

Raidef=

1

2log (1 + giPi) =

1

2log 1 +

�2S + �2i

�2i= I(Xi;Zi); for all 1 � i � L (30)

where fZig’s are auxiliary random variables Zi in the Gaussian testchannels (22) with variances f�2i : 1 � i � Lg chosen according to(29). Thus, we have

i2A

Rai =i2A

I(Xi;Zi)

� I(XA;ZA); for all A � IL

where the last step follows from Lemma 2.5. Introduce

Ra D; �21 ; . . . ; �2L

= (R1; . . . ; RL)i2A

Ri � I(XA;ZA) for all A � IL :

(31)

The Markov chain ZA ! XA ! ZA implies

I(XA;ZA) = I(XA; ZA ;ZA)

= I(XA;ZA jZA ) + I(XA;ZA )

� I(XA;ZA jZA ):

In the last step, strict inequality should hold for some A � IL. Oth-erwise I(XA;ZA ) = 0 for all A which leads to �2i = +1 for all i[c.f. (22)]. This is possible only if D � �2S , which results in a trivialdistortion level [see (23) and (24)].

Comparing (25) and (31) shows that for any choices of(�21 ; . . . ; �

2L) 2 �(D);R

a(D; �21 ; . . . ; �2L) � R(D; �

21 ; . . . ; �

2L).

Fig. 6. Comparison of rate-distortion and power-distortion regions achieved bythe separate source and channel coding and the uncoded transmission when thetotal number of sensorsL = 2 and source–channel coding rate� = 1. Other pa-rameters: � = 9; D = 0:7; � = 1:6; � = 0:8; g = 0:8; g = 1. Theregions R(D);R (D);P(D);P (D) have solid boundaries, and are to theright and above their boundaries. The four regions with dashed boundaries cor-respond to four subsets ofR(D);R (D);P(D);P (D), which are achievedthrough specific pairs of auxiliary Gaussian random variables (Z ;Z ) withvariances � = 0:5300; � = 0:3793. Varying all feasible pairs of (� ; � ) in�(D) [c.f. (23)] gives the whole rate or power distortion regions.

This is illustrated in the left plot of Fig. 6, where

Ra(D)def=

(� ;...;� )2�(D)

Ra D; �21 ; . . . ; �2L :

Thus Ra(D) � R(D), which further implies that Pa(D) � P(D)(also see the right plot of Fig. 6).

IV. CONCLUSION

In this correspondence, we considered the multiterminal source-channel communication problem for which the goal is to characterizethe optimal cost–distortion tradeoff. By extending the work of Berger

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007 3263

and Tung [3], [23], we first gave a multiple-letter characterization ofthe rate distortion region for the multiterminal source coding. We thenestablished a multiterminal source–channel separation theorem byassuming orthogonal multiple-access protocols for the multiple-accesschannel.

Our result implies that Shannon’s separation theorem, which wasestablished for a point-to-point communication system, can be ex-tended to the multiterminal source–channel communication systemwhen the multiple-access channel has an orthogonal structure. Espe-cially for the Gaussian case, the so-called quadratic Gaussian sensornetwork has been studied in [10], [12]. It was shown therein that whenany nonorthogonal multiple access is allowed and the transmittersare perfectly synchronized, an analog uncoded (amplify-and-forward)transmission significantly outperforms the separate source and channelcoding, and achieves a nearly optimal power–distortion region. Ourresult implies that for the orthogonal multiple-access case, the digitalapproach using separate source and channel coding however performsoptimally. In fact, this optimal digital approach strictly outperformsthe analog uncoded transmission. This result is in sharp contrast tothat obtained in [10], [12].

APPENDIX

A. Properties of R(D) and R�(D)

Let R(D) and R�(D) be defined in Section II-A. For D � 0, wesayD is achievable ifR(D) 6= ;, and we sayD is a strictly achievableif there exists an � > 0 such that R(D � �) 6= ;.

Lemma 4.1 (Property of R(D)): For any achievable distortion D,the following is true.

i) R(D) is a closed and convex set for any D. Also, for anyD(1) � D(2)(which means D(1)i � D

(2)i for all i), it holds that

R(D(1)) � R(D(2)).ii) R(D) is convex2 in D in the sense that if D = 1

2(D(1)+D(2)),

then 12(R(D(1) +R(D(2))) � R(D).

iii) R(D) is right continuous at D in the sense that

R(D+)def= lim

�!0; ��0R(D+ �) = R(D): (32)

Moreover, if D is strictly achievable, then R(D) is left contin-uous at D in the sense that

R(D�)def= lim

�!0; ��0R(D� �) = R(D): (33)

Proof: In i), the closedness ofR(D) follows from the definition[c.f. (4)], and its convexity can be shown by standard time sharingargument for fixed D. The proof of ii) also follows from a similartime-sharing argument but with different distortion levels D(1) andD(2). To prove (32), we first notice that property i) impliesR(D+) �R(D) since for any � > 0;R(D + �) � R(D). To show the re-verse containment, suppose R 2 R(D+), then R 2 R(D + �) forany � > 0. This means that for any � > 0, there exists a sequence ofsource-coding schemes such that the required rates is upper boundedby R+ �, while the achieved distortion is not more than D+ �+ �. Let� ! 0 and � ! 0 simultaneously, then from the operational meaningof the rate-distortion region, we obtain that R 2 R(D). It remains

2In this case,R(D) is treated as as a functional which maps a distortion vectorD to a rate-distortion region.R(D) is convex inD denotes the fact thatR(D)is a convex functional in its variableD. This is different from the statement thatR(D) itself is a convex set for each D.

to prove (33). Notice that if D is strictly achievable, then there exists� > 0 such that R(D � �) 6= ;. Then for any 0 < � � �, it followsfrom ii) that

1

2(R(D) +R(D� 2�)) � R(D� �):

For any R 2 12(R(D) +R(D�)), by the definition ofR(D�), there

holds R 2 12(R(D)+R(D� 2�)) for any 0 < � � �. Thus, we have

R 2 R(D � �). Let � ! 0, we obtain R 2 R(D�). So

1

2(R(D) +R(D�)) � R(D�)

which further implies thatR(D) � R(D�). SinceR(D�) � R(D)holds trivially, it follows thatR(D�) = R(D) as claimed.

Lemma 4.2 (Property of R�(D)): For any D � 0, the followingholds true:

i) R�(D) is a closed and convex set;ii) for any D(1) � D(2), it holds thatR�(D(1)) � R�(D(2));

iii) R�(D) is right continuous at D in the sense that

R�(D+)def= lim

�!0; ��0R�(D + �) = R�(D): (34)

Proof: Properties i) and ii) are obvious in light of the definitionof R�(D). To prove (34), we only need to show R�(D+) � R�(D)since the containment R�(D) � R�(D+) follows directly from (ii).Fix a rate vector R 2 R�(D+). Then R 2 R�(D + �) and thus(R;D + �) 2 QX for any � > 0 [c.f. (7)]. Furthermore, we obtain(R;D) 2 QX due to the convex hull operation. So R 2 R�(D) andtherefore R�(D) � R�(D+).

B. Optimal Power Allocation in the Uncoded Transmission

By Theorem 3.7, the optimal power allocation problem (27) can berewritten as

minP ;P ;...;P

L

i=1

Pi

s.t.1

�2S+

L

i=1

1

�2i +� +�

g P

�1

D

Pi � 0; 1 � i � L:

Define

aidef= (�2S + �

2i ) =gi: (35)

Let sidef= 1

� +

= 1� +a P

. Then Pi =a

s ��, and the

problem in (35) reduces to

mins ;s ;...;s

L

i=1

a2isi

1� �2i si

s.t.1

�2S+

L

i=1

si �1

D

0 � si <1

�2i; 1 � i � L:

The above problem is convex in si, and can be solved analytically fromits Lagrangian and the Karush–Kuhn–Tucker conditions [5]. The solu-tion is stated as follows. Without loss of generality, we assume that

3264 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 9, SEPTEMBER 2007

a1 � a2 � � � � � aL where ai is given by (35), and define

f(M)def= aM

M

i=1

ai�2i

�1

1

�2S

�1

D+

M

i=1

1

�2i

;

for 1 �M � L:

It is easy to verify that for any M , if f(M) � 1, then f(M + 1) � 1.Let L1 be the smallest integer such that

f(L1) < 1 and f(L1 + 1) � 1:

SuchL1 always exists and is unique except for the following two cases:1) If f(1) > 1, then D > �2S , we take L1 = 0; 2) If f(L) < 1, wetake L1 = L. Also define

�0 =1

�2S

�1

D+

L

i=1

1

�2i

�1L

i=1

ai�2i

:

Then

sopti

=1

�2i

1�ai�0

+

; 1 � i � L

where (x)+ equals 0 when x < 0, and otherwise is equal to x. Thusthe optimal transmission power for node i is given as

P opti

=a2i

1=sopti

� �2i

=ai�i

(�0 � ai)+ ; 1 � i � L:

It is easy to see that

P opti

=0; for i � L1 + 1ai�i(�0 � ai; ) for i � L1.

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