the capacity of several new classes of semi-deterministic relay channels

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 10, OCTOBER 2011 6397

The Capacity of Several New Classes ofSemi-Deterministic Relay Channels

Hon-Fah Chong, Member, IEEE, and Mehul Motani, Member, IEEE

Abstract—The relay channel consists of a transmitter input ,a relay input , a relay output , and a receiver output . Inthis paper, we establish the capacity of three new classes of semi-deterministic relay channels: 1) a class of degraded semi-deter-ministic relay channels, 2) a class of semi-deterministic orthogonalrelay channels, and 3) a class of semi-deterministic relay chan-nels with relay-transmitter feedback. For the first class of relaychannels, the output of the relay depends on a deterministicfunction of the transmitter’s input , i.e., on , ratherthan on directly. In addition, the relay channels satisfy the con-dition that forms a Markov chain for allinput probability distributions . Hence, the first class ofrelay channels includes, but is strictly not limited to, the class ofdegraded relay channels previously considered by Cover and ElGamal. The partial decode-and-forward strategy achieves the ca-pacity of the class of degraded semi-deterministic relay channels.Next, we consider the class of semi-deterministic orthogonal relaychannels where there are orthogonal channels from the relay tothe receiver and from the transmitter to the receiver. In addition,the output of the relay is a deterministic function of , and, i.e., . The class of semi-deterministic or-

thogonal relay channels is a generalization of the class of determin-istic relay channels considered byKim. The compress-and-forwardstrategy achieves the capacity of the class of semi-deterministic or-thogonal relay channels. For the third class of relay channels, thereis a causal and noiseless feedback from the relay to the transmitter.In addition, similar to the second class of relay channels, the outputof the relay is a deterministic function of , , and . Boththe generalized strategy ofGabbai andBross and the hash-and-for-ward strategy of Kim achieve the capacity of the class of semi-de-terministic relay channels with relay-transmitter feedback.

Index Terms—Capacity, compress-and-forward, decode-and-forward, feedback, hash-and-forward, relay channel, semi-deterministic.

I. INTRODUCTION

T HE discrete-memoryless relay channel consists of foursets— , , , —and a collection of conditional

probability mass functions on , one foreach . The transmitter input is denoted

Manuscript received April 09, 2009; revised April 21, 2010; acceptedNovember 18, 2010. Date of current version October 07, 2011. This work wassupported by the National University of Singapore under Grant NUS WBSR-263-000-579-112. The material in this paper was presented in part at the2008 IEEE International Symposium on Information Theory.H.-F. Chong was with the Electrical and Computer Engineering Department,

National University of Singapore, Singapore 117576. He is now with the Insti-tute for Infocomm Research, Modulation and Coding Department, Singapore138632 (e-mail: [email protected]).M. Motani is with the Electrical and Computer Engineering Department, Na-

tional University of Singapore, Singapore 117576, (e-mail: [email protected]).Communicated by M. Gastpar, Associate Editor for Shannon Theory.Digital Object Identifier 10.1109/TIT.2011.2165131

by , the relay input by , the relay output byand the receiver output by .

A code for a relay channel without feedback con-sists of a set of integers , an encodingfunction

a set of relay functions such that

and a decoding function

The relay is causal in nature. Hence, the input of the relayis allowed to depend only on the past outputs of the relay

. If the message is sent, let

denote the conditional probability of error. The average proba-bility of error is defined by

The probability of error is calculated under the uniform distribu-tion over the codewords . The rate is said to be achiev-able by the relay channel if there exists a sequence of

codes with as . The capacity of a relaychannel is the supremum of the set of achievable rates.For a relay channel with causal and noiseless relay-trans-

mitter feedback, the only difference is that the transmitter con-sists of a set of encoding functions such that

The relay channel was first introduced by van der Meulen in[1], [2]. Cover & El Gamal established two fundamental codingtheorems for the relay channel in an important paper [3]. In ad-dition, these two coding theorems were combined in the samepaper to give the best lower bound for the capacity of a generalrelay channel [3, Th. 7]. Recently, Chong et al. determined a po-tentially larger achievable rate in [4, Th. 2]. In particular, theydetermined that the following rate is achievable for any relaychannel:

(1)

0018-9448/$26.00 © 2011 IEEE

6398 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 10, OCTOBER 2011

where the supremum is taken over all joint probability massfunctions of the form

(2)

and subject to the constraint

(3)

The capacity of the relay channel has been determined for thefollowing special cases:1) the degraded relay channel, the reversely degraded relaychannel and the relay channel with causal noiseless feed-back from the receiver to the relay [3];

2) the semi-deterministic relay channel [5];3) a class of relay channels with orthogonal components [6];4) a class of modulo-sum relay channels [7];5) a class of deterministic relay channels [8].However, the capacity of the general relay channel remains

unknown. The achievability of the above classes of relay chan-nels follows directly from appropriate substitutions for the aux-iliary random variables in [3, Th. 7]. Moreover, except for theclass of modulo-sum relay channels [7], the capacity of all theother classes of relay channels meet the cut-set upper bound.The question remains as to whether there exists other classes ofrelay channels where the lower bound given by [3, Th. 7] meetsthe cut-set upper bound.In this paper, we answer this question affirmatively and es-

tablish the capacity of three new classes of semi-deterministicrelay channels. In Section II, we consider the class of degradedsemi-deterministic relay channels which strictly includes theclass of degraded relay channels. In Section III, we considerthe class of semi-deterministic orthogonal relay channels wherethere are orthogonal channels from the transmitter to the re-ceiver and from the relay to the receiver. Furthermore, the classof semi-deterministic orthogonal relay channels satisfy the con-dition that the output of the relay is a deterministic functionof , and . In Section IV, we consider the class of relaychannels with causal and noiseless relay-transmitter feedback.Similar to the second class of relay channels, the output of therelay is a deterministic function of , and .

II. DEGRADED SEMI-DETERMINISTIC RELAY CHANNELS

We first describe the class of degraded semi-deterministicrelay channels as shown in Fig. 1.

Definition 1: Let be a deterministic function of, i.e.,

(4)

Hence, we have .We define the class of degraded semi-deterministic relay

channels as those channels which satisfy the followingconditions:• The conditional probability mass function of the channelcan be expressed as

(5)

Fig. 1. Degraded semi-deterministic relay channel.

We emphasize that and use the notation forthe sake of brevity.

• In addition, we require that the following Markov chain:

(6)

holds true for all input probability distributions .

Remark 1: A particular instance when condition (6) holdstrue for all input probability distributions is when is a deter-ministic function of and , i.e., . We can alsosee by inspection from (5) that the following Markov chain:

(7)

holds true for all input probability distributions. More specifi-cally, this follows from the equalities:

(8)

where(a) follows from the fact that is a deterministic function of

.Hence, the output of the relay depends (probabilistically) on, and as shown in Fig. 1.The main contribution of this paper for the class of degraded

semi-deterministic relay channels described in Definition 1 isgiven by the following theorem:

Theorem 1: The capacity of the degraded semi-deterministicrelay channel is given by

(9)

Proof:1) Achievability: This follows directly from substituting

, and into [3, Th. 7].

CHONG AND MOTANI: CAPACITY OF SEVERAL NEW CLASSES OF SEMI-DETERMINISTIC RELAY CHANNELS 6399

2) Converse: The converse follows from the cut-set upperbound [3, Th. 4]. Hence, is upper bounded by

(10)

The first term can be expressed as

(11)

where:(a) follows from the fact that is a deterministic function of

;(b) follows from the fact that forms a

Markov chain [condition (6)];(c) follows from the fact that forms

a Markov chain [condition (7)].

Corollary 1: If is a deterministic function of and , thecapacity is given by

(12)

Proof: The proof follows directly from Theorem 1 and thefact that is a deterministic function of and .

III. SEMI-DETERMINISTIC ORTHOGONAL RELAY CHANNELS

In [8], Kim considered a class of relay channels where thereare orthogonal channels from the transmitter/relay to the re-ceiver. The channel from the relay to the receiver is a noiselessone and the output of the relay is a deterministic functionof and . The cut-set upper bound is maximized by inde-pendent input probability distributions and is achievable by thecompress-and-forward strategy. We can extend the result to alarger class of relay channels. We state this formally below.

Theorem 2: If independent input probability distributions at-tain the cut-set upper bound and if is a deterministic functionof , and , the capacity of the relay channel is given bythe cut-set upper bound.

Proof: Refer to Appendix I.In this section, we consider a class of relay channels which

also satisfies the conditions of Theorem 2. The class of semi-de-terministic orthogonal relay channels as shown in Fig. 2 is ageneralization of the relay channels considered by Kim. Thereis a noisy link from the relay to the receiver and the output ofthe relay is a deterministic function of , and . More-over, the output of the relay and the output of the receiver

Fig. 2. Semi-deterministic orthogonal relay channel.

from the relay depends (probabilistically) on an additionalnoise component which is independent of the inputs and .The main part of the proof consists of showing that indepen-dent input probability distributions maximize the cut-set upperbound for this class of relay channels.We first define the class of semi-deterministic orthogonal

relay channels below.

Definition 2: The channel output of the receiver for thesemi-deterministic orthogonal relay channel is given by

. There is an orthogonal channelfrom the transmitter to the receiver (the output is denoted by

) and from the relay to the receiver (the output isdenoted by ). Furthermore, the output of the relayand the output of the receiver from the relay is dependenton a noise component .More specifically, we define the class of semi-deterministic

orthogonal relay channels as those channels which satisfy thefollowing conditions:• There is a noise component (with values ) inde-pendent of the inputs and . We also require thatsatisfies the following relationship

(13)

where is a deterministic function.• The conditional probability mass function describing thechannel can be expressed as

(14)

We emphasize that and use the notationfor the sake of brevity.

• In addition, we require that satisfies the followingrelationship

(15)

where is a deterministic function.The main contribution of this paper for the class of semi-

deterministic orthogonal relay channels described in Definition2 is given by the following theorem:


Fig. 3. Semi-deterministic modulo-sum relay channel.

Theorem 3: The capacity of the semi-deterministic orthog-onal relay channel is given by

(16)

Proof: Refer to Appendix II.

Remark 2: We may verify that the class of relay channelsconsidered by Kim satisfies the conditions of Definition 2 (set

and , where we have ). Wecan also combine the class of degraded semi-deterministic relaychannels with the class of semi-deterministic orthogonal relaychannels to obtain a new class of relay channels whose capacitycan also be determined (see [9]).

Example 1: We consider the discrete semi-deterministicmodulo-sum relay channel shown in Fig. 3. This channel isspecified by seven finite sets with the same cardinality, i.e.,

(17)

where is an arbitrary positive integer greater than 1.The conditional probability mass function

is determined from the following equations:

(18)

(19)

(20)

where denotes addition modulo ; and are positive in-tegers; and and are independent noise variables definedover and , respectively, with probability mass-functionsgiven by

(21)

(22)

We note that when , the discrete semi-deterministicmodulo-sum relay channel is just a reversely degraded relaychannel as forms a Markov chain.

Fig. 4. Relay channels with causal noiseless relay-transmitter feedback.

However, in general, when , the discrete semi-determin-istic modulo-sum relay channel does not fall under any class ofrelay channels whose capacity has been previously proven.Furthermore, we may easily verify that this class of

modulo-sum relay channels satisfies the conditions of Def-inition 2 ( ). Hence, the capacity of the class ofsemi-deterministic modulo-sum relay channels follows fromTheorem 3.

IV. SEMI-DETERMINISTIC RELAY CHANNELS WITH CAUSALNOISELESS RELAY-TRANSMITTER FEEDBACK

Finally, we consider a class of relay channels with causalnoiseless relay-transmitter feedback as shown in Fig. 4.

Definition 3: We define the class of semi-deterministic relaychannels with causal noiseless relay-transmitter feedback asthose channels which satisfy the condition that the output ofthe relay is a deterministic function of , and , i.e.,

(23)

The relay channel with causal noiseless relay-transmitterfeedback was studied by Gabbai and Bross in [10]. In [10,Th. 3], Gabbai and Bross proved that the following rate isachievable for a discrete memoryless relay channel with causalnoiseless relay-transmitter feedback:

(24)

where the supremum is taken over all probability mass functionsof the form

(25)

We can obtain the capacity of the class of relay channelsin Definition 3 by appropriate substitutions for the auxiliaryrandom variables in [10, Th. 3]. In particular, the capacity ofthe class of relay channels in Definition 3 is given by the fol-lowing theorem:Theorem 4: The capacity of the semi-deterministic relay

channel with causal noiseless relay-transmitter feedback asshown in Fig. 4, satisfying condition (23), is given by

(26)


Fig. 5. Semi-deterministic modulo-sum relay channel with causal noiseless relay-transmitter feedback.

Proof:1) Achievability: We note that rate is achievable by

substituting and in [10, Th. 3]. To see this,we note that the first term gives us . For thesecond term, we obtain

(27)

where:(a) follows from condition (23).

Hence, the rate is achievable. An alternative proof of achiev-ability may be shown using hash-and-forward [8] (see [11]).

2) Converse: The proof for the converse follows directlyfrom the cut-set upper bound for the relay channel withoutfeedback since it is also an upper bound for the relay channelwith causal noiseless receiver-transmitter, relay-transmitter andreceiver-relay feedback [3]. Therefore, the cut-set upper boundfor the relay channel without feedback is also an upper boundfor the relay channel with causal noiseless relay-transmitterfeedback.

Remark 3: This class of relay channels is also closely relatedto the class of deterministic relay channels solved by Kim in [8].In [8], the output of the relay is a deterministic function ofand but there is a noiseless link of rate from the relay

to the receiver. In our case, the output of the relay is a deter-ministic function of , and but there is a causal noiselessfeedback from the relay to the transmitter. In both cases, the ca-pacity may be achieved by both the hash-and-forward schemeor the compress-and-forward scheme.

Example 2: We consider a discrete semi-deterministicmodulo-sum relay channel with causal noiseless relay-trans-mitter feedback as shown in Fig. 5. This channel is specified by

(28)

(29)

where , and are positive integers. The conditional prob-ability mass function is determined from thefollowing equations:

(30)

(31)

where denotes addition modulo ; a is a positive integer; isa random variable which takes on the value with probabilityand the value with probability ; and is a noise

variable defined over with probability mass function givenby

(32)

If or , we readily see that this is neither a de-graded relay channel ( does not form aMarkov chain) nor a reversely degraded relay channel (

does not form a Markov chain). Moreover, wenote that the output of the receiver is not a deterministic func-tion of , and . Therefore, the transmitter only has feed-back from the relay but not from the receiver. On the other hand,we note that the output of the relay is a deterministic functionof , and . Its capacity is thus given by Theorem 4.

APPENDIX IPROOF OF THEOREM 2

1) Achievability: Substituting and in [3, Th.7] gives us the achievable rate

(33)

subject to the constraint

(34)

where the supremum is taken over all joint probability massfunctions of the form

(35)


If , we immediately obtain. If , we have

(36)

where:

(a) follows from the fact that is a deterministic function of, , ;

(b) follows from the fact that if ,there exists probability mass functions such that

.

2) Converse: The converse follows from the cut-set upperbound and the assumption that the cut-set upper bound is max-imized by independent input probability distributions.

APPENDIX IIPROOF OF THEOREM 3

We first make some observations on the properties of thesemi-deterministic orthogonal relay channel that will be usefullater on.

Property 1: For all input probability distributions, forms a Markov

chain.Property 2: For all input probability distributions

, forms a Markov chain.Property 3: For all input probability distributions

, forms aMarkov chain.

Property 4: For independent input probability distribu-tions, i.e., ,

forms a Markov chain.Property 5: For independent input probability distri-

butions, i.e., , and areindependent.

Property 6: The terms andfor the semi-deterministic orthogonal relay channel is maxi-mized by the marginal input probability distribution .The term for the semi-deterministic orthogonalrelay channel is maximized by the marginal input probabilitydistribution .

We may readily prove Property 1–Property 4 from Definition2. Property 5 can be shown from the following equalities:

(37)

where(a) follows from the fact that is independent of and(b) follows from the fact that is a deterministic function of

and [condition (13)].To prove Property 6, we first consider the term

which depends only on the probability distribution (whichis independent of and ) and the conditional probabilitydistribution . We note that the conditional proba-bility distribution depends on the marginal prob-ability distribution and not on the joint probability distri-bution from the following equalities:

(38)

where:(a) follows from the fact that is independent of and ;(b) follows from the fact that

forms a Markov chain (Property 3).Hence, the term is maximized by the mar-

ginal input probability distribution . Similarly, theterm is maximized by the marginal inputprobability distribution . Therefore,is maximized by the marginal input probability distribution


. We can likewise prove that is maximizedby the marginal input probability distribution from thefact that forms a Markov chain (Property1). We can also prove that is maximized by themarginal input probability distribution from the fact that

forms a Markov chain (Property 2).1) Achievability: This follows from the compress-and-for-

ward strategy. If , fol-lowing the proof of Theorem 2, we immediately obtain

(39)

where:(a) follows from the fact that forms a

Markov chain (Property 2);(b) follows from the fact that is a deterministic function of

and [condition (13)];(c) follows from the fact that

forms a Markov chain for all input probabilitydistributions (Property 3) and that

forms a Markov chain for independent inputprobability distributions (Property 4).

If , following the proof ofTheorem 2, we have

(40)

where:(a) follows from the fact that forms a

Markov chain (Property 1);(b) follows from the fact that and are independent if

the input probability distributions are independent (Prop-erty 5);

(c) follows from the fact that formsa Markov chain (Property 2).

Hence, the compress-and-forward strategy achieves the rate

(41)

for the semi-deterministic orthogonal relay channel.2) Converse: The converse follows from the cut-set upper

bound. We will show that the cut-set upper bound is maximized

by independent input probability distributions. For the first termin (10), we obtain

(42)

(a) follows from the fact that is a deterministic function ofand [condition (13)];

(b) follows from the fact that conditioning reduces entropy;(c) follows from the fact that forms a

Markov chain (Property 2);(d) follows from the fact that

forms a Markov chain (Property 3).We note that is maximized by the marginalinput probability distribution (Property 6). Furthermore,we note that (b) can be replaced by an equality as long as theinput probability distributions are independent (Property 4).Hence, independent input probability distributions maximizethe first term in (10). For the second term in (10), we obtain

(43)

(a) follows from the fact that conditioning reduces entropy;(b) follows from the fact that

forms a Markov chain (Property 1) and from the fact thatforms a Markov chain (Property

2).We also note that is maximized by the marginal

input probability distribution and similarly, ismaximized by the marginal input probability distribution(Property 6). Furthermore, (a) can be replaced by an equalityas long as the input probability distributions are independent(Property 5). Hence, independent input probability distributionsachieve capacity.

ACKNOWLEDGMENT

We are grateful to the anonymous reviewers for their con-structive comments and suggestions.


REFERENCES[1] E. C. van der Meulen, “Transmission of Information in a -Terminal

Discrete Memoryless Channel,” Ph.D. dissertation, Univ. California,Berkeley, CA, 1968.

[2] E. C. van der Meulen, “Three-terminal communication channels,” Adv.Appl. Prob., vol. 3, pp. 120–154, 1971.

[3] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,”IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sep. 1979.

[4] H. F. Chong, M. Motani, and H. K. Garg, “Generalized backward de-coding strategies for the relay channel,” IEEE Trans. Inf. Theory, vol.53, no. 1, pp. 394–401, Jan. 2007.

[5] A. E. Gamal and M. Aref, “The capacity of the semideterministic relaychannel,” IEEE Trans. Inf. Theory, vol. 28, no. 3, p. 536, May 1982.

[6] A. E. Gamal and S. Zahedi, “Capacity of a class of relay channels withorthogonal components,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp.1815–1817, May 2005.

[7] M. Aleksic, P. Razaghi, andW. Yu, “Capacity of a class of modulo-sumrelay channels,” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 921–930,Mar. 2009.

[8] Y.-H. Kim, “Capacity of a class of deterministic relay channels,” IEEETrans. Inf. Theory, vol. 54, no. 3, pp. 1328–1329, Mar. 2008.

[9] H. F. Chong andM.Motani, “The Capacity of a Class ofMixture Semi-Deterministic Relay Channels” Nat. Univ. Singapore, 2010, Tech. Rep.NUS-TR1001-20100330.

[10] Y. Gabbai and S. I. Bross, “Achievable rates for the discrete memory-less relay channel with partial feedback configurations,” IEEE Trans.Inf. Theory, vol. 52, no. 11, pp. 4989–5007, Nov. 2006.

[11] H. F. Chong and M. Motani, “The capacity regions of some classesof deterministic relay channels,” in Proc. IEEE Int. Symp. Inf. Theory,Toronto, ON, Canada, Jul. 2008, pp. 344–348.

Hon-Fah Chong (M’09) received the Bachelor’s degree, the Masters degree,and the Ph.D degree in Electrical and Computer Engineering from the NationalUniversity of Singapore in 2000, 2002, and 2008, respectively. Currently, he isworking as a Research Fellow at the Institute for Infocomm Research in Singa-pore. His main research interests are information theoretical problems relatedto the broadcast channel, the relay channel and the interference channel.

MehulMotani (M’00) received the B.S. degree from Cooper Union, NewYork,the M.S. degree from Syracuse University, Syracuse, NY, and the Ph.D. degreefrom Cornell University, Ithaca, NY, all in electrical and computer engineering.He is currently a Visiting Fellow with Princeton University, Princeton, NJ,

and an Associate Professor with the Electrical and Computer Engineering De-partment, National University of Singapore. Previously, he was a Research Sci-entist at the Institute for Infocomm Research in Singapore for three years anda Systems Engineer at Lockheed Martin in Syracuse, NY, for over four years.His research interests are in the area of wireless networks. Recently he has beenworking on research problems which sit at the boundary of information theory,communications and networking, including the design of wireless ad hoc andsensor network systems.Dr. Motani was awarded the Intel Foundation Fellowship for work related

to his Ph.D. research, which focused on information theory and coding forCDMA systems. He has served on the organizing committees of ISIT, WiNC,and ICCS, and the technical program committees of MobiCom, Infocom,ICNP, SECON, and several other conferences. He participates actively inIEEE and ACM and has served as the secretary of the IEEE InformationTheory Society Board of Governors. He is currently an Associate Editor for theIEEE TRANSACTIONS ON INFORMATION THEORY and an Editor for the IEEETRANSACTIONS ON COMMUNICATIONS.

the capacity of several new classes of semi-deterministic relay channels

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