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Bounds on Capacity and MinimumEnergy-Per-Bit for AWGN Relay Channels

Abbas El Gamal, Mehdi Mohseni and Sina ZahediInformation Systems Lab

Department of Electrical EngineeringStanford University, Stanford, CA 94305-9510

{abbas, mmohseni, szahedi }@stanford.edu

Abstract

Upper and lower bounds on the capacity and minimum energy-per-bit for generaladditive white Gaussian noise (AWGN) and frequency division additive white Gaussiannoise (FD-AWGN) relay channel models are established. First the max-ow min-cutbound and the generalized block Markov coding scheme are used to derive upper andlower bounds on capacity. These bounds are never tight for the general AWGN modeland are tight only under certain conditions for the FD-AWGN model. The gap betweenthe upper and lower bounds is the largest when the gain of the channel to the relay iscomparable or worse than that of the direct channel. To obtain tighter lower bounds,

two coding schemes that do not require the relay to decode any part of the messageare investigated. First the side information coding scheme is shown to outperformthe block Markov coding scheme. It is shown that the achievable rate of the side-information coding scheme can be improved via time-sharing and a general expressionfor the achievable rate with side information is found for relay channels in general. Inthe second scheme, the relaying functions are restricted to be linear. A simple sub-optimal linear relaying scheme is shown to signicantly outperform the generalizedblock Markov and the side-information coding schemes for the general AWGN modelin some cases. It is shown that the optimal rate using linear relaying can be obtained bysolving a sequence of non-convex optimization problems. The problem is reduces to asingle-letter non-convex optimization problem for the FD-AWGN model. The paperestablishes a relationship between the minimum energy-per-bit and capacity of theAWGN relay channel. This relationship together with the lower and upper bounds oncapacity are used to establish corresponding lower and upper bounds on the minimumenergy-per-bit for the general and FD-AWGN relay channels. The bounds are veryclose and do not differ by more than a factor of 1.45 for the FD-AWGN relay channelmodel and 1.7 for the general AWGN model.

Index Terms Relay channel, channel capacity, minimum energy-per-bit, additive white Gaus-sian noise channels.

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1 Introduction

The relay channel, rst introduced by van der Meulen [1] in 1971, consists of a sender-receiver pair whose communication is aided by a relay node. In [1] and [2], simple lower andupper bounds on the capacity of the discrete-memoryless relay channel were established.In [3] and [4], capacity theorems were established for: (i) physically degraded and reverselydegraded discrete memoryless relay channels, (ii) physically degraded and reversely degradedadditive white Gaussian noise (AWGN) relay channels with average power constraints, (iii)deterministic relay channels, and (iv) relay channels with feedback. A max-ow min-cutupper bound and a general lower bound based on combining the generalized block Markovand side information coding schemes were also established in [4]. In [5], the capacity of therelay channel with one deterministic component was established. It is interesting to notethat in all special cases where the relay channel capacity is known, it is equal to the max-owmin-cut bound. Generalizations of some of the single relay channel results to channels withmany relays were given in [6]. In [7], Aref established the capacity for a cascade of degraded

relay channels. The relay channel did not receive much attention and no further progresswas made toward establishing its capacity for a long time after this early work.

Recent interest in multi-hop and ad hoc wireless networks has spurred the interest instudying additive white Gaussian noise (AWGN) relay channels. In [8] achievable ratesfor AWGN channels with two relays were investigated. In [9], the capacity of a class of orthogonal relay channels was established. In [10] upper and lower bounds on the capacityof AWGN channels were established. The capacity of AWGN relay networks as the numberof nodes becomes very large were investigated in (e.g., [11]- [14]). Motivated by energyconstraints in sensor and mobile networks, recent work has also investigated the saving intransmission energy using relaying [16]. In [18], upper and lower bounds on the capacity of

AWGN relay channels were used to establish bounds on the minimum energy-per-bit that donot differ by more than a factor of 2. The capacity and minimum energy-per-bit for AWGNrelay channels, however, are not known in general.

In this paper, we provide detailed discussion and several extensions of the bounds oncapacity and minimum energy-per-bit for AWGN relay channels presented in [10] and [18],including correcting an error in the capacity with linear relaying result reported in [10]. Weconsider the two discrete-time AWGN relay channel models depicted in Figure 1. In thegeneral model, Figure 1(a), the received signal at the relay and at the receiver at time i 1are given by

Y 1i = aX i + Z 1i , and Y i = X i + bX 1i + Z i ,

where X i and X 1i are the signals transmitted by the sender and by the relay, respectively. Thereceivers noise processes {Z 1i}and {Z i}are assumed to be independent white Gaussian noiseprocesses each with power N , and the constants a, b > 0 represent the gain of the channelsfrom the sender to the relay and from the relay to the receiver, respectively, relative to thegain of the direct channel (which is assumed to be equal to one). The frequency-divisionAWGN (FD-AWGN) model is motivated by the constraint that in practice the relay cannotsend and receive information within the same frequency band (or at the same time). Tosatisfy this constraint, one can either split the channel from the sender to the relay and the

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receiver into two bands or alternatively split the channel to the receiver from the sender andthe relay. The capacity of the rst model has been established in [9]. In this paper, we focuson the second FD-AWGN model, depicted in Figure 1(b), the received signal at time i isgiven by Y i = {Y Di , Y Ri }, where Y Di = X i + Z Di is the received signal from the sender andY Ri = bX 1i + Z Ri is the received signal from the relay, and

{Z Di

}and

{Z Ri

}are independent

white Gaussian noise processes each with average noise power N .

X X

a a

Z 1 Z 1

Y 1 Y 1X 1 X 1b

bZ

Y 1 1

(a) (b)

Z R

Y R

Y D

Y

Z D

Figure 1: (a) General AWGN relay channel model. (b) FD-AWGN relay channel model.Path gains are normalized to 1 for the direct channel, a > 0 for the channel to the relay, andb > 0 for the channel from the relay to the receiver.

The paper establishes upper and lower bounds on the capacity and minimum energy-per-bit for the general and FD-AWGN relay channel models. In the following discussion wesummarize our main results and provide an outline for the rest of the paper.

Bounds on capacity: In Section 2, we use the max-ow min-cut upper bound [4] and thegeneralized block Markov lower bound [4, 5] on the capacity of the relay channel to deriveupper and lower bounds on the capacity of the general and FD-AWGN relay channels (seeTable 1). The bounds are not tight for the general AWGN model for any a, b > 0 and aretight only for a restricted range of these parameters for the FD-AWGN model. We nd thatthe gap between the upper and lower bounds is the largest when the channel to the relay isnot much better than the channel from the sender to the receiver, i.e., a is close to 1. Weargue that the reason for the large gap is that in the generalized block Markov coding schemethe relay is either required to decode the entire message or is not used at all. For a close to 1,this severely limits the achievable rate. Motivated by this observation, in Sections 4 and 5,we investigate achievable rates using two schemes where the relay cooperates in sending themessage but without decoding any part of it. In Section 4, we explore achievable rates usingthe side information coding scheme in [4]. We nd that this scheme can outperform block

Markov coding and in fact becomes optimal as b . We show that the achievable rate canbe improved via time-sharing and provide a general expression for achievable rate with sideinformation for relay channels in general (see Theorem 2). In Section 5, we investigate theachievable rates when the relay is restricted to sending linear combinations of past receivedsignals. We show that when a is close to 1, a simple sub-optimal linear relaying scheme cansignicantly outperform the more sophisticated block Markov scheme (see Example 1). Weshow that the capacity with linear relaying functions can be found by solving a sequence of non-convex optimization problems. One of the main results in this paper is showing that

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this formulation can be reduced to a single-letter optimization problem for the FD-AWGNmodel (see Theorem 3).

Bounds on minimum energy-per-bit: In Section 3, we establish a general relationship betweenthe minimum energy-per-bit and capacity (see Theorem 1). We use this relationship togetherwith the lower bound on capacity based on the generalized block Markov coding scheme andthe max-ow min-cut upper bound to establish upper and lower bounds on the minimumenergy-per-bit (see Table 2). These bounds can be very close and do not differ by more thana factor of two. For the FD-AWGN model the upper and lower bounds coincide when thechannel from the relay to the receiver is worse than the direct channel, i.e., b 1. For thegeneral AWGN model, the bounds are never tight. Using the lower bounds on capacity inSections 4 and 5, we are able to close the gap between the lower and upper bounds to lessthan a factor of 1.45 for the FD-AWGN model and 1.7 for the general AWGN model.

2 Basic Bounds on Capacity

As in [4], we dene a (2nR n , n) code for the relay channel to consist of, (i) a set M={1, 2, . . . , 2nR n }of messages, (ii) a codebook {xn (1), xn (2), . . . , x n (2nR n )}consisting of code-words of length n, (iii) a set of relay functions {f i}ni=1 such that x1i = f i(y11 , y12, . . . , y1i 1),and (iv) a decoding rule D(yn ) M. The average probability of decoding error is denedin the usual way as

P (n )e =1

2nR n

2nR n

k=1

P{D(yn ) = k|xn (k) transmitted }.We assume the average transmitted power constraints

n

i=1

x2i (k) nP, for all kM, andn

i=1

x21i nP, for all yn1 IRn , 0.A rate R is said to be achievable if there exists a sequence of (2 nR , n) codes satisfying theaverage power constraints P on X and P on X 1, such that P

(n )e 0 as n . Thecapacity C (P,P ) of the AWGN relay channel with average power constraints is dened as

the supremum of the set of achievable rates. To distinguish between the capacity for thegeneral AWGN and the FD model, we label the capacity of the FD-AWGN relay channel asC FD (P,P ).

In this section, we evaluate the max-ow min-cut upper bound and the block Markovlower bounds on the capacity of the general and frequency-division AWGN channels. Inparticular we show that the lower bounds achieved with the block Markov encoding and thegeneralized block Markov encoding schemes are the same for AWGN relay channels.

First note that the capacity of both the general and FD-AWGN relay channels are lowerbounded by the capacity of the direct link

CP N

12

log 1 +P N

.

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Next, we use bounds from [4] on the capacity C of the discrete-memoryless relay channel toderive the upper and lower bounds on C (P,P ) and C FD (P,P ) given in Table 1.

Generalized Block Markov Lower Bound Max-ow Min-cut Upp er Bound

R ( P,P ) =8>>>>>>>>>>>:

C 0B@bq ( a 2 1)+ q a 2 b 2

2P

a 2 N 1CA, if a

21+ b 2

1

C max {1 ,a 2 }P N , otherwiseC ( P,P ) =

8>>>>>>>>>>>:

C 0B@ab + q 1+ a 2 b 2

2P

(1+ a 2 ) N 1CA, if a

2b 2

> 1

C (1+ a 2 ) P N , otherwise

R FD ( P,P ) =8>>>>>>>:

C P N + C b2 P N , if a 21+ b 2 P N +1 1

C max {1 ,a 2 }P N , otherwiseC FD ( P,P ) =

8>>>>>>>:

C P N + C b2 P N , if a 2b 2 P N +1 1

C (1+ a 2 ) P N , otherwise

Table 1: Max-ow min-cut upper bounds and generalized block Markov lower bounds onC (P,P ) and C FD (P,P ).

The upper bounds are derived using the max-ow min-cut bound on the capacity of the discrete-memoryless relay channel (Theorem 4 in [4])

C maxp(x,x 1 ) min{I (X, X 1; Y ), I (X ; Y, Y 1|X 1)}. (1)This bound is the tightest upper bound to date on the capacity of the relay channel. Forcompletion, derivations of the upper bounds are given in Appendix A.

The lower bounds in the table are obtained using a special case of Theorem 7 in [4] thatyields

C maxp(u,x,x 1 ) min{I (X, X 1; Y ), I (U ; Y 1|X 1) + I (X ; Y |X 1, U )}, (2)This lower bound is achieved using a generalized block Markov coding scheme, where in eachblock the relay decodes part of the new message (represented by U ) and cooperatively sendsenough information with the sender to help the receiver decode the previous message ( U then X ). Note that if we set U = X , we obtain the rate for the block Markov scheme,which is optimal for the physically degraded relay channel, while if we set U = , the boundreduces to the capacity of the direct channel, which is optimal for reversely degraded relaychannels. In addition to these two special cases, this bound was also shown to be tight forsemi-deterministic relay channels [5] and more recently for a class of relay channels with

orthogonal components [9].The bounds on the capacity of the general AWGN channel in the table are not tight for

any a, b > 0. In particular, when a < 1, the generalized block Markov coding bound yields

C(P/N ), which is simply the capacity of the direct link. For the FD-AWGN, the boundscoincide for a2 1 + b2 1 + P N . In Sections 4 and 5, we show that side information andlinear relaying coding schemes can provide much tighter lower bounds than the generalizedblock Markov bound for small values of a.

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Derivation of the lower bounds: Consider the lower bound for the general AWGN case. Notethat R(P,P ) is in fact achievable by evaluating the mutual information terms in (2) usinga jointly Gaussian ( U,X,X 1). We now show that the lower bound in (2) with the powerconstraints is upper bounded by R(P,P ) in Table 1. It is easy to verify that

I (X, X 1; Y ) C (1 + b2

+ 2 b )P N ,where is the correlation coefficient between X and X 1. Next consider

I (U ; Y 1|X 1) + I (X ; Y |X 1, U ) = h(Y 1|X 1) h(Y 1|X 1, U ) + h(Y |X 1, U ) h(Y |X, X 1, U )

12

logE Var( Y 1|X 1)

N h(Y 1|X 1, U ) + h(Y |X 1, U )

12

loga2 E (X 2) E (XX 1 )

2

E (X 21 )+ N

N h(Y 1|X 1, U ) + h(Y |X 1, U )

C a2

(1 2

)P N h(Y 1|X 1, U ) + h(Y |X 1, U ).We now nd an upper bound on h(Y |X 1, U ) h(Y 1|X 1, U ). Note that

12

log(2eN ) h(Y 1|X 1, U ) 12

log(2e(a2P + N )).

Therefore, there exists a constant 0 1 such that h(Y 1|X 1, U ) = 12 log(2e(a2P + N )).First assume a < 1. Using the entropy power inequality we obtain

h(aX + Z 1|X 1, U ) = h a X +Z 1a X 1, U

= h X +Z 1a

X 1, U + log a

= h ( X + Z + Z |X 1, U ) + log a

12

log 22h(X + Z |X 1 ,U ) + 2 2h(Z |X 1 ,U ) + log a

=12

log 22h(X + Z |X 1 ,U ) + 2 e1a2 1 N + log a

=12

log 22h(Y |X 1 ,U ) + 2 e1a2 1 N + log a,

where Z N 0, 1a2 1 N and is independent of Z . Since h(aX + Z 1|X 1, U ) = 12 log(2e(a2P +N )), we obtainh(Y |X 1, U )

12

log (2e(P + N )) ,

andh(Y |X 1, U ) h(Y 1|X 1, U )

12

logP + N

a2P + N 12

logP + N

a2P + N .

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Hence, for a < 1,

I (U ; Y 1|X 1) + I (X ; Y |X 1, U ) 12

loga2(1 2)P + N

N +

12

logP + N

a2P + N CP N

.

For a > 1, note that h(Y 1

|X 1, U ) = h(aX + Z 1

|X 1, U ) = h(aX + Z

|X 1, U )

h(X + Z

|X 1, U ) =

h(Y |X 1, U ) and henceI (U ; Y 1|X 1) + I (X ; Y |X 1, U ) C

a2(1 2)P N

.

Note that the above bounds are achieved by choosing ( U, X 1, X ) to be jointly Gaussian withzero mean and appropriately chosen covariance matrix. Performing the maximization over gives the lower bound result in Table 1. This completes the derivation of the lower bound.The lower bound for the FD-AWGN case can be similarly derived.

3 Basic Bounds on Minimum Energy-per-BitIn this section we establish a general relationship between minimum energy-per-bit andcapacity with average power constraints for the discrete time AWGN relay channel. We thenuse this relationship and the bounds on capacity established in the previous section to ndlower and upper bounds on the minimum energy-per-bit.

The minimum energy-per-bit can be viewed as a special case of the reciprocal of thecapacity per-unit-cost [19], when the cost is average power. In [22], Verdu established arelationship between capacity per-unit-cost and channel capacity for stationary memorylesschannels. He also found the capacity per-unit cost region for multiple access and interferencechannels. Here, we dene the minimum energy-per-bit directly and not as a special case of capacity per unit cost. We consider a sequence of codes where the rate Rn 1/n can varywith n. This allows us to dene the minimum energy-per-bit in an unrestricted way. Theenergy for codeword k is given by

E (n )(k) =n

i=1

x2i (k).

The maximum relay transmission energy is given by

E (n )r = maxyn

1

n

i=1

x21i .

The energy-per-bit for the code is therefore given by

E (n ) =1

nR nmax

k E (n )(k) + E (n )r .An energy-per-bit E is said to be achievable if there is a sequence of (2 nR n , n) codes withP (n )e 0 and limsup E (n ) E . We dene the minimum energy-per-bit as the energy-per-bitthat can be achieved with no constraint on the rate. More formally

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Denition 1 The minimum energy-per-bit E b is the inmum of the set of achievable energy-per-bit values.To distinguish between the minimum energy-per-bit for the general and FD-AWGN channelmodels, we label the minimum energy-per-bit for the FD-AWGN relay channel by

E FDb . In

the discussion leading to Theorem 1, we derive a relationship between C (P,P ) and E b. Thestatements and results including the theorem apply with no change if we replace C (P,P )by C FD (P,P ) and E b by E FDb .

First note that C (P,P ) can be expressed as

C (P,P ) = supk

C k(P,P ) = limk

C k(P,P ), (3)

whereC k(P,P ) =

1k

supP X k , {f i }ki =1 :

Pki =1 E (X

2i ) kP,

max y n1 Pki =1 x21i kP

I (X k ; Y k), (4)

where x1i = f i(y11 , . . . , y1(i 1) ), and X k = [X 1, X 2, . . . , X k]T . Note that by a standardrandom coding argument, any rate less than C k(P,P ) is achievable. It is easy to argue thatkC k(P,P ) as dened in (4) is a super-additive sequence in k, i.e., (k + m)C k+ m (P,P ) kC k(P,P ) + mC m (P,P ) for any k, m 1. Hence the supremum in (3) can be replacedwith the limit. We now establish the following properties of C (P,P ) as a function of P .Lemma 1 The capacity of the AWGN relay channel with average power constraints satisesthe following:

(i) C (P,P ) > 0 if P > 0 and approaches as P .(ii) C (P,P ) 0 as P 0.

(iii) C (P,P ) is concave and strictly increasing in P .

(iv) (1+ )P C (P,P ) is non-decreasing in P , for all P > 0.

The proof of this lemma is provided in Appendix D.

We are now ready to establish the following relationship between the minimum energy-per-bit and capacity with average power constraint for the AWGN relay channel.

Theorem 1 The minimum energy-per-bit for the AWGN relay channel is given by

E b = inf 0 limP 0(1 + )P C (P,P )

. (5)

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Proof: We establish achievability and weak converse to show that

E b = inf 0 inf P > 0(1 + )P C (P,P )

.

From part (iv) of Lemma 1, the second inf can be replaced by lim.

To show achievability, we need to show that any E > inf 0 inf P > 0 (1+ )P C (P,P ) is achievable.First note that there exist P > 0 and 0 such that E > (1+ )P C (P , P ) = inf 0 inf P > 0 (1+ )P C (P,P ) +

for any small > 0. Now we set R = (1 + )P / E . Using standard random coding withpower constraints argument, we can show that R < C (P , P ) is achievable, which provesthe achievability of E .

To prove the weak converse we need to show that for any sequence of (2 nR n , n) codeswith P (n )e 0,

lim inf

E (n )

E b = inf

0inf P > 0

(1 + )P

C (P,P ).

Using Fanos inequality, we can easily arrive at the bound

Rn C (P n , n P n ) +1n

H (P (n )e ) + Rn P(n )e ,

where P n > 0 is the maximum average codeword power and n P n is the maximum averagetransmitted relay power. Thus

Rn C (P n , n P n ) + 1n H (P

(n )e )

1 P (n )e.

Now, by denition the energy-per-bit for the code is E (n ) P n (1 + n )/R n . Using the boundon Rn we obtain the bound

E (n ) P n (1 + n )(1 P

(n )e )

C (P n , n P n ) + 1n H (P(n )e )

=P n (1 + n )(1 P

(n )e )

C (P n , n P n )(1 + 1n H (P(n )e )/C (P n , n P n ))

E b(1 P (n )e )

1 + 1n

H (P (n )e )/C (P n , n P n ).

Now since P(n )e 0 and H (P(n )e ), C (P n , n P n ) > 0, we get that lim inf E (n ) E b.

We now use the above relationship and the bounds on capacity to establish lower andupper bounds on the minimum energy-per-bit. First note that the minimum energy-per-bitfor the direct channel, given by 2 N ln 2, is an upper bound on the minimum energy-per-bitfor both relay channel models considered. Using Theorem 1 and the bounds on capacitygiven in Table 1, we obtain the lower and upper bounds on the minimum energy-per-bit

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Normalized Minimum Energy-per-bit Max-ow Min-cut Lower Bound Block Markov Upper Bound

Eb / 2N ln 21+ a 2 + b 2

(1+ a 2 )(1+ b 2 )min 1, a

2 + b 2

a 2 (1+ b 2 )ff

E FDb / 2N ln 2 min 1,a 2 + b 2

b 2 (1+ a 2 )ffa 2 + b 2 1

a 2 b 2, if a,b > 1

1, otherwise

Table 2: Lower and upper bounds on E b/ 2N ln2 and E FDb / 2N ln2.

provided in Table 2. The bounds in the table are normalized with respect to 2 N ln2 andtherefore represent the reduction in the energy-per-bit using relaying.

It is easy to verify that the ratio of the upper bound to the lower bound for each channelis always less than 2. This maximum ratio is approached for a = 1 as b , i.e., when therelay and receiver are at the same distance from the transmitter and the relay is very closeto the receiver. Note that for the FD-AWGN channel, the lower and upper bounds coincideand are equal to 2 N ln2 for b 1, and therefore relaying does not reduce the minimumenergy-per-bit. For the general AWGN channel, the ratio is very close to 1 when a or b aresmall, or when a is large. We now derive the upper and lower bounds for the general AWGNmodel. The bounds for the FD-AWGN relay channel can be similarly established.

Derivation of bounds : Using Theorem 1 and the bounds on capacity derived in section 2,we now establish the bounds on the minimum energy-per-bit of the general AWGN relaychannel given in table, i.e.,

1 + a2 + b2

(1 + a2

)(1 + b2

) E b

2N ln 2 min 1,

a2 + b2

a2

(1 + b2

)

.

To prove the lower bound we use the upper bound C (P,P ) on capacity in Table 1 andthe relationship of Theorem 1 to obtain the bound. Substituting the upper bound given inTable 1 and taking limits as P 0, we obtain the expression

E b 2N ln 2 min min0 < a 2b2(1 + )(1 + a2)

ab + 1 + a2 b2 2 , min

a 2b2

1 + 1 + a2

.

To complete the derivation of the lower bound, we analytically perform the minimization.

For a2/b 2, it is easy to see that the minimum is achieved by making as small as possible,i.e., = a2/b 2, and the minimum is given by a 2 + b2b2 (1+ a2 ) . On the other hand, if < a 2/b 2, theminimum is achieved when = a2b2/ (a2 + b4 + 2 b2 + 1) < a 2/b 2 and is given by 1+ a

2 + b2(1+ a 2 )(1+ b2 ) .

Now, since 1+ a2 + b2

(1+ a 2 )(1+ b2 ) 2, this is a non-convex optimization problem. Thus nding C L(P,P ) involves solvinga sequence of non-convex optimization problems, a daunting task indeed! Interestingly, wecan show that even a sub-optimal linear relaying scheme can outperform generalized blockMarkov coding for small values of a.

Example 1: Consider the following linear relaying scheme for the general AWGN relay channelwith block length k = 2 (See Figure 2). In the rst transmission, the senders signal isX 1N (0, 2P ), for 0 < 1, and the relays signal is X 11 = 0. The received signal at thereceiver and the relay receiver are Y 1 = X 1 + Z 1 and Y 11 = aX 1 + Z 11 , respectively. In thesecond transmission, the senders signal is X 2 = (1 )/X

1, i.e., a scaled version of X 1with average power 2(1 )P . The relay cooperates with the sender by relaying a scaledversion of Y 11 , X 12 = dY 11 , where d = 2P / (2a2P + N ) is chosen to satisfy the relaysender power constraint. Thus, = 2 P (1 ) (1 ) 1

, D = 0 0d 0 .

The received signal after the second transmission is given by Y 2 = X 2 + dbY 11 + Z 2.

It can be easily shown that the rate achieved by this scheme is given by

r L(P,P ) =12

I (X 1, X 2; Y 1, Y 2) = max0 1

12C

2P N

1 + 1 + abd2

1 + b2d2, (15)

where d = 2P / (2a2P + N ).This scheme is not optimal even among linear relaying schemes with block length k = 2.However, as demonstrated in Figure 3, it achieves higher rate than the generalized block

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a b

1X 1 X 2

X 12

Y 1 Y 2

Y 11

Figure 2: Sub-optimal linear relaying scheme for general AWGN relay channel with blocklength k = 2.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Upper BoundBlockMarkovLinear Relaying (2)Side Information

P/N

Rate(Bits/T

rans)

Figure 3: Comparison of achievable rates based on the generalized block Markov, linearrelaying and side information encoding schemes for the general AWGN relay channel fora = 1 and b = 2.

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Markov coding when a is small and can also outperform the more sophisticated side infor-mation coding scheme when all distributions are Gaussians.

Next we consider the FD-AWGN relay channel with linear relaying functions. Sincethe channel from the relay to the receiver uses a different frequency band than the chan-nel from the sender, without loss of generality we assume that ith relay transmission candepend on all received signals up to i (instead of i 1). With this relabeling, for blocklength k, the transmitted vector X k = [X 1, X 2, . . . , X k]T , the transmitted relay vectorX k1 = [X 11 , X 12 , . . . , X 1k]T = D Y k1 , where D is a lower triangular weight matrix withpossibly non-zero diagonal elements, and the received vector Y k = [Y kD

T , Y kRT ]T , where

Y kD = [Y D 1, Y D 2, . . . , Y Dk ]T and Y kR = [Y R 1, Y R 2, . . . , Y Rk ]T . The capacity with linear relayingcan be expressed as in (12), where

C FD Lk (P,P ) =1k

supP X k , D : Pki =1 E (X 2i ) kP,

max y n1 Pki =1 x21i kP

I (X k ; Y k). (16)

It can be easily shown that the above maximization is achieved when X k is Gaussian. De-noting the covariance of X k by , we can reduce (16) to

C FD Lk (P,P ) =1

2kmax ,D

log

+ NI abD T abD a2b2DD T + N (I + b2DD T )

N I 00 I + b2DD T , (17)

where the maximization is subject to 0, tr() kP , tr( a2DD T + NDD T ) kP anddij = 0 for i > j . This is again a non-convex optimization problem and nding C FD L(P,P )reduces to solving a sequence of such problems indexed by k. Luckily in this case we canreduce the problem to a single-letter optimization problem. Before proceeding to provethis result, consider the following simple amplify-and-forward scheme.

Example 2: We consider block length k = 1. It is easy to show that Equation (17) reducesto

C FD L1 (P,P ) = CP N

1 +a2b2P

(a2 + b2 )P + N , (18)

which can be achieved by the simple amplify-and-forward scheme depicted in Figure 4, withX 1N (0, P ) and X 11 =

P

a2 P + N Y 1.

It can be shown that if ab < 1/ 2, then C FD L1 (P,P ) is a concave function of P . Otherwise, it is convex for small values of P and concave for large values of P . Theinterpretation is that as P is decreased, linear relaying injects more noise than signal, thusbecoming less helpful. In such cases, the performance of this scheme can be improved bytime-sharing between amplify-and-forward for a fraction 0 1 of the time and directtransmission, where only the sender node transmits, for the rest of the time. The followinglower bound can be established using this scheme and can be shown to be concave in P forall parameter values and all P > 0 (See Figure 6 for an illustration).

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ab

1

X 1 X 2

Y 11 Y 12 X 11 X 12

Y R 2Y R 1

Y D 1 Y D 2

Figure 4: Linear relaying scheme for FD-AWGN relay channel with block length k = 1.

C FD L(P,P ) max0

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0 P

C FD L1 (P,P ) with time-sharing

C FD L1 (P,P )

Figure 6: Comparison of achievable rates using amplify-and-forward and time-sharing be-tween amplify-and-forward and direct transmission only.

where = [ 0, 1, . . . , 4], = [0, 1, . . . , 4], = [1, 2, . . . , 4], subject to j , j 0,j > 0, 4j =0 j = 4j =0 j = 1 , and 4j =1 j (a2j P + N j ) = P .Proof: We rst outline the proof. The main difficulty in solving the optimization problem

is that it is not concave in and D. However, for any xed D, the problem is concave in and the optimal solution can be readily obtained using convex optimization techniques [27].We show that for any matrix D, there exists a diagonal matrix L whose associated optimalcovariance matrix is also diagonal and such that the value of the objective function for thepair (D, ) is the same as that for the pair ( L, ). Hence the search for the optimal solutioncan be restricted to the set of diagonal D and matrices. The reduced optimization problem,however, remains non-convex. We show, however, that it can be reduced to a non-convexconstrained optimization problem with 14 variables and 3 equality constraints.

Simplifying the expression for C FD Lk (P,P ) in (17), the optimization problem can beexpressed as

C FD Lk (P,P ) = max ,D1

2klog I +

1N

I ab(I + b2DD T ) 1/ 2D

I ab(I + b2DD T ) 1/ 2D

T

,

(21)subject to 0, tr() kP and tr( a2DD T + NDD T ) kP , where D is a lowertriangular matrix.

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Finding the optimal for any xed D is a convex optimization problem [27]. Dening

G =1

N I

ab(I + b2DD T ) 1/ 2D ,

and neglecting the 12k

factor, the Lagrangian for this problem can be expressed as

L( , , , ) = log |I + GGT | tr() + (tr() kP ) + (tr( a2DD T + NDD T ) kP ).The Karush-Kuhn-Tucker (KKT) conditions for this problem are given by

I + a2D T D = GT I + GGT 1 G + ,

0 tr() = 0 ,tr() kP (tr() kP ) = 0 ,

tr( a2DD T + NDD T ) kP (tr( a2DD T + NDD T ) kP ) = 0 ,

where , 0. Now if UHV T

is the singular value decomposition of G and F LQT

is thesingular value decomposition of D, then

V H 2V T = GT G

=1N

Q(I + a2b2L(I + b2L2) 1L)QT .

Columns of matrices V and Q are eigenvectors of the symmetric matrix GT G. Hence V T Qand QT V are permutation matrices. As a result, V T D T DV = V T QL2QT V = L2 is adiagonal matrix and therefore, D T D = V V T where = L2. Therefore, the rst KKTcondition can be simplied to

I + a2 = H I + HV T V H 1

H + V T V.

Since the KKT conditions are necessary and sufficient for optimality, it follows that musthave the same set of eigenvectors V , i.e., = V V T , where is a diagonal matrix. Thedual matrix can be chosen to have the same structure, i.e., = V V T and tr() =tr() = 0, hence satisfying the KKT conditions. As a result, the expression in (21) can besimplied to

C FD Lk (P,P ) =1

2kmax,

log I +1N

(I + a2b2(I + b2) 1) , (22)

where and are diagonal matrices. Since this expression is independent of V , we can setV = I . Hence, the search for the optimal matrices D and can be limited to the spaceof diagonal matrices and . In particular, if D T D = V V T , then the diagonal matrixL = .

Thus, the maximization problem can be simplied to the following

C FD Lk (P,P ) =1

2kmax

1 , . . . , k 1 , . . . , k

logk

i=1

1 +1N

i 1 +a2b2 i

1 + b2 i, (23)

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subject to i 0, i 0, for i = 1 , 2, . . . , k , ki=1 i kP , and ki=1 i(a2i + N ) kP .First it is easy to see that at the optimum point, ki=1 i = kP and

ki=1 i(a2i +

N ) = kP . Note that the objective function is an increasing function of i . Therefore,if ki=1 i (a2i + N ) < kP , we can always increase the value of objective function by

increasing j assuming that j = 0. Now we show thatki=1 i = kP . It is easy to

verify that j 1 +1N j 1 +

a2 b2 j1+ b2 j is positive along the curve j (a

2j + N ) = const. .

Therefore, while keeping ki=1 i(a2i + N ) xed at kP , we can always increase the objectivefunction by increasing j and hence ki=1 i = kP .

Note that at the optimum, if j = 0, then j = 0. However, if j = 0, the value of jis not necessarily equal to zero. Without loss of generality, assume that at the optimum, j = 0 for the rst 0 k0 k indices, and that the total power assigned to the k0 indicesis given by k0j =1 j = 0kP , for 0 0 1. Then by convexity

logk0

j =11 + 1N j 1 + a

2

b2

j1 + b2 j= log

k0

j =11 + 1N j k0 log 1 + 0kP k0N ,

where the upper bound can be achieved by redistributing the power as j = 0 kP k0 , for1 j k0. In Appendix E, we show that at the optimum, there are no more than four(j , j ) distinct pairs such that j > 0 and j > 0. Including the case where j = 0, wetherefore conclude that there are at most ve distinct ( j , j ) pairs for any k 5. Thus, ingeneral, the capacity can be expressed as

C FD Lk (P,P ) =1

2k

maxk , ,

log 1 +0kP

k0N

k0 4

j =1

1 +j kP

kj N

1 +a2b2j

1 + b2

j

kj

, (24)

where k = [k0, k1, . . . , k4], = [0, 1, . . . , 4], = [1, 2, . . . , 4], subject to j 0, j > 0,4j =0 kj = k,

4j =0 j = 1, and

4j =1 kj j a2

k j P kj + N = kP .

To nd C FD L(P,P ) we need to nd limk C FD Lk (P,P ). Taking the limit of theabove expression as k , we obtain

C FD L(P,P ) = max , ,

0C0P 0N

+4

j =1

j Cj P j N

1 +a2b2j

1 + b2j, (25)

where = [0, 1, . . . , 4], subject to j , j 0, j > 0, 4j =0 j = 4j =0 j = 1, and4j =1 j (a2j P + N j ) = P . This completes the proof of the theorem.

We have shown that the capacity with linear relaying can be computed by solving asingle-letter cosntrained optimization problem. The optimization problem, however, is non-convex and involves 14 variables (or 11 if we use the three equality constraints). Findingthe solution for this optimization problem in general requires exhaustive search, which iscomputationally extensive for 11 variables. Noting that the problem is convex in each set of

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variables , and if we x the other two, the following fast Monte Carlo algorithm canbe used to nd a good lower bound to the solution of the problem. Randomly choose initialvalues for the three sets of variables, x two of them and optimize over the third set. Thisprocess is continued by cycling through the variables sets, until the rate converges to a localmaximum. The process is repeated many times for randomly chosen initial points and local

maximas are found.Figure 7 compares the lower bound on the capacity of the FD-AWGN relay channel with

linear relaying to the max-ow min-cut upper bound and the generalized block Markov andside-information coding lower bound. Note that when a is small, the capacity with linearrelaying becomes very close to the upper bound. Further, as b , the capacity withlinear relaying becomes tight. On the other hand, as a , the generalized block Markovlower bound becomes tight. Note that if a2 1 + b2 1 + P N , the capacity is given by theblock Markov lower bound. This is similar to the result reported in [17].

5.1 Upper bound on minimum energy-per-bit

Linear relaying can improve the upper bound on the minimum energy-per-bit for the generalAWGN relay model. To demonstrate this, consider the achievable rate by the scheme inExample 1. It can be shown that the rate function (15) is convex for small P and therefore,as in Example 2, the rate can be improved by time-sharing. The rate function with time-sharing can be used to obtain an upper bound on the minimum energy-per-bit. Figure 8plots the ratio of the best upper to lower bounds on the minimum energy-per-bit. Notethat the simple scheme in Example 1 with time-sharing reduces the maximum ratio to 1.7.The minimum energy-per-bit using the linear relaying is usually lower than the minimumenergy-per-bit using the side-information scheme for the general AWGN relay channel.

Using the relationship between the minimum energy-per-bit and capacity in Theorem 1and the capacity with linear relaying for the FD-AWGN model established in Theorem 3,we can readily obtain an upper bound on minimum energy-per-bit of the FD-AWGN relaychannel. Linear relaying improves the ratio for large b and a 1. The maximum ratio usingthe linear relaying can be reduced to 1.87. The maximum ratio can be further reduced byusing the side information coding scheme to 1.45. Figure 9 plots the ratio of the best upperbound on the minimum energy-per-bit to the lower bound in Table 2 for the FD-AWGNrelay channel.

6 Conclusion

The paper establishes upper and lower bounds on the capacity and minimum energy-per-bitfor general and FD-AWGN relay channel models. The max-ow min-cut upper bound andthe generalized block Markov lower bound on capacity of the relay channel are rst used toderive corresponding upper and lower bounds on capacity. These bounds are never tight forthe general AWGN model and are tight only under certain conditions for the FD-AWGN

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0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

Upper BoundBlockMarkovLinear RelayingSide Information

P/N

Rate(Bits/Trans.)

(a) a = 1, and b = 2

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

Upper BoundBlockMarkovLinear RelayingSide Information

P/N

Rate(Bits/Trans.)

(b) a = 2, and b = 1

Figure 7: Comparison of achievable rates based on the generalized block Markov, linearrelaying, and side information encoding schemes for the FD-AWGN relay channel.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

b=50

20

10

5

2

a

RatioofUppertoLowerBou

ndsonE

b

Figure 8: Ratio of the best upper bound to the lower bound of Theorem 2 for various valuesof a and b for general AWGN relay channel.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

b=2

5

10

20

50

a

RatioofUppertoLowerBoundsonE

b

Figure 9: Ratio of the best upper bound to the lower bound of Theorem 2 for various valuesof a and b for FD-AWGN relay channel.

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model. The gap between the upper and lower bounds is largest when the gain of the channelto the relay is comparable or worse than that of the direct channel. We argue that the reasonfor this large gap is that in the generalized block Markov scheme, the relay either decodesthe entire message or it is not used at all. When a 1 or less than 1, this restricts theachievable rate to be close to the capacity of the direct channel. To obtain tighter lowerbounds for this case, two coding schemes are investigated where the relay cooperates withthe sender but without decoding any part of the message. First the side information codingscheme is shown to outperform the block Markov coding when the gain of the channel to therelay is comparable to that of the direct channel. We show that the achievable rate can beimproved via time-sharing and provide a general expression for the achievable rate using sideinformation coding for relay channels in general. In the second scheme, the relaying functionsare restricted to be linear. For the general AWGN model, a simple linear-relaying scheme isshown to signicantly outperform the more sophisticated generalized block Markov and sideinformation schemes in some cases. It is shown that the capacity with linear relaying can befound by solving a sequence of non-convex optimization problems. One of our main resultsin the paper is reducing this formulation to a single-letter expression for the FD-AWGNmodel. Figures 3 and 7 compare the rates for the different schemes.

The paper also established a general relationship between the minimum energy-per-bitand the capacity of the AWGN relay channel. This relationship together with the lower andupper bounds on capacity are used to establish corresponding lower and upper bounds onthe minimum energy-per-bit for the general and FD-AWGN relay channels. The bounds arevery close and do not differ by more than a factor of 1.45 for the FD-AWGN relay channelmodel and by 1.7 for the general AWGN model.

Two open problems are suggested by the work in this paper. The rst is to nd thedistribution on ( Q,X,X 1, Y ) that optimizes the achievable rate using side information coding

given in Theorem 2. Our bounds are obtained with the assumption that ( X, X 1, Y ) isGaussian and with specic choices of the time-sharing random variable Q. The second openproblem is nding a single-letter characterization of the capacity with linear relaying forthe general AWGN relay model. We have been able to nd such characterization only forthe FD-AWGN case.

In conclusion, the upper and lower bounds for the capacity and minimum energy-per-bitestablished in this paper are still not tight for the general AWGN relay model and are onlytight under certain conditions for the FD-AWGN relay channel. Establishing capacity andthe minimum energy-per-bit is likely to require a combination of new coding schemes and atighter upper bound than the max-ow min-cut bound.

References

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

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[2] H. Sato, Information transmission through a channel with relay, The Aloha System,University of Hawaii, Honolulu, Tech. Rep B76-7, March 1976.

[3] A. El Gamal, Results in Multiple User Channel Capacity , PhD. Thesis, Stanford Uni-versity, May 1978.

[4] T. M. Cover and A. El Gamal, Capacity Theorems for the Relay Channel, IEEE Transactions on Information Theory , Vol. 25, No. 5, pp. 572-584, September 1979.

[5] A. El Gamal and M. Aref, The Capacity of the Semi-Deterministic Relay Channel,IEEE Transactions on Information Theory, Vol. IT-28, No. 3, pp. 536, May 1982.

[6] A. El Gamal, On Information Flow in Relay Networks, IEEE National Telecommu-nications Conference, Vol. 2, pp. D4.1.1 - D4.1.4, November 1981.

[7] M. Aref, Information ow in relay networks , PhD. Thesis, Stanford University, October1980.

[8] B. Schein and R. G. Gallager, The Gaussian parallel relay network, Proc IEEE Int Symp Info Theory , Sorrento, Italy, p. 22, July 2000.

[9] A. El Gamal and S. Zahedi, Capacity of a Class of Relay Channels with OrthogonalComponents, IEEE Transactions on Information Theory , Vol. IT-51, No. 5, pp. 1815-1817, May 2005.

[10] S. Zahedi, M. Mohseni and A. El Gamal, On the Capacity of AWGN Relay Chan-nels with Linear Relaying Functions, Proc. International Symposium on Information Theory , June 27-July 2, 2004, Chicago, IL, P. 399.

[11] P. Gupta and P.R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory , Vol. 46, pp. 388-404, March 2000.

[12] P. Gupta and P. R. Kumar, Towards an information theory of large networks: Anachievable rate region, Proc IEEE Int Symp Info Theory , Washington DC, p. 159,June 2001.

[13] M. Gastpar and M. Vetterli, On The Capacity of Wireless Networks: the relay case,Proc IEEE INFOCOM 2002 , New York city, NY, vol. 3, pp. 1577-1586, June 2002.

[14] M. Gastpar and M. Vetterli, On The Asymptotic Capacity of Gaussian Relay Chan-nels, Proc IEEE Int Symp Info Theory , Lausanne, Switzerland, June 30-July 5 2002.

[15] G. Kramer, M. Gastpar and P. Gupta, Cooperative strategies and capacity theoremsfor relay networks, to appear in IEEE Transactions on Information Theory .

[16] V. Rodoplu and T.H. Meng, Minimum energy mobile wireless networks, IEEE Journal on Selected Areas in Communications , pp. 1333 -1344, August 1999.

[17] Y. Liang and V.V. Veeravalli, Gaussian frequency division relay channels: optimalbandwidth allocation and capacity, In Proc CISS , Princeton, NJ, March 2004.

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[18] A. El Gamal and S. Zahedi, Minimum Energy Communication Over a Relay Chan-nel, Proc. International Symposium on Information Theory , June 29-July 4, 2003,Yokohama, Japan, P. 344.

[19] R.J. McEliece, The Theory of Information and Coding , Addison-Wesley Publishing

Company, Reading, Massachusetts, 1977.[20] R. G. Gallager, Energy Limited Channels:Coding, Multiaccess, and Spread Spectrum,

Tech. Report LIDS-P-1714, LIDS, MIT, Cambridge, Mass., November 1987.

[21] R. G. Gallager, Energy limited channels, coding, multiaccess and spread spectrum,Proc. Conf. Inform. Sci. Syst. , p. 372, Princeton, NJ, Mar. 1988.

[22] S. Verdu, On Channel Capacity per Unit Cost, IEEE Transactions on Information Theory , Vol. 36, pp. 1019-1030, September 1990.

[23] S. Verdu, G. Caire and D. Tuninetti, Is TDMA optimal in the low power regime?

Proc IEEE Int Symp Info Theory , Lausanne, Switzerland, p. 193, July 2002.[24] A. Hst-Madsen and J. Zhang: Capacity Bounds and Power Allocation for Wireless

Relay Channel, submitted to IEEE Trans. Information theory .

[25] S. Verdu, Spectral efficiency in the wideband regime, IEEE Transactions on Infor-mation Theory , Vol. 48, pp. 1319 -1343, Jun 2002.

[26] T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley, New York,1991.

[27] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press,

2004.

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Appendix A

In this appendix, we evaluate upper bounds on the capacity of the general and FD-AWGNrelay channels. The max-ow min-cut bound gives the

C maxp(x,x 1 ) min{I (X, X 1; Y ), I (X ; Y, Y 1|X 1)}.First consider the general AWGN relay channel. To prove the upper bound we begin withthe rst bound. Using standard arguments, it can be easily shown that

I (X, X 1; Y ) = h(Y ) h(Y |X, X 1)= h(Y )

12

log(2eN )

12

logVar( Y )

N

12 log 1 +

Var( X ) + b2Var( X 1) + 2 bE (XX

1)

N

12

log 1 +(1 + b2 + 2 b )P

N ,

where we dene =

E (XX 1)

E (X 2)E (X 21 ).

Now, consider the second bound

I (X ; Y, Y 1

|X 1) = h(Y, Y 1

|X 1)

h(Y, Y 1

|X, X 1)

h(Y, Y 1|X 1) log2eN = h(Y |X 1) + h(Y 1|Y, X 1) log2eN

12

log2eE Var( Y |X 1) +12

log2eE Var( Y 1|Y, X 1i) log2eN

12

log 2e E (X 2) (E (XX 1))2

E (X 21 )+ N

+12

log 2e(1 + a2) E (X 2) (E (XX 1 ))

2

E (X 21 )N + N 2

E (X 2) (E (XX 1 ))2

E (X 21 )+ N log2eN

= 12 log (1 + a2) E (X 2) (E (XX 1))

2

E (X 21 )N + N 2

12

log 1 +(1 + a2)(1 2)P

N .

Therefore, the upper bound can be expressed as

C max0 1 min C(1 + a2)(1 2)P

N , C

(1 + b2 + 2 b )P N

.

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Performing the maximization over , we can easily obtain the upper bound given in Table 1.

Now consider the bound for the FD-AWGN relay channel. Substituting Y by {Y D , Y R}in (1) yields the max-ow min-cut upper bound on capacity of the FD-AWGN channel. Notethat

I (X ; Y 1, Y D , Y R |X 1) = I (X ; Y 1|X 1) + I (X ; Y D |X 1, Y 1) + I (X ; Y R |X 1, Y 1, Y D )= I (X ; Y 1|X 1) + I (X ; Y D |X 1, Y 1)= h(Y 1|X 1) h(Y 1|X, X 1) + h(Y D |X 1, Y 1) h(Y D |X, X 1, Y 1) h(Y 1|X 1) + h(Y D |X 1, Y 1) log2eN = h(Y 1|X 1) + h(Y D |Y 1) log2eN

12

log2eVar( Y 1|X 1) +12

log2eVar( Y D |Y 1) log2eN = C

a2P (1 2)N

+ CP

a2P + N .

Similarly, it can be shown that

I (X, X 1; Y D , Y R ) = I (X ; Y D , Y R ) + I (X 1; Y D , Y R |X )= I (X ; Y D ) + I (X ; Y R |Y D ) + I (X 1; Y R |X ) + I (X 1; Y D |X, Y R )= I (X ; Y D ) + I (X ; Y R |Y D ) + I (X 1; Y R |X ) C

P N

+ Cb2 2NP

(b2P (1 2) + N )(P + N )+ C

b2P (1 2)N

.

Again both terms are maximized for = 0. As a result the following upper bound oncapacity can be established

C min CP N

+ Cb2P

N , C

(1 + a2)P N

.

Upper and lower bounds in Table 1 can be readily established.

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Appendix B

We show that the lower bound on the capacity of the discrete-memoryless relay channel of Theorem 6 in [4] can be recast into the bound

C maxp(x)p(x1 )p(y1 |y1 ,x 1 ) min I (X ; Y, Y 1|X 1), I (X, X 1; Y ) I (Y 1; Y 1|X, X 1) . (26)

Achievability of any rate R I (X ; Y, Y 1|X 1) subject to the constraint I (X 1; Y ) I (Y 1; Y 1|X 1, Y ), for any distribution p(x, x 1, y, y1, y1) = p(x)p(x1)p(y, y1|x, x 1)p(y1|y1, x1)was proved in [4]. We show the converse, i.e., any rate satisfying the original conditionsR I (X ; Y, Y 1|X 1) and I (X 1; Y ) I (Y 1; Y 1|X 1, Y ), also satises (26). ConsiderR = I (X ; Y, Y 1|X 1)

= H (Y, Y 1|X 1) H (Y, Y 1|X, X 1)= H (Y |X 1) + H (Y 1|X 1, Y ) H (Y 1|X, X 1) H (Y |X, X 1, Y 1)= H (Y |X 1) + H (Y 1|X 1, Y ) H (Y 1|X, X 1) H (Y |X, X 1)= H (Y ) I (X 1; Y ) + I (Y 1; Y 1|X 1, Y ) + H (Y 1|X 1,Y,Y 1) H (Y 1|X, X 1) H (Y |X, X 1) H (Y ) + H (Y 1|X 1,Y,Y 1) H (Y 1|X, X 1) H (Y |X, X 1)= H (Y ) + H (Y 1|X 1, Y 1) H (Y 1|X, X 1) H (Y |X, X 1)= H (Y ) + H (Y 1|X, X 1, Y 1) H (Y 1|X, X 1) H (Y |X, X 1)= I (X, X 1; Y ) I (Y 1; Y 1|X, X 1).

It is not difficult to show the achievability, i.e., that any rate satisfying the above inequality

also satises the original conditions.

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Appendix C

Achievability of any rate

R

max

p(q)p(x |q)p(x1 |q)p(y1 |y1 ,x 1 ,q)min I (X ; Y D , Y R , Y 1

|X 1, Q), I (X, X 1; Y D , Y R

|Q)

I (Y 1; Y 1

|X, X 1, Q) ,

was shown in section 4. We now evaluate the mutual information terms for the AWGNrelay channel. The optimal choice of probability mass functions are not known. We assumethe random variable Q has cardinality 2 and takes values in {0, 1}. We further assumeP{Q = 1}= . Consider X as a Gaussian random variable with variance P if Q = 1 andzero otherwise. Furthermore, assume X 1 is a Gaussian random variable with variance P irrespective of the value of random variable Q and independent of X . Dene the randomvariable Y 1 = 0 i f Q = 0, and Y 1 = (Y 1 + Z ) if Q = 1, where is a constant andZ N (0, N ), is independent of Q, X , X 1, Z , and Z 1.

Now consider

h(Y 1|X 1, Y D , Q = 1) = E X 1 ,Y D12

log2eVar( Y 1|yD , x1)=

12

log2e2 N + N +a2P N

P + N ,

h(Y 1|Y 1, Q = 1) = E Y 112

log2eVar( Y 1|y1)=

12

log2e2N ,

h(Y 1|

X, X 1, Q = 1) = E

X,X 1

1

2log2eVar( Y

1|x, x

1)

=12

log2e2(N + N ).

Using the above results we can easily show that

I (X ; Y D , Y R , Y 1|X 1, Q) = I (X ; Y D |X 1, Q) + I (X ; Y 1|X 1, Y D , Q) + I (X ; Y R |X 1, Y 1, Y D , Q)= I (X ; Y D |X 1, Q) + I (X ; Y 1|X 1, Y D , Q)= C

P N

+ Ca2P N

(P + N )(N + N ),

I (X, X 1; Y D , Y R

|Q) = I (X, X 1; Y D

|Q) + I (X, X 1; Y R

|Y D , Q)

= I (X ; Y D |Q) + I (X 1; Y D |X, Q ) + I (X 1; Y R |Y D , Q) + I (X ; Y R |X 1, Y D , Q)= I (X ; Y D |Q) + I (X 1; Y R |Y D , Q)= I (X ; Y D |Q) + I (X 1; Y R |Q)= C

P N

+ Cb2P

N ,

I (Y 1; Y 1|X, X 1, Q) = CN N

.

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Combining the above results, it can be shown that any rate

R max0

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Appendix D

Lemma: The capacity of the AWGN relay channel with average power constraints satisesthe following:

(i) C (P,P ) > 0 if P > 0 and approaches as P .(ii) C (P,P ) 0 as P 0.

(iii) C (P,P ) is concave and strictly increasing in P .

(iv) (1+ )P C (P,P ) is non-decreasing in P , for all P > 0.

Proof:

(i) This follows from the fact that

C(P/N ), which is less than or equal to C (P,P ), is

strictly greater than zero for P > 0, and approaches innity as P .(ii) This follows from the fact that C (P,P ) in table 1 which is greater than or equal to

C (P,P ) approaches zero as P 0.(iii) Concavity follows by the following time-sharing argument. For any P, P > 0 and

> 0, there exists k and k such that C (P,P ) < C k(P,P ) + and C (P , P ) 0, i > 0, for i = k0+1 , . . . , k , ki= k0 +1 i = k(10)P , and ki= k0 +1 i(a2i+N ) = kP .For a given 0 and k0, this optimization problem is equivalent to nding the maximum

of

logk

i= k0 +1

1 +1N

i 1 +a2b2 i

1 + b2 i,

subject to i > 0, i > 0, for i = k0+1 , . . . , k , ki= k0 +1 i = k(10)P , and ki= k0 +1 i(a2i+N ) = kP .To nd the optimality condition for this problem, we form the Lagrangian

L(kk0 +1 , kk0 +1 , , ) =k

i= k0 +1

log 1 + 1N

i 1 + a2b2 i

1 + b2 i+

k

i= k0 +1

i + k

i= k0 +1

i(a2i+ N )

where and are Lagrange multipliers for the two equality constraints (the Lagrangemultipliers for the inequality constraints are all equal to zero at the optimum, since byassumption i > 0 and i > 0 for all i > k 0).

At the optimum, we must have

L i

= 0 , and L i

= 0 , for all i = k0 + 1 , . . . , k .

Computing the derivatives, we obtain the conditions

1 + (1 + a2)b2 i)1 + b2 i + i(1 + b2 i(1 + a2))

+ + a2 i = 0 ,

anda2b2i

(1 + b2 i)(1 + b2 i + i(1 + b2 i(1 + a2)))+ (a2i + N ) = 0 .

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Solving this set of equations, we obtain

i =N (1 + b2 i)(1 + (1 + a2)b2 i))

a2b2 a2 2a2b2 i (1 + a2)a2b4 2i,

where is are the positive roots of the fourth order polynomial equation

C 4Z 4 + C 3Z 3 + C 2Z 2 + C 1Z + C 0 = 0 ,

with coefficients

C 0 = a2b6 2(1 + a2)(N (1 + a2) a2)C 1 = (1 + a2)b4 ((1 + a2)(Nb2 a2b2 + a2N ) a 2b2 + 2 a2N a4 ) 2a4b4 2C 2 = (1 + a2)b2 (3a2b2 + 3 Nb2 + Na 2b2 + 2 a2N ) + a2 2b2N 3a2 2b2(a2 + )C 3 = (1 + a2)b2(a 2b2 + 2 N a2 ) + a2b2(2 + (a2 3) 2b2) + Nb 2 a4 2C 4 = a2(1 + )(b2

) + N.

This polynomial equation has at most four distinct roots for any given channel gain coeffi-cients a and b. Denote the roots by 1, 2, 3, 4. Substituting in the optimality conditions,we obtain at most four distinct values of , which we denote by by 1, 2, 3, 4. Note thatonly pairs such that j > 0 and j > 0 are feasible.