THE relay channel, whereby point-to-point communication

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 63, NO., FEBRUARY The Approximate Capacity of the MIMO Relay Channel Xianglan Jin, Member, IEEE, Young-Han Kim, Fellow, IEEE Abstract Capacity bounds are studied for the multipleantenna complex Gaussian relay channel with t 1 transmitting antennas at the sender, r receiving t transmitting antennas at the relay, r 3 receiving antennas at the receiver. It is shown that the partial decode forward coding scheme achieves within mint 1, r ) bits from the cutset bound at least one half of the cutset bound, establishing a good approximate characterization of the capacity. A similar additive gap of mint 1 + t, r 3 ) + r bits is shown to be achieved by the compress forward coding scheme. The corresponding results for time-division half-duplex relay channels are also established. Index Terms Additive gap, channel capacity, compress forward, cutset bound, multiplicative gap, partial decode forward. I. INTRODUCTION THE relay channel, whereby point-to-point communication between a sender a receiver is aided by a relay, is a canonical building block for cooperative wireless communication. Introduced by van der Meulen [1], this channel model has been studied extensively in the literature, including the now classical paper by Cover El Gamal []. The problem of characterizing the capacity in a computable expression, however, remains open even for simple channel models, consequently a large body of the literature has been devoted to the study of upper lower bounds on the capacity. Reminiscent of the max-flow min-cut theorem [3], the cutset bound was established by Cover El Gamal [], which sets an intuitive upper limit on the capacity. On the other direction, there are a myriad of coding schemes, typically referred to as * forward [4], each establishing a lower bound on the capacity. Among these, the two most versatile coding schemes are partial decode forward [, Th. 7] compress forward [, Th. 6], which are complementary to each other providing digital-to-digital analog-to-digital relays, respectively) have been successfully extended to Manuscript received September 7, 015; revised May 1, 016; accepted July 5, 016. Date of publication November 3, 016; date of current version January 18, 017. This work was supported in part by the Korean National Research Foundation under the Basic Science Research Program under Grant NRF-014R1A1A in part by the National Science Foundation under Grant CCF This work was presented at the 014 IEEE International Symposium on Information Theory, Honolulu, Hawaii. X. Jin is with the Department of Electrical Computer Engineering, Pusan National University, Busan 4641, South Korea jinxl77@pusan.ac.kr). Y.-H. Kim is with the Department of Electrical Computer Engineering, University of California at San Diego, La Jolla, CA USA yhk@ucsd.edu). Communicated by S. Avestimehr, Associate Editor for Communications. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIT general relay networks for unicast, multicast, broadcast, multiple access [5] [9]. The Gaussian relay channel, whereby the signals from the sender the relay are corrupted by additive white Gaussian noise, is one of the most basic channel models studied in the literature. The capacity of the Gaussian relay channel, however, is again unknown for any nondegenerate channel parameters. Instead, the following results have been established for single-antenna Gaussian relay channels. Partial decode forward, which is a superposition of decode forward direct transmission, reduces to the better of the two [10]. Partial decode forward achieves within one bit from the cutset bound [6], [11] a similar 1-bit gap result can be also obtained using compress forward [1]). These results establish simple approximate expressions of the capacity, which are particularly useful at high signal-to-noise ratio SNR). A natural question arises on how these results can be extended to multiple-antenna also known as multiple-input multiple-output or MIMO) Gaussian relay channels. Capacity bounds for MIMO relay channels have been studied in several papers. By convex optimization techniques [13], Wang et al. [14] derived upper lower bounds based on looser versions of the cutset bound the decode forward bound. These results have been improved by more advanced coding schemes partial decode forward compress forward) with suboptimal decoding rules by Simoens et al. [15] Ng Foschini [16]. The usual focus of this line of work, however, has been on the optimization of resources power bwidth) for practical implementations on numerical computation of resulting capacity bounds. In the same vein, a recent study by Gerdes et al. [17] established the optimal input distribution of the partial decode forward lower bound for the Gaussian MIMO relay channel. The most relevant results to our main question are established in [6], [9], [18]. In the award-winning paper, Avestimehr et al. [6] proposed a powerful approach to studying wireless networks, which, inter alia, establishes an approximate capacity of the Gaussian MIMO relay network using the quantize map forward scheme. Among several extensions of this seminal work, one of the tightest results to date for MIMO relay networks has been established by Kolte et al. [18], who carefully compared the noisy network coding lower bound for the general relay network [8] with the cutset bound. In an alternative direction, the distributed decode forward scheme [9], which generalizes partial decode forward to general relay networks, can be used to approximate the capacity of the IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 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2 1168 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 63, NO., FEBRUARY 017 Gaussian MIMO relay network. These results [6], [9], [18] can be readily specialized to the 3-node relay channel yield constant gap results in terms of the number of antennas. This paper provides more direct tighter answers to our main question through an elementary yet careful analysis of the partial decode forward compress forward lower bounds for the MIMO relay channel. The main contributions are summarized as follows. Fig. 1. The MIMO relay channel. For the complex Gaussian relay channel with t 1 transmitting antennas at the sender, r receiving t transmitting antennas at the relay, r 3 receiving antennas at the receiver, we show that the partial decode forward achieves within mint 1, r ) bits of the cutset bound Theorem 1). This gap is somewhat relaxed when noncoherent transmission is employed Proposition 4). Unlike the single-antenna counterpart, partial decode forward can achieve rates arbitrarily higher than the better of decode forward direct transmission in MIMO relay channels Proposition 5). To complement the additive gap result, we show that both coherent noncoherent partial decode forward coding schemes achieve at least half the cutset bound Theorem ). We show that compress forward achieves mint 1 + t, r 3 ) + r bits within the cutset bound Theorem 3). In conclusion, the paper establishes simple approximate expressions of the capacity, which are particularly useful at high low SNR. Beyond these analytical results, we also discuss how these expressions can be computed efficiently. Several models for half-duplex relay channels have been studied in the literature, in which the relay or relay antenna cannot transmit receive simultaneously; see, for example, [6], [10], [16], [19] [3]. Among these, arguably the most interesting model was proposed by Kramer [1], whereby the relay state is switched between the transmitting receiving modes based on the channel output at the relay. As an extension, Cardone et al. [3] introduced an individual antenna switching strategy at the relay for MIMO relay networks established the constant gap result of 1.96 bits per antenna using the noisy network coding scheme. This result extends their earlier work on the single-antenna relay channel case [] that shows partial decode forward achieves the capacity within 1 bit compress forward achieves the capacity within 1.61 bits. Our main results for the full-duplex case can be easily adapted to the half-duplex MIMO relay channel with t 1 transmitting antennas at the sender, a halfduplex antennas at the relay, r 3 receiving antennas at the receiver as follows. Partial decode forward achieves within mint 1, a ) bits of the cutset bound as well as at least half the cutset bound Proposition 13). Compress forward achieves within mint 1 + a, r a ) bits of the cutset bound Proposition 14). The rest of the paper is organized as follows. In the next section, we formally define the channel model review the cutset upper bound, the partial decode forward lower bound, the compress forward lower bound on the capacity. The main results on additive multiplicative gaps for partial decode forward compress forward are also stated therein. The proofs of these results are given in Sections III IV. Section V is devoted to the computational aspects of our results, namely, how the capacity bounds can be computed efficiently via appropriate convex optimization formulations. Using these computational tools, the main results are verified by numerical simulations. In Section VI, half-duplex MIMO relay channels are discussed. Throughout the paper, we use the following notation. The superscript ) H denotes the complex conjugate transpose of a complex) matrix; tr ) denotes the trace of a matrix; I n denotes the n n identity matrix where the subscript n is omitted when it is irrelevant or clear from the context); denotes a set of n m complex matrices; A B denotes that A B is hermitian positive semidefinite for hermitian matrices A B, E ) denotes the expectation with respect to the rom variables in the argument. C n m II. PROBLEM SETUP AND MAIN RESULTS We model the point-to-point communication system with a relay as a MIMO relay channel with sender node 1, relay node, receiver node 3; see Fig. 1. Throughout the paper, we assume the complex signal model, but corresponding results for the real case can be easily obtained; see the conference version [4] of the current paper for some results on the real model. The relay the receiver have r r 3 receiving antennas with respective channel outputs Y = G 1 X 1 + Z, Y 3 = G 31 X 1 + G 3 X + Z 3, 1) where G 1 C r t 1, G 31 C r 3 t 1,G 3 C r 3 t are complex channel gain matrices, X 1 C t 1 X C t are the respective inputs at the sender the relay, Z CN0, I r ) Z 3 CN0, I r3 ) are independent complex Gaussian noise components. For simplicity, we will often use the shorth notation G 3 = [ [ ] ] G1 G 31 G 3 G 1 =. G 31 We assume average power constraints P across the t 1 t transmitting antennas at the sender the relay, respectively. As in the stard relay channel model [], the encoder is defined by x1 nm), the relay encoder is defined by x iy i 1 ), i = 1,...,n, the decoder is defined by ˆmy 3 n ). We assume that the message M is uniformly distributed over the message set. The average probability of error is defined as P e n) = P{ ˆM = M}. ArateR is said to be achievable for the relay

3 JIN AND KIM: APPROXIMATE CAPACITY OF THE MIMO RELAY CHANNEL 1169 channel if there exists a sequence of nr, n) codes such that lim n P e n) = 0. The capacity C of the relay channel is the supremum of all achievable rates. The following upper bound on the capacity is well known. Proposition 1 Cutset Bound [, Th. 4]): The capacity C of the MIMO relay channel is upper bounded by decode forward lower bound: R NPDF = sup min { I X 1, X ; Y 3 ), I U; Y X ) + I X 1 ; Y 3 X, U) } 9) = sup min { I X 1, X ; Y 3 ), I X 1 ; U, Y 3 X ) I X 1 ; U X, Y ) } 10) R CS = sup min { I X 1, X ; Y 3 ), I X 1 ; Y, Y 3 X ) } ) Fx 1,x ) = max min{ log I r3 + G 3 KG H 3, K log I r +r 3 + G 1 K 1 G H 1 } 3) = max min{ log I r3 +[G 31 G 3 ]K [G 31 G 3 ] H, K log I t1 + G H 1 G 1 + G H 31 G 31)K 1 } 4) where the supremum in ) is over all joint distributions Fx 1, x ) such that EX H j X j ) P, j = 1,, the maxima in 3) 4) are over all t 1 + t ) t 1 + t ) matrices [ ] K1 K K = 1 K1 H K 0 5) such that trk j ) P, j = 1,, K 1 = K 1 K 1 K 1 K H 1. The equality in 4) is justified by the following simple fact that will be used repeatedly throughout the paper. Lemma 1: For γ [0, 1], ar t matrix G, a t t matrix K 0, I r + γ GKG H = I t + γ G H GK γ mint,r) I r + GKG H. 6) We compare the cutset bound with two lower bounds on the capacity. The first lower bound is based on the partial decode forward coding scheme, in which the relay recovers part of the message forwards it. Proposition Partial Decode Forward Bound [, Th. 7]): The capacity C of the MIMO relay channel is lower bounded by R PDF = sup min { I X 1, X ; Y 3 ), I U; Y X ) + I X 1 ; Y 3 X, U) } 7) = sup min { I X 1, X ; Y 3 ), I X 1 ; U, Y 3 X ) I X 1 ; U X, Y ) } 8) where the suprema are over all joint distributions Fu, x 1, x ) such that EX H j X j ) P, j = 1,. Remark 1: The equivalence between 7) 8) is due to the fact that U X 1 Y form a Markov chain. It can be readily checked that the partial decode forward lower bound does not increase by coded) time sharing. The partial decode forward lower bound can be relaxed in several directions. First, by limiting the input distribution to a more practical product form, we obtain the noncoherent partial where the suprema are over all product distributions Fu, x 1 )Fx ) such that EX H j X j ) P, j = 1,. Second, by setting U = X 1, which is equivalent to having the relay recover the entire message, we obtain the decode forward lower bound: R DF = sup min { I X 1, X ; Y 3 ), I X 1 ; Y X ) } 11) = max min{ log I r3 + G 3 KG H 3, K log I r + G 1 K 1 G H 1 } 1) where the supremum in 11) is over all distributions Fx 1, x ) such that EX H j X j ) P, j = 1,, the maximum in 1) is over all t 1 + t ) t 1 + t ) matrices K 0 of the form 5) such that trk j ) P, j = 1,. Third, by setting U = X = 0, we obtain the direct-transmission lower bound: R DT = sup I X 1 ; Y 3 ) = max log I r3 + G 31 K 1 G H 31 13) K 1 where the supremum is over all distributions Fx 1 ) such that EX1 HX 1) P the maximum is over all t 1 t 1 matrices K 1 0suchthattrK 1 ) P. Remark : Since decode forward direct transmission schemes are two special cases of partial decode forward, we have in general R PDF maxr DF, R DT ). 14) Next, we present another important lower bound, in which the relay compresses its noisy observation instead of recovering the message. Proposition 3 Compress Forward Bound [, Th. 6], [10]): The capacity C of the MIMO relay channel is lower bounded by R CF = sup I X 1 ; Ŷ, Y 3 X ) 15) where the supremum is over all conditional distributions Fx 1 )Fx )Fŷ y, x ) such that EX H j X j ) P, j = 1, I X ; Y 3 ) I Y ; Ŷ X, Y 3 ). This lower bound can be expressed equivalently as R CF = sup min { I X 1, X ; Y 3 ) I Y ; Ŷ X 1, X, Y 3 ), I X 1 ; Ŷ, Y 3 X ) } 16) where the supremum is over all conditional distributions Fx 1 )Fx )Fŷ y, x ) such that EX H j X j ) P, j = 1,. Remark 3: The compress forward lower bound before taking the supremum in 15) or 16) is not a convex function of the conditional distribution Fx 1 )Fx )Fŷ y, x ) in general

4 1170 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 63, NO., FEBRUARY 017 can be potentially improved by coded) time sharing [5, Remark 16.4]. Remark 4: By setting Ŷ =, compress forward reduces to direct transmission thus R CF R DT. Remark 5: It is worthwhile to compare the cutset bound in ), the partial decode forward lower bound in 8), the compress forward lower bound in 16). The first term in the partial decode forward lower bound is identical to the cooperative multiple access channel MAC) bound I X 1, X ; Y 3 ), but the second term differs from the cooperative broadcast channel BC) bound I X 1 ; Y, Y 3 X ) in that U replaces Y there is a rate loss of I X 1 ; U X, Y ). In comparison, the second term of the compress forward lower bound is of the same form to the cooperative BC bound except for Ŷ in place of Y ), but the first term differs from the cooperative MAC bound in that there is a rate loss of I Y ; Ŷ X 1, X, Y 3 ) in addition to the loss from using the noncoherent input distribution. We are now ready to state the main results of the paper. Theorem 1: For every G 1, G 31, G 3,P, PDF := R CS R PDF mint 1, r ). 17) This result improves upon the constant gap of t 1 + t + r + r 3, which is achieved by distributed decode forward [9, Remark 5] when specialized to the 3-node relay channel. Theorem 1 will be proved in Subsection III-A. As a supplement to the additive gap result in Theorem 1, which is useful in approximating the capacity at high SNR, we establish the following multiplicative gap to provide a tighter approximation in low SNR, which will be proved in Subsection III-C. Theorem : For every G 1, G 31, G 3, P, R CS. 18) R PDF In other words, partial decode forward always achieves at least half the capacity. The above results can be relaxed by using the noncoherent partial decode forward. Proposition 4: For every G 1, G 31, G 3, P, NPDF := R CS R NPDF max [ mint 1, r ), mint 1 + t, r 3 ) ] 19) R CS. 0) R NPDF Proposition 4 will be proved in Subsections III-B III-C. For the single-antenna case, the partial decode forward lower bound can be shown [10, Sec. II] to be equal to the maximum of the decode forward direct transmission lower bounds; cf. 14). With multiple antennas, however, partial decode forward is in general much richer than decode forward direct transmission. Proposition 5: If t 1, t, r, r 3, [ sup RPDF maxr DF, R DT ) ] =. G 1,G 31,G 3,P Proposition 5 will be proved in Subsection III-D. As mentioned earlier, several generalizations of the compress forward coding scheme have been shown to achieve the capacity of general MIMO relay networks within a finite number of bits, which can be then specialized back to the 3-node relay channel. In particular, the quantize map forward scheme by Avestimehr et al. [6] achieves within 1r +r 3 ) + 3t 1 + t ) bits from the cutset bound. This bound can be improved by specializing a recent result [18, Th. 1] for general MIMO relay networks making it channel-independent. Proposition 6 Kolte et al. [18]): For every G 1, G 31, G 3, P, CF := R CS R CF DOF log 1 + t ) 1 + t DOF [ ] r + r 3 + min σ σ log e + DOF log1 + σ ) 1) where DOF = max[mint 1 + t, r 3 ), mint 1, r + r 3 )]. In this paper, we tighten this result further as follows. Theorem 3: For every G 1, G 31, G 3, P, CF min max[ mint 1 + t, r 3 ) + r log1 + 1/σ ), σ mint 1, r ) log1 + σ ) ] ) mint 1 + t, r 3 ) + r. 3) This theorem will be proved in Section IV. No multiplicative gap is known between the compress forward lower bound the cutset bound. This follows partly from the fact that the optimal distribution for 15) or 16) is rather difficult to characterize. It can be shown, however, that when restricted to Gaussian distributions, the compress forward lower bound even with time sharing) may have an unbounded multiplicative gap from the cutset bound. As a compromise, we state the following simple consequence of Remark 4 the proof of Theorem. Proposition 7: For every G 1, G 31, G 3, P, R CS. 4) maxr DF, R CF ) III. PARTIAL DECODE FORWARD In this section, we establish the results on partial decode forward stated in the previous section Theorem 1, Theorem, Proposition 4, Proposition 5). A. Partial Decode Forward Proof of Theorem 1) We evaluate the partial decode forward lower bound in 7) with X 1, X ) CN0, K ),wherek 0 is of the form in 5), U = G 1 X 1 + Z, 5) where Z CN0, I r ) is independent of X 1, X, Z, Z 3 ). Note that X 1, X, U, Y 3 ) has the same distribution as X 1, X, Y, Y 3 ). The first term of the minimum in 7) is I X 1, X ; Y 3 ) = log Ir3 + G 3 KG H 3. 6)

5 JIN AND KIM: APPROXIMATE CAPACITY OF THE MIMO RELAY CHANNEL 1171 For the second term, since CovX 1 U, X ) = CovX 1 Y, X ) = K 1 K 1 G H 1 I r + G 1 K 1 G H 1 ) 1 G 1 K 1 ) 1, = K 1 I t1 + G H 1 G 1K 1 we have I U; Y X ) + I X 1 ; Y 3 X, U) = log I r 3 + G 31 CovX 1 U, X )G H 31 I r + G 1 CovX 1 U, X )G H 1 + log I r + G 1 K 1 G H 1 = log I t1 + G H 1 G 1 + G H 31 G 31)K 1 + log I t 1 + G H 1 G 1K 1 I t1 + G H 1 G 7) 1K 1 log I t1 + G H 1 G 1 + G H 31 G 31)K 1 mint 1, r ) 8) where the last inequality follows by Lemma 1. Since mina, b) minc, d) maxa c, b d), comparing 6) 8) with the cutset bound in 4) completes the proof of Theorem 1. We can prove Theorem 1 alternatively using the following result that is applicable to a more general class of relay channels follows by setting pu x 1, x ) = p Y X 1,X u x 1, x ) in the second form of the partial decode forward lower bound in 8). Proposition 8: For a discrete memoryless relay channel py, y 3 x 1, x ) = py x 1, x )py 3 x 1, x ), PDF := R CS R PDF max I X 1; U X, Y ) px 1,x ) where pu x 1, x ) = p Y X 1,X u x 1, x ). Now, we apply Proposition 8 to the MIMO relay channel by setting U as the form in 5). Then, PDF sup I X 1 ; U X, Y ) Fx 1,x ) = sup hu X, Y ) hz ) Fx 1,x ) = max log I r + G 1 CovX 1 Y, X )G H 1 9) K = max log I t 1 + G H 1 G 1K 1 I t1 + G H 1 G 1K 1 ) 1 K = max log I t 1 + G H 1 G 1K 1 K I t1 + G H 1 G 1K 1 mint 1, r ), where the suprema are over all joint cdfs Fx 1, x ) such that EX H j X j ) P, j = 1,, the maxima are over all jointly Gaussian pairs X 1, X ) their covariance matrices K of the form 5). Here the equality in 9) is due to the maximum differential entropy lemma see, for example, [5, Sec..]). B. Noncoherent Partial Decode Forward Proof of the First Statement of Proposition 4) We use the following fact. Lemma : Let K 0 be of the form in 5). Then, for every G 31 G 3, we have G 31 K 1 G H 31 + G 3K G H 3 G 3K1 H GH 31 + G 31K 1 G H 3. Proof: Consider 30) [ ] [ ] K G31 G 1 K 1 [G31 ] H 3 G K1 H K 3 0. To prove Proposition 4, let K 0 be of the form in 5). Let X 1 CN0, K 1 ) X CN0, K ) be independent of each other, define U as in 5). Then, by Lemmas 1, the first term of the minimum in the noncoherent partial decode forward lower bound in 9) is I X 1, X ; Y 3 ) = log I r3 + G 31 K 1 G H 31 + G 3K G H 3 31) logi r3 + 1 G 31K 1 G H 31 + G 3K G H 3 + G 3 K1 H GH 31 + G 31K 1 G H 3 ) 3) log I r3 + G 3 KG H 3 mint 1 + t, r 3 ). 33) Following steps similar to the coherent case in Subsection III-A, we have CovX 1 U, X ) = CovX 1 U) = K 1 I t1 + G H 1 G 1K 1 ) 1 I U; Y X ) + I X 1 ; Y 3 X, U) = log I r 3 + G 31 CovX 1 U)G H 31 I r + G 1 CovX 1 U)G H 1 + log I r + G 1 K 1 G H 1 = log I t1 + G H 1 G 1 + G H 31 G 31)K 1 + log I r + G 1 K 1 G H 1 I t1 + G H 1 G 1K 1 log I t1 + G H 1 G 1 + G H 31 G 31)K 1 mint 1, r ). 34) Recalling K 1 K 1 comparing 33) 34) with the cutset bound in 4) completes the proof. C. Multiplicative Gap Proofs of Theorem the Second Statement of Proposition 4) We establish the factor-of-two gap of noncoherent partial decode forward, which in turn implies the factor-of-two gap of coherent) partial decode forward in Theorem. By setting U = X 1 or in 9) specializing 1) to independent X 1, X ), it can be readily checked that R NPDF maxr DF, R DT ) are simultaneously lower bounded by { max max min log I r3 + G 31 K 1 G H 31 + G 3K G H 3, K 1,K log I r + G 1 K 1 } G H 1 ), max log I r3 + G 31 K 1 G H 31 K 1 = max K 1,K min { log I r3 + G 31 K 1 G H 31 + G 3K G H 3, max log I r + G 1 K 1 G H 1, log I r3 + G 31 K 1 G H 31 )}. 35)

6 117 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 63, NO., FEBRUARY 017 We further bound each term in 35) from below. By 3) the fact that I + A/ = I + A + A /4 I + A for any A 0, we have log I r3 + G 31 K 1 G H 31 + G 3K G H 3 logi r3 + 1 G 31K 1 G H 31 + G 3K G H 3 +G 3 K1 H GH 31 + G 31K 1 G H 3 ) 1 log I r 3 +[G 31 G 3 ]K [G 31 G 3 ] H. 36) Similarly, since I + A I +B I + A+B for any A, B 0 see [6]), max { log I t1 + G H 1 G 1K 1, log I t1 +G H 31 G 31K 1 } 1 ) log I t1 +G H 1 G 1K 1 +log I t1 +G H 31 G 31K 1 1 log It1 + G H 1 G 1 + G H 31 G 31)K 1. 37) Comparing 36) 37) with the cutset bound in 4) establishes that R PDF maxr NPDF, R DF, R DT ) min{r NPDF, maxr DF, R DT )} 1 R CS. 38) We can establish Theorem directly based on the following result that is applicable to a more general class of relay channels. Proposition 9: For a discrete memoryless relay channel py, y 3 x 1, x ) = py x 1, x )py 3 x 1, x ), Proof: Consider R CS R PDF. I X 1 ; Y, Y 3 X ) = I X 1 ; Y X ) + I X 1 ; Y 3 X, Y ) I X 1 ; Y X ) + I X 1 ; Y 3 X ) max I X 1 ; Y X ), I X 1 ; Y 3 X ) ), 39) where the first inequality follows since Y X 1, X ) Y 3 form a Markov chain. By setting U = X 1 U = in 7), using the inequality in 39), we have R PDF max max{ min I X 1, X ; Y 3 ), I X 1 ; Y X ) ), px 1,x ) I X 1 ; Y 3 X ) } = max min{ I X 1, X ; Y 3 ), px 1,x ) max I X 1 ; Y X ), I X 1 ; Y 3 X ) )} 1 max min{ I X 1, X ; Y 3 ), px 1,x ) max I X 1 ; Y X ), I X 1 ; Y 3 X ) )} 1 max min I X 1, X ; Y 3 ), I X 1 ; Y, Y 3 X ) ) px 1,x ) = 1 R CS. D. Decode Forward Direct Transmission Proof of Proposition 5) Consider the MIMO relay channel with G 31 = diagg, 1), G 1 = diag1, g), G 3 = diagg, g), g > 1, which is equivalent to a product of two mismatched single-antenna relay channels, one with the direct channel stronger than the sender-to-relay channel the other in the opposite direction. Setting K 1 = K 1 = K = P/)I in 6) 8), we have { ) R PDF min log 1 + g P g ) P ), log g ) P ) } = log g ) P ). 40) In comparison, R DF = R DT = max log1 + P 1)1 + g P ) P 1 +P P log1 + P)1 + g P). 41) Therefore, we have g ) P ) R PDF maxr DF, R DT ) log 1 + P)1 + g P) which tends to infinity as g. Intuitively, partial decode forward can choose different operations per antenna, the flexibility of which is missing in decode forward direct transmission. Based on this example, more examples of larger dimensions can be constructed. IV. COMPRESS FORWARD We prove Theorem 3. Let K 0 be of the form in 5). Let X 1 CN0, K 1 ) X CN0, K ) be independent of each other, Ŷ = Y + Ẑ 4) where Ẑ CN0,σ I r ) is independent of X 1, X, Z, Z 3. Then, I Y ; Ŷ X 1, X, Y 3 ) = hŷ X 1, X, Y 3 ) hŷ X 1, X, Y, Y 3 ) = r log1 + 1/σ ) 43) I X 1 ; Ŷ, Y 3 X ) [ 1 + σ ] )I r 0 + G 0 I 1 K 1 G H 1 r3 = log [ 1 + σ )I r 0 0 I r3] [ = log I 1 r +r 3 + G ] [ 1+σ 1 1 K G ] H 1 1+σ 1 G 31 G 31 [ 1 1 = log I A 1+σ r +r 3 + B ] 1+σ 1 1+σ BH D = log I r3 + D + logi r + 1 A 1 + σ BIr3 + D) 1 B H) 44)

7 JIN AND KIM: APPROXIMATE CAPACITY OF THE MIMO RELAY CHANNEL 1173 log I r3 + D +log Ir + A BI r3 + D) 1 B H log1 + σ ) mint 1,r ) 45) [ ] = log A B I r +r 3 + mint B H D 1, r ) log1 + σ ) 46) = log I t1 + G H 1 G 1 + G H 31 G 31)K 1 mint 1, r ) log1 + σ ), 47) where A = G 1 K 1 G H 1, B = G 1K 1 G H 31, D = G 31K 1 G H 31, 44) 46) are due to [Ã ] B = D Ã B D C D 1 C, 45) follows by Lemma 1. The first statement of Theorem 3 is now established by substituting 33), 43), 47) in the compress forward lower bound in 16) comparing it with the cutset bound in 4). Setting σ = 1 in ) yields the second statement in 3). V. COMPUTATION OF THE CAPACITY BOUNDS A. Formulations of Optimization Problems 1) Cutset Bound: Computing the cutset upper bound in 3) can be formulated as the following convex optimization problem [16]: where maximize R CS over R CS 0, K 0, K 1 0 subject to R CS log I r3 + G 3 KG H 3 R CS log Ir +r 3 + G 1 K 1 G H 1 tra H 1 KA 1) P, tra H KA ) P K A 1 K 1 A H ) A 1 = [ It1 0 t t 1 ] A = [ ] 0t1 t. I t The optimization problem in 48) can be solved by stard convex optimization techniques or packages, e.g., [7]. ) Partial Decode Forward Lower Bound: Since direct computation of 7) or 8) is intractable, we instead consider three lower bounds on R PDF, namely, R DF, R DT, the special case of R PDF evaluated by 5), take the maximum of the three. Note that all three lower bounds can be viewed as the partial decode forward lower bound evaluated by 5) with a more general choice of Z CN0,σ I r ),where σ = 0,, 1, respectively. Considering more values of σ can further improve the bound at the cost of complexity. As with the cutset bound, both R DF R DT can be computed efficiently as a convex optimization problem. The third bound, characterized by 6) 7), is nonconvex. Thus, we evaluate the bound with the optimal solution to the convex optimization problem defined by 6) 8). A similar approach can be taken for computation of R NPDF. Fig.. The additive gaps between the cutset bound the partial decode forward compress forward lower bounds for romly generated MIMO relay channels. 3) Compress Forward Lower Bound: We consider two convex lower bounds on R CF, namely, the special case of R CF evaluated by 4) with σ = 1, namely, max min K 1,K { log I r3 + G 31 K 1 G H 31 + G 3K G H 3 r, 1 } logi t1 + GH 1 G 1 + G H 31 G 31 )K 1 R DT which corresponds to σ = ). As in the case of partial decode forward, considering more values of σ can further improve the bound at the cost of complexity. B. Numerical Results We consider the additive multiplicative gaps on 000 MIMO relay channels with rom channel gains independently distributed according to CN0, 1). The gaps are evaluated by relaxed bounds discussed in the previous subsection. The maximum average of the additive gaps plotted against the power constraint P at the sender the relay are shown in Fig. similar multiplicative gaps are shown in Fig. 3. The simulation results are consistent with the theoretical predictions in Theorems 1,, 3, Proposition 4. VI. HALF-DUPLEX MIMO RELAY CHANNELS Half-duplex relay channel models are often investigated to study wireless communication systems in which relays cannot send receive in the same time slot or frequency b. In this section, we consider the time-division halfduplex relay channel model proposed by Kramer [1], which has been recently extended to MIMO relay networks by Cardone et al. [3], with minor modifications. In this model, the relay has a total of a antennas, each of which can either transmit or receive at a given time. Let S i {0, 1}, i = 1,...,a, denote the operation of antenna i at the relay, say, S i = 0 means the antenna is in the receiving mode S i = 1 means the antenna is in the transmitting mode. Then, the channel outputs at the relay the receiver can be expressed as Y = DS)G 1 X 1 + Z, Y 3 = G 31 X 1 + G 3 DS)X + Z 3, 49)

8 1174 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 63, NO., FEBRUARY 017 The partial decode forward coding scheme can achieve the following lower bound. Proposition 11 PartialDecode ForwardLowerBound [1]): The capacity C of the time-division half-duplex MIMO relay channel is lower bounded by R PDF = sup min{i X 1, X, S; Y 3 ), I U; Y X, S) + I X 1 ; Y 3 X, U, S) } = sup min { I X 1, X, S; Y 3 ), I X 1 ; U, Y 3 X, S) I X 1 ; U X, Y, S) } Fig. 3. The multiplicative gaps between the cutset bound the partial decode forward compress forward lower bounds for romly generated MIMO relay channels. where G 1 C a t 1, G 31 C r 3 t 1, G 3 C r 3 a are channel gain matrices, X 1 C t 1 X C a are the respective inputs at the sender the relay, Z CN0, I a ) Z 3 CN0, I r3 ) are independent complex Gaussian noise components. Here henceforth, we use the shorth notation DS) = diags) DS) = I a DS) to represent the antenna mode vector S = S 1,...,S a ) its complement in matrix forms. Assume that the relay can dynamically choose the antenna modes based on its channel outputs. Then this model can be viewed as a special case of the general relay channel with the equivalent channel input X, S) at the relay. This observation leads to the following upper bound on the capacity. Proposition 10 Cutset Bound [1]): The capacity C of the time-division half-duplex MIMO relay channel in 49) is upper bounded by R CS = sup min { I X 1, X, S; Y 3 ), I X 1 ; Y, Y 3 X, S) } = sup min { I S; Y 3 ) + I X 1, X ; Y 3 S), I X 1 ; Y, Y 3 X, S) }, 50) where the suprema are over all pmfs ps) conditional cdfs Fx 1, x s) such that EX H 1 X 1) P EX H DS)X ) P. This bound can be further upper bounded by sup ps),k s) mina + J 1, J ), 51) where the supremum is over all pmfs ps) t 1 + a ) t 1 + a ) matrix-valued functions K s) such that [ ] K1 s) K K s) = 1 s) K1 H s) K 0, s {0, 1} s), 5) E[trK 1 S))] = ps) trk 1 s)) P, 53) s E[trDS)K S))] = ps) trds)k s)) P. 54) s Here, J 1 = E log I r3 +[G 31 G 3 DS)]K S)[G 31 G 3 DS)] H ), J = E log I t1 + G H 1 DS)G 1 + G H 31 G 31)K 1 S) ), K 1 S) = K 1 S) K 1 S)K 1 S)K 1 H S). where the suprema are over all pmfs ps) conditional cdfs Fu, x 1, x s) such that EX1 HX 1) P EX HDS)X ) P. Similarly, the compress forward coding scheme can achieve the following lower bound. Proposition 1 Compress Forward Lower Bound []): The capacity C of the time-division half-duplex MIMO relay channel is lower bounded by R CF = sup min{i X 1, X, S; Y 3 ) I Y ; Ŷ X 1, X, S, Y 3 ), I X 1 ; Ŷ, Y 3 X, S)} sup min{i X 1, X ; Y 3 S) I Y ; Ŷ X 1, X, S, Y 3 ), I X 1 ; Ŷ, Y 3 X, S)}, 55) where the suprema are over all pmfs ps) conditional cdfs Fx 1 s)fx s)fŷ y, x, s) such that EX H 1 X 1) P EX H DS)X ) P. Comparing the cutset upper bound the partial decode forward lower bound, we can establish the following gap result. Proposition 13: For every G 1,G 31,G 3, P, PDF := C R PDF mint 1, a ) 56) R CS. 57) R PDF Proof: The multiplicative gap in 57) follows immediately from Proposition 9. For the additive gap in 56), we adapt Proposition 8 by setting U = DS)G 1 X 1 + Z,whereZ CN0, I a ) is independent of X 1, X, S, Z, Z 3 ). Then, PDF sup I X 1 ; U X, Y, S) ps)fx 1,x s) = sup hu X, Y, S) hu X 1, X, Y, S) ps)fx 1,x s) = sup hu X, Y, S) hz ) ps)fx 1,x s) = sup ps)fx 1,x s) ps)hu X, Y, S = s) hz )). s

9 JIN AND KIM: APPROXIMATE CAPACITY OF THE MIMO RELAY CHANNEL 1175 Now following arguments similar to those in Subsection III-A, we have for each s hu X, Y, S = s) hz ) sup log I t 1 + G H Ds)G 1 1 K 1 s) K s) I t1 + G H Ds)G 1 1 K 1 s) mint 1, a ), where the supremum is over all covariance matrices K s) of the form 5). This completes the proof of 56). An additive gap that is somewhat weaker than 56) can be established by comparing the cutset upper bound the compress forward lower bound, which still improves upon the existing gap result of 1.96t 1 + a + r 3 ) in [3]. Proposition 14: For every G 1,G 31,G 3, P, CF = C R CF [ min max min t 1 + max [0, a log σ ] ) σ 1 + σ, r 3 + a log1 + 1/σ )), ] mint 1, a ) log1 + σ ) 58) mint 1 + a, a log 3 + r 3 ). 59) Proof: Let Ŷ = DS)Y +Ẑ,whereẐ CN0,σ I a ) is independent of X 1, X, S, Z, Z 3. Consider any ps), K s)) satisfying 5) 54). Given {S = s}, letx 1 CN0, K 1 s)) X CN0, K s)) be conditionally independent. Then, we have I X 1 ; Ŷ, Y 3 X, S = s) = log I r +r 3 + [ Ds)G 1 1+σ G 31 ] K 1 s) [ Ds)G 1 1+σ G 31 ] H log I t1 + G H 1 Ds)G 1 + G H 31 G 31)K 1 s) mint 1, r ) log1 + σ ), 60) where t = a i=1 s i denotes the number of transmitting antennas r = a t denotes the number of receiving antennas when s {0, 1} a is the antenna mode vector. Similarly, I X 1, X ; Y 3 S = s) = log I r3 + G 31 K 1 s)g H 31 + G 3Ds)K s)ds)g H 3 log I r3 +[G 31 G 3 Ds)]K s)[g 31 G 3 Ds)] H mint 1 + t, r 3 ), 61) I Y ; Ŷ X 1, X, Y 3, S = s) = r log1 + 1/σ ). 6) Subtracting 6) from 61) comparing it with a + J 1 in 51) before taking the expectation, we can obtain the gap mint 1 + t, r 3 ) + r log1 + 1/σ ) + a min t 1 + t log σ ) 1 + σ, r 3 + a log1 + 1/σ )) min t 1 + max [0, a log σ ] ) 1 + σ, r 3 + a log1 + 1/σ )). 63) Similarly, comparing 60) with J in 51), we obtain the gap mint 1, a ) log1 + σ ). 64) Combining 63) 64) considering all possible σ, we can establish 58). Finally, setting σ = in 58) yields 59). As a final remark, we note that similar additive multiplicative gap results can be established for fixed timedivision frequency-division half-duplex relay channel models [6], [10], [5, Secs ]. We refer the reader to [6, Th. 8.3] an earlier conference version [4] of this paper for such results. ACKNOWLEDGMENTS The authors would like to thank the Associate Editor the anonymous reviewers for many valuable comments that helped improve the presentation of this paper. REFERENCES [1] E. C. van der Meulen, The discrete memoryless channel with two senders one receiver, in Proc. nd Int. Symp. Inf. Theory, Tsahkadsor, Armenian, 1971, pp [] T. M. Cover A. A. El Gamal, Capacity theorems for the relay channel, IEEE Trans. Inf. Theory, vol. 5, no. 5, pp , Sep [3] L. R. Ford, Jr., D. R. Fulkerson, Maximal flow through a network, Canad. J. Math., vol. 8, no. 3, pp , [4] Y.-H. Kim. 014). Dictionary of Relaying Schemes. [Online]. Available: [5] G. Kramer, M. Gastpar, P. Gupta, Cooperative strategies capacity theorems for relay networks, IEEE Trans. Inf. Theory, vol. 51, no. 9, pp , Sep [6] A. S. Avestimehr, S. N. Diggavi, D. N. C. Tse, Wireless network information flow: A deterministic approach, IEEE Trans. Inf. Theory, vol. 57, no. 4, pp , Apr [7] M. H. Yassaee M. R. Aref, Slepian Wolf coding over cooperative relay networks, IEEE Trans. Inf. Theory, vol. 57, no. 6, pp , Jun [8] S. H. Lim, Y.-H. Kim, A. El Gamal, S.-Y. Chung, Noisy network coding, IEEE Trans. Inf. Theory, vol. 57, no. 5, pp , May 011. [9] S.H.Lim,K.T.Kim,Y.-H.Kim, Distributeddecode forward for relay networks, IEEE Trans. Inf. Theory. [Online]. Available: [10] A. El Gamal, M. Mohseni, S. Zahedi, Bounds on capacity minimum energy-per-bit for AWGN relay channels, IEEE Trans. Inf. Theory, vol. 5, no. 4, pp , Apr [11] A. S. Avestimehr, S. N. Diggavi, D. N. C. Tse, A deterministic approach to wireless relay networks, in Proc. 45th Allerton Conf. Commun. Control Comput., Monticello, IL, USA, Sep. 007, pp [1] W. Chang, S.-Y. Chung, Y. H. Lee. Nov. 010). Gaussian relay channel capacity to within a fixed number of bits. [Online]. Available: [13] S. Boyd L. Venberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 004. [14] B. Wang, J. Zhang, A. Høst-Madsen, On the capacity of MIMO relay channels, IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 9 43, Jan [15] S. Simoens, O. Munoz-Medina, J. Vidal, A. del Coso, On the Gaussian MIMO relay channel with full channel state information, IEEE Trans. Signal Process., vol. 57, no. 9, pp , Sep [16] C. T. K. Ng G. J. Foschini, Transmit signal bwidth optimization in multiple-antenna relay channels, IEEE Trans. Commun., vol. 59, no. 11, pp , Nov. 011.

10 1176 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 63, NO., FEBRUARY 017 [17] L. Gerdes, C. Hellings, L. Weil, W. Utschick, On the maximum achievable partial decode--forward rate for the Gaussian MIMO relay channel, IEEE Trans. Inf. Theory, vol. 61, no. 1, pp , Dec [18] R. Kolte, A. Özgür, A. E. Gamal, Capacity approximations for Gaussian relay networks, IEEE Trans. Inf. Theory, vol. 61, no. 9, pp , Jun [19] A. Host-Madsen J. Zhang, Capacity bounds power allocation for wireless relay channels, IEEE Trans. Inf. Theory, vol. 51, no. 6, pp , Jun [0] Y. Liang V. V. Veeravalli, Gaussian orthogonal relay channels: Optimal resource allocation capacity, IEEE Trans. Inf. Theory, vol. 51, no. 9, pp , Sep [1] G. Kramer, Models theory for relay channels with receive constraints, in Proc. 4nd Annu. Allerton Conf. Commun. Control Comput., Monticello, IL, USA, Sep./Oct. 004, pp [] M. Cardone, D. Tuninetti, R. Knopp, U. Salim, On the Gaussian half-duplex relay channel, IEEE Trans. Inf. Theory, vol. 60, no. 5, pp , May 014. [3] M. Cardone, D. Tuninetti, R. Knopp, Gaussian MIMO halfduplex relay networks: Approximate optimality of simple schedules, in Proc. IEEE Int. Symp. Inf. Theory, Hong Kong, Jun. 015, pp [4] X. Jin Y.-H. Kim, The approximate capacity of the MIMO relay channel, in Proc. IEEE Int. Symp. Inf. Theory, Honolulu, HI, USA, Jun./Jul. 014, pp [5] A. El Gamal Y.-H. Kim, Network Information Theory. Cambridge, U.K.: Cambridge Univ. Press, 011. [6] M. Fiedler, Bounds for the determinant of the sum of hermitian matrices, Proc. Amer. Math. Soc., vol. 30, no. 1, pp. 7 31, [7] M. Grant S. Boyd. Sep. 013). CVX: MATLAB Software for Disciplined Convex Programming, Version.0 Beta. [Online]. Available: Xianglan Jin M 1) received her B.S. M.S. degrees in telecommunications engineering from Beijing University of Posts Telecommunications, China, in , respectively, her Ph.D. degree in electrical engineering computer science from Seoul National University, Korea, in 008. She was an engineer at Beijing Samsung Telecommunications R&D center from April 00 to June 004. From 008 to 010, she was a postdoctoral scholar in the Department of Electrical Engineering Computer Science, Seoul National University. From 010 to 014, she was a Research Assistant Professor in the Department of Information Communication Engineering at Dongguk University, Korea. She visited the Department of Electrical Computer Engineering at the University of California, San Diego from 015 to 016. Currently, she is a Research Professor in the Department of Electrical Computer Engineering at Pusan National University, Korea. Her research interests include MIMO, cooperative communication, network information theory. Young-Han Kim S 99 M 06 SM 1 F 15) received his B.S. degree with honors in electrical engineering from Seoul National University, Korea, in 1996 his M.S. degrees in electrical engineering in statistics, his Ph.D. degree in electrical engineering from Stanford University in 001, 006, 006, respectively. In 006, he joined the University of California, San Diego, where he is currently an Associate Professor in the Department of Electrical Computer Engineering. His research interests are in information theory, communication engineering, data science. He coauthored the book Network Information Theory Cambridge University Press 011). He is a recipient of the 008 NSF Faculty Early Career Development CAREER) Award, the 009 US-Israel Binational Science Foundation Bergmann Memorial Award, the 01 IEEE Information Theory Paper Award, the 015 IEEE Information Theory Society James L. Massey Research Teaching Award for Young Scholars. He served as an Associate Editor of the IEEE TRANSACTIONS ON INFORMATION THEORY a Distinguished Lecturer for the IEEE Information Theory Society.