Optimized Capacity Bounds for the Half-Duplex Gaussian MIMO Relay Channel

Transcription

1 Optimize Capacity Bouns for the Half-Duplex Gaussian MIMO elay Channel Lennart Geres an Wolfgang Utschick International ITG Workshop on mart Antennas (WA) February 211 c 211 IEEE. Personal use of this material is permitte. However, permission to reprint/republish this material for avertising or promotional purposes or for creating new collective works for resale or reistribution to servers or lists, or to reuse any copyrighte component of this work in other works must be obtaine from the IEEE. Technische Universität München Associate Institute for ignal Processing

2 Optimize Capacity Bouns for the Half-Duplex Gaussian MIMO elay Channel Lennart Geres, Wolfgang Utschick Associate Institute for ignal Processing Technische Universität München, 829 München, Germany {geres, Abstract In this paper, the optimization of the cut-set boun (CB) an the achievable ecoe-an-forwar (DF) rate for the half-uplex Gaussian multiple-input multiple-output (MIMO) relay channel is consiere. In particular, it is shown that evaluating the cut-set boun an the maximal achievable DF rate is equivalent to solving convex optimization problems if perfect channel state information (CI) is available at all noes. Our work therefore extens results for the full-uplex relay channel, which we also revisit here, an it emonstrates that it is possible to efficiently etermine (generally loose) bouns on the capacity of the half-uplex MIMO relay channel. Inex Terms elay channel, MIMO, half-uplex, cut-set boun, ecoe-an-forwar, convex optimization. I. INTODUCTION elaying has attracte a lot of interest in recent years ue to the ability to provie increase throughput an extene coverage to a growing number of mobile users. cenarios in which relaying may be employe inclue for example multihop wireless networks an sensor networks where noes have limite transmit power. In this work, we consier the simplest relay channel where one source transmits information to one estination with the help of a single relay. It is assume that this relay has no own information to transmit so that its only purpose is to assist the communication between the source an the terminal. A moel for this particular relay channel was first stuie by van er Meulen as early as 1971 [1], but its general capacity is still unknown. However, substantial avances towars the information theoretic unerstaning of the relay channel were mae by Cover an El Gamal. In [2], they erive upper an lower bouns on the capacity of the full-uplex relay channel base on a then new cut-set boun (CB) as well as two funamental coing schemes that are now referre to as ecoe-an-forwar (DF) an compress-an-forwar (CF), respectively. The DF strategy requires the relay to ecoe the whole source message, which is then re-encoe an sent to the estination. When the relay uses CF, it reliably forwars an estimate, a compresse version of its receive signal, to the estination. In [3], these two basic strategies were generalize to various relay channel moels that inclue multiple sources, relays, or estinations. Another relaying strategy of lower complexity than both DF an CF is calle amplify-an-forwar (AF), which is consiere in [4] for example. When the relay uses AF, it is restricte to perform linear operations on its receive signal. All information theoretic results that have been erive for the relay channel hol whether the noes are equippe with one or multiple antennas. However, multi-antenna noes may employ precoing, which evices having a single antenna are not capable of. The aitional egrees of freeom in the multiple-input multiple-output (MIMO) relay channel are of course reflecte in the corresponing optimization problems. Assuming Gaussian channel inputs, maximizing the achievable DF rate for the MIMO relay channel requires to solve an optimization problem with respect to the source covariance matrix, the relay covariance matrix, an their cross covariance matrix, for example. If all noes have a single antenna, on the other han, the same objective is maximize with respect to a scalar correlation coefficient ρ [, 1]. urprisingly, there are only few contributions that have focusse on optimizing bouns on the capacity of the MIMO relay channel. In [5], it is prove that Gaussian input istributions maximize the CB an the achievable DF rate for the full-uplex relay channel. Furthermore, the authors provie an upper boun on the CB that is loose in general. uboptimal lower bouns are also given base on point-topoint transmission (source to estination) an the cascae relay channel (source to relay, relay to estination). Two partial ecoe-an-forwar (PDF) strategies, where the relay only ecoes part of the source message, using superposition or irty-paper coing [6] at the source are presente in [7]. While the achievable rates are shown to improve on the lower bouns of [5], the authors only formulate but o not solve the corresponing rate maximization problems for the general case. We were able to show that, for the full-uplex case, the cut-set boun as well as the maximal achievable DF rate are obtaine as the solutions of convex optimization problems [8]. In this paper, we exten these results to the half-uplex relay channel, where a practical half-uplex constraint is impose on all noes. imilar work (for both the full-uplex an the halfuplex case) is presente in [9]. However, it can be verifie that the expressions resulting from those erivations are only upper bouns to the optimal solutions. The remainer of this paper is organize as follows. The system moels for the full-uplex an the half-uplex MIMO relay channel are introuce in ection II. ection III revisits the optimization of the CB for the full-uplex case an shows how these results can be extene to the half-uplex

3 elay elay H H D H (1) ource Figure 1. H D Full-uplex MIMO relay channel. Destination ource H D (a) elay receive phase. Destination elay case. The maximization of the achievable DF rate for both the full-uplex an the half-uplex relay channel is aresse in ection IV. Numerical results an a iscussion of these results are presente in ection V before we conclue in ection VI. II. YTEM MODEL In the full-uplex relay channel, which is illustrate in Figure 1, the signals receive at the relay an the estination can be expresse as y = H x + n, y D = H D x + H D x + n D, where H C N N, H D C ND N, H D C ND N represent the channel gain matrices assume to be perfectly known at all noes, an n N C (, I N ), n D N C (, I ND ) enote complex white Gaussian noise of unit variance. Let an enote the covariance matrices of the zero-mean source an relay inputs, then the source an the relay are subject to transmit power constraints given by tr( ) P an tr( ) P, respectively. Furthermore, the joint transmit covariance matrix of the source an relay inputs is etermine by [ [x ] [ ] ] H [ ] x = E = x x H. (2) Note that, by efining the two selection matrices D = [ I N N N ], D = [ N N I N ], (3) both the source an the relay transmit covariance matrices can be expresse as linear functions of : (1) = D D H, = D D H. (4) Let us now consier the more practical case where a halfuplex constraint is impose on all noes. In the half-uplex relay channel, which is epicte in Figure 2, a relay receive phase is followe by a relay transmit phase. These two phases are specifie by an y (1) = H x (1) + n (1), y (1) D = H Dx (1) + n (1) D, (5) y (2) D = H Dx (2) + H D x (2) + n(2) D, (6) where the noise n (1) N C(, I N ), n (1) D N C(, I ND ), an n (2) D N C (, I ND ) is inepenent of n (1) D. Note that the ource Figure 2. (2) H D (b) elay transmit phase. H D Half-uplex MIMO relay channel. Destination source transmits an the estination listens uring both phases, which means that the half-uplex constraint basically affects only the relay. Furthermore, without loss of generality H D is assume to be the same for both phases, an all channels are assume to be perfectly known at all noes again. ince only the source transmits in the first phase, we have [ (1) = E x (1) x(1),h ] = (1). (7) The joint transmit covariance matrix of the source an relay inputs for the relay transmit phase is given by [ ][ ] H [ ] (2) = E x (2) x (2) x (2) (2) x (2) = (2) (2),H (2). (8) Moreover, we have (2) = D (2) D H, (2) = D (2) D H. (9) Like in the full-uplex case, the source an the relay are subject to power constraints of the form tr( (1) ) P (1), tr( (2) ) P (2), an tr( (2) ) P (2), respectively. III. CUT-ET BOUND First, let us revisit the optimization of the cut-set boun for the full-uplex case. Cover an El Gamal [2] prove that the capacity of the full-uplex relay channel is upper boune by C FD CB = maxmin{ I(X ; Y Y D X ), I(X X ; Y D ) }, (1) where the maximization is with respect to the joint istribution of the source an relay signals. Furthermore, it is known that the source an relay inputs that optimize the CB are Gaussian [5]. With x N C (, ) an x N C (, ), it follows that I(X ; Y Y D X ) = log et ( I + H 1 H1 H ), (11) I(X X ; Y D ) = log et ( I + H 2 H2 H ), (12)

4 where H 1 = [ H H HD] H H, H2 = [ ] H D H D, an = H is the conitional covariance matrix of X given X. Here enotes the Moore-Penrose pseuoinverse of. emark 1: Note that is equal to the covariance matrix of the ranom vector X ˆX (X ) in this case, where ˆX (X ) = X is the linear minimum mean square error (MME) estimator of X given X. However, this is only because we have Gaussian inputs an the linear MME estimator is equal to the MME estimator for Gaussian ranom vectors. For general inputs, it oes not hol that the conitional covariance matrix of X given X is equal to H. We see that computing the cut-set boun for the fulluplex MIMO relay channel requires to solve the nonconvex optimization problem s.t. max CFD CB CB log et ( I + H 1 H1 H ), CCB FD log et ( I + H 2 H2 H ), tr(d D H ) P, tr(d D) H P, C FD (13) where D an D enote the selection matrices efine in (3). Observe that the nonconvexity is cause only by the conitional covariance matrix in the first inequality constraint. However, by means of introucing a slack variable Q an applying the chur complement conition for positive semi-efinite matrices [1, A.5.5], it is shown in [8] that the following is an equivalent optimization problem: CCB FD = max min{ log et ( I + H 1 QH H ) 1, Q, log et ( I + H 2 H2 H )} (14) s.t. tr(d D H ) P, tr(d D H ) P, Q, D H QD. Note that the last two constraints imply. As the objective function is the pointwise minimum of two concave functions (in Q, ), it is also concave [1]. Furthermore, all constraints are affine so that this optimization problem, which etermines the cut-set boun, is convex. Now, consier the optimization of the cut-set boun for the half-uplex case. In [11], it was shown that the CB for the half-uplex relay channel reas as CCB HD = maxmin{ τ 1 I(X (1) ; Y (1) Y (1) D ) + τ 2I(X (2) ; Y (2) D X(2) ), τ 1 I(X (1) ; Y (1) D ) + τ 2I(X (2) X(2) ; Y (2) D )}, (15) where τ 1 an τ 2 enote the urations of the relay receive an the relay transmit phase, respectively. The same arguments as in the full-uplex case may be put forwar to show that is also maximize by Gaussian source an relay inputs. Assuming x (1) N C (, (1) ), x(2) N C (, (2) ), an x (2) N C (, (2) ), the optimize cut-set boun is hence CB given by max C (1), (2) CB HD, τ 1,τ 2,τ 1+τ 2 1 s.t. CB τ 1 log I + H 1 (1) H1 + τ 2 log I + H D (2) H D, CB τ 1 log I + H D (1) HD + τ 2 log I + H 2 (2) H2, tr( (1) ) P (1), tr( (2) ) P (2), tr( (2) ) P (2). (16) This is again a nonconvex optimization problem, where the nonconvexity is cause by the conitional covariance matrix (2) = (2) (2) (2), (2),H in the first constraint on CCB HD. However, after reformulating this problem using the slack variable Q an the properties of the generalize chur complement, we obtain the following equivalent optimization problem: max CCB HD s.t. (1), (2),Q,τ 1,τ 2 CB τ 1 log I + H 1 (1) H1 + τ 2 log I + HD QHD, CB τ 1 log I + H D (1) HD + τ 2 log I + H 2 (2) H2, Q, (1), (2) D H QD, tr( (1) ) P (1), tr( (2) ) P (2), tr( (2) ) P (2), τ 1, τ 2, τ 1 + τ 2 1. (17) ince the nonnegative weighte sum of concave functions is concave [1], we can follow the same arguments as for the full-uplex case an conclue that this problem is convex (in Q, (1), (2), where the latter is implie by Q an (2) D HQD ) for fixe τ 1, τ 2. However, joint convexity in Q, (1), (2), τ 1, an τ 2 cannot be claime in this formulation. Nevertheless, letting τ 1 = τ an τ 2 = 1 τ, the optimal solution can be foun by sampling over one scalar parameter τ [, 1] an convex programming. Beyon that, it is also possible to formulate an equivalent optimization problem that is jointly convex in all parameters. To this en, we introuce the new variables C (1) = τ 1 (1), C (2) = τ 2 (2), Q = τ 2 Q. (18) The resulting optimization problem hence reas as max C C (1),C (2) CB HD s.t.,q,τ 1,τ 2 CB τ 1 log I + H C (1) 1 H1 H τ 1 + τ 2 log I + H Q D HD H τ 2, CB τ 1 log I + H C (1) D HD H τ 1 + τ 2 log I + H C (2) 2 H2 H τ 2, Q, C (1), C (2) D H Q D, tr(c (1) ) τ 1P (1), tr(c (2) ) τ 2P (2), tr(c (2) ) τ 2P (2), τ 1, τ 2, τ 1 + τ 2 1. (19) Note that g : (C (1) C, τ 1 ) τ 1 log I +H (1) 1 τ 1 H1 H efine on omg = {(C (1), τ 1 ) : C (1), τ 1 > } is the perspective of the function f : C (1) log I + H 1 C (1) H1 H efine on

5 the cone of positive semi-efinite matrices. Furthermore, f is concave in C (1). The convexity preserving property of the perspective operation [1, ec ], which was use in [9] to convexify a very similar problem, thus implies that g is jointly convex in C (1) an τ 1 on its entire omain. Analogous results hol for the other log-et functions. In aition, observe that all other constraints are affine. Therefore, we have obtaine an optimization problem that is jointly convex in all parameters Q, C (1), C (2), τ 1, τ 2. emark 2: The perspective function is efine only for positive real numbers. However, if we assume the convention that log x = for all x, which is commonly use in information theory base on continuity arguments [12, p. 19], then it is not necessary to exclue the cases τ 1 = an τ 2 = as the problematic terms vanish. IV. ACHIEVABLE DECODE-AND-FOWAD (DF) ATE A lower boun on the capacity of the relay channel is given by the rate that can be achieve with the ecoe-anforwar protocol erive by Cover an El Gamal [2]. Before we aress the half-uplex case, let us first revisit the fulluplex relay channel again. If the relay uses DF, all achievable rates in the full-uplex case satisfy FD DF maxmin { I(X ; Y X ), I(X X ; Y D ) }, (2) where the maximization is again with respect to the joint istribution of the source an relay signals. Observe that DF FD iffers from CCB FD only in the first mutual information term, whereas the secon one is the same. Beyon that, it has also been prove that Gaussian input istributions optimize the achievable DF rate [5]. Therefore, letting x N C (, ) an x N C (, ), it follows that I(X ; Y X ) = log et ( I + H H H ), (21) ] HD H H where the only ifference to (11) is that H 1 = [ H H is replace by H. Using the same arguments as for the calculation of the cut-set boun, it is thus shown in [8] that DF FD max min{ log et ( I + H QH H ), Q, log et ( I + H 2 H2 H )} (22) s.t. tr(d D H ) P, tr(d D H ) P, Q, D H QD. Consequently, the maximal achievable DF rate for the fulluplex MIMO relay channel is also obtaine as the solution of a convex optimization problem. A rather straightforwar application of the ecoe-anforwar scheme to the half-uplex relay channel yiels that the best DF rate is given by DF HD = maxmin { τ 1 I(X (1) ; Y (1) ) + τ 2I(X (2) ; Y (2) D X(2) ), τ 1 I(X (1) ; Y (1) D ) + τ 2I(X (2) X(2) ; Y (2) D )}, (23) where τ 1 an τ 2 again enote the lengths of the time slots allocate to the relay receive an the relay transmit phase, respectively. emark 3: This assumes that the source may transmit new information to the estination in the relay transmit phase. As the relay oes not receive (an thus ecoe) all the information the source communicates ue to the impose half-uplex constraint, this strategy is sometimes terme a partial DF scheme in the literature, incluing [9] for example. We agree with [11], however, an consier this as a DF strategy since the relay ecoes everything it can receive. an HD DF iffer only in one term. For Gaussian source an relay inputs x (1) N C (, (1) ), x(2) N C (, (2) ), an x(2) N C (, (2) ), the achievable DF rate is hence maximize by Like in the full-uplex case, CB max (1), (2) DF HD s.t.,q,τ 1,τ 2 DF τ 1 log I + H (1) H + τ 2 log I + HD QHD, DF τ 1 log I + H D (1) HD + τ 2 log I + H 2 (2) H2, HD HD Q, (1), (2) D H QD, tr( (1) ) P (1), tr( (2) ) P (2), tr( (2) ) P (2), τ 1, τ 2, τ 1 + τ 2 1, (24) where we have alreay introuce the slack variable Q an reformulate the problem. This optimization problem is basically obtaine by replacing H 1 in (17) by H, an like problem (17), it is convex in (1), (2), an Q for fixe time allocation parameters τ 1 an τ 2. If we further introuce the variables C (1), C (2), an Q as efine in (18), then, not surprisingly, it is also possible to express the maximal achievable DF rate for the halfuplex MIMO relay channel as the solution of an optimization problem that is jointly convex in all parameters. We state it here for reasons of completeness: max C (1),C (2),Q DF HD s.t.,τ 1,τ 2 DF τ 1 log I + H C (1) H H τ 1 + τ 2 log I + H Q D HD H τ 2, DF τ 1 log I + H C (1) D HD H τ 1 + τ 2 log I + H C (2) 2 H2 H τ 2, HD HD Q, C (1), C (2) D H Q D, tr(c (1) ) τ 1P (1), tr(c (2) ) τ 2P (2), tr(c (2) ) τ 2P (2), τ 1, τ 2, τ 1 + τ 2 1. (25) V. NUMEICAL EULT AND DICUION In this section, we provie numerical results for the cut-set boun an the best achievable DF rate for both the full-uplex an the half-uplex case an several antenna configurations. Consiering the full-uplex relay channel, we compare the CB to the upper boun that was erive in [5], an we iscuss conitions uner which this generally loose upper boun is equal to the cut-set boun. For the half-uplex relay channel, we compare our results to those presente in [9]. Finally, we comment on why it is not possible to irectly apply stanar

6 Figure Line Network. semi-efinite program (DP) solvers like DPT3 or eumi to the half-uplex optimization problems. The example scenario we consier here is the line network epicte in Figure 3. This is a commonly use geometry where D = 1 is fixe an the relay is positione on the line connecting the source an the estination such that = an D = 1. The entries of the channel gain matrices H, H D, an H D are assume to be inepenent an ientically istribute complex Gaussian ranom variables with zero mean an variance α, α D, an α D, respectively, where the pass loss exponent is chosen as α = 2. Note that all numerical results are average over a number of inepenent channel realizations, where perfect CI at all noes is assume for every realization. The upper boun for the full-uplex MIMO relay channel presente in [5] is given by C FD UB = max,,ρ min{ log et ( I + (1 ρ 2 )H 1 H H 1 inf log et( I + (1 + ρ2 a> a )H D HD H + (1 + a)h D HD H )} s.t. tr( ) P, tr( ) P,,, ρ 1. (26) In orer to arrive at this result, the authors introuce the scalar parameter ρ [, 1] to capture the cross correlation between the source an relay inputs (instea of the matrix ) an subsequently mae smart use of matrix inequalities. Although the erivation is very elegant, the boun suffers from several restrictions that are pointe out in [9]. Most importantly, this boun is loose in general. Moreover, it is only vali for antenna configurations that satisfy N N, which will most likely be violate if the source is a base station an the relay is some evice of lower complexity for example. However, in orer to compare our results to this upper boun, we restrict our scenarios to those which satisfy this conition. For the simplest case of single antenna noes, i.e., when all channels are scalars, the cross covariance matrix boils own to a scalar. As a consequence, C FD UB D ), is equal to CFD CB in this case [5]. Figure 4 thus compares only the cut-set boun with the achievable DF rate given that N = N = N D = 1. We can observe the well known result that the DF scheme achieves the CB when the relay is very close to the source, i.e., when the probability that the source-relay link is the bottleneck goes to zero. The value of ρ that maximizes CUB FD (an thus CCB FD ) is close to 1 in this case, which means that source an relay can realize multi-antenna transmission C CB DF ρ Figure 4. Comparison of cut-set boun an maximal achievable DF rate for full-uplex relay channel, N = N = N D = 1 an P = P = 1 (average over 5 realizations) C CB DF ρ Figure 5. Comparison of cut-set boun, upper boun (from [5]), an maximal achievable DF rate for full-uplex relay channel, N = N = N D = 2 an P = P = 1 (average over 1 realizations) C CB DF ρ Figure 6. Comparison of cut-set boun, upper boun (from [5]), an maximal achievable DF rate for full-uplex relay channel, N = N = 2, N D = 3, an P = P = 1 (average over 1 realizations).

7 When some noes are equippe with multiple antennas, on the other han, then CUB FD is not tight for all channel conitions. This is illustrate in Figures 5 an 6 for N = N = N D = 2 as well as N = N = 2, N D = 3, respectively. We see that CUB FD = CFD CB only if ρ =. In fact, it can be shown that the upper boun is equal to the cut-set boun if the optimal solution requires that the source an relay inputs are inepenent, i.e., if ρ = an = are optimizers of the respective optimization problems. This is because the structure of the optimal correlation is completely capture by the scalar ρ = in this case. In general, this oes not hol, which explains why the upper boun is not always tight. For the half-uplex MIMO relay channel, no such upper boun that relies on the introuction of a scalar correlation parameter an utilizes matrix inequalities has been erive. However, a ifferent upper boun is presente in [9], which reas as max C (1), (2) UB HD, τ 1,τ 2,τ 1+τ 2 1 s.t. UB τ 1 log I + H 1 (1) H1 + τ 2 log I + H D (2) H D, UB τ 1 log I + H D (1) HD + τ 2 log I + H 2 (2) H2, tr( (1) ) P (1), tr( (2) ) P (2), tr( (2) ) P (2). (27) emark 4: Note that a boun of the same structure was also presente for the full-uplex case in [9]. However, as the ifferences to our results are of the same kin, we iscuss all properties on the basis of the half-uplex relay channel. Observe that the conitional covariance matrix (2) in (16) must be replace by (2) to obtain (27), which is the only ifference between these two optimization problems. Because log et(i +HH H ) is an increasing function in the transmit covariance matrix given the channel gain matrix H, an since (2) = (2) (2) (2), (2),H (2), it follows that CCB HD CHD UB. Therefore, CHD UB is loose in general, an CCB HD = CHD UB only if the optimal solution of (16) requires that (2) =, i.e., if the optimal source an relay inputs are inepenent. Note that this conition for the tightness of the upper boun is the same as for the boun erive in [5] for the full-uplex case, although the reasons for the bouns not being tight in general are completely ifferent. Irrespective of these properties, (16) poses a much more complicate optimization problem than (27). In particular, (27) is alreay convex in the transmit covariance matrices for fixe τ 1 an τ 2. It is hence not necessary to introuce a slack variable an to apply the chur complement conition. ather, only the convexity preserving property of the perspective function is neee to reformulate (27) as a convex optimization problem. Figures 7 9 compare CCB HD, CHD UB, an HD DF for ifferent antenna configurations, which are actually the same as for the full-uplex case. First, observe that there is no antenna configuration for which CUB HD = CHD CB in general, not even the single antenna case. In fact, the relative gap between CCB HD C CB DF Figure 7. Comparison of cut-set boun, upper boun (from [9]), an maximal achievable DF rate for half-uplex relay channel, N = N = N D = 1 an P (1) = P (2) = P (2) = 1 (average over 5 realizations) C CB DF Figure 8. Comparison of cut-set boun, upper boun (from [9]), an maximal achievable DF rate for half-uplex relay channel, N = N = N D = 2 an P (1) = P (2) = P (2) = 1 (average over 2 realizations) C CB DF Figure 9. Comparison of cut-set boun, upper boun (from [9]), an maximal achievable DF rate for half-uplex relay channel, N = N = 2, N D = 3, an P (1) = P (2) = P (2) = 1 (average over 2 realizations).

8 an UB among the three configurations consiere here is largest for N = N = N D = 1. Furthermore, it can be seen that the gap between the cut-set boun an the upper boun ecreases with. From this, we can conclue that the better the relay-estination link is compare to the source-estination link, the less correlate the optimal source an relay inputs must be. Moreover, the simulation results shown in Figures 8 an 9 suggest that the gap between CB an CHD UB ecreases faster with for larger N D if the number of source an relay antennas is not change, which means that more egrees of freeom at the estination also favor less correlate source an relay input istributions. Comparing the full-uplex to the half-uplex results, it stans out that the relative gap between increases less than that between CCB FD CB an HD DF an FD DF when the relay is move closer to the estination, especially for the single antenna case. This may be explaine by the fact that in the half-uplex case the optimization is with respect to the source an relay inputs an the time-sharing between the relay receive an the relay transmit phase. The optimal time allocation for the achievable DF rate nee not be the same as for the cut-set boun. ince the source transmits uring both phases an the source-estination channel oes not change, a weak source-relay channel can thus partly be compensate for by prolonging the relay receive phase. In the full-uplex case, this is not possible as both the source an the relay transmit the whole time. Across-the-boar, all simulation results show that the rates achieve using the DF scheme approach the CB when the relay is close to the source. Furthermore, it can be seen that substantial rate gains can be achieve when multiple antennas are use at each noe without increasing the power at the source or the relay. These results are certainly not surprising as they are in accorance with previous knowlege about ecoean-forwar relaying an MIMO channels. Before we conclue this section, a few comments about problems (19) an (25) are in orer. Note that these two convex optimization problems o not satisfy the ruleset of iscipline convex programming (DCP) [13]. Problems that ahere to this ruleset can automatically be verifie as convex an converte to solvable form, which allows to irectly use stanar DP solvers. This oes not apply to problems that violate the ruleset, even if they are convex. That is not to say that there oes not exist a suitable reformulation of such problems. However, if no applicable reformulation can be foun, other solution methos nee to be consiere. VI. CONCLUION In this paper, we exten previous results for the full-uplex MIMO relay channel an show that the cut-set boun an the achievable DF rate can also be obtaine as the solutions of convex optimization problems in the half-uplex case. It is therefore possible to efficiently etermine these (generally loose) upper an lower bouns on the capacity of the MIMO relay channel, which may serve as benchmarks for achievable rates of various relay strategies for example. Aitionally, we iscuss how our results for the cut-set boun compare to the upper bouns presente in [5] an [9]. In particular, we point out why these upper bouns are not tight in general, an we ientify cases in which they are equal to the CB. Finally, simulation results for various antenna configurations emonstrate the importance of our work as they reveal nonnegligible gaps between the CB an the aresse upper bouns. EFEENCE [1] E. 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