The Optimal Input Distribution for Partial Decode-and-Forward in the MIMO Relay Channel

The Optimal Input Distribution for Partial Decode-and-Forward in the MIMO Relay Channel

Abstract

This paper considers the partial decode-and-forward (PDF) strategy for the Gaussian multiple-input multiple-output (MIMO) relay channel. Unlike for the decode-and-forward (DF) strategy or point-to-point (P2P) transmission, for which Gaussian channel inputs are known to be optimal, the input distribution that maximizes the achievable PDF rate for the Gaussian MIMO relay channel has remained unknown so far. For some special cases, e.g., for relay channels where the optimal PDF strategy reduces to DF or P2P transmission, it could be deduced that Gaussian inputs maximize the PDF rate. For the general case, however, the problem has remained open until now. In this work, we solve this problem by proving that the maximum achievable PDF rate for the Gaussian MIMO relay channel is always attained by Gaussian channel inputs. Our proof relies on the channel enhancement technique, which was originally introduced by Weingarten et al. to derive the (private message) capacity region of the Gaussian MIMO broadcast channel. By combining this technique with a primal decomposition approach, we first establish that jointly Gaussian source and relay inputs maximize the achievable PDF rate for the aligned Gaussian MIMO relay channel. Subsequently, we use a limiting argument to extend this result from the aligned to the general Gaussian MIMO relay channel.

{IEEEkeywords}

Gaussian relay channel, MIMO, partial decode-and-forward, optimal channel input distribution, channel enhancement.

1 Introduction

This work considers the Gaussian multiple-input multiple-output (MIMO) relay channel, a three-node network where one source wants to convey information to one destination with the help of a single relay. All three nodes may be equipped with multiple antennas and they are connected by additive Gaussian noise channels. Furthermore, it is assumed that the relay does not have own information to transmit or receive so that its only purpose is to assist the communication from the source to the destination.

The concept of relaying traces back to van der Meulen [1], who introduced the first information theoretic model for the relay channel. While the capacity of the relay channel is still unknown, substantial advances towards its information theoretic understanding have since been made. The most important work on the relay channel is by Cover and El Gamal [2], who derived a capacity upper bound and achievable rates based on a then new cut-set bound (CSB) and two coding schemes that are nowadays referred to as decode-and-forward (DF) and compress-and-forward (CF), respectively. The DF strategy requires the relay to decode the entire source message, which is then re-encoded and, in cooperation with the source, transmitted to the destination. When using CF, the relay reliably forwards an estimate, i.e., a compressed version of its received signal, to the destination. In [3], these two basic strategies were generalized to various relay channel models that include multiple sources, relays, or destinations.

In their pioneering work, Cover and El Gamal also proposed a more general coding scheme that combines the DF and CF strategies [2, Theorem 7]. If the relay uses this strategy, it decodes only a part of the source message and compresses the remainder. The partial decode-and-forward (PDF) scheme is a special case of this mixed strategy where the relay only forwards information about the part of the source message it has decoded. We remark that PDF in turn includes the DF strategy and point-to-point (P2P) transmission from source to destination as special cases. Since the PDF scheme allows to optimize the amount of information the relay must decode, it provides the possibility to tradeoff sending information via the relay versus sending it over the direct link. In particular, equipping all nodes with multiple antennas creates spatial degrees of freedom which the PDF scheme may exploit to outperform the DF scheme.

Upper and lower bounds on the capacity of the Gaussian MIMO relay channel were first studied in [4], where it was shown that Gaussian channel inputs maximize both the CSB and the achievable DF rate. Furthermore, a generally loose upper bound on the CSB was established and evaluated, and different lower bounds on the capacity based on suboptimal DF strategies or P2P transmission were also derived. Using the fact that Gaussian channel inputs maximize the CSB and the achievable DF rate, it was then independently shown in [5] and [6] that, if perfect channel state information (CSI) is available at all nodes, the corresponding optimal values can be determined as the solutions of convex optimization problems.

Employing PDF in the Gaussian MIMO relay channel was first considered in [7], where the strategy was termed “transmit-side message splitting”. The authors formulated the PDF rate maximization problem for jointly Gaussian source and relay inputs, but they did not solve the resulting nonconvex problem. In addition, no attempt was made to characterize the input distribution that maximizes the achievable PDF rate. Rather, proper complex Gaussian channel inputs were assumed as part of the system model. If the channel inputs are restricted to be complex Gaussian, it was then shown in [8] that jointly proper source and relay inputs are indeed optimal. For the general case, however, the optimal input distribution has been unknown so far. Consequently, it has not been possible to characterize the maximum achievable PDF rate for the general Gaussian MIMO relay channel.

This is in contrast to the CSB, the DF rate, and the P2P capacity, for which it is well known that Gaussian inputs are optimal, cf. [9, 4]. The maximum achievable PDF rate for the Gaussian MIMO relay channel can thus be characterized whenever the optimal PDF strategy is equivalent to the DF strategy or P2P transmission, or if PDF achieves the CSB. Such special cases include the (physically) degraded and the reversely degraded relay channel [2], the semideterministic relay channel [10], the relay channel with orthogonal components [11], as well as the stochastically degraded and the reversely stochastically degraded relay channel [12].1 Moreover, the achievable PDF rate is also maximized by Gaussian channel inputs if the row spaces of the source-to-relay and the source-to-destination channel gain matrices are disjoint [13].

In this paper, we generalize these previous results by showing that the maximum achievable PDF rate for the Gaussian MIMO relay channel is always attained by jointly Gaussian source and relay inputs. To this end, we first establish that jointly Gaussian source and relay inputs maximize the achievable PDF rate for the aligned Gaussian MIMO relay channel, which constitutes the main challenge of the proof. Subsequently, we use a limiting argument to extend this result from the aligned to the general Gaussian MIMO relay channel. We remark that the idea to first consider an aligned channel was introduced by Weingarten et al. [14] to derive the (private message) capacity region of the Gaussian MIMO broadcast channel.

The proof that Gaussian channel inputs maximize the achievable PDF rate for the aligned Gaussian MIMO relay channel requires a large variety of ingredients. For the achievability part, we simply use jointly Gaussian source and relay inputs. The converse is based on a channel enhancement argument, which, like the idea to first consider the aligned channel, goes back to [14]. More specifically, the enhanced aligned relay channel we consider in the converse is stochastically degraded. As a result, its maximum achievable PDF rate is attained by a pure DF strategy [12], for which Gaussian channel inputs are known to be optimal [4]. Finally, the key to proving that achievability and converse meet is a primal decomposition approach, which we use to split the complicated PDF rate maximization into subproblems. In a slightly different context, this decomposition was first proposed in our previous work [8]. Therein, it enabled us to show and exploit the mathematical equivalence between one of the resulting subproblems and a sum rate maximization problem for a Gaussian MIMO broadcast channel with dirty paper coding. For the proof presented in this paper, however, we instead obtain a subproblem that is mathematically equivalent to the problem of finding the secrecy capacity of the aligned Gaussian MIMO wiretap channel (vector Gaussian wiretap channel) under shaping constraints [15, Section II-A]. To facilitate the proof that jointly Gaussian source and relay inputs maximize the achievable PDF rate for the aligned Gaussian MIMO relay channel, we can thus adopt considerations that were used in the derivation of [15, Theorem 2].

Notation: stands for the set of nonnegative real numbers. Matrices are denoted by bold capital letters, vectors by bold lowercase characters. The identity matrix and the all-zeros matrix/vector are represented by and , respectively, where the dimensions are indicated by subscripts if necessary. , , , , and denote the conjugate transpose, inverse, Moore-Penrose pseudoinverse, determinant, and trace of matrix , while and mean that is positive semidefinite (nonnegative definite) and positive definite, respectively. is the expectation operator and means that is a zero-mean proper (circularly symmetric) complex Gaussian random vector with covariance matrix . Finally, is the conditional mutual information of and given , and denotes the conditional differential entropy of given .

2 System Model

The channel model for the Gaussian MIMO relay channel, which is illustrated in Figure 1, is obtained by applying the linear MIMO model to the considered relay scenario. The receive signal vectors of the relay and the destination can thus be expressed as

 \mathbityR=\mathbitHSR\mathbitxS+\mathbitnR,\mathbitnR∼NC(0,\mathbitZR),\mathbityD=\mathbitHSD\mathbitxS+\mathbitHRD\mathbitxR+\mathbitnD,\mathbitnD∼NC(0,\mathbitZD), (1)

where , , and represent the channel gain matrices of appropriate dimensions, which are assumed to be perfectly and instantaneously known at all nodes. Moreover, and denote zero-mean proper complex Gaussian noise vectors with nonsingular covariance matrices and . The noise vectors are independent of each other and independent of the transmit signals and . Finally, perfectly synchronized transmission and reception between all nodes is assumed, and it is implicit in (1) that the relay operates in full-duplex mode and is able to completely cancel its own self-interference.

Without further conditions on the channel inputs and , the capacity of the Gaussian MIMO relay channel is infinite. That is because one can then choose infinite subsets of inputs arbitrarily far apart so that they are distinguishable at the outputs with arbitrarily small probability of error, cf. [16, Chapter 9]. We therefore impose the transmit power constraints

 E[\mathbitxHS\mathbitxS]≤PS,E[\mathbitxHR\mathbitxR]≤PR (2)

on the channel inputs, where and denote the power budgets available at the source and the relay, respectively.

Note that, without loss of generality, we can restrict our attention to zero-mean channel inputs as it is clear that the optimal and are always zero-mean. The reason for this is that channel inputs with nonzero mean consume more transmit power than the corresponding zero-mean signals, but they cannot convey more information since translations do not change the differential entropy of continuous random vectors, cf. [16, Theorem 8.6.3]. As a consequence, the covariance matrices of the source and relay inputs are given by and so that the transmit power constraints can equivalently be expressed as

 tr(\mathbitCS)≤PS,tr(\mathbitCR)≤PR. (3)

3 Partial Decode-and-Forward (PDF)

The partial decode-and-forward (PDF) strategy can be viewed as a generalization of the well-known decode-and-forward (DF) scheme. When using DF, the relay is required to decode the entire message transmitted by the source, even if this means that the source-to-relay link becomes the bottleneck of the communication. One way to overcome this problem is to allow the relay to partially decode the source message. For this purpose, the message that is to be transmitted from the source to the destination is split into two independent parts and , of which the relay is only required to decode . By constructing separate codebooks for and and using superposition coding at the source, a PDF scheme that achieves all rates smaller than or equal to

 RPDF=maxp(\mathbitu,\mathbitxS,\mathbitxR)min{I(\mathbitu;\mathbityR|\mathbitxR)+I(\mathbitxS;\mathbityD|\mathbitu,\mathbitxR),I(\mathbitxS,\mathbitxR;\mathbityD)}s.\,t.\mathbitu↔(\mathbitxS,\mathbitxR)↔(\mathbityD,\mathbityR),tr(\mathbitCS)≤PS,tr(\mathbitCR)≤PR (4)

is obtained as shown in [17, Section 9.4.1]. Here, is an auxiliary variable representing the part of the source message the relay must decode. In addition to the power constraints, the maximization over the joint distribution of , , and is subject to the constraint that forms a Markov chain.

It is easy to see that DF is a special case of PDF for which . In particular, choosing in (4) yields that

 RDF=maxp(\mathbitxS,\mathbitxR)min{I(\mathbitxS;\mathbityR|\mathbitxR),I(\mathbitxS,\mathbitxR;\mathbityD)}s.\,t.tr(\mathbitCS)≤PS,tr(\mathbitCR)≤PR (5)

is achievable by means of the PDF strategy, where denotes the maximum achievable DF rate for the Gaussian MIMO relay channel, cf. [2, Theorem 1]. Moreover, choosing yields the point-to-point (P2P) capacity of the source-to-destination link

 RP2P=maxp(\mathbitxS)I(\mathbitxS;\mathbityD|\mathbitxR=0)s.\,t.tr(\mathbitCS)≤PS. (6)

Therefore, it is clear that PDF always achieves at least the maximum of the rates that are achievable by means of DF and by means of direct transmission from source to destination, i.e.,

 RPDF≥max{RDF,RP2P}. (7)

In contrast to , , and the cut-set bound (CSB), which is given by [2, Theorem 4]

 CCSB=maxp(\mathbitxS,\mathbitxR)min{I(\mathbitxS;\mathbityR,\mathbityD|\mathbitxR),I(\mathbitxS,\mathbitxR;\mathbityD)}s.\,t.tr(\mathbitCS)≤PS,tr(\mathbitCR)≤PR, (8)

we cannot simply invoke the entropy maximizing property of the zero-mean proper (circularly symmetric) complex Gaussian distribution (cf. [9, 18]) to argue that is maximized by Gaussian channel inputs. The reason for this is that the entropy maximizing property cannot be applied to the term

 I(\mathbitu;\mathbityR|\mathbitxR)+I(\mathbitxS;\mathbityD|\mathbitu,\mathbitxR)=h(\mathbitHSR\mathbitxS+\mathbitnR|\mathbitxR)−h(\mathbitnD)+h(\mathbitHSD\mathbitxS+\mathbitnD|\mathbitu,\mathbitxR)−h(\mathbitHSR\mathbitxS+\mathbitnR|\mathbitu,\mathbitxR), (9)

which includes the difference of two conditional differential entropies involving , , and .

For , the maximization of such a difference over the conditional probability distribution subject to shaping constraints on the conditional covariance matrix was analyzed in [19, Theorem 8], where it is proved that the optimal distribution is Gaussian. However, the term , and hence the difference , is only one part of the objective function of the PDF rate maximization problem given in (4). Therefore, we cannot directly apply [19, Theorem 8] to prove that the achievable PDF rate is maximized by Gaussian channel inputs. Rather, we need to establish achievability and converse for the whole objective function.

4 Aligned Gaussian MIMO Relay Channel

As a first step towards a characterization of the maximum achievable PDF rate for the Gaussian MIMO relay channel, we consider the aligned Gaussian MIMO relay channel. The results for this special case are then generalized in the following section.

Definition 1.

The Gaussian MIMO relay channel is said to be aligned if and .

The channel model for the aligned Gaussian MIMO relay channel is hence given by

 \mathbityR=\mathbitxS+\mathbitnR,\mathbitnR∼NC(0,\mathbitZR),\mathbityD=\mathbitxS+\mathbitHRD\mathbitxR+\mathbitnD,\mathbitnD∼NC(0,\mathbitZD). (10)

As the theorem below reveals, Gaussian channel inputs maximize the achievable PDF rate for this particular relay channel.

Theorem 1.

For the aligned Gaussian MIMO relay channel, the maximum achievable PDF rate is attained by jointly proper complex Gaussian source and relay inputs.

Proof.
\red

Achievability: Let , , be independent, , and such that with . Then,

 RPDF≥RNCPDF=max\mathbitC% Q,\mathbitCV,\mathbitCR,\mathbitAmin{log|\mathbitCQ+\mathbitCV+\mathbitZR||\mathbitCV+\mathbitZR|+log|\mathbitCV+\mathbitZD||\mathbitZD|,log|\mathbitCQ+\mathbitC%V+(\mathbitHRD+\mathbitA)\mathbitCR(\mathbitHRD+\mathbitA)H+\mathbitZ%D||\mathbitZD|}s.\,t.\mathbitCQ,\mathbitCV,\mathbitCR⪰0,% tr(\mathbitCQ+\mathbitCV+\mathbitA\mathbitCR\mathbitAH)≤PS,tr(\mathbitCR)≤PR, (11)

where denotes a PDF rate that is achievable with proper complex Gaussian channel inputs. \redBy introducing an auxiliary variable , this achievable rate can equivalently be expressed as

 Missing or unrecognized delimiter for \Biggr (12)

If we now apply the primal decomposition approach that was already considered in [8] to this problem, we obtain

 RNCPDF=max\mathbitSRNCPDF(\mathbitS)%s.t.\mathbitS⪰0,tr(\mathbitS)≤PS (13)

with

 Missing or unrecognized delimiter for \Biggr (14)

Note that only appears in the constraints and , i.e., it is a slack variable and can be eliminated, after which the equality constraint becomes . Furthermore, only contributes to the second summand of the first term inside the minimum of the objective function so that is equal to

 Missing or unrecognized delimiter for \Biggr (15)

In order to \redfurther simplify this expression, consider the inner maximization problem

 max\mathbitCVlog|\mathbitCV+\mathbitZD||\mathbitCV+\mathbitZR|s.\,t.0⪯\mathbitCV⪯\mathbitS, (16)

which up to the additive constant is mathematically equivalent to the problem that yields the secrecy capacity of the aligned Gaussian MIMO wiretap channel (vector Gaussian wiretap channel) under shaping constraints, cf. [15, Section II-A].2 Following the proof of [15, Theorem 2], which carries over to the complex-valued setting under consideration here, we can determine the optimal value of the inner problem (16).

To this end, first note that the Karush–Kuhn–Tucker (KKT) conditions are necessary for problem (16) since the Abadie constraint qualification is automatically satisfied if all constraints are linear [20, Section 5.1] and since the KKT conditions readily extend to problems with generalized inequalities such as positive semidefiniteness constraints [21, Section 5.9.2]. Thus, any optimizer of problem (16) must satisfy

 (\mathbitC⋆V+\mathbitZD)−1+\mathbitΛ1 =(\mathbitC⋆V+\mathbitZ%R)−1+\mathbitΛ2, (17) \mathbitC⋆V\mathbitΛ1 =0, (18) (\mathbitS−\mathbitC⋆V)\mathbitΛ2 =0, (19)

where and denote the Lagrangian multipliers corresponding to the (generalized) inequality constraints and , respectively. Now, let such that

 (\mathbitC⋆V+\mathbitZ)−1=(\mathbitC⋆V+\mathbitZD)−1+\mathbitΛ1. (20)

It then follows from (18) that an explicit expression for (as a function of and the Lagrangian multiplier ) is given by

 \mathbitZ=(\mathbitZ−1D+\mathbitΛ1)−1. (21)

Since , we can conclude that . Furthermore, (17) and the definition of in (20) imply that

 (\mathbitC⋆V+\mathbitZ)−1=(\mathbitC⋆V+\mathbitZR)−1+\mathbitΛ2. (22)

By means of the variable , we can characterize the optimal value of problem (16) as follows. First, note that

 (\mathbitC⋆V+\mathbitZ)\mathbitZ−1=\mathbitC⋆V\mathbitZ−1+I\mathbitC⋆V(\mathbitZ−1D+\mathbitΛ1)+I\mathbitC⋆V\mathbitZ−1D+I=(\mathbitC⋆V+\mathbitZ% D)\mathbitZ−1D, (23)

which implies

 |\mathbitC⋆V+\mathbitZD||\mathbitZD|=|\mathbitC⋆V+\mathbitZ||\mathbitZ|. (24)

Similarly, it holds that

 (\mathbitS+\mathbitZ)(\mathbitC⋆V+\mathbitZ)−1=(\mathbitS−\mathbitC⋆V+\mathbitC⋆V+\mathbitZ)(\mathbitC⋆V+\mathbitZ)−1=(\mathbitS−\mathbitC⋆V)(\mathbitC⋆V+\mathbitZ)−1+I(\mathbitS−\mathbitC⋆V)((\mathbitC⋆V+\mathbitZR)−1+\mathbitΛ2)+I(\mathbitS−\mathbitC⋆%V)(\mathbitC⋆V+\mathbitZ% R)−1+I=(\mathbitS−\mathbitC⋆V+\mathbitC⋆V+\mathbitZR)(\mathbitC⋆V+\mathbitZR)−1=(\mathbitS+\mathbitZR)(\mathbitC⋆V+\mathbitZR)−1, (25)

from which we obtain

 |\mathbitC⋆V+\mathbitZ||\mathbitC⋆V+\mathbitZR|=|\mathbitS+\mathbitZ||\mathbitS+\mathbitZR|. (26)

The optimal value of problem (16) can be therefore calculated as

 log|\mathbitC⋆V+\mathbitZD||\mathbitC⋆V+\mathbitZR|=log|\mathbitC⋆V+\mathbitZD||\mathbitZD|−log|\mathbitC⋆V+\mathbitZR||\mathbitZD|log|\mathbitC⋆V+\mathbitZ||\mathbitZ|−log|\mathbitC⋆V+\mathbitZR||\mathbitZD|=log|\mathbitC⋆V+\mathbitZ||\mathbitC⋆V+\mathbitZR|−log|\mathbitZ||\mathbitZD|log|\mathbitS+\mathbitZ||\mathbitS+\mathbitZR|−log|\mathbitZ||\mathbitZD|=log|\mathbitS+\mathbitZ||\mathbitZ|−log|\mathbitS+\mathbitZR||\mathbitZ% D|. (27)

Using this result, it is straightforward to verify that is equal to

 RNCPDF(\mathbitS)=max\mathbitCR,\mathbitAmin{log|\mathbitS+\mathbitZ||\mathbitZ|,log|\mathbitS+(\mathbitHRD+\mathbitA)\mathbitCR(\mathbitHRD+\mathbitA)H+\mathbitZD||\mathbitZD|}s.\,t.\mathbitCR⪰0,tr(\mathbitA\mathbitCR\mathbitAH)≤PS−tr(\mathbitS),tr(\mathbitCR)≤PR (28)

with from (21).

Converse: The converse of the proof is based on the so-called channel enhancement technique, which was originally introduced in [14]. From (20), (22), and the positive semidefiniteness of the Lagrangian multipliers , it follows that

 \mathbitZ⪯\mathbitZD,\mathbitZ⪯\mathbitZR. (29)

Consequently, we can use to define an enhanced aligned Gaussian MIMO relay channel. In particular, let and

 ~\mathbityR=\mathbitxS+~\mathbitnR% ,~\mathbitnR∼NC(0,~\mathbitZR),\mathbityD=\mathbitxS+\mathbitHRD\mathbitxR+\mathbitnD,\mathbitnD∼NC(0,\mathbitZD). (30)

Since , is a stochastically degraded version of so that \redfor all feasible . Therefore,

 RPDF≤~RPDF=maxp(\mathbitu,\mathbitxS,\mathbitxR)min{I(\mathbitu;~\mathbityR|\mathbitxR)+I(\mathbitxS;\mathbityD|\mathbitu,\mathbitxR),I(\mathbitxS,\mathbitxR;\mathbityD)}s.\,t.\mathbitu↔(\mathbitxS,\mathbitxR)↔(\mathbityD,~\mathbityR),tr(\mathbitCS)≤PS,tr(\mathbitC% R)≤PR, (31)

which explains why we call the relay channel defined in (30) enhanced.

\red

Moreover, given , is a stochastically degraded version of as well. In fact, since , the enhanced aligned Gaussian MIMO relay channel belongs to the class of stochastically degraded relay channels according to the definition in [12]. From [12, Proposition 1], it hence follows that the optimal PDF strategy for the enhanced relay channel (30) is equivalent to DF, i.e., with

 ~RDF=maxp(\mathbitxS,\mathbitxR)min{I(\mathbitxS;~\mathbityR|\mathbitxR),I(\mathbitxS,\mathbitxR;\mathbityD)}s.\,t.tr(\mathbitCS)≤PS,tr(\mathbitCR)≤PR. (32)

However, the maximum achievable DF rate for the Gaussian MIMO relay channel is attained by jointly proper complex Gaussian channel inputs [4], which essentially follows from the fact that the zero-mean proper (circularly symmetric) complex Gaussian distribution maximizes the differential entropy, cf. [18, 9].

Therefore, the achievable PDF rate for the enhanced aligned Gaussian MIMO relay channel is maximized by letting and be independent and such that with , i.e.,

 ~RPDF=max\mathbitCQ,\mathbitCR,\mathbitAmin{log|\mathbitCQ+\mathbitZ||\mathbitZ|,log|\mathbitCQ+(\mathbitHRD+\mathbitA)\mathbitCR(\mathbitHRD+\mathbitA)H+\mathbitZD||\mathbitZD|}s.\,t.\mathbitCQ,\mathbitCR⪰0,tr(\mathbitCQ% +\mathbitA\mathbitCR\mathbitAH)≤P%S,tr(\mathbitCR)≤PR. (33)

Using a primal decomposition again, this maximization problem can equivalently be written as

 ~RPDF=max\mathbitCQ~RPDF(\mathbitCQ)s.\,t.\mathbitCQ⪰0,tr(\mathbitCQ)≤PS, (34)

where

 ~RPDF(\mathbitCQ)=max\mathbitCR,\mathbitAmin{log|\mathbitCQ+\mathbitZ||\mathbitZ|,log|\mathbitCQ+(\mathbitHRD+\mathbitA)\mathbitCR(\mathbitHRD+\mathbitA)H+\mathbitZD||\mathbitZD|}s.\,t.\mathbitCR⪰0,tr(\mathbitA\mathbitCR\mathbitAH)≤PS−tr(\mathbitCQ),tr(\mathbitCR)≤PR. (35)

Comparing (28) to (35), we notice that for , from which we can directly conclude that as the constraints in (13) and (34) are the same. But since in general, it follows that . ∎

5 General Gaussian MIMO Relay Channel

In this section, we extend the result that jointly proper complex Gaussian source and relay inputs maximize the achievable PDF rate to the general Gaussian MIMO relay channel. The main idea for extending the proof from the aligned to the general case is adopted from [14]. First, we write the channel model of the Gaussian MIMO relay channel (1) in an equivalent form with square channel gain matrices. In a second step, we use the singular value decomposition (SVD) to enhance and by adding small perturbations to their singular values such that the resulting channel gain matrices are invertible. Finally, we show that the maximum achievable PDF rate for the original Gaussian MIMO relay channel can be obtained by a limit process on the maximum achievable PDF rate for the enhanced (perturbed) relay channel.

Theorem 2.

For the Gaussian MIMO relay channel, the maximum achievable PDF rate is attained by jointly proper complex Gaussian source and relay inputs.

Proof.

Without loss of generality, we may assume that with . If this were not the case, we could augment the matrices with zeros to obtain square channel gain matrices without changing the achievable PDF rate. Furthermore, we may also assume that since any Gaussian MIMO relay channel with nonsingular noise covariances can be transformed into one with additive white Gaussian noise by means of a noise whitening operation, cf. [22].

\red

Achievability: Let , , be independent, , and such that with . Then, the PDF rate

 RNCPDF=max\mathbitCQ,\mathbitCV,\mathbitC