Energy Efficiency Optimization in RelayAssisted MIMO Systems with Perfect and Statistical CSI
Abstract
A framework for energyefficient resource allocation in a singleuser, amplifyandforward (AF), relayassisted, multipleinputmultipleoutput (MIMO) system is devised in this paper. Previous results in this area have focused on rate maximization or sum power minimization problems, whereas fewer results are available when bits/Joule energy efficiency (EE) optimization is the goal. Here, the performance metric to optimize is the ratio between the system’s achievable rate and the total consumed power. The optimization is carried out with respect to the source and relay precoding matrices, subject to qualityofservice (QoS) and power constraints. Such a challenging nonconvex optimization problem is tackled by means of fractional programming and alternating maximization algorithms, for various channel state information (CSI) assumptions at the source and relay. In particular the scenarios of perfect CSI and those of statistical CSI for either the sourcerelay or the relaydestination channel are addressed. Moreover, sufficient conditions for beamforming optimality are derived, which is useful in simplifying the system design. Numerical results are provided to corroborate the validity of the theoretical findings.
Energy Efficiency, Resource allocation, RelayAssisted communications, Multipleantenna systems, Fractional programming, Statistical CSI.
I Introduction
Wireless relaying is a wellknown technique to provide reliable transmission, high throughput, broad coverage and agile frequency reuse in modern wireless networks [1, 2]. In a cellular environment, relays are usually deployed in areas where a significant shadowing effect is present such as tunnels or the inside of buildings, as well as in areas that are far away from the transmitter and that otherwise would not be covered. In this context, AF is one of the most widely used choices because it does not require the relays to decode and know the users’ codebooks, thus allowing a faster and simpler design and placement of the relays. This relaying strategy is also one candidate approach in the standard LTEAdvanced and is usually referred to as layer1 relaying [3]. Another keyfactor in modern communication systems is the use of multiple antennas. It is established that the use of multiple antennas grants higher data rates and lower bit error rates [4]. As a result, recently a great deal of research has focused on MIMO relaying, where a multiple antenna, nonregenerative relay precodes the signal received from the source by an AF matrix, and then forwards it to the destination.
Most previous papers in this research direction consider source and relay precoding matrix allocation for the optimization of traditional performance measures such as achievable rate and minimum mean square error [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and references therein. However, the consideration that in mobile networks the nodes are typically batterypowered, thus having a limited lifetime, as well as the concerns for sustainable growth due to the increasing demand for energy, have garnered a great deal of interest on an efficient use of energy in wireless networks [15, 16] both in academia and in industry. Information and communication technologies (ICT) consume about 2 of the entire world energy consumption, and the situation is likely to reach the point wherein ICT equipments in large cities will require more energy than it is actually available [17]. One approach in this sense is to consider the minimization of the transmit power subject to QoS constraint [18, 19]. However, the tradeoff between achieving high data rates and limiting energy consumptions is mathematically more thoroughly described by considering the optimization of new, fractional performance measures, which are measured in bit/Joule and thus naturally represent the efficiency with which each Joule of energy drained from the battery is being used to transmit information. Resource allocation for bit/Joule EE optimization has been extensively analyzed in singleantenna, onehop networks and several performance metrics have been proposed. In [20, 21, 22, 23] and references therein, the EE is defined as the ratio between the achieved throughput and the consumed power. Instead, in [24, 25] the ratio between the achievable rate and the consumed power has been considered. As for onehop, multipleantenna networks, fewer results are available. In [26], the ratio between the throughput and the consumed power is optimized, but the simplifying assumption of singlestream transmission is made. In [27] the EE is defined as the ratio between the goodput and the transmit power and the problem of transmit covariance matrix allocation is studied. In [28] a broadcast MIMO channel is considered and uplinkdownlink duality is exploited to come up with a transmit covariance matrix allocation algorithm so as to maximize the ratio between the system capacity and the consumed power. Few results are available for relayassisted singleantenna networks, too. In [29] competitive power control algorithms for EE maximization in relayassisted singleantenna multiple access networks are devised, while [30] extends the results of [29] to interference networks.
All of the previously cited works assume that perfect CSI is available for resource allocation purposes, which might not be feasible in realworld systems. Indeed, a significant research trend is to devise resource allocation algorithms that require only statistical CSI, thus reducing the amount of communication overhead. As far as MIMO systems are concerned, contributions in this direction have mainly focused on the optimization of traditional performance measures such as achievable rate [31, 32, 33] and mean square error [34]. Instead, less attention has been given to the problem of EE optimization with statistical CSI. In [35, 36], a onehop MIMO link is considered and the transmit covariance matrix is allocated so as to maximize the ratio between the ergodic capacity and the consumed power. Instead, no results are available for bit/Joule EE optimization in relayassisted MIMO systems, and even the simpler case in which perfect CSI is assumed is an almost unexplored field. Indeed, to the best of our knowledge, the first contribution in this direction is the conference paper [37], where preliminary results on EE maximization in MIMO relayassisted systems with perfect CSI are provided.
Motivated by this background, this work is aimed at providing a thorough investigation of energyefficient resource allocation in MIMO relayassisted systems, with perfect and statistical CSI. The EE is defined as the ratio between the system’s achievable rate and the consumed power. In the definition of the consumed power, not only the transmit power, but also the circuit power dissipated in the devices’ electronic circuitry is accounted for. In such a scenario, energyefficient resource allocation algorithms that jointly allocate the source and relay precoding matrix subject to power and QoS constraints have been devised. In particular, the following cases have been considered.

Perfect CSI is available for both sourcetorelay and relaytodestination channel.

Perfect CSI is available only for the relaytodestination channel, while the sourcetorelay channel is only statistically known.

Perfect CSI is available only for the sourcetorelay channel, while the relaytodestination channel is only statistically known.
In all three cases, the fractional, nonconvex optimization problem to be solved has been tackled by means of a twostep approach. First, the optimal source and relay transmit directions have been determined in closed form. Next, plugging the optimal transmit directions in the objective function, it has been shown that the resulting problem is separately pseudoconcave in the source and relay power allocation vectors. Thus, the alternating maximization algorithm coupled with fractional programming tools have been used to complete the resource allocation process. Moreover, with reference to scenarios 2) and 3) sufficient conditions for the optimality of source beamforming transmission have been derived, which allows to reduce the complexity of the resource allocation phase. Otherwise stated, sufficient conditions under which the optimal power allocation at the source is to concentrate all the available power on just one data stream have been derived.
The rest of the paper is organized as follows. Section II describes the considered scenario, formally stating the problem to be tackled. In Section III the energyefficient resource allocation problem is solved assuming perfect CSI is available for both the sourcetorelay and the relaytodestination channel. Section IV addresses the case in which only statistical CSI for the sourcetorelay channel is available, while Section V tackles the opposite case in which the relaytodestination channel is statistically known. In Section VI the optimality of beamforming transmission is investigated for the scenarios considered in Sections IV and V. Numerical results are provided in Section VII, while concluding remarks are provided in Section VIII. Some lemmas which are instrumental to the derivation of the theoretical results are provided in the Appendix.
Notation: In the sequel, is the statistical expectation operator, denotes an identity matrix, , , , and denote Hermitian, trace, determinant, and pseudoinversion of a matrix, respectively. denotes a diagonal matrix with as diagonal elements. while denotes the space of , Hermitian, positive semidefinite matrices. Matrix inequalities will be intended in the Löwner sense^{1}^{1}1For any two Hermitian, positive semidefinite matrices and , means by definition that is positive semidefinite.. The acronym EVD and SVD stand for eigenvalue decomposition and singular value decomposition, respectively, and, without loss of generality, in all EVDs and SVDs, the eigenvalues and singular values will be assumed to be arranged in decreasing order.
Ii System Model and Problem Statement
Consider a relayassisted MIMO system consisting of one source , one halfduplex AF relay and one destination , which are equipped with , and antennas, respectively. Let be the source’s unitnorm symbol vector, and , with being the source transmit covariance matrix. Let us also denote by and the sourcerelay and relaydestination channels, and by the AF relay matrix. Then, the signals and received at the relay and destination respectively, can be written as and , with and being the thermal noise at relay and destination, modeled as zeromean complex circular Gaussian vectors with covariance matrices and , respectively.
The goal of the resource allocation process is to maximize the efficiency with which the system nodes employ the energy supply at their disposal to transmit information. The efficiency of any physical system is usually defined by the benefitcost ratio, and in communication systems two natural measures of benefit and cost are the achievable rate and the consumed energy. The ratio between the achievable rate in a communication system and the consumed energy is commonly referred to as the global energy efficiency (GEE) of the system. Clearly, a tradeoff exists between ensuring high achievable rates and saving as much energy as possible. Therefore, the maximization of the GEE is not trivial and fundamentally different from achievable rate maximization, since the resource allocation algorithm should aim at striking the optimal balance between high datarates and low consumed energy.
In the considered system, the achievable rate is expressed in bits/s/Hz as [10]
(1) 
with being the overall noise covariance matrix, and the factor stemming from the fact that the signal vector is transmitted in two time slots. Then, denoting by the total transmission time, the amount of information that can be reliably transmitted in the timeinterval is bits/Hz, with the source and relay transmit power constraints
(2) 
wherein and denote the maximum feasible transmit powers at and , respectively.
In a halfduplex relay channel, each node has three operation modes: transmission, reception and idle mode [38]. The power consumptions in these modes are denoted by , and , respectively, where is the transmit power, is the power amplifier efficiency, , , and are the circuit power consumption in transmission, reception, and idle mode, respectively. We assume that , and are modeled as constant terms independent of the data rate [38], [39]. In the first time slot, , and are in transmission mode, reception mode and idle mode, respectively. In the second time slot, , and are in idle mode, transmission mode and reception mode, respectively. Then, the amount of energy consumed in the timeinterval can be expressed as
(3) 
where is the total circuit power dissipated in the network nodes. For notational ease, and without loss of generality, in the following we assume . Then, the GEE is defined as
(4) 
Note that (4) is measured in , thus representing a natural measure of the efficiency with which each Joule of energy is used. The problem to be tackled is that of GEE maximization subject to the power constraints (2) and to the QoS constraint , with being the minimum acceptable achievable rate. Such problem will be addressed with reference to the following scenarios:

and have perfect CSI for both channels and .

and have perfect knowledge of the relaytodestination channel , but only statistical CSI for the sourcetorelay channel .

and have perfect knowledge of the sourcetorelay channel , but only statistical CSI for the relaytodestination channel .
In all scenarios it is assumed that has perfect knowledge of both channels and .
Iii GEE maximization with perfect CSI
Assume both and have perfect CSI on and . For future reference, denote the SVDs of the channels by , , while the EVD of and SVD of are given by and . The resource allocation problem can be formulated as the maximization problem
(5) 
Problem (5) is a complex fractional problem which is not jointly convex in . It should also be remarked that, while the numerator of the GEE is wellknown to be maximized by diagonalizing the channel matrices and arranging the eigenvalues of and in decreasing order, the same allocation of and would actually maximize the denominator, which is instead minimized by arranging the eigenvalues of and in increasing order. Therefore, it is not straightforward to conclude that diagonalization is optimal when maximizing the GEE. In order to show that diagonalization is indeed optimal, the following result provides a change of variables that allows to rewrite the GEE as a fraction whose numerator and denominator will be shown to be simultaneously maximized and minimized, respectively, by diagonalization.
Proposition 1
Consider Problem (5). The optimal and are such that , , and .
Proof:
We start by rewriting the objective function as
(6) 
Now, defining the variables and , (6) can be expressed as
(7) 
Defining by the SVD of and by the EVD of , by virtue of Lemma 1 in Appendix, it follows that the first and second summand in the denominator of (7) are minimized when and , respectively. Moreover, exploiting Lemma 2, it can also be seen that the numerator is maximized for . Therefore, such choices for , , and simultaneously maximize the numerator and minimize the denominator of (7). Moreover, they are also feasible because the numerator of the objective is also the LHS of the QoS constraint, while the first and second summand in the denominator are the LHS of the power constraints. Next, from the expression of we have , from which it follows that in order to achieve , the relation needs to hold. Similarly, for we have . Thus, in order to achieve and , the relations , and need to hold. \qed
Remark 1
In the proof of Proposition 1 it has been implicitly assumed that both and are tall fullrank matrices. However, this assumption has been made only for notational ease and the result of Proposition 1 can be readily extended to the case of generic matrices and , too. For example, assume and are wide fullrank^{2}^{2}2The case of nonfullrank channel matrices is of little practical relevance since and will be fullrank with probability 1. However, the method reported here can be applied also to rankdeficient matrices. matrices. Thus we have and defining the matrices and , the objective of (5) can be rewritten as equation (8),
(8) 
wherein is the left diagonal block of , is the left diagonal block of , is the upperleft block of , while is a matrix containing the first rows of . Moreover, for any and we have and . Therefore, it is seen that the entries of and that are not contained in and should be set to zero since they do not affect the numerator of (8) and only increase the consumed power. Therefore, Problem (5) can be recast in terms of only and , and thus can be solved by means of Proposition 1. In the sequel of the paper, similarly to Proposition 1, some results will implicitly assume tall, fullrank channel matrices. Such assumptions cause no loss of generality since they can be relaxed with similar techniques as shown here for Proposition 1.
As a consequence of Proposition 1, denoting by , , , and , the generic entry of the matrices , , , and , respectively, and by and the vectors and , Problem (5) can be expressed as
(9) 
Problem (9), although being a vectorvalued, simpler problem than (5), is still nonconvex. However, it can be tackled using the tools of fractional programming and the alternating maximization algorithm [40], as shown in the following. We start by recalling the following result.
Proposition 2
Consider the fractional function . If is a concave function and is a linear function, then is a pseudoconcave function. Moreover, consider the function defined as
(10) 
is continuous, convex and strictly decreasing, while, for fixed , the maxmization problem in (10) is a strictly convex optimization problem. Moreover, the problem of maximizing is equivalent to the problem of finding the positive zero of .
Thus, a pseudoconcave problem can be solved by finding the zero of the auxiliary function . This can be done with a superlinear convergence by means of Dinkelbach’s algorithm [42]. Since pseudoconcave functions have the pleasant property to have no stationary point other than global maximizers [41], the output of Dinkelbach’s algorithm is guaranteed to be the global solution of the problem, assuming the constraint set of the problem is a convex set.
Now, it is seen by inspection that the objective of Problem (9) is pseudoconcave in for fixed and pseudoconcave in for fixed . Therefore, one convenient way to solve (9) is to employ the alternating maximization algorithm [40], according to which Problem (9) can be alternatively solved with respect to , for fixed and with respect to , for fixed , until the objective converges. Denoting by the value of the GEE achieved after the th iteration of the algorithm, the formal procedure can be stated as follows.
Convergence of Algorithm 1 is ensured by the observation that after each iteration the objective is not decreased and that the objective is upperbounded. It should also be mentioned that, while the global solution of each subproblem in Algorithm 1 is found thanks to Dinkelbach’s algorithm, in general it can not be guaranteed that the overall Algorithm 1 converges to the global optimum of the GEE because the GEE is not jointly pseudoconcave in , and because these two vectors are optimized alternatively. However, if it holds that for all , then each summand in the numerator of the objective can be approximated by , which is a strictly jointly concave function of and . As a consequence, since strictly pseudoconcave functions enjoy the property to have only one stationary point, which is the function’s global maximizer, it is likely that Algorithm 1 converges to the GEE global maximizer. Indeed, the numerical results that will be presented in Section VII confirm such conjecture. Algorithm 1 can be implemented either centrally or in a distributed fashion. In the former case, it could be implemented at the relay, which then feeds back the resulting to the source. In the latter scenario the algorithm should be run in parallel at and , which, at the end, will automatically learn their respective precoding matrices.
Iv GEE maximization with partial CSI on
Assume that the relaytodestination channel is perfectly known but that only statistical CSI is available for the sourcetorelay channel in the form of covariance feedback. This scenario is realistic in all situations in which the relaytodestination channel is slowly timevarying, whereas the sourcetorelay channel is rapidly timevarying. Indeed, a rapidly varying channel is more difficult to estimate and a resource allocation that depends on such an estimate would have to be updated very frequently, which results in a significant amount of overhead. A typical example is the uplink of a communication system, in which the relay and destination are usually fixed, while the source is a mobile terminal.
Specifically, in this section the channel matrix is expressed according to the Kronecker model [43], as
(11) 
where is a random matrix with independent, zeromean, unitvariance, proper complex Gaussian entries, whereas and are the positive semidefinite receive and transmit correlation matrices associated to . The matrices and are assumed known whereas the matrix is unknown at the source and relay. The covariance feedback model has been widely used in the literature, [32, 31], [44, 45], and applies for example to scenarios in which relay and base station are surrounded by local scatterers that induce the matrices and , and are separated by a rich multipath environment that is modeled by the matrix . We also remark that by letting the transmit and receive correlation matrices be identity matrices, the special notable case in which is completely unknown and modeled as a random matrix with independent, zeromean, unitvariance, proper complex Gaussian entries is obtained. For future reference, let us define
(12) 
As for the performance measure to optimize, since only statistical knowledge of is available, it is not possible to optimize the instantaneous GEE (4). Instead, the GEE of the considered system should be defined recalling the original definition of the GEE which is the ratio between the benefit and cost of the system. For the case at hand, the benefit is given by the ergodic achievable rate, while the cost is the average consumed energy, which leads to the definition
(13) 
It should be mentioned that another approach would be to consider the maximization of the average of (4) with respect to , namely
(14) 
However, (14) can not be considered a proper GEE since it is not the ratio between the benefit produced by the system and the cost incurred to achieve such benefit. Thus, (14) does not represent the efficiency with which the resources are being used to produce the necessary goods, as instead does (13). Therefore, (14) will not be considered as performance measure and the focus will be on (13). The optimization problem at hand can be formulated as follows
(15) 
The following proposition determines the optimal source eigenvector matrix .
Proposition 3
Consider Problem (15). For any AF matrix , the optimal is such that .
Proof:
To begin with, let us rewrite the objective as
(16) 
Next, plugging (11) and (12), (16) can be expressed as in (17),
(17) 
where it has been exploited the fact that multiplying , from left or right, by a unitary matrix does not change its distribution. Next, defining , the numerator of (17) can be written as the concave function , shown in (18).
(18) 
At this point, defining as in the proof of Lemma 3 in Appendix, we have (19), where it has been exploited that and commute because they are both square diagonal matrices, and that is a unitary matrix and thus the random matrix has the same distribution as .
(19) 
Hence, by virtue of Lemma 3, it holds that is maximized when is diagonal. Next, we show that this choice for is also optimal as far as the denominator of (17) is concerned. Indeed, the part of the denominator of (17) that depends on is the function
(20) 
which is linear in . Moreover, it is easy to check that . Thus, employing again Lemma 3, it follows that we can set without affecting . Moreover, this choice is also feasible since it maximizes the LHS of the QoS constraint, while leaving unaffected the LHS of the power constraints. Finally, from we obtain . \qed
Next, we tackle the optimization with respect to the left and right eigenvector matrices of .
Proposition 4
Consider Problem (15). For any source covariance matrix with the optimal structure , the optimal is such that and , if either or is a scaled identity matrix^{3}^{3}3The proof also holds under more general assumptions as explained next..
Proof:
Plugging the optimal into (17) and defining by the th column of for all , the statistical mean at the denominator can be computed in closedform as follows.
(21) 
Next, exploiting again that multiplying from left or right by a unitary matrix does not change its distribution, and defining the auxiliary variable , after some elaborations the objective can be expressed as in (22).
(22) 
(23) 
which is concave in by virtue of Lemma 4 and because of the concavity and monotonicity of the function . Moreover, consider a generic matrix as in the proof of Lemma 3, and also define the matrix^{4}^{4}4Here we are assuming , but the proof can be extended to the case with similar arguments as those used in Remark 1, and in this case .