Layered Capacity-Based Relay-and-Antenna Joint Selection for MIMO-AF-Multiple-Relay Systems in Correlated Channels

Abstract

A computationally efficient two-stage greedy capacity maximization (GCM) relay-and-antenna joint selection is proposed for a dual-hop nonregenerative amplify-and-forward (AF) multiple-input multiple-output (MIMO) multiple-relay system with multiple antennas equipped at each node in correlated fading channels. This modified GCM (MGCM) antenna selection strategy selects a subset of antenna pairs from available relays based on the concept of channel capacity maximization subject to an optimal power allocation constraint across the activated antenna pairs. In order to reduce system hardware complexity, antenna selection schemes are performed at the destination node as well. Finally, simulations are conducted to compare the channel capacity of the proposed two-layered antenna selection technique with other existing antenna selection algorithms for half-duplex AF-MIMO multiple-relay systems.

1. Introduction

Multiple-input multiple-output (MIMO) systems using multielement antennas (MEA) at the transmitter and the receiver improve substantially capacity and reliability of wireless links [1, 2]. As a simplified version of spatial multiplexing (SM) systems, the vertical Bell Laboratories layered space-time (V-BLAST) architecture [3] exploits the channel capacity and diversity gain of MEA systems over the rich-scattering wireless channel. A new communication paradigm that incorporates MIMO technologies into cooperative relay networks [4, 5] has attracted significant interest recently. In the cooperative communications and networking, the source node communicates with the destination node by making use of wireless relays. With the aid of node cooperation, wireless relay networks are able to obtain spatial diversity in a distributed manner. In general, there are two categories of cooperative relaying strategies, such as nonregenerative amplify-and-forward- (AF-) relaying protocols [5, 6] and regenerative decode-and-forward- (DF-) relaying protocols [5, 7]. An AF-mode protocol shows a simple relaying approach and possesses a low implementation complexity. However, the limitation on the implementation of MIMO systems is the cost and complexity of multiple radio frequency (RF) chains associated with multiple antennas. The increase in system hardware and signal processing complexity has inhibited the widespread adoption of MIMO systems. Fortunately, the utilization of antenna subset selection approaches for MIMO systems has successfully become a low-cost alternative due to the reduced complexity without significant performance degradation compared to the full-complexity (FC) MIMO system [8]. These approaches alleviate system hardware complexity by using fewer RF chains than the actual number of antenna elements.

So far most of antenna selection techniques applied to non-regenerative wireless relay networks are focusing on single-antenna relay models [9] or single-stream MIMO-relay networks [10]. An opportunistic relaying approach, namely, distributed orthogonal relay selection (DORS), for dual-hop noncoherent AF-MIMO relay networks are proposed in [11] based on the harmonic mean of dual-hop subchannel gains. In [12], a cross-entropy- (CE-) based method is introduced to perform the optimal relay subset selection for two-way AF-MIMO relaying systems in order to reduce computational complexity while still maximizing the achievable sum rate. However, only single-antenna relays are dealt with for one-or two-way dual-hop AF-wireless MIMO-relay networks. A fast antenna subset selection algorithm (FASSA) [13] is proposed for two-hop AF- and DF-MIMO relay systems to maximize system capacity. The selection criterion of the FASSA algorithm for AF relay is designed to maximize a lower bound of the capacity, but not the exact capacity. The authors in [14] present a two-stage antenna subset selection scheme to maximize the channel capacity of a conventional three-node AF-MIMO relay system. Moreover, a low-complexity near-optimal antenna selection algorithm is considered based on a constrained cross-entropy optimization (CCEO) method to maximize the achievable rate for MIMO-AF single-relay networks in [15]. However, only single relay is dealt with in a non-regenerative multiantenna relay network. In [16], a greedy mean squared error (MSE) minimization antenna selection is developed under an equal-power allocation across the multistream data for AF-MIMO relay systems. The fact that only single-antenna pair can be activated on each relay node in the greedy MSE-based minimization (GMM) antenna selection should be noted. Additionally, the computational complexity of the GMM antenna selection remains high and is thus of limited practical use. A greedy capacity maximization (GCM) antenna selection technique proposed in [17] is applied to an identical scenario employed in [16]. Four joint relay and antenna selection strategies are investigated in terms of the outage probability in [18] with the use of either full or partial CSI for dual-hop MIMO-AF multiple-relay networks. A comprehensive performance analysis framework is provided based on practical transmission impairments in [19] for dual-hop MIMO-AF relay networks with hop-by-hop beamforming (i.e., both source and relay perform beamforming). In [20], the outage probability and the average symbol-error-rate (SER) performance of three transmit antenna selection (TAS) strategies are analyzed for dual-hop MIMO ideal channel-assisted AF-relay networks. Although these antenna selection schemes are effective, most of these works have not taken spatial correlation between antennas into account. Various measurements have shown that the realistic MIMO-channel capacity is significantly lower than that in the ideal case owing to the spatial correlation of the MIMO channel. The correlation between the signal transmitted/received at different antenna elements is due to the lack of spacing between antennas as well as the existence of small angular spread [21]. Therefore, to mitigate the effect of spatial correlation, a correlated channel model has to be employed to replace the ideal identically independent distribution (iid.) one.

In the paper, a reduced-complexity two-stage relay-and-antenna joint selection is proposed for a dual-hop non-regenerative AF-MIMO multiple-relay system with multiple antennas equipped at each node over correlated fading channels. This relay-and-antenna selection strategy selects a subset of antenna pairs from multiple relays based on channel capacity maximization under an optimal power allocation constraint across the activated antenna pairs. To reduce further system hardware and signal processing complexity, antenna selection schemes are also performed at the destination node. In addition, all wireless channel links are characterized as spatially correlated random variables based on the Kronecker correlation model [22]. As a consequence, the use of the proposed two-layer relay-and-antenna joint selection is capable of trading in a relatively small fraction of loss in system performance, as compared with existing antenna selection schemes for half-duplex AF-MIMO relay networks, for a considerable reduction in system hardware and signal processing complexity.

The remainder of the paper is organized as follows. Section 2 describes the system and channel model. Section 3 presents a two-stage modified GCM (MGCM) antenna selection technique for a dual-hop non-regenerative MIMO-AF multiple-relay system in correlated channels. Both the optimal and equal-power allocation constraints across the activated 𝐾 antenna pairs in the half-duplex AF-MIMO multiple-relay system are provided in Section 4. To further alleviate hardware cost and implementation complexity of a dual-hop AF-wireless MIMO multiple-relay system, destination antenna selection schemes are utilized and analyzed in Section 5. Two kinds of existing relay-and-antenna joint selection criteria are briefly reviewed in Section 6. Numerical results and conclusions are presented in Sections 7 and 8, respectively.

Notation 1. Symbols for matrices (vectors) are denoted by boldface upper- (lower-) case letters. The superscripts (⋅)^⊤,(⋅)^H, and (⋅)⁻¹ are transposition, Hermitian transposition, and matrix inversion operations, respectively. ‖⋅‖_F indicates the matrix/vector Frobenius norm. 𝐸{⋅} represents the expected-value operator. 𝐈_𝑀 is an 𝑀×𝑀 identity matrix. tr(⋅) and [⋅]_𝑖,𝑗 denote, respectively, the trace and the (𝑖,𝑗)th entry of a matrix.(⋅)^1/2 is the square root of a positive semidefinite matrix. Finally, (𝑥)₊ stands for max(x,0).

2. System and Channel Model

A dual-hop non-regenerative AF-wireless MIMO multiple-relay system is considered in correlated fading channels, as illustrated in Figure 1. The information-bearing signals are sent from an 𝑁_S-antenna source terminal to an 𝑁_D-antenna destination terminal with the aid of 𝑁 half-duplex relay terminals each with 𝑁_R antennas. Note that no direct link is considered due to the long distance or the fading obstacle between the source and the destination terminals. In order to provide sufficient degrees of freedom for signal detection, the condition of 𝑁_S≤min{𝑁_R,𝑁_D} is preserved in what follows. In addition, elements of all channel links such as the source-to-relay (backward) channels 𝐇_SR𝑖∈𝐶^𝑁R×𝑁S and the relay-to-destination (forward) channels 𝐆_R𝑖D∈𝐶^𝑁D×𝑁R for all 𝑖∈[1,𝑁] are modeled as spatially correlated random variables based on the Kronecker correlation model [22], which are given as follows: 𝐇_SR𝑖=𝚺^1/2_SR𝑖𝐇^𝑤_SR𝑖Ψ^1/2_SR𝑖=𝐡^⊤_𝑖,1,𝐡^⊤_𝑖,2,…,𝐡^⊤_𝑖,𝑁R^⊤,𝑖=1,2,…,𝑁_S,𝐆_R𝑖D=𝚺^1/2_R𝑖D𝐆^𝑤_R𝑖DΨ^1/2_R𝑖D=𝐠_𝑖,1,𝐠_𝑖,2,…,𝐠_𝑖,𝑁R,𝑖=1,2,…,𝑁_D, (1) where 𝐇^𝑤_SR𝑖∈𝐶^𝑁R×𝑁S and 𝐆^𝑤_R𝑖D∈𝐶^𝑁D×𝑁R are, respectively, the backward and the forward spatially white complex Gaussian random matrices, in which elements are i.i.d. zero-mean circularly symmetric complex Gaussian (ZMCSCG) random variables with unit covariance. 𝚺_SR𝑖∈𝐶^𝑁R×𝑁R and 𝚺_R𝑖D∈𝐶^𝑁D×𝑁D are the receive-side spatial correlation matrices and Ψ_SR𝑖∈𝐶^𝑁S×𝑁S and Ψ_R𝑖D∈𝐶^𝑁R×𝑁R are the transmit-side correlation matrices for the backward channel 𝐇_SR𝑖 and the forward channel 𝐆_R𝑖D, respectively. 𝐡_𝑖,𝑗 and 𝐠_𝑖,𝑗, 𝑗=1,2,…,𝑁_R, represent, respectively, the 1×𝑁_S backward channel vector from the source node to the 𝑗th antenna of the 𝑖th relay and the 𝑁_D×1 forward channel vector from the 𝑗th antenna of the 𝑖th relay to the destination node.

Figure 1 

Block diagram of an AF-approach MIMO multiple-relay system in which the source node, each of N half-duplex AF-relay nodes, and the destination node are equipped with 𝑁S, 𝑁R, and 𝑁D antennas, respectively.

In what follows, it should be noted that the correlation between different MIMO-channel elements in the Kronecker channel model is modeled under the assumption that the correlation among receive antennas is independent of the correlation between transmit antennas. The approximation of the spatial cross-correlation function 𝑟(𝑑), that determines the correlation between two adjacent antenna elements separated in space by a distance 𝑑 [23], is given as follows: 𝑟(𝑑)≈exp−23𝐴²𝑑𝜆², (2) where 𝜆 is the wavelength and 𝐴 is the angular spread parameter. The angular spread parameter of 𝐴 is defined by [24] 𝐴=𝜃²+2cos𝜃−2,2𝜃 (3) where 𝜃∈[0,2𝜋] denotes the width of the sector of arriving multipath power.

During the first time slot, a subset of 𝐾 antennas are selected from 𝑁 relay nodes each with 𝑁_R antennas, that is, a total of 𝑁𝑁_R antenna selection choices. In the sequel, let the subscript of the pair (𝑅(𝑘),𝑏(𝑘)),𝑘=1,2,…,𝐾, represent the 𝑘th selected antenna index 𝑏(𝑘) and the corresponding relay index 𝑅(𝑘) in time-slot I. The 𝐾-activated antenna pairs generate a 𝐾×𝑁_S compound backward channel matrix 𝐇 given by 𝐇=𝐡^⊤_{()𝑅(1),𝑏(1)},𝐡^⊤_{()𝑅(2),𝑏(2)},…,𝐡^⊤_{()𝑅(𝐾),𝑏(𝐾)}^⊤. (4) The received signal vector at the 𝐾 antenna pairs can be expressed as 𝐲=𝑦₁,𝑦₂,…,𝑦_𝐾^⊤=𝐇𝐱+𝐧_R, (5) where 𝑦_𝑖=𝐡_{(𝑅(𝑖),𝑏(𝑖))}𝐱+𝑛_𝑖, 𝑖=1,2,…,𝐾, is the 𝑖th received signal component in 𝐲 and 𝑛_𝑖 is the corresponding noise random variable in 𝐧_R during the first-time slot. In (5), 𝐱∈𝐶^𝑁S×1 stands for the information-bearing signal vector whose elements are chosen independently from the same constellation and satisfy 𝐑_𝐱=𝐸{𝐱𝐱^H}=𝜎²_x𝐈_𝑁S and 𝐧_R∈𝐶^𝐾×1 denotes an i.i.d. ZMCSCG noise vector with the covariance matrix 𝐑_𝐧R=𝐸{𝐧_R𝐧^H_R}=𝜎²_𝑛R𝐈_𝐾. Additionally, the power constraint at the source is written as 𝐸{tr(𝐱𝐱^H)}≤𝑃_S. Under the case of the equal power allocation, 𝜎²_x=𝑃_S/𝑁_S is attained. In the second-time slot, the received signal vector 𝐲 is amplified by a 𝐾×𝐾 relay weighting matrix 𝐖 and subsequently the relay forwards the precoded signal vector to the destination. The amplified signal vector denoted by 𝐩 is given by 𝐩=𝐖𝐲. (6) Note that the total transmit power of 𝐾 activated antenna pairs selected from 𝐿_R relays is bounded by 𝑃_R as follows: 𝐸tr𝐩𝐩^H=tr𝐖²𝜎²_x𝐇𝐇^H+𝜎²_𝑛R𝐈_𝐾≤𝑃_R. (7)

In the paper, the perfect synchronization is assumed to be achieved at all channel links in the MIMO-AF multiple-relay system. Thus, the received signal vector 𝐫 at the destination node is given by 𝐫=𝐆𝐩+𝐧_D=𝐆𝐖𝐇𝐱+𝐆𝐖𝐧_R+𝐧_D, (8)(9) where 𝐆=[𝐠_{(𝑅(1),𝑓(1))},𝐠_{(𝑅(2),𝑓(2))},…,𝐠_{(𝑅(𝐾),𝑓(𝐾))}]∈𝐶^𝑁D×𝐾 in (8) denotes the compound forward channel matrix which is generated by the selected 𝐾 antennas at the second-time slot. Note that the selection of a set of 𝐾 antennas is performed on a total of 𝐾𝑁_R possible antenna choices which is composed by 𝐿_R candidate relays with a total of the 𝐾-activated antennas in time-slot I each candidate relay containing 𝑁_R corresponding antenna selection choices. The subscript of the pair (𝑅(𝑘),𝑓(𝑘)),𝑘=1,2,…,𝐾, represents the selected antenna index 𝑓(𝑘) and the corresponding relay index 𝑅(𝑘) in time-slot II. 𝐧_D∈𝐶^𝑁D×1 denotes the ZMCSCG noise vector with the covariance matrix 𝐑_𝐧D=𝐸{𝐧_D𝐧^H_D}=𝜎²_𝑛D𝐈_𝑁D. Remarkably, with the aid of the proposed two-layer antenna selection scheme, an exhaustive antenna-pair search from a pool of 𝑁𝑁²_R can be obviated successfully. Finally, (9) can be reformulated as 𝐳=𝐇_eq𝐱+𝐧_eq, (10) where 𝐇_eq=𝐆𝐖𝐇 and 𝐧_eq=𝐆𝐖𝐧_R+𝐧_D are, respectively, the equivalent channel matrix and the compound noise vector. The capacity of the equivalent channel matrix 𝐇_eq from the source to the destination is represented as C_SD𝐇_eq=12log₂det𝐈_𝑁D+𝑃_s𝑁_s𝐇_eq𝐇^H_eq𝚼⁻¹, (11) where 𝚼=𝐸{𝐧_eq𝐧^H_eq}=𝜎²_𝑛D𝐈_𝑁D+𝜎²_𝑛R𝐆𝐖(𝐆𝐖)^H denotes the covariance matrix of the compound noise vector 𝐧_eq and the factor 1/2 is multiplied in (11) due to the use of a two-hop relaying transmission.

3. Modified Greedy Capacity Maximization Algorithm

An efficient two-layered greedy capacity maximization (GCM) antenna selection technique is proposed for a dual-hop non-regenerative MIMO-AF multiple-relay system with multiple antennas equipped at each node in correlated fading channels. This modified GCM (MGCM) antenna selection strategy is employed to select a subset of antenna pairs from multiple relays based on the channel capacity maximization criterion subject to an optimal power allocation across the activated antenna pairs. The use of the proposed two-stage MGCM antenna selection is capable of trading in a relatively small fraction of loss in system performance, as compared with existing antenna selection schemes for half-duplex AF-MIMO relay networks, for a considerable reduction in system hardware and signal processing complexity. Let 𝐇_𝑖=[𝐡^⊤_{(𝑅(1),𝑏(1))},𝐡^⊤_{(𝑅(2),𝑏(2))},…,𝐡^⊤_{(𝑅(𝑖),𝑏(𝑖))}]^⊤ denote the selected partial backward channel matrix including the 𝑖 selected antennas. At the (𝑖+1)th step, the capacity of the channel matrix 𝐇_𝑖+1 with the inclusion of the channel vector 𝐡_{(𝑅𝑘,𝑚)} from the source node to the 𝑚th candidate antenna of the 𝑅_𝑘th relay can be reexpressed as 𝐶_SR𝐇_𝑖+1=log₂det𝐈_𝑖+𝑃_s𝑁_s𝐇_𝑖𝐇^H_𝑖+log₂𝑃det1+^s𝑁_s𝐡_{(𝑅𝑘,𝑚)}−𝑃_s𝑁_s𝐡_{(𝑅𝑘,𝑚)}×𝐇^H_𝑖𝐈_𝑖+𝑃_s𝑁_s𝐇_𝑖𝐇^H_𝑖⁻¹𝐇_𝑖𝐡^H_{𝑅𝑘,𝑚}. (12) The (𝑖+1)th antenna is selected based on the maximization of (13) as follows: (𝑅(𝑖+1),𝑏(𝑖+1))=argmax_{𝑅𝑘,𝑚}×𝑃1+^s𝑁_s𝐡_{(𝑅𝑘,𝑚)}−𝑃_s𝑁_s𝐡_{(𝑅𝑘,𝑚)}×𝐇^H_𝑖𝐈_𝑖+𝑃_s𝑁_s𝐇_𝑖𝐇^H_𝑖⁻¹𝐇_𝑖𝐡^H_{𝑅𝑘,𝑚}. (13) This selection procedure is repeated until a total of 𝐾 backward antennas are determined. A total of 𝑁𝑁_R possible antenna choices need to be examined at the first stage. At the (𝑖+1)th step, the equivalent channel capacity resulted from the antenna pair (𝑅_𝑘,𝑚,𝑛) can be expressed as C_SD𝐇_𝑖+1=12log₂det𝚼_𝑖+𝑤²_{𝑅𝑘,𝑚,𝑛}𝐠_{(𝑅𝑘,𝑛)}𝐠^H_{𝑅𝑘,𝑛}+𝑃_s𝑁_s𝐐_𝑖𝐐^H_𝑖−12log₂det𝚼_𝑖+𝑤²_{𝑅𝑘,𝑚,𝑛}𝐠_{(𝑅𝑘,𝑛)}𝐠^H_{𝑅𝑘,𝑛}, (14) where 𝐐_𝑖=𝐆_𝑖𝐖_𝑖𝐇_𝑖+𝑤²_{(𝑅𝑘,𝑚,𝑛)}𝐠_{(𝑅𝑘,𝑛)}𝐡_{(𝑅𝑘,𝑚)}. According to (15), the antenna pair that is capable of maximizing the capacity C_SD is selected. For simpler selection, we rewrite the selection criterion and select the antenna pair which satisfies 𝑅(𝑖+1),𝑏(𝑖+1),𝑓(𝑖+1)=argmax_{𝑅𝑘,𝑚,𝑛}det×𝐀_𝑖+⁴_𝑎=1𝑠_𝑎𝐪_𝑎𝐪^H_𝑎−det𝚼_𝑖−1+𝑤²_{𝑅𝑘,𝑚,𝑛}𝐠^H_{𝑅𝑘,𝑛}𝚼_𝑖⁻¹𝐠_{(𝑅𝑘,𝑛)}, (15) where 𝐪₁=𝐮_{(𝑖,𝑅𝑘,𝑚,𝑛)}+𝐠_{(𝑅𝑘,𝑛)}, 𝐪₂=𝐠_{(𝑅𝑘,𝑛)}, 𝐪₃=𝐮_{(𝑖,𝑅𝑘,𝑚,𝑛)}, 𝐪₄=𝑤²_{(𝑅𝑘,𝑚,𝑛)}((𝑃_s/𝑁_s)‖𝐡_{(𝑅𝑘,𝑚)}‖²+1)𝐠_{(𝑅𝑘,𝑛)}, [𝑠₁,𝑠₂,𝑠₃,𝑠₄]=[1,−1,−1,1], 𝐮_{(𝑖,𝑅𝑘,𝑚,𝑛)}=(𝑃_s/𝑁_s)𝐆_𝑖𝐖_𝑖𝐇_𝑖𝑤²_{(𝑅𝑘,𝑚,𝑛)}𝐡^H_{(𝑅𝑘,𝑚)}, and 𝐀_𝑖=𝚼_𝑖+(𝑃_s/𝑁_s)𝐆_𝑖𝐖_𝑖𝐇²_𝑖𝐖^H_𝑖𝐆^H_𝑖. Note that only 𝐾𝑁_R antenna combinations have to be evaluated at the second stage. It is readily known that the GCM technique proposed in [17] is able to produce an optimal set of active antennas by performing an exhaustive search from a pool of 𝑁𝑁²_R antenna combinations to maximize (11). However, the computational cost incurred by such a brute-force search is very expensive and thus leads to a limited usage in practice. This computational requirement is, in general, prohibitive for low-power and cost-effective MIMO-relaying networks. Fortunately, the MGCM approach obviates the necessity of an exhaustive relay and antenna joint search and thus a substantial saving in complexity can be achieved. The fact that in contrast to the GCM approach, the antenna combinations to be treated in the MGCM algorithm are reduced greatly from 𝑁𝑁²_R to (𝑁+𝐾)𝑁_R should be pointed out.

4. Optimal Power Allocation

In this section, both the optimal and equal-power allocation constraints across the activated 𝐾 antenna pairs in the half-duplex AF-MIMO multirelay system are considered in what follows. To obtain the optimal-power allocation (OPA) of the relay-weighting matrix 𝐖, the singular value decomposition (SVD) technique is employed to perform factorizations of both the compound backward and forward channel matrices, that is, 𝐇=𝐔_HΛ_H𝐕^H_H,𝐆=𝐔_GΛ_G𝐕^H_G, (16) where 𝐔_H∈𝐶^𝐾×𝐾 and 𝐔_G∈𝐶^𝑁D×𝑁D denote the left singular matrices of compound matrices 𝐇 and 𝐆, respectively. 𝐕_H∈𝐶^𝑁S×𝑁S and 𝐕_G∈𝐶^𝐾×𝐾 indicate the right singular matrices of compound matrices 𝐇 and 𝐆, respectively. Λ_H and Λ_G are diagonal singular-value matrices of compound matrices 𝐇 and 𝐆, respectively. 𝜎_H,𝑖, 𝑖=1,2,…,𝐾, and 𝜎_G,𝑖, 𝑖=1,2,…,𝐾, represent, respectively, the 𝑖th diagonal elements of Λ_H and Λ_G, which are arranged in a decreasing order. Note that the condition of 𝐾≤min{𝑁_S,𝑁_D} is preserved in what follows. According to [25], the SNR at the destination node is readily shown as SNR=tr𝐆𝐖𝐇𝜎²_x𝐇^H𝐖^H𝐆^Htr𝐆𝐖𝜎²_𝑛R𝐖^H𝐆^H+𝜎²_𝑛D. (17) The SNR optimization problem under the total transmit power constraint on 𝐾-activated antenna pairs can be equivalently formulated as min_{𝑤2𝑖,𝑖}−^𝐾_𝑖=1𝜎²_G,𝑖𝜎²_H,𝑖𝜎²_x𝑤²_𝑖,𝑖𝜎²_G,𝑖𝜎²_𝑛R𝑤²_𝑖,𝑖+𝜎²_𝑛Ds.t.^𝐾_𝑖=1𝑤²_𝑖,𝑖𝜎²_x𝜎²_H,𝑖+𝜎²_𝑛R≤𝑃_R (18) The optimum values of 𝑤²_𝑖,𝑖,𝑖=1,2,…,𝐾, can be attained based on the Karush-Kuhn-Tucker (KKT) optimality conditions [26], which is given by 𝑤²_𝑖,𝑖=𝜎_G,𝑖𝜎_H,𝑖𝜎_x𝜎_𝑛R/𝜆^1/2𝜎²_x𝜎²_H,𝑖+𝜎²_𝑛R−𝜎²_𝑛D𝜎²_G,𝑖𝜎²_𝑛R+,𝑖=1,2,…,𝐾, (19) where the Lagrangian multiplier 𝜆 is chosen to meet the total transmit-power constraint over 𝐾 antenna pairs in (18), which is determined by means of an iterative water-filling technique as follows: 𝜆=^𝐾_𝑖=1𝜎_G,𝑖𝜎_H,𝑖𝜎_x𝜎_𝑛D𝜎²_H,𝑖𝜎²_x−𝜎²_𝑛D𝑃_R𝜎²_G,𝑖𝜎²_𝑛R𝜎²_x𝜎²_H,𝑖+𝜎²_𝑛R². (20)

When the equal-power allocation (EPA) constraints across the activated 𝐾 antenna pairs is considered, the values of 𝑤²_𝑖,𝑖,𝑖=1,2,…,𝐾, can be attained as 𝑤²_𝑖,𝑖=𝑃_R𝐾𝜎²_x𝐡_{(𝑅(𝑖),𝑏(𝑖))}²_F+𝜎²_𝑛R,𝑖=1,2,…,𝐾. (21)

5. Destination Antenna Subset Selection Criteria

To alleviate hardware cost and implementation complexity of a dual-hop AF-wireless MIMO multiple-relay system, antenna selection schemes are also performed at the destination node. Through the use of (11), the capacity of the equivalent channel matrix 𝐇_eq from the source to the destination can be reformulated as C_SD𝐇_eq=−12log₂det𝐈_𝑁D+𝑃_s𝑁_s𝐇_eq𝐇^H_eq𝚼⁻¹⁻¹. (22) The MSE for the 𝑗th detection stage of the dual-hop AF-MIMO multiple-relay system can be expressed as [27] MSE_𝑘𝑗=𝑃_s𝑁_s𝐈_𝑁D+𝐇_eq𝑗𝐇^H_eq𝑗𝚼⁻¹_eq𝑗⁻¹_{𝑘𝑗,𝑘𝑗}, (23) where the matrix 𝐇_eq𝑗 in (23) is obtained by setting all the 𝑘₁th,𝑘₂th,…,𝑘_(𝑗−1)th columns of 𝐇_eq to zeros. By recursively substituting the MSE_𝑘𝑗 of (23) at stage 𝑘_𝑗, 𝑘_𝑗=1,2,…,𝑁_D, into (22), the capacity of the dual-hop AF-MIMO multiple-relay system at the final detection stage 𝑁_D can be reexpressed as [27] C_SD𝐇_eq𝑃=−^s2𝑁_slog₂^𝑖_𝑘𝑗=1MSE_𝑘𝑗×det𝐈_𝑁D−𝑖+𝐇_eq𝑖𝐇^H_eq𝑖𝚼⁻¹_eq𝑖⁻¹𝑃=−^s2𝑁_slog₂^𝑁D_𝑘𝑗=1MSE_𝑘𝑗. (24) Thus, according to (24), to maximize the capacity of the selected antenna subset 𝑐_𝑙 for the dual-hop AF-MIMO multiple-relay system is equivalent to minimize the products of MSEs at all stages (i.e., MSE_𝑘𝑗,𝑘_𝑗=1,2,…,𝑁_D), for 𝑐_𝑙. In what follows, let 𝑐_𝑙, for 𝑙=1,2,…,^𝑁D_𝐿, denote all possible combinations of 𝐿 antennas out of the complete set of 𝑁_D receive antennas. In the sequel, the symbol detection is performed by the V-BLAST MMSE detection scheme.

6. Existing Antenna Subset Selection Criteria

In this section, two categories of joint relay-and-antenna selection algorithms, namely, distributed orthogonal relay selection (DORS) [11] and fast antenna subset selection algorithm (FASSA) [13], are briefly reviewed.

6.1. Distributed Orthogonal Relay Selection

The first relay-based antenna selection algorithm is called “distributed orthogonal relay selection” (DORS) [11]. The authors in [11] propose a selection process that combines projection and harmonic mean to select antenna pairs from multiple-relay nodes. In order to maximize the channel capacity, two selection criteria play significant roles. At first, the equivalent channel matrix has to be made as diagonal as possible to minimize the interferences among the parallel data streams. Secondly, the equivalent channel gains need to be as large as possible to maximize the SNR per data stream. Let 𝐇_𝑖=[𝐡_{(𝑅(1),𝑏(1))},𝐡_{(𝑅(2),𝑏(2))},…,𝐡_{(𝑅(𝑖),𝑏(𝑖))}]^⊤ and 𝐆_𝑖=[𝐠_{(𝑅(1),𝑓(1))},𝐠_{(𝑅(2),𝑓(2))},…,𝐠_{(𝑅(𝑖),𝑓(𝑖))}] denote the backward and forward channel matrices including the 𝑖-selected antennas-respectively, and the index (𝑅_𝑘,𝑚,𝑛) represents the 𝑚th backward candidate antenna and the 𝑛th forward candidate antenna at the 𝑅_𝑘th relay. The (𝑖+1)th antenna pair is determined by 𝑅(𝑖+1),𝑏(𝑖+1),𝑓(𝑖+1)=argmax_{𝑅𝑘,𝑚,𝑛}𝑓𝐡^⟂_{𝑅𝑘,𝑚F},𝐠^⟂_{𝑅𝑘,𝑛F}, (25) where 𝑓(‖𝐚‖_F,‖𝐛‖_F) denotes a harmonic mean of vectors 𝐚 and 𝐛 given by 𝑓(‖𝐚‖_F,‖𝐛‖_F)=(2‖𝐚‖_F⋅‖𝐛‖_F)/(‖𝐚‖_F+‖𝐛‖_F). Additionally, the projection vectors 𝐡^⟂_{(𝑅𝑘,𝑚)} and 𝐠^⟂_{(𝑅𝑘,𝑛)} are derived by means of Gram-Schmidt (GS) orthogonalization. The backward and forward candidate antenna vectors are projected into the spaces spanned by 𝐇_𝑖 and 𝐆_𝑖, respectively, which are given by 𝐡^⟂_{𝑅𝑘,𝑚}=𝐡_{(𝑅𝑘,𝑚)}−^𝑖_𝑗=1𝐡_{(𝑅𝑘,𝑚)}𝐡^H_{(𝑅(𝑗),𝑏(𝑗))}𝐡_{(𝑅(𝑗),𝑏(𝑗))}²_F𝐡_{(𝑅(𝑗),𝑏(𝑗))}𝐠^⟂_{𝑅𝑘,𝑛}=𝐠_{(𝑅𝑘,𝑛)}−^𝑖_𝑗=1𝐠_{(𝑅𝑘,𝑛)}𝐠^H_{(𝑅(𝑗),𝑓(𝑗))}𝐠_{(𝑅(𝑗),𝑓(𝑗))}²_F𝐠_{(𝑅(𝑗),𝑓(𝑗)).} (26) The DORS antenna selection algorithm runs until a subset of 𝐾 antennas are selected.

6.2. Modified Fast Antenna Subset Selection Algorithm

In [13], a fast antenna subset selection algorithm (FASSA) is introduced for a two-hop MIMO-AF relay system to avoid the intensive complexity of an exhaustive search algorithm and make antenna subset selection practical for applications. The FASSA algorithm for MIMO-AF relay attempts to maximize a lower bound of the capacity instead of the exact capacity. According to [13], the capacity C_SD in (11) can be further simplified and lower bounded by C_SD=12log₂det𝐈_K+𝑃_s𝑁_s𝐇𝐇^H𝐆^H𝐆𝐈_K+𝜌𝐆^H𝐆⁻¹=12log₂det𝐈_𝐾+𝑃_s𝑁_s𝐇𝐇^H𝐆^H𝐆×𝐈_K−𝐈_K+𝜌𝐆^H𝐆⁻¹≥12log₂𝑃_s/𝑁_s^𝐾det𝐇𝐇^Hdet𝐆^H𝐆∏^K_𝑎=11/𝜌+𝐠_{(𝑅(𝑎),𝑓(𝑎))}², (27)(28)(29) where the coefficient 𝜌 keeps the total transmit power of the 𝐿_R-selected relays equal to 𝑃_R. It should be noticed that the lower bound of the capacity C_SD is derived based on the Hadamard inequality [28]. In the (𝑛+1)th step of the FASSA algorithm, the index set of (𝑅(𝑖+1),𝑏(𝑖+1),𝑓(𝑖+1)) is determined in order to maximize the lower bound given in (29), which satisfies 𝑅(𝑖+1),𝑏(𝑖+1),𝑓(𝑖+1)=argmax_{𝑅𝑘,𝑚,𝑛}det𝐇∪𝐡_{(𝑅𝑘,𝑚)}𝐇∪𝐡_{(𝑅𝑘,𝑚)}^H⋅det𝐆∪𝐠_{(𝑅𝑘,𝑛)}^H𝐆∪𝐠_{(𝑅𝑘,𝑛)}∏^𝑖+1_𝑎=11/𝜌+𝐠_{(𝑅(𝑎),𝑓(𝑎))}². (30) However, the FASSA scheme in [13] ignores the effects of relay weighting matrix when executing antenna selection so that the accuracy of channel capacity is affected seriously. Thus, by taking the relay weighting matrix into account, the modified FASSA (MFASSA) algorithm can be operated in more comprehensive and accurate way. Let 𝐖_𝑖 denote the compound weighting matrix at the 𝑖th step and then be substituted into (29). After the completeness of the 𝑖th antenna selection, the lower bound for C_SD shown in (29) can be reformulated as C_SD≥12log₂𝑃_s/𝑁_s^𝑖det𝐇_𝑖𝐇^H_𝑖det𝐆_𝑖𝐖_𝑖^H𝐆_𝑖𝐖_𝑖∏^𝑖_𝑎=1𝜎²_𝑛D𝐈_𝑖+𝜎²_𝑛R𝐆_𝑖𝐖_𝑖^H𝐆_𝑖𝐖_{𝑖𝑎,𝑎}. (31)

7. Numerical Results

In this section, computer simulations are conducted to demonstrate the system-capacity performance of the proposed antenna-pair selection scheme for a dual-hop AF-MIMO multiple-relay system in correlated channels. Let the notation of [𝑁_S,𝑁_R,𝑁_D,𝑁,𝐾] denote a subset of 𝐾 antenna pairs selected from an AF-mode MIMO 𝑁-relay system equipped with 𝑁_S, 𝑁_R, and 𝑁_D antennas, respectively, at the source, each relay, and the destination. A Kronecker correlated channel model [22] with the use of 𝑑_S=𝑑_R𝑖=𝑑_D and 𝜃_S=𝜃_R𝑖=𝜃_D is considered in the correlated MIMO channel. Here 𝑑 denotes the antenna spacing and 𝜃 stands for the angular spread. Throughout simulations, both the total transmit power of 𝐾-activated antenna pairs selected from 𝐿_R relays (i.e., 𝐿_R≤𝐾) 𝑃_R and the power constraint at the source 𝑃_S are set to 10 dB. In addition, it is assumed that the full channel state information (CSI) of all channel links is available and the perfect synchronization can be achieved. Here, 10000 Monte Carlo runs are conducted for each relay deployment.

In Figure 2, the system-capacity comparisons of various antenna subset selection schemes are provided for an AF-MIMO multiple-relay system with the use of [𝑁_S,𝑁_R,𝑁_D,𝐾]=[10,2,10,4] and two different spatial-correlation scenarios, that is, 𝑑_S=𝑑_R𝑖=𝑑_D=0.8𝜆 and 𝜃_S=𝜃_R𝑖=𝜃_D=100^∘ in (a) and 𝑑_S=𝑑_R𝑖=𝑑_D=0.4𝜆 and 𝜃_S=𝜃_R𝑖=𝜃_D=40^∘ in (b), in terms of the number of candidate relays. It is observed from both figures that the system-capacity performance degrades significantly when all antenna-pair selection schemes operate in a high spatial-correlation channel. Among those five antenna selection algorithms, the GCM method accomplishes the best capacity performance at the expense of the most heavy computational complexity. Remarkably, it is evident that the proposed MGCM antenna selection scheme is capable of achieving similar capacity level to the GCM algorithm at a much lower complexity cost. It is also seen from Figure 2 that the proposed MGCM method produces uniformly better performance than those of the FASSA, the MFASSA, and the DORS techniques. Moreover, the DORS scheme produces a worst capacity performance among those antenna selection schemes due to ignorance of the effects of the matrix 𝚼 during the antenna selection process. Furthermore, it should be emphasized that all antenna-pair selection algorithms with the aid of an optimal power allocation outperform significantly the ones with an equal power allocation in system capacity performance no matter under the low or high correlated channel.

(a)

(b)

Figure 2 

System-capacity performance comparisons of various relay antenna selection algorithms in terms of the number of candidate relays N for an AF-MIMO multiple-relay system with the use of [𝑁S,  𝑁R,𝑁D,   K] = [10,2,10,4] and the (a) low and (b) high spatial-correlation scenarios under either an optimal or equal-power allocation.

Figure 3 evaluates system-capacity comparisons of various destination antenna selection algorithms for an AF-MIMO multiple-relay system with [𝑁_S,𝑁_R,𝑁_D,𝑁,𝐾,𝐿]=[10,2,10,10,4,4] and a fixed relay antenna-pair selection process in the low (a) and high (b) spatial-correlation channels in terms of SNR. In the figure, “full complexity” means that all the available receive antennas at the destination node are employed for demodulation in an AF-MIMO multiple-relay system (i.e., the full set of antennas at the destination node). “Random” denotes the receive antenna subset is chosen randomly from an entire set of receive antennas. “MinmaxMSE” indicates that the desired receive antennas are selected based on the performance-based criterion in Section 5 which is determined by the calculated MSEs in (24). “Optimal” represents that an exhaustive search over all possible receive-antenna combinations is performed based on the capacity-based criterion. As the results in Figure 2, the system-capacity performance degrades significantly when all destination antenna selection schemes operate in a high spatial-correlation channel. In addition, from both figures, the proposed antenna selection algorithm at destination can maintain good performance no matter in high or low correlation channel. The fact that with the use of moderate antenna selection criteria, the selected receive antenna subset is able to produce an improved channel capacity over the random antenna selection should be emphasized.

(a)

(b)

Figure 3 

System-capacity performance comparisons of various destination antenna selection algorithms in terms of SNR for an AF-MIMO multiple-relay system with [𝑁S,𝑁R,𝑁D,N,K,L] = [10,2,10,10,4,4] and a fixed relay antenna-pair selection process in the (a) low and (b) high spatial-correlation channels.

8. Conclusions

In this paper, a reduced-complexity layered MGCM antenna subset selection scheme is proposed for a dual-hop AF-MIMO multiple-relay system over correlated fading channels. From simulation results, it is obvious that a considerable reduction in system hardware cost and signal processing complexity is achieved by means of the modified GCM architecture under either optimal or equal-power allocation. In addition, the modified GCM method provides similar capacity performance to the GCM scheme and achieves better capacity performance than those of the FASSA, the MFASSA, and the DORS techniques. As a consequence, this computationally-efficient modified GCM-based antenna-pair selection strategy is suitable for applications in the non-regenerative AF-MIMO multiple-relay wireless systems.

Acknowledgment

This work was supported by Taiwan National Science Council under Grant NSC98-2221-E-194-018-MY2.