Sparse Channel Estimation for MIMO-OFDM Two-Way Relay Network with Compressed Sensing

Abstract

Accurate channel impulse response (CIR) is required for equalization and can help improve communication service quality in next-generation wireless communication systems. An example of an advanced system is amplify-and-forward multiple-input multiple-output two-way relay network, which is modulated by orthogonal frequency-division multiplexing. Linear channel estimation methods, for example, least squares and expectation conditional maximization, have been proposed previously for the system. However, these methods do not take advantage of channel sparsity, and they decrease estimation performance. We propose a sparse channel estimation scheme, which is different from linear methods, at end users under the relay channel to enable us to exploit sparsity. First, we formulate the sparse channel estimation problem as a compressed sensing problem by using sparse decomposition theory. Second, the CIR is reconstructed by CoSaMP and OMP algorithms. Finally, computer simulations are conducted to confirm the superiority of the proposed methods over traditional linear channel estimation methods.

1. Introduction

Two-way relay network (TWRN) has attracted great attention because it can improve spectral efficiency unlike a one-way relay network [1]. In a typical TWRN, two terminals, 𝕋₁ and 𝕋₂, exchange information through the assistance of an intermediate relay ℝ via amplify-and-forward (AF). The TWRN requires two time slots to exchange their information. At the first time slot, both terminals send their data to the relay simultaneously, and then ℝ amplifies the superimposed signal and broadcasts it to the end terminals during the second time slot. After receiving the signal from ℝ, the end nodes remove their own interference and perform a coherent detection process to recover the data transmitted from the other node. Compared with the basic single-antenna relay channel, the multiple-input multiple-output (MIMO) relay channel provides the high capacity of MIMO communication with the coverage extension capability of relay nodes. MIMO relay channel is one of the most promising solutions as it increases channel capacity and network reliability and effectively combats multipath fading. Furthermore, the orthogonal frequency-division multiplexing (OFDM) modulation technique provides efficient bandwidth utilization and robustness against time-dispersive channels. Therefore, the MIMO-OFDM-TWRN is a promising technique for next-generation wireless communication systems.

The MIMO-OFDM-TWRN system faces more challenges because channel estimation is required not only for data detection but also for self-data cancellation at the two terminals. Linear channel estimation methods have been proposed for MIMO-OFDM-TWRN. In [2], channel estimation by using least squares (LS) was proposed, and the block-based training design method was considered to achieve the optimal linear channel estimator. In addition, channel information estimation which uses the expectation conditional maximization (ECM) algorithm was proposed in [3]. However, both proposed methods do not take advantage of channel sparsity, and they result in performance loss of MIMO-OFDM-TWRN. In the development of channel modeling, numerous measurements demonstrated that multipath wireless channels tended to exhibit a cluster or sparse structure in which majority of the channel taps end up being either zero or below the noise floor, especially when operating at large bandwidths and signaling durations and/or with numbers of antenna elements [4–6]. In addition, with the theoretical development of compressed sensing (CS) [7], numerous researchers have proposed sparse channel estimation methods for point-to-point (P2P) communication systems, including single-antenna [8] or multiple-antenna [9] systems. Sparse channel estimation methods [10, 11] on cooperative networks have also been investigated. All channel estimation methods are limited to either single-antenna or multiple-antenna systems for traditional P2P transmission and single-antenna relay channel. To the best of our knowledge, corresponding work has not been conducted for MIMO-OFDM-TWRN.

In this study, we focus on the TWRN with multiple antennas at relay ℝ and both end nodes, which is modulated by OFDM. We introduce the MIMO-OFDM-TWRN channel model and formulate composite channels estimation as a compressed sensing problem by using sparse decomposition theory. Then, the composite channel is reconstructed by CoSaMP [12] and OMP [13] algorithms. Finally, we verify the proposed methods via computer simulations. This study contributes to the field by introducing a sparse channel estimation technique with compressed sensing for MIMO-OFDM-TWRN and the use of the sparse structure information in the CIR by the end users.

The remainder of this paper is organized as follows. Section 2 describes the system model. Section 3 provides details on the proposed algorithm. The simulation results are reported in Section 4, and the conclusions are provided in Section 5.

2. System Model

2.1. Relay Transmission Model

Figure 1 shows a typical TWRN with two source nodes, 𝕋₁ and 𝕋₂, and one relay node, ℝ. The two terminal users exchange information with the assistance of the relay node. 𝕋₁ and 𝕋₂ each have 𝑁_𝑇 antennas, and ℝ has 𝑁_𝑅 antennas. The bidirectional communication is performed in two phases, as shown in Figure 1. In phase I, 𝕋₁ and 𝕋₂ send their signal simultaneously to the relay node ℝ. Then, the relay amplifies the received signals and broadcasts them to the source nodes 𝕋₁ and 𝕋₂ in phase II.

Figure 1 

MIMO-OFDM-TWRN: (1) multiaccess stage, (2) broadcasting stage.

The channel between the 𝑚th antenna of 𝕋_𝑖 and the 𝑟th antenna of ℝ is assumed to be a quasi-static frequency-selective fading channel with an impulse response denoted by 𝐡^𝑖_𝑟,𝑚=[ℎ^𝑖_𝑟,𝑚(0),ℎ^𝑖_𝑟,𝑚(1),…,ℎ^𝑖_𝑟,𝑚(𝐿−1)]^𝑇∈𝐶^𝐿×1, where 𝑖=1,2 and 𝐿 is the channel length. All channels 𝐡^𝑖_𝑟,𝑚,(𝑖=1,2;𝑚=1,…,𝑁_𝑇;𝑟=1,…,𝑁_𝑅) are assumed as zero-mean circularly symmetric complex Gaussian random variables with variance (𝜎^𝑖_𝑟,𝑚)² and are independent of each other. The average transmission powers of 𝕋₁, 𝕋₂, and ℝ are 𝑃₁, 𝑃₂, and 𝑃_𝑅, respectively. For the time being, we assume perfect synchronization among three terminals.

2.2. Received Signals at the Relay Node and the End Users

The training signal vectors transmitted from the 𝑚th antenna of 𝕋₁ and 𝕋₂ are denoted by 𝑠^𝑖_𝑚. We assume that 𝐸{‖𝐬^𝑖_𝑚‖²}=𝑃_𝑖. In phase I, the signal vectors 𝐬¹_𝑚 and 𝐬²_𝑚 are processed by an inverse fast Fourier transform as 𝐬^𝑖_𝑚̃=𝐅^𝐻𝐬^𝑖_𝑚=[𝑠^𝑖_𝑚̃(0),𝑠^𝑖_𝑚̃(1),…,𝑠^𝑖_𝑚̃(𝑁−1)], where 𝐹 is the discrete Fourier transformation (DFT) matrix with the (𝑚,𝑛)th entity given by 𝐅=1/√𝑁𝑒^{−𝑗2𝜋𝑚𝑛/𝑁},(𝑚,𝑛=0,1,…,𝑁). To avoid inter-block interference, 𝐬^𝑖_𝑚̃ is the cyclic prefix (CP) of length 𝐿_𝑝 added before being transmitted, and 𝐿_𝑝 should satisfy 𝐿_𝑝≥𝐿−1. The received signal vector at the node ℝ after removing CP can be written as 𝐲_𝑅=𝐇¹𝑠¹̃+𝐇²𝑠²̃+𝐰, (1) where 𝐬¹̃=𝑠̃¹₁^𝑇,𝑠̃¹₂^𝑇,…𝑠̃¹_𝑁𝑇^𝑇𝑇,𝐬²̃=𝑠̃²₁^𝑇,𝑠̃²₂^𝑇,…𝑠̃²_𝑁𝑇^𝑇𝑇,𝐇¹=𝐇¹_1,1𝐇¹_1,2⋯𝐇¹_1,𝑁𝑇𝐇¹_2,1𝐇¹_2,2⋯𝐇¹_2,𝑁𝑇𝐇⋮⋮⋮⋮_1𝑁𝑅,1𝐇¹_𝑁𝑅,2⋯𝐇¹_{𝑁𝑅,𝑁𝑇},𝐇²=𝐇²_1,1𝐇²_1,2⋯𝐇²_1,𝑁𝑇𝐇²_2,1𝐇²_2,2⋯𝐇²_2,𝑁𝑇𝐇⋮⋮⋮⋮_2𝑁𝑅,1𝐇²_𝑁𝑅,2⋯𝐇²_{𝑁𝑅,𝑁𝑇},𝐰=𝐰^𝑇₁𝐰^𝑇₂⋯𝐰^𝑇_𝑁𝑅^𝑇. (2)

The matrix 𝐇^𝑖_𝑟,𝑚 is an 𝑁×𝑁 circulant matrix in which the first column takes the form 𝐡^𝑖_𝑟,𝑚=[ℎ^𝑖_𝑟,𝑚(0)ℎ^𝑖_𝑟,𝑚(1)⋯ℎ^𝑖_𝑟,𝑚(𝐿−1)0_{1×(𝑁−𝐿)}]^𝑇. The noise vector 𝐰 is a complex Gaussian random vector with zero mean and covariance matrix 𝐸{𝐰𝐰^𝐻}=𝑁₀𝐈_𝑁.

In phase II, the vector 𝑦_𝑅 is amplified by a real factor 𝛽, and 𝛽²𝐸{𝐲^𝐻𝐲}=𝑁_𝑅×𝑃_𝑅. The factor 𝛽 is given by 𝛽=𝑃_𝑅∑^𝑁𝑅_𝑟=1∑^𝑁𝑇_𝑚=1∑^𝐿−1_𝑙=0𝜎¹_𝑟,𝑚(𝑙)+𝜎²_𝑟,𝑚(𝑙)+𝑁₀, (3) where 𝜎^𝑖_𝑟,𝑚(𝑙)=𝐸[|𝐡^𝑖_𝑟,𝑚(𝑙)|²],𝑖=1,2.

The vector 𝛽𝐲_𝑅 is CP-added before being transmitted back to 𝕋₁ and 𝕋₂. Only the channel estimation problem at 𝕋₁ is considered in this study. A similar procedure can be applied at 𝕋₂. The received signal at 𝕋₁, which is from relay ℝ after removing CP, can be written as [3] 𝐲_𝑇1=𝛽𝐆𝐇¹𝑠¹̃+𝐇²𝑠²̃+𝐧_𝑇1, (4) where 𝐇𝐆=^11,1𝐇¹_2,1⋯𝐇¹_𝑁𝑅,1𝐇¹_1,2𝐇¹_2,2⋯𝐇¹_𝑁𝑅,2𝐇⋮⋮⋮⋮_11,𝑁𝑇𝐇¹_2,𝑁𝑇⋯𝐇¹_{𝑁𝑅,𝑁𝑇} (5) and 𝐧_𝑇1=𝛽𝐆𝐰+𝐧,𝐧=[𝑛^𝑇₁𝑛^𝑇₂⋯𝑛^𝑇_𝑁𝑅]^𝑇.

According to the matrix theory, matrices 𝐇^𝑖_𝑟,𝑚 can be decomposed as 𝐇^𝑖_𝑟,𝑚=𝐅^𝐻Λ^𝑖_𝑟,𝑚𝐅 [14], where Λ^𝑖_𝑟,𝑚=diag{𝐇^𝑖_𝑟,𝑚(0),𝐇^𝑖_𝑟,𝑚(1),…,𝐇^𝑖_𝑟,𝑚(𝑁−1)} and 𝐇^𝑖_𝑟,𝑚(𝑐)=∑^𝐿−1_𝑙=0𝐡^𝑖_𝑟,𝑚𝑒^{−𝑗(2𝜋𝑙𝑐/𝑁)},(𝑐=0,1,…,𝑁−1)𝛽𝐇¹_𝑟1,𝑚1𝐇¹_𝑟2,𝑚2=𝐅^𝐻𝛽Λ¹_𝑟1,𝑚1Λ¹_𝑟2,𝑚2𝐅,𝛽𝐇¹_𝑟1,𝑚1𝐇²_𝑟2,𝑚2=𝐅^𝐻𝛽Λ¹_𝑟1,𝑚1Λ²_𝑟2,𝑚2𝐅, (6) where 𝑟₁ and 𝑟₂=1,2,…,𝑁_𝑅, and 𝑚₁ and 𝑚₂=1,2,…,𝑁_𝑇. Equations (6) are two circulant matrices that have the first columns of [𝛽(𝐡¹_𝑟1𝑚1∗𝐡¹_𝑟2𝑚2)0_{1×(𝑁−2𝐿+1)}]^𝑇 and [𝛽(𝐡¹_𝑟1𝑚1∗𝐡²_𝑟2𝑚2)0_{1×(𝑁−2𝐿+1)}]^𝑇. Thus, by normalizing DFT of 𝐲_𝑇1, system model (4) can be rewritten as 𝐲₁=(𝐈⊗𝐅)𝑦_𝑇1=^𝑁𝑅𝑁_𝑟=1^𝑇_𝑚=1𝛽Λ¹_𝑟1,𝑚1Λ¹_𝑟2,𝑚2𝑠¹+Λ²_𝑟2,𝑚2𝑠²+𝐅𝑛_𝑇1. (7)

From (7), we define composite channels 𝑔 and 𝑞, which are given, respectively, as 𝐠=𝛽𝐡¹_𝑟,𝑚⊗𝐡¹_𝑟,𝑛,𝛽Λ¹_𝑟1,𝑚1Λ¹_𝑟2,𝑚2(),𝐡=diag𝐖𝐠𝐪=𝛽¹_𝑟,𝑚⊗𝐡²_𝑟,𝑛,𝛽Λ¹_𝑟1,𝑚1Λ²_𝑟2,𝑚2=diag(𝐖𝐪). (8) Then, (7) can be expressed as 𝐲₁=𝐒𝐤+𝐅𝐧_𝑇1, (9) where 𝐒=[𝐬¹𝐬²],𝐤=[𝐠^𝑇𝐪^𝑇], and 𝐖 is a matrix which takes the first (2𝐿−1) columns of √𝑁𝐅.

3. Sparse Channel Estimation

3.1. Overview of Compressed Sensing

Compressed sensing (CS) describes a new signal acquisition theory in which sparse high dimensional vectors can be accurately recovered from a small number of linear observations. CS has been applied in various areas, such as imaging, radar, speech recognition, and data acquisition. In communications, an immediate application of CS is wireless sparse multipath channel estimation. Detailed descriptions can be found in [7].

In this paper, we consider the linear model as (9). According to the CS, if an unknown signal vector satisfies the sparse or approximate sparse requirements, the conditions under which CS succeeds depends on the structure of the measurement matrix 𝐒. Thus, these kinds of unknown signals can be robustly reconstructed from observation signal 𝐲₁. However, the sparsest solution is always a nondeterministic polynomial-time hard (NP-hard) problem. According to recent theoretical results, the observation signal can be used to efficiently recover any “sparse enough” signal provided that the matrix 𝐒 satisfies the so-called restricted isometric property (RIP) [15, 16]. We suppose that 𝐒 is a 𝑛×𝑝 complex-valued measurement matrix that has unit ℓ₂-norm columns. The 𝐒 satisfies the RIP of order 𝑑 with parameter 𝛿_𝑑∈(0,1), which can satisfy the inequality 1−𝛿_𝑑‖𝐤‖²₂≤‖𝐒𝐤‖²≤1+𝛿_𝑑‖𝐤‖²₂, (10) where ‖𝐤‖²₂ denotes the ℓ₂-norm, which is given by ‖𝐤‖²₂=∑|𝐡^𝑖_𝑟,𝑚|². If (10) is satisfied, the training sequence is said to satisfy the RIP of order 𝑑, and the accurate channel estimator with high probability can be obtained by using CS methods. Although verifying whether a given matrix satisfies this condition is difficult, many matrices satisfy the restricted isometry constant (RIC) with high property and few measurements. In particular, it has been shown exponentially with high probability that the random Gaussian, Bernoulli, and partial Fourier matrices satisfy the RIC with a number of measurements that are nearly linear in the sparsity level.

3.2. Compressed Channel Estimation (CCE)

Since the channel impulse responses 𝐡^𝑖_𝑟,𝑚 are sparse enough, their cooperation convoluted channel 𝑔 and 𝑞 have been verified to be sparse or approximate sparse [17]. In this case, we use two greedy algorithms, that is, orthogonal matching pursuit (OMP) [13] and compressive sampling matching pursuit (CoSaMP) [12], which select each dominant coefficient in channel through iteration. We also present the LS channel estimator for comparison. The LS estimator (known position) is given by numerous practical algorithms for channel estimation. The LS estimator is expressed as 𝐒𝐤=^†𝑇𝐲₁,(𝐤),,𝑇⊆supp0,others (11) where supp(𝐤) denotes the nonzero taps that support the channel vector 𝐤, 𝐒_𝑇 is the submatrix constructed from the columns of 𝐒, and 𝑇 denotes the selected subcolumns which correspond to the nonzero index set of the convoluted channel vector ℎ. The mean square error (MSE) of the LS estimator 𝐤 is given by 𝐤≜𝐒^†𝐲₁=𝐤+𝐒^†𝐅𝐧_𝑇, (12) where 𝐒^† is the pseudo-inverse of 𝐒 and is given by 𝐒^†=(𝐒^𝐻𝐒)⁻¹𝐒^𝐻. The MSE of the LS channel estimate is given by MSE=𝐸{‖𝐤−𝐤‖²}=𝐸{‖𝐒^†𝐅𝐧_𝑇‖²}, which can also be written as MSE𝐤=𝜎²_𝑛tr𝐒^†𝐒⁻¹. (13)

By utilizing CS recovery algorithms for compressive channel estimation, we propose CCE-OMP and CCE-CoSaMP. The two estimation methods for convoluted channels are described as follows.

3.2.1. CoSaMP Estimator 𝐤_CoSaMP

Given 𝑦₁, 𝐹 and 𝑊, and training signal matrix 𝐒=diag(𝐅𝐬^𝑖)𝐖, the maximum number of dominant channel coefficients is assumed as 𝑑. The CCE-CoSaMP is performed as follows.

Initialization. We set the nonzero coefficient index as 𝑇₀=∅, the residual estimation error 𝐫₀=𝐲₁, and the initialize iteration counter as 𝑙=1.

Identification. We select a column index 𝑛_𝑙 of 𝐒 that is most correlated with the residual 𝑛_𝑙=𝑟_𝑙−1,𝐒_𝑛,𝑇_𝑙=𝑇_𝑙−1∪𝑛_𝑙. (14) We use the LS method to calculate a channel estimator as 𝐓_LS=argmin‖𝐲₁−𝐒𝐤‖ and select a maximum of 𝑇 dominant taps denoted by 𝐤_LS. The positions of the selected dominant taps in this substep are denoted by 𝑇_LS.

Merge. The positions of the dominant taps are merged by 𝑇_𝑙=𝑇_LS∪𝑇_𝑙.

Estimation. We compute the best coefficient for approximating the channel vector with chosen columns 𝐤_𝑙=argmin_𝑘𝐲−𝐒_𝑇𝑙𝐤₂. (15)Pruning. We select the 𝑇_𝑙 largest channel coefficients 𝑘_𝑙=[𝑘]_𝑑 (16) and replace the left taps 𝑇⧵𝑇_𝑙 by zero.

Iteration. We update the estimation error 𝐫_𝑙=𝐲−𝐗_𝑇𝑙𝐤_𝑙. (17) We increase the iteration counter 𝑘. We repeat (14) to (17) until the stopping criterion holds, and then set 𝐤_CoSaMP=𝐤_𝑙.

3.2.2. OMP Estimator 𝐤_OMP

Given the received signal 𝐲_𝟏, 𝐖, 𝐅, and 𝐬^𝑖, the CCE-OMP estimator runs as follows.

Initialize. We set the nonzero coefficient index as 𝑇₀=∅, the residual estimation error 𝐫₀=𝐲₁, and the initialize iteration counter as 𝑙=1.

Identification. We select a column index 𝑛_𝑙 of 𝐬^𝑖 that is most correlated with the residual 𝑛_𝑙=𝐫_𝑙−1,𝐒_𝑛,𝑇_𝑙=T_𝑙−1∪𝑛_𝑙,𝐤_𝑙=argmin_𝑘𝐲₁−𝐒_𝑇𝑙. (18)Estimation. We compute the best coefficient for approximating the channel vector with chosen columns 𝐤_𝑙=argmin_𝑘𝐲₁−𝐒_𝑇𝑙𝐤₂. (19)Iteration. We update the estimation error 𝐫_𝑙=𝐲₁−𝐒_𝑇𝑙𝐤_𝑙. (20) We increase the iteration counter 𝑙. We repeat (18) to (20) until the stopping criterion holds, and then set 𝐤_OMP=𝐤_𝑙.

4. Simulation Results

In this section, we present the simulation results and analyze the performance of compressive channel estimation in a MIMO-OFDM two-way relay network. We compare the performance of the proposed estimators with that of an LS-based linear estimator and adopt 10,000 independent Monte Carlo runs for averaging. We consider the MIMO relay network with 𝑁_T=𝑁_𝑅=2 antennas; the number of carriers is 128. All channels have the same length (𝑃=32), and the positions of the nonzero channel taps are randomly generated. QPSK modulation is used. Transmit power is set as 𝑃₁=𝑃₂=𝑃, and AF relay power is set as 𝑃_𝑅=𝑃. The signal-to-noise ratio is defined as 𝑎 as 10log(𝑃/𝜎²_𝑛). When the number of nonzero taps in cooperation channels 𝐡^𝑖_𝑟,𝑚,(𝑖=1,2;𝑟=1,2,…,𝑁_𝑅;𝑚=1,2,…,𝑁_𝑇) is changed, the simulation results are shown in Figures 2 to 5.

Figure 2 

MSE performance of channel estimation at 𝕋1 for the case when 𝑑=2.

The channel estimators are evaluated via the average MSE, which is defined by ()=𝐤averageMSEΔ𝐤𝐤−²₂𝑀(),2𝐿−1 (21) where 𝐤 and 𝐤 denote the channel vector and its estimator, respectively, 𝑀 is the number of Monte Carlo runs, and (2𝐿−1) is the overall length of channel vector 𝑘. In Figure 2, the number of nonzero taps of 𝐡^𝑖_𝑟,𝑚,(𝑖=1,2) is set to 2, and the cooperation convoluted channel also has sparsity. Figure 2 shows that the performance of the proposed CCE methods is significantly better than that of the LS estimator and is close to the ideal LS estimator by using the known position of the channel. In Figure 3, the number of nonzero taps of 𝐡^𝑖_𝑟,𝑚,(𝑖=1,2) is set as 4.

Figure 3 

MSE performance of channel estimation at 𝕋1 when 𝑑=4.

A comparison between the simulation results in the two figures (Figures 2 and 3) shows that the proposed estimators can exploit the channel sparseness. Notably, if channels are dense rather than sparse, all proposed estimators will have the same performance as the LS estimator.

We also compare the performance of the proposed CCE-CoSaMP estimator with that of the ECM estimator algorithm in this section. Figures 4 and 5 show that when channel impulse responses 𝐡^𝑖_𝑟,𝑚 are sparse enough, the proposed CCE estimator performs significantly better than the ECM algorithm. However, as the number of nonzero taps of all the channels increases, the performance of CCE-CoSaMP is closer to that of the ECM algorithm.

Figure 4 

MSE performance of channel estimation at 𝕋1 for when channel impulse responses 𝐡𝑖𝑟,𝑚 are sparse enough (𝑑=2).

Figure 5 

MSE performance of channel estimation at 𝕋1 for when channel impulse responses 𝐡𝑖𝑟,𝑚 are sparse (𝑑=4).

5. Conclusion

This paper investigated the channel estimation problem in sparse multipath MIMO two-way relay networks that adopt the OFDM technique. To address the shortcomings of conventional linear channel estimation methods, we proposed compressed channel estimation methods for MIMO-OFDM two-way relay networks under the AF protocol. The sparseness of convoluted sparse channels was demonstrated by a measure function. The proposed methods exploited the sparsity in the MIMO TWRN channel. The simulation results confirmed the superior performance of the proposed method compared with conventional linear methods, for example, LS and ECM.

Acknowledgment

This study is funded by the National Natural Science Foundation of China (Grant nos. 61071175, 61202499, 61271421, and 61152004).