Sparse Channel Estimation for OFDM

Sparse Channel Estimation for OFDM Transmission over Two-Way Relay Networks Peng Cheng†⇤ , Lin Gui†⇤ , Meixia Tao† , Y Jay Guo⇤ , Xiaojing Huang⇤ and Yun Rui‡

Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China ⇤ CSIRO ICT Centre, Marsfield NSW 2122, Australia ‡ Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, China Email: {cp2001cp, guilin, mxtao}@sjtu.edu.cn {jay.guo, xiaojing.huang}@csiro.au †

Abstract—Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. CS enables the recovery of high-dimensional sparse signals from much fewer samples than usually required. Further, quite a few recent channel measurement experiments show that many wireless channels also tend to exhibit sparsity. In this case, CS theory can be applicable to sparse channel estimation and its effectiveness has been validated in point-to-point (P2P) communication. In this work, we study sparse channel estimation for two-way relay networks (TWRN). Unlike P2P systems, applying CS theory to sparse channel estimation in TWRN is much more challenging. One issue is that the equivalent channels (terminal-relay-terminal) may be no longer sparse due to the linear convolutional operation. On this basis, novel schemes are proposed to solve this problem and effectively improve the accuracy of TWRN channel estimation when using CS theory. Extensive numerical results are provided to corroborate the proposed studies.

I. I NTRODUCTION Relaying is one of the key technologies to meet the strong capacity of future wireless broadband communication systems. Two-way relaying networks (TWRN) [1] is known to exploit the radio resources more efficiently than one-way relaying [2], in that both terminals are allowed to exchange information simultaneously with the assistance of an intermediate relay node and make explicit use of the bidirectional nature of communication [3]. Both amplify-and-forward (AF) and decode-and-forward (DF) protocols are developed in TWRN. In contrast to the DF protocol, the AF one requires minimal processing at the relay which can be transparent to the employed modulation and coding. The research on two-way AF relaying has gained much attention in recent years, such as optimal beamforming process at relay node [4] and joint optimal source-relay precoding design [5]. Followed by previous achievements on two-way AF relaying which have assumed that perfect channel state information (CSI) is available at the terminals, the impact of imperfect CSI on the performance of TWRN is then investigated over the flat fading channels [6]. Afterwards, the research work This paper is supported by the National Natural Science Foundation of China under grant 60902019, the Innovation Program of Shanghai Municipal Education Commission under grant 11ZZ19, the funds of MIIT of China under grant 2011ZX03001-007-03, National 863 Program of China under grant No. 2011AA01A105, Shanghai Committee of Science and Technology (11DZ1500500) and the DIISR Australia-China special fund CH080270.

on two way relay channel estimation is extended to the frequency selective channel in corporation with the wellknown orthogonal frequency division multiplexing (OFDM) [7]. However, traditional channel estimation methods are usually proposed for rich multipath channel. In parallel, increasing channel measurement experiments showed that many wireless channels encountered in practice tend to be sparse, i.e., impulse responses dominated by a relatively small number of clusters of significant paths, especially when operating at large bandwidths. Recently, advance in sampling theory, known as compressed sensing (CS), is proposed in [8]. The CS breaks through the restriction of well-known Nyquist Sampling Theory for signal recovery, and is capable of perfectly recovering original signal with much fewer samples than usually required, as long as the signal itself or its transform domain is sparse. As such, inspired by CS theory, this particular approach promises to estimate the sparse channel with far fewer additional communication resources consumption, whose effectiveness has been confirmed in [9] for P2P communication systems. However, unlike P2P systems, applying CS theory for sparse channel estimation in TWRN is much more challenging. One issue is that even the individual channels between the terminals and the relay are sparse, the resultant equivalent channels (terminalrelay-terminal) may be no longer sparse due to the linear convolutional operation. For the first time, this paper attempts to tackle key challenges in the pilot assisted sparse channel estimation scheme for TWRN by applying the CS theory. In specific, the effectiveness of CS recovery theory highly depends on three factors, namely, Restricted Isometry Property (RIP) for the measurement matrix, the sparsity of unknown signal, and an effective recovery algorithm. For the last one, in less dependence, there have been many recovery methods, such as Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) algorithms, could be applied for signal recovery with stable performance. Further, we will show that the first factor, however, notably depends on the pattern design of pilot symbols. While the second one is largely affected by the reduction in the sparsity of the equivalent channels, an inherited property in TWRN. On this basis, new schemes are proposed to redesign the pattern of pilot symbols so as to nearly meet the RIP criterion;

meanwhile, in attempting to eliminate the effect introduced by reduced channel sparsity. Simulation results indicate that the proposed new schemes effectively exploit the CS theory, thereby enabling higher accuracy for sparse channel estimation with much smaller number of pilot symbols. The reduced overhead could translate into higher spectral efficiency for TWRN. The rest of the paper is organized as follows. Section II presents the system model of two-way relay network. Section III gives an overview of conventional channel estimation methods. The compressed sensing based algorithms are provided in Section IV. In Section V, simulation results are demonstrated to validate the effectiveness of proposed scheme. Finally, conclusion are made in Section VI. 1 II. S YSTEM D ESCRIPTION We consider a wireless relaying system where two terminals A and B exchange data with the assistance of a half-duplex relay R. The relay node and both two terminals are equipped with one antenna each. Given a specific frame, the fading channel from A to R is modeled as a frequency selective chanT nel, and denoted as hA = [hA (1) , hA (2) , . . . , hA (L1 )] 2 L1 C with length of L1 . The similar assumption is made T for hB = [hB (1) , hB (2) , . . . , hB (L2 )] 2 CL2 from B to R. In time-division-duplexing (TDD) mode, we assume that the channel from R to A remains the same as that from A to R. In particular, these two channels herein are characterized by sparse multipath channel due to much fewer distinct arrivals; as such, it means that the most taps in hA and hB are negligible while one nonzero tap hA (i) is modeled as independent and identically distributed (i.i.d) 2 random variables according to CN 0, A . The transmitted N symbols (subcarriers) at terminal A and B are deT noted as SA = [SA (1) , SA (2) , . . . , SA (N )] 2 CN and T N SB = [SB (1) , SB (2) , . . . , SB (N )] 2 C , respectively. ⇥ ⇤ The power⇤ constraints of the transmission are E SH A SA = ⇥ H E SB SB = PT , where PT is the average transmitting power of A and B without loss of generality. Furthermore, a set of pilot symbols dA and dB , are embedded in SA and SB in order to probe channel impulse responses with a set of indices P A = {a1 , a2 , . . . aP } \ {1, . . . , N } and P B = {b1 , b2 , . . . bP } \ {1, . . . , N }, of cardinality |P A | = |P B | = P , respectively. Here, P is the number of used pilots. It is clear that the condition P A \ P B = ; is sufficient to avoid the interference, which is also adopted in long term evolution (LTE) system for multiple antenna transmission. 1 Notations: (·) 1 , (·)† , (·)T and (·)H denote inverse, pseudo inverse, transpose, and conjugate transpose, respectively. The linear convolution and the element-wise division between a and b are denoted as a ⇤ b and a ↵ b. E [·] denotes expectation operation. kak denotes the Euclidean norm of the vector a. diag (a) changes a vector a into a diagonal matrix ⇤ while diag (⇤) changes a diagonal matrix ⇤ into a vector a. IN is the N ⇥ N identity matrix. W is an N point Fourier matrix with the entries given by [W]p,q = e j2⇡(p 1)(q 1)/N . CN 0, 2 denotes the complex normal distribution with independent real and imaginary parts each with mean zero and variance 2 /2. Besides, CM ⇥N represents the set of M ⇥ N matrices in complex field.

The information exchange is divided into multiple access (MAC) and broadcast (BC) phase, which will be specified as follows. A. Multiple Access Phase During this phase, inverse discrete Fourier transform (IDFT) is performed over SA and SB ; accordingly, the resultant timedomain signal vector, sA = WH SA and sB = WH SB , are then added by cyclic prefix (CP) with the length of LCP before being transmitted.. Clearly LCP > max{LA 1, LB 1} should be satisfied to avoid interblock interference. The received signal at relay R after CP removal are r = HA sA + HB sB + nR

(1)

in which ⇥HA and HB are matrices with ⇤T circulant ⇥ ⇤T the first columns, hTA , 01⇥(N LA ) and hTB , 01⇥(N LB ) , respecT tively. nR = [n1 , n2 , ..., nN ] 2 CN denotes the noise vector whose entries are i.i.d. random variables according to CN (0, N0 ). B. Broadcast Phase Before entering into BC phase, a linear precoding scheme at relay R can be employed for the received signal r over AFbased transmission. In general, the precoding matrix should be carefully designed, in order to optimize system throughput or error probability. In this paper, for simplicity, we only consider that an amplification coefficient ↵ is used in R, such as s Pr ↵= (2) PLA 1 2 PLB 1 2 PT l=0 l=0 A,l + PT B,l + N0 where Pr is the average power of R. Note that CP of length LCP should be re-added before broadcasting to both source terminals. Due to symmetry, we only consider the channel estimation problem at terminal A. The received signal vector at A after CP removal and DFT operation can be written as RA = ↵WHA r + WnA = ↵WHA HA WH SA + ↵WHA HB WH SB + NA (3) in ⇥ which ⇤NA = ↵WHA nR + WnA and one has 2 H E NH A NA =N0 ↵ HA HA + IN under the assumption that nA and nR share the same statistics. C. Coherent Data Detection Based on the property of circulant matrices, (3) can be simplified as RA = ↵⇤A ⇤A SA + ↵⇤A ⇤B SB + NA ,

(4)

where the frequency diagonal channel matrix ⇤A = WHA WH and ⇤A = diag (HA (1) , HA (2) , . . . , HA (N )) whose can be calculated as HA (i) = PN 1 entries j2⇡(i 1)(k 1)/N h (k) e , i = 1, · · · , N. The A k=0 similar calculation is made for ⇤B . As the terminal A knows its own transmitted vector SA and thus it is feasible to cancel it so as to detect the desired signal SB . The maximum

likelihood (ML) detection for SB is then carried out in the sense that ˆ B = arg S

min

SB (i)2CB

{kRA

↵⇤AA SA

↵⇤AB SB k}

(5)

in which CB is the signal constellation. The structure of diagonal matrices ⇤AA = ⇤A ⇤A and ⇤AB = ⇤A ⇤B leads to parallel decoding for each SB (i), free from intel-symbol interference (ISI). Here, we simply realize that the estimation of frequency channel ⇤A and ⇤B is not really necessary; instead, that of the equivalent channels ⇤AA and ⇤AB are more desirable to the data detection. III. C ONVENTIONAL C HANNEL E STIMATION S CHEME The DFT theory tells us that the result of circular convolution operation between the channel hA and hB is preserved as does the linear convolution, in the case of LA + LB 1  N . In the following, defining two equivalent channels hAA = hA ⇤hA 2 C2LA 1 and hAB = hA ⇤hB 2 CLA +LB 1 with the condition of N max {2LA 1, LA + LB 1}. Then, channel estimation problems for hAA and hAB are formed in terms of the determined pilots patterns of P A and P B as follows {P A }

= DA WAA hAA + NA

{P B }

= DB WAB hAB + NA

RA RA

{P A }

(6)

{P B }

(7)

in which DA = diag (↵dA ) and DB = diag (↵dB ). WAA = [wAA (1) , . . . , wAA (2LA 1)] 2 CP ⇥(2LA 1) and WAB = [wAB (1) , . . . , wAB (LA + LB 1)] 2 CP ⇥(LA +LB 1) are two submatrics of W with indices {P A , 1 : 2LA 1} and {P B , 1 : LA + LB 1}, respectively. Clearly P A and P B decide which rows are selected from W. It is then straightforward to show the conventional least square (LS) estimation for hAA , given by †

{P A }

hLS AA = (DA WAA ) RA

,

processing and computational harmonic analysis. Consider the following classical linear measurement model

(8)

in the case of N > P max {2LA 1, LA + LB 1} . Similar estimated result is calculated for hLS AB . It is true that linear minimum mean square error (LMMSE) estimator is also available here with better performance. LS Once hLS AA and hAB are obtained, the corresponding freLS quency composed channels ⇤LS AA and ⇤AB are readily to LS be obtained as ⇤LS = diag W h and ⇤LS AA AA AA AB = diag WAB hLS , respectively. On the other hand, in the case AB LS of P < min {2LA 1, LA + LB 1}, ⇤LS and ⇤ AA AB can only be estimated by frequency interpolation based method. In the following, we only consider the channel estimation problem in the case of P ⌧ min {LA , LB }, which is dramatically differs from the conventional methods and yields the key innovation of this work. IV. P ROPOSED N OVEL C HANNEL E STIMATION S CHEME A. Background of Compressed Sensing Compressed sensing is a novel sampling theory at the intersection of multiple research fields such as statistics signal

Y=

(9)

✓+⌘

in which Y 2 CM denotes the observed vector and ⌘ 2 CM is either a noise or a perturbation vector. 2 CM ⇥N represents a measurement matrix, whose goal is to reliably reconstruct unknown ✓ 2 CN based on the knowledge of Y and , under the case of M