IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. X, NO. XX, XXXXX 2011
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Frequency Offset and Channel Estimation in Cooperative Relay Networks Kyeong Jin Kim, Member, IEEE, R. A. Iltis, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract—A new joint frequency offset and channel estimator for cooperative relay networks operating over time-varying multipath channels based on Gauss-Hermite integration (GHI) and approximate Rao-Blackwellization (GHI-ARB) is proposed. The resulting algorithm incorporates parallel Schmidt-Kalman Filters (SKFs) to separate all interfering links at the destination receiver. In each SKF, Rao-Blackwellization is conditioned on a GHI integration point. Using the Rao-Blackwellization for the frequency offset in the Gauss-Hermite filter (GHF), a joint channel and frequency offset can be efficiently estimated by the proposed GHI-approximate Rao-Blackwellization-based Schmidt-Kalman Filter (GHI-ARB-SKF) in the relay network. Simulation results show the superiority of the proposed GHI-ARB-SKF over a RaoBlackwellized unscented-Kalman Filter based approach (GHIARB-UKF). Index Terms—Approximate Rao-Blackwellization, adaptive decode-forward (ADF) relay protocol, Gauss-Hermite integration, Gauss-Hermite filter, joint channel and frequency offset estimation, relay networks, parallel Schmidt-Kalman filters.
I. I NTRODUCTION UE to limitations on bandwidth and transmit power, it is often necessary to enhance point-to-point communications between a source and destination using relays. If relays are located carefully, the resulting channels will be of better quality than the direct channel between the source and terminal [1], [2]. Relay systems are also required in networks with large cell sizes, which suffer from higher consumption of transmitter power, intercellular interference, and unreliable transmission at the cell edge due to large path losses and shadowing fading. The addition of relays can also address these issues by providing cooperative diversity [3], [4]. One of the main objectives of this form of cooperative communications is to form a virtual antenna array steered to the destination [5]. Such
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Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected]. Manuscript received October 11, 2010; revised April 27, 2011; accepted June 14, 2011. The editor coordinating the review of this paper and approving it for publication was Prof. Andrea Tonello. K. J. Kim is with the UWB Wireless Communications Research Center, Inha University, Korea (e-mail:
[email protected]). R. A. Iltis is with Department of Electrical & Computer Engineering, University of California, Santa Barbara, CA 93106 (e-mail:
[email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 (e-mail:
[email protected]). This research was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(No.20100018116), and in part by the U.S. National Science Foundation under Grant CNS-09-05398.
fixed relay systems are being incorporated into the WiMAX [6] and LTE-Advanced [7] standards. In cooperative relay systems, the individual relay links may have different symbol timings, carrier frequency offsets and path loss behavior [8], [9]. To compensate for frequency offset, several approaches [5], [10]–[12] have been proposed for single carrier systems based on minimum mean square error (MMSE) estimation. In [11], delay diversity is used in frequency offset estimation. For distributed MIMO flat-fading channels, iterative expectation conditional maximization and space-alternating generalized expectation-maximization algorithms are used in [12]. To combat the effects of frequency selective channels, orthogonal frequency-division multiplexing (OFDM) is also sometimes employed in relay systems [13]–[17], in which accurate frequency offset estimation and correction is required. The design of a robust estimator is thus especially critical in the relay system since each link has a different frequency offset. Several schemes have been proposed to perform frequency offset estimation in cooperative relay systems, e.g. [5], [10], [13] and [15]–[18]. Despite their good performance, these existing schemes assume that the intervening channels are time-invariant or the mobility of the terminals is relatively small. Since relatively large Doppler frequencies are expected at the destination due to high user mobility, these approaches may not be suitable for time-varying channels. Therefore development of a joint frequency offset and channel estimator is of considerable interest for cooperative relay networks operating in the timevarying channel environment. Parallel extended Kalman Filters (EKFs) [19] may be used for the joint channel and frequency offset estimation. However, the EKF approach suffers from divergence caused by the highly nonlinear dependence of the received signal on the frequency offsets. In addition, it is not easy to decompose the received signal into a form suitable for the parallel EKFs. More recently, Rao-Blackwellized particle filtering (RBPF) has been developed for joint linear and nonlinear state estimations [20]– [22] based on resampling techniques [20], [21], [23], [24]. In this paper, the channel coefficients and the frequency offset are recognized as linear and nonlinear channel parameters, respectively. Compared to these previous works, we propose a GHI-approximate Rao-Blackwellization-based SchmidtKalman Filter (GHI-ARB-SKF). Our main contributions are summarized as follows:
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. X, NO. XX, XXXXX 2011
1) We approximate the posterior distribution of the frequency offset using the Gauss-Hermite Filter (GHF) [25], [26], which uses numerical quadrature to approximate a weighted integral. Only linear channel parameters, e.g., a set of channel coefficients, can be estimated by using the GHF [25], [26]. Compared to [25] and [26], a Rao-Blackwellization is employed in the GHF to estimate the linear and nonlinear channel parameters simultaneously. 2) We introduce a parallel bank of Schmidt-Kalman Filters (SKFs) [22], [23], [27], [28] in the GHF. By decomposing the joint channel and frequency offset estimator into a form of parallel bank of SKFs, the dimensionality of the state vector can be significantly reduced. In [22], Jacobian matrices for frequency offsets are derived and then used in each Schmidt extended Kalman filter (SEKF). Each SEKF is related to one link in the uplink orthogonal frequency-division multiple access (OFDMA) system, in which each joint subestimation problem uses a Rao-Blackwellized particle filter. Motivated by the approach in [22], each SKF will be employed for the desired relay link in the GHI-ARBSKF. However, compared to [22], Gauss-Hermite integration (GHI) [29]–[31] will be used to avoid the need for linearlization with respect to (w.r.t.) the frequency offset of the desired link. 3) We provide a systematic approximation to the exact Bayesian filter in the Rao-Blackwellized GHI for the joint frequency offset and channel estimation in the cooperative relay systems without using of resampling techniques [20], [21], [23], [24]. In the GHI-ARB-SKF, the desired link’s frequency offset is estimated through the GHF, whereas the channel coefficients are updated via a Bayesian approach conditioned on sample points generated by the GHI [29]–[31]. As a result, the number of required sample points for the GHI-ARB-SKF can be reduced. Moreover, the SKF can reduce interference from the interfering links in the receiver at the destination, which results in the improved performance of the GHI-ARB-SKF over the SEKF. The outline of this paper is as follows. Our system and signal models are presented in Section II. The proposed joint frequency offset and channel estimation algorithm is discussed in Section III. Section IV provides simulation parameters and results. Section V contains conclusions. Notation: The superscripts T and H denote transposition and conjugate transposition, respectively. E{·} denotes expectation; Diag{x} denotes a diagonal matrix with x on its main diagonal; IN denotes the N × N identity matrix; 0M ×N denotes an M -by-N all-zeros matrix; ( ) the i-th element of vector x is denoted by (x)i ; CN µ, σ 2 denotes the complex 2 Gaussian ( )distribution with mean µ and variance σ , while 2 N µ, σ denotes the real Gaussian distribution with mean µ and variance σ 2 ; real and imaginary parts of a complex number are denoted as Re (·) and Im (·); blkdiag(A, B) denotes a block diagonal concatenation of matrices A and B; Cm×n denotes the vector space of all m × n complex matrices, while
Rm×n denotes the vector space of all m × n real matrices. II.
SYSTEM AND SIGNAL MODELS
In the considered relay system, we make the following assumptions: • Adaptive decode-and-forward (ADF) and half duplex relay protocols are used [32] since the fixed DF (FDF) relay cannot achieve diversity due to error-propagation [3]. Compared to the FDF relay protocol, the employed ADF can eliminate the propagation of errors to the destination [32]. A training symbol based relay system is also assumed in cooperative relay systems, thus a data detection is not considered in this paper. • There are one source (S), M relays, denoted by Ri , i = 1, .., M , and one destination (D) in the considered system. There is a direct path between the source and destination. • The source and relays are assumed to be in the fixed locations, so that the channels between them are assumed to be known exactly in the system. A. Half Duplex Cooperative Relay System We assume separate transmission intervals from the source to relays and relays to the destination. An information bit stream is Gray mapped to quadrature phase-shift keying (QPSK) symbols b(2n′ ). In the first (2n′ ) time interval, the source broadcasts modulated transmission symbols d(2n′ ) ∈ CN ×1 to all relays and the destination, where d(2n′ ) = W b(2n′ ) with the inverse discrete Fourier transform (IDFT) matrix W . Transmission symbols satisfy E{d(2n′ )} = 0 H and E{d(2n′ ) (d(2m′ )) } = Es IN δ(2n′ − 2m′ ), where δ(·) denotes the discrete-time Dirac delta function. Each transmission packet forms multiple OFDM symbols each having N subcarriers. The received vector signal at relay Rm is given by √ ′ y m (2n′ ) = p(lS−m )D(2n′ )hS−m (2n′ ) + nm (1) 1 (2n ) ′ T where hS−m (2n′ ) = [hS−m (2n′ ), · · · , hS−m 0 Nfm −1 (2n )] is the multipath channel vector for the link between the source and relay Rm with p(lS−m ), an allocated transmission power, and Nfm , the multipath channel length. The power decreases with distance lS−m between the source and relay Rm . The matrix m D(2n′ ) ∈ CN ×Nf is formed by the first Nfm columns of the circulant matrix with first row given by d(2n′ ). The frequency offset of the source is assumed to be corrected at all relays and thus does not appear in (1). The additive noise at relay m ′ is circular Gaussian with nm 1 (2n ) ∼ CN (0N ×1 , IN ). Employing the ADF protocol [32], the source-to-relay channels are assumed to be error-free. That is, the reconstructed symbols D(2n′ + 1) are transmitted to the destination (D) through the channels {hm−D }. The received signal at the l destination in the presence of frequency offsets is shown in [33]–[35] to be given by M √ ∑ ( p(lm−D ) ∆(εm−D (2n′ + 1))× y(2n′ + 1) = m=1 ) D(2n′ + 1)hm−D (2n′ + 1) + n2 (2n′ + 1)
(2)
KIM et al.: FREQUENCY OFFSET AND CHANNEL ESTIMATION IN COOPERATIVE RELAY NETWORKS
where p(lm−D ) is the received power over the m-th relayto-destination channel, hm−D (2n′ + 1) = [hm−D (2n′ + 0 m−D ′ T ′ 1), · · · , hNf −1 (2n +1)] and n2 (2n +1) ∼ CN (0N ×1 , IN ). Note that index m = 1 denotes the direct link between the source and destination. The maximum channel length from all channels is denoted by Nf = max{Nfm }. Let εm−D = δf m−D /Ts be the normalized carrier frequency offset w.r.t. the subcarrier spacing 1/Ts between the transmit antenna in the m-th relay and receive antenna at the destination. The OFDM symbol interval is given by Ts . In ′ general, εm−D ̸= εm −D if m ̸= m′ . The diagonal matrix ∆(εm−D (2n′ + 1)) is defined in terms of this normalized frequency offset. To simplify our notation, m denotes a link between relay Rm and destination. Also, (n) denotes the n-th transmission interval consisting of two distinctive slots, (2n′ ) △ △ and (2n′ + 1), so that n=(2n′ ) and n=(2n′ + 1) w.l.o.g. depending on even and odd slots. For example, △
′
∆(εm (n))=∆(εm−D (2n′ + 1)) = ejθ (2n +1) × ( ) 2πεm (2n′ +1) 2π(N −1)εm (2n′ +1) N N Diag 1, ej , · · · , ej . (3) m
∑n′ −1 △ In (3), θm (2n′ + 1)=2π l=1 εm (2l + 1). In the sequel, we focus only on time intervals (2n′ + 1) and write the received signal in compact form as △
z(n) = y(2n′ + 1) =
M ∑
∆(εi (n))D(n)hi (n) + n(n). (4)
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B. Channel and Frequency Offset Models We model the fading channel f˜li (n), the l-th element of i ˜ f (n), by a second order autoregressive (AR) process f˜li (n) = −a1 f˜li (n − 1) − a2 f˜li (n − 2) + wli (n) (6) √ where a1 = −2rd cos(2πfd / 2), a2 = rd2 , and fd is the normalized Doppler frequency. Also, wli (n) is the independent and identically distributed (i.i.d.) driving Gaussian process distributed as wli (n) ∼ N (0, 1). The radius rd is chosen to have a peak near the Doppler frequency of the fading process. We adopt the AR model [36]–[38] to characterize the timevarying frequency offset and channel responses. Note that the channel is assumed to be time-invariant over each OFDM symbol. To simplify our approach, we assume that the speed of the destination is known exactly and we use the same AR coefficients for all channels in the system. When we assume f˜li (n) as Gaussian process with E{f˜li (n)} = 0 and E{(f˜li (n))2 } = 2 σf2˜i , then the covariance of wli (n) ∼ N (0, σw ) is given by l
2 σw = σf2˜i l
(1 − a2 )((1 + a2 )2 − a21 ) . 1 + a2
(7)
From (7), channels with different channel gains can be also described by (6) with available AR coefficients a1 and a2 . According to the employed channel model, the dynamic equations for the channel vector and frequency offset are given, respectively, by f i (n) = Ff f i (n − 1) + wfi (n) and
i=1
Note that D(n) = D(2n′ + 1); h1 (n) = p(lS−D )hS−D (2n′ ); ε1 (n) = εS−D (2n′ ); hi (n) is the convolved channel between p(lS−i )hS−i (2n′ ) and p(li−D )hi−D (2n′ + 1) for i = 2, · · · , M ; εi (n) = εi−D (2n′ + 1), i = 2, · · · , M . Recall that (n) denotes the n-th transmission interval comprising even (2n′ ) and odd (2n′ + 1) slots. The received signal (4) is converted to a real valued vector to facilitate the estimator derivation. Thus, let △ r(n)=[Re{z(n)}T , Im{z(n)}T ]T yielding
εi (n) = αε εi (n − 1) + wεi (n) where the transition −a1 INf △ 0Nf ×Nf Ff = INf 0Nf ×Nf
(8)
matrix Ff is defined by 0Nf ×Nf −a1 INf 0Nf ×Nf INf
−a2 INf 0Nf ×Nf 0Nf ×Nf 0Nf ×Nf
0Nf ×Nf −a2 INf . 0Nf ×Nf 0Nf ×Nf
(9)
Distributions for wfi (n) and wεi (n) are defined, respectively, as
] M [ ∑ (10) wfi (n) ∼ N (04Nf ×1 , Qf ) and wεi (n) ∼ N (0, qε ). Re(∆(εi (n))D) −Im(∆(εi (n))D) × i i Im(∆(ε (n))D) Re(∆(ε (n))D) ( ) △ [ i=1 i ] [ ] In (10), Qf =blkdiag 21 I2Nf , 02Nf ×2Nf . The AR parameRe(h (n)) Re(n(n)) ters for the frequency offset are assumed to be fixed with + Im(hi (n)) Im(n(n)) αe approaching unity, and 0 < qe
