Doubly selective channel estimation for amplify-and-forward relay networks

In this paper, the estimation of doubly selective channel is considered for amplify-and-forward (AF) relay networks. The complex exponential basis expansion model (CE-BEM) is chosen to describe the time-varying channel, from which the infinite channel parameters are mapped onto finite ones. Since direct estimation of these coefficients encounters high computational complexity and large spectral cost, we develop an efficient estimator targeting at some specially defined channel parameters. The training sequence design that can minimize the channel estimation mean-square error is also proposed.


I. INTRODUCTION
Wireless relay networks have attracted a lot of attention since the pioneer work [1]. Like any other wireless communication system, the relay network performs better with better channel estimates. Channel state information (CSI) is usually estimated and tracked by periodic training signals [2]. Flat-fading channel estimation was discussed in [3] and frequency-selective channel estimators were developed in [4], respectively. All these estimators assumed that channels are time-invariant during a certain period.
However in many practical cases, the source node, the relay node and the destination node can all be mobile. The relative motion between any two nodes causes Doppler shift and thus makes the channel time-varying [5]. Therefore, the relay network is expected to operate under doubly selective channels. To our best knowledge, estimation techniques for this case have not yet been developed. This motivates our current work.
The doubly selective channel is typically approximated in two ways: the autoregressive (AR) process [2] or the basis expansion model (BEM) [6]. AR model describes the channel variation through a symbol-by-symbol update manner, while BEM expresses the doubly selective channel as the superpositions of time-varying basis functions weighted by This  time-invariant coefficients. In this paper, we adopt the complex exponential BEM (CE-BEM) [7] and develop an efficient estimator to find the channel parameters, which can sufficiently aid the data detection. The optimal training sequence design that minimizes parameter mean-square error (MSE) is also proposed.

II. SYSTEM MODEL
Consider an amplify-and-forward (AF) relay network with one source node , one relay node ℝ and one destination node . Let ℎ( ; ) denote the channel between and ℝ, ( ; ) denote the channel between ℝ and , respectively. The lengths of both channels are assumed as + 1 without loss of generality. According to the CE-BEM analysis in [6], [7], we can express the doubly selective channels as where 0 ≤ ≤ − 1, 0 ≤ ≤ , ( = 1, 2) is the number of basis, and is the number of symbols during one transmission. The value of is 2⌈ ⌉ where 1 means the maximum Doppler shift of the link → ℝ, while 2 denotes the maximum Doppler shift of the link ℝ → . The CE-BEM coefficients ℎ ( ) and ( ) are assumed as zeromean, complex Gaussian random variables with variance 2 ℎ, , and 2 , , respectively.
To simplify the notation, we assume 1 = 2 and 1 = 2 = . Further denote = 2 ( − /2)/ and define We propose a new transmission scheme as shown in Fig. 1. Each transmission block that contains symbols is divided into subblocks. Assume the th subblock contains symbols, of which symbols are data and are represented by s , while symbols are pilots and are represented by b . The total number of data symbols is = ∑ =1 and the total number of pilots is = ∑

=1
. With such a structure, we can represent the whole block as a vector During the first phase, ℝ receives where 1 ( ) is the additive complex white Gaussian noise with mean zero and variance 2 1 . During the second phase, ℝ amplifies ( ) with a constant factor and then re-transmits it to . The signal obtained by is where ( ) is the combined noise.

A. Channel Partition
Following the channel partition method in [7], we can split the channel matrix H into three matrices, namely, H , H , and H¯, which are shown in Fig. 2. Similarly, the channel H , the th (1 ≤ ≤ ) part of H corresponding to the th sub-block input of [s , b ], can also be partitioned into three matrices H , H and H¯, as shown in Fig. 3.
We then have contains the first and the last entries of b for all 1 ≤ ≤ , and w 1 , w 1 denote the corresponding noise vectors.
Repeat the same partition process for G and G , that is, split G into G , G¯and G , while split G , the th component of G, into G , G¯and G . We then obtain , r¯contains the first and the last entries of r for all 1 ≤ ≤ , w 2 and w 2 denote the corresponding noise vectors.
Combining (13) and (15) produces where w is defined as the corresponding item. It can be readily checked that (16) is equivalent to , the training length for the th subblock should satisfy ≥ 2 + 1.

B. Estimation Algorithm
Based on these definitions, we then have the following two lemmas: Lemma 1: where There is where * denotes linear convolution. Proof: Proved from straightforward calculations. According to these definitions and (1), we obtain where Φ is a lower triangular Toeplitz matrix with the first Similarly, based on (2), we obtain where Ω is a lower triangular Toeplitz matrix with the first column Combining (22) and (24) gives where and Ξ , , is defined as the corresponding item. Using Lemma 1 and Lemma 2, Ξ , , can be simplified as Since T ( , ) is a Toeplitz matrix, we obtain Unfortunately, it remains challenging to estimate , from (30). A direct way to estimate all , requires to be no less than 2 + ( + 1) 2 (2 + 1), which is too large and the transmission efficiency will be reduced. To solve this problem, we choose to estimate other type of channel information that requires smaller training length but can guarantee the data detection. Let us introduce two variables , and defined as respectively. It can be readily checked that . Then we can combine those items that satisfy + = in (30) and obtain Substituting (34) into (17) provides a simpler model where Instead of estimating the coefficients h and g , we could estimate fromˆ= andˆis obtained from the corresponding section inˆfor each ∈ [0, 2 ].

C. Data Detection
Substituting (12) into (14) yields Lemma 3: Among all training choices that lead to identical covariance matrix of the channel estimation error, if the training length is greater than 4 + 1 and if the training has the first 2 and the last 2 entries equal to zero, then the interference to the data detection is minimized.
Proof: By setting the first 2 and the last 2 entries zero, the second item in (39) becomes zero and the fourth item in (39) will only contain the noise item w1 , which indicates the minimum interference for data detection.
Following Lemma 3, we can simplify (39) as which is equivalent to Define U (h ) as a Toeplitz matrix generated by the vector h in the following way: According to (1) and (2), we obtain Next it can be found that . Using (43) and (44), it can be derived that where and , are defined in (28) and (29) respectively. Substituting (48) into (47), we can obtain Clearly, given the estimates of , G H can be reconstructed from (49). Then, the data detection for s can be performed.

D. Training Sequence Design
The estimation error of can be expressed as The correlation matrix of w is found from (24) as Thus the mean square error of e is / 2 . According to [8], we know that 2 in (52) is lower bounded as where the equality holds if and only if (Ψ Ψ ) is a diagonal matrix. We then need to design the training sequence that can diagonalize (Ψ Ψ ).
Based on the definition of Ψ (37), the optimal training sequence that can minimizes the 2 requires the following conditions to be satisfied for where is the power allocated to the training sequence and 0 2 +1 is a (2 + 1) × (2 + 1) matrix with all zero entries.
Observing the structure of B (b ) , we know that (54) can be fulfilled if the following conditions are satisfied: With conditions (C1) and (C2), we can further simplify (55) as ∑ It can be readily checked that the sufficient conditions to achieve (58) is

IV. SIMULATION RESULTS
We assume that the carrier frequency = 900 MHz, one symbol period = 50 and the mobility speed is 90 km/hour. Thus the maximum Doppler shift is 75 Hz and = 3.75 × 10 −3 . Suppose one block contains 360 symbols, i.e., = 360. Then = 2⌈ ⌉ = 4. Assume that ℎ( ; ) and ( ; ) has 3 taps, i.e., = 2. The doubly selective channels are generated directly from the CE-BEM (1) and (2). Thus we know that ≥ (2 + 1) = 9 and ≥ (4 + 1) = 81. First we set the total number of pilots = 120 and use three types of training: (i) equi-powered and equi-spaced (our optimal design); (ii) equi-powered but with random length; (iii) equi-spaced but with random power. For performance comparison, the total power of each type of training is the same. For each type of training, we find the MSE of our specially defined channel . The estimation MSEs for all three types of training versus SNR are plotted in Fig. 4. The lower bound of 2 (53) is also displayed for comparison.
We also examine the performance of the suggested estimation and detection methods under real channel situations. Three different number of bases are chosen as 4, 6, and 8 respectively, and hence the corresponding number of data symbols is 279, 243, and 207. The BER versus SNR is plotted in Fig. 5. For comparison, the BER curve under perfect channel knowledge at the receiver is also displayed. Clearly, the proposed methods yield effective data detection.
V. CONCLUSION In this paper, doubly selective channel estimation was considered for AF-based relay networks. Based on CE-BEM, we designed an efficient method to estimate equivalent channel parameters. The optimal training sequence that can minimize the estimation MSE was also derived.