Tree-based backward pilot generation for sparse channel estimation C. Qi and L. Wu A scheme of tree-based backward pilot generation for sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems is proposed. Instead of straightforward searching for the best from all possible pilot subsets, the scheme iteratively removes a subcar- rier from all OFDM subcarriers in a backward manner. At each time, every subcarrier is tested by deleting it and calculating the coherence of the resulting matrix, where the subcarrier with the smallest coher- ence is picked up. Finally the subcarriers left make up the optimal pilot subset. Considering the greedy essence of the scheme, a tree struc- ture is also incorporated to avoid the locally-optimal but globally- incorrect selections. Simulation results demonstrate that the proposed scheme can obtain substantial improvement for sparse channel estimation. Introduction: Recently, compressed sensing (CS), which receives a great deal of attention, has been successfully applied for sparse channel estimation [1]. Many CS algorithms including matching pursuit (MP), orthogonal matching pursuit (OMP) and basis pursuit (BP) have been adopted to explore the channel sparsity, which results in more accurate channel estimation and less pilot overhead than the standard least squares (LS) [2]. However, there are few works discussing the optimal pilot pattern for sparse channel estimation. Although it is well-known that equi-spaced pilots are optimal under various criterions [3], there is no general theory on the optimal pilot pattern for channel estimation using CS algorithms. In [4], a deterministic pilot selection scheme is proposed for sparse channel estimation using the Dantzig selector. In [5], a clustered pilot design is presented in the study of underwater acoustic (UWA) channel estimation. And a scheme using UWA channel data to off-line train the pilots and to search for the opti- mised pilot placements at the transmitter is proposed in [6]. In this Letter, we propose a scheme of tree-based backward pilot gen- eration for sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems. Instead of straightforward searching for the best from all possible pilot subsets, the scheme iteratively removes a subcarrier from all OFDM subcarriers in a backward manner. At each time, every subcarrier is tested by deleting it and calcu- lating the coherence of the resulting matrix, where the subcarrier with the smallest coherence is picked up. Finally the subcarriers left make up the optimal pilot subset. Considering the greedy essence of the scheme, a tree structure is also incorporated to avoid the locally- optimal but globally-incorrect selections. System model: Considering an OFDM system with N subcarriers, we use K (K ≤ N ) comb-type pilot subcarriers denoted as P 1 , P 2 , ..., P K (1 ≤ P 1 , P 2 , ... , P K ≤N ) for frequency-domain channel esti- mation. The transmit pilot symbols and the receive pilot symbols are denoted as X(P 1 ), X(P 2 ), ..., X(P K ) and Y(P 1 ), Y(P 2 ), ..., Y(P K ), respect- ively. Then the channel estimation is formulated as y = DFh + h (1) where y ¼ [Y(P 1 ), Y(P 2 ), ..., Y(P K )] T , D ¼ diag{X(P 1 ), X(P 2 ), ..., X(P K )} is a diagonal matrix of pilots, h ¼ [h(1), h(2), ... , h(L)] T is the sampled channel impulse response (CIR) with length L, h ¼ [h(1), h(2), ... ,h(K )] T ≏ CN(0, s 2 I K ) is an additive white Gaussian noise (AWGN) term, and F is a K by L sub-matrix indexed by [P 1 , P 2 , ... , P K ] in row and [1, 2, ... , L] in column from a standard N by N discrete Fourier transform (DFT) matrix [7]. Considering the fact that the sampling interval at the receiver is usually much smaller than the channel delay spread, most components of h are either zero or nearly zero, which means that h is a sparse vector. In particular, we can improve the data rate by using less pilots than the unknown channel coefficients, e.g. K , L, where CS algorithms are employed to reconstruct h instead of the standard LS channel estimation. We denote A ¼ DF. Then (1) is reformulated as y = Ah + h (2) So the sparse channel estimation is essentially using y and A to recon- struct h with the perturbation h. Pilot generation: The well-known restricted isometry property (RIP) indicates that the sparse h can be reconstructed using the measurement y and the dictionary matrix A if A satisfies RIP [8]. However, there is no known method to test in polynomial time whether a given matrix satisfies RIP. Alternatively, we adopt an approach to minimise the coher- ence of A [9]. We define the coherence of a matrix to be the maximum absolute correlation between two different columns, denoted as m(P)= max 0≤i,l≤L−1 |kA(i), A(l ))l| where A(i ) represents the ith column of A, and P ¼ [P 1 ,P 2 ... , P K ] is a pilot subset of all OFDM subcarriers. The objective function is to mini- mise m(P) and the optimal pilot subset is P opt = arg min P m(P) (3) Suppose all OFDM pilot symbols are equi-powered that |X (P 1 )| 2 = |X (P 2 )| 2 =···=|X (P K )| 2 , (3) is simplified to be the problem of optimal row selection from the N by L DFT submatrix denoted as M = 1  N √ 1 1 1 1 1 v ··· v L−1 . . . . . . . . . . . . 1 v L−1 ··· v (N−1)(L−1) ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ where v ¼ e 2j2p/N . Notice that the columns of M are orthonormal, the coherence of M equals zero. Instead of straightforward searching for the best P from all possible N K   pilot subsets, we iteratively remove one row from M with the objective to minimise the coherence of the result- ing matrix in a backward manner. At each time, we test every row by deleting it and calculating the coherence of the resulting matrix. Then we pick up the row with the smallest coherence. We repeat it (N 2 K ) times. Finally the K rows left make up the optimal pilot subset. In prac- tice, the calculation of coherence of M can be fast implemented by searching the largest off-diagonal component of M H M, where the super- script H is short for Hermitian according to the conventions. Essentially, the above scheme belongs to the greedy algorithms that usually make a sequential locally-optimal choice in an effort to deter- mine a globally-optimal solution. However, the greedy choice in every step not necessarily guarantees global-optimal. Therefore we also consider using a tree structure to avoid locally-optimal but glob- ally-incorrect selections. Fig. 1 illustrates a two-branch tree where the rows to be removed at each step are marked in shadow. Rather than itera- tively removing only one row from M, we always consider to separately remove two rows. So in the next steps we always have to refine two from four row selections. In this way, we keep two parallel branches till the last step, where we finally choose one with the smallest coherence. It is also noticed that the same row may be simultaneously selected by two branches and the deserted rows are likely to be selected again. Only the selections in the branching path are unique. 1 3 5 9 9 4 9 5 7 8 3 6 4 2 Fig. 1 Tree-based backward pilot generation Simulation results: We consider an OFDM system with N ¼ 256 sub- carriers, where K ¼ 16 pilot subcarriers are employed for frequency- domain channel estimation. A sparse multipath channel is generated to be a zero CIR vector h with L ¼ 50, where S ¼ 5 positions are randomly selected to be nonzero channel taps. The attenuation of each tap satisfies the independent and identically distributed (I.I.D.) CN(0,1). As shown in Table 1, we use a different number of branches of the tree for pilot gener- ation. The coherence reduces from 5.5717 to 4.9189 when the number of ELECTRONICS LETTERS 26th April 2012 Vol. 48 No. 9