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 subcarrier 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 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 globallyincorrect 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 optimised pilot placements at the transmitter is proposed in [6]. In this Letter, we propose a scheme of tree-based backward pilot generation 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 calculating 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 locallyoptimal but globally-incorrect selections. System model: Considering an OFDM system with N subcarriers, we use K (K ≤ N ) comb-type pilot subcarriers denoted as P1 , P2 , . . ., PK (1 ≤ P1 , P2 , . . . , PK ≤N ) for frequency-domain channel estimation. The transmit pilot symbols and the receive pilot symbols are denoted as X(P1), X(P2), . . ., X(PK) and Y(P1), Y(P2), . . ., Y(PK), respectively. Then the channel estimation is formulated as y = DFh + h
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 [P1 ,P2. . . , PK] is a pilot subset of all OFDM subcarriers. The objective function is to minimise m(P) and the optimal pilot subset is P opt = arg min m(P)
(3)
P
Suppose all OFDM pilot symbols are equi-powered that |X (P1 )|2 = |X (P2 )|2 = · · · = |X (PK )|2 , (3) is simplified to be the problem of optimal row selection from the N by L DFT submatrix denoted as ⎡ ⎤ 1 1 1 1 v ··· vL−1 ⎥ 1 ⎢ ⎢1 ⎥ M = √ ⎢ . ⎥ . . .. .. ⎦ N ⎣ .. .. . 1 vL−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. of straightforward searching for the Instead
N best P from all possible pilot subsets, we iteratively remove one K row from M with the objective to minimise the coherence of the resulting 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 practice, the calculation of coherence of M can be fast implemented by searching the largest off-diagonal component of MHM, where the superscript 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 determine 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 globally-incorrect selections. Fig. 1 illustrates a two-branch tree where the rows to be removed at each step are marked in shadow. Rather than iteratively 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.
2
1
(1)
where y ¼ [Y(P1), Y(P2), . . ., Y(PK)]T, D ¼ diag{X(P1), X(P2), . . ., X(PK)} 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, s2IK) is an additive white Gaussian noise (AWGN) term, and F is a K by L sub-matrix indexed by [P1 , P2 , . . . , PK] 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
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 coherence of A [9]. We define the coherence of a matrix to be the maximum absolute correlation between two different columns, denoted as
(2)
So the sparse channel estimation is essentially using y and A to reconstruct h with the perturbation h.
3
6
5
7
9
4
8
9
4
3
9
5
Fig. 1 Tree-based backward pilot generation
Simulation results: We consider an OFDM system with N ¼ 256 subcarriers, where K ¼ 16 pilot subcarriers are employed for frequencydomain 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 generation. The coherence reduces from 5.5717 to 4.9189 when the number of
ELECTRONICS LETTERS 26th April 2012 Vol. 48 No. 9
branches grows from 1 to 11, which verifies the effectiveness of our scheme to avoid locally-optimal but globally-incorrect selections. Notice that the 1-branch tree is actually the original scheme without assistance of the tree structure. The complexity is also compared in terms of the CPU running time using MATLAB v7.9 (R2009b) running on a Lenovo laptop with an Intel Core 2 Duo CPU at 2.5 GHz and 2GB of memory. We observe that it grows linearly with the number of the branches of the tree. However, since the pilot generation is off-line performed before the transmission of OFDM signals, they are still acceptable.
Table 1: Comparisons of pilot generation using different number of branches of tree Number Coherence Running of branches time (s) 1 5.5717 18.02 2
5.4896
36.42
3
5.0499
55.44
7
4.9710
126.95
11
4.9189
199.86
3, 12, 21, 82, 89, 117, 137, 141, 145, 158, 161, 174, 183, 187, 236, 247 30, 33, 51, 61, 65, 86, 91, 106, 119, 122, 130, 171, 208, 213, 222, 246 8, 30, 38, 81, 85, 100, 120, 125, 135, 150, 162, 165, 171, 228, 232, 238 5, 19, 31, 35, 40, 44, 76, 85, 100, 116, 120, 185, 213, 224, 238, 242 11, 21, 26, 30, 41, 49, 73, 96, 156, 160, 175, 191, 199, 240, 249, 253
MSE
100
10–1
10–2
10–3 5
10
15
20
25
# The Institution of Engineering and Technology 2012 2 January 2012 doi: 10.1049/el.2012.0010 C. Qi and L. Wu (School of Information Science and Engineering, Southeast University, Nanjing 210096, People’s Republic of China) E-mail:
[email protected] References
randomly-generated pilots pilots generated using 1-branch tree pilots generated using 3-branch tree pilots generated using 11-branch tree
0
Acknowledgments: The authors thank G. Yue of NEC Laboratories America for valuable comments and suggestions. The work is supported by the National Natural Science Foundation of China (NSFC) under grant 60872075, the Scientific Research Foundation of Southeast University under the grant Seucx201116, and the Huawei Innovative Research Plan under CAST8804009011.
Optimal pilot subset
As shown in Fig. 2, we also compare the channel estimation performance in terms of mean squares errors (MSEs) for different pilot subsets. The CS algorithm we adopt here is the orthogonal matching pursuit (OMP) [2]. It is observed that the pilot subset generated by our scheme significantly outperforms the randomly-generated pilots. In particular, using more branches of the tree is more beneficial, while the rate of improvement is getting slower. 101
Conclusion: We have proposed a scheme that iteratively removes a subcarrier from all OFDM subcarriers in a backward manner. Considering the greedy essence of the scheme, a tree structure has also been incorporated to avoid the locally-optimal but globally-incorrect selections. The effectiveness of the proposed scheme has been demonstrated in simulations where substantial improvement for sparse channel estimation is achieved.
30
SNR, dB
1 Bajwa, W.U., Haupt, J., Sayeed, A.M., and Nowak, R.: ‘Compressed channel sensing: a new approach to estimating sparse multipath channels’, Proc. IEEE, 2010, 98, (6), pp. 1058– 1076 2 Berger, C.R., Wang, Z., Huang, J., and Zhou, S.: ‘Application of compressive sensing to sparse channel estimation’, IEEE Commun. Mag., 2010, 48, (11), pp. 164– 174 3 Tong, L., Sadler, B.M., and Dong, M.: ‘Pilot-assisted wireless transmissions: general model, design criteria, and signal processing’, IEEE Signal Process. Mag., 2004, 21, (6), pp. 12–26 4 Applebaum, L., Bajwa, W.U., Calderbank, A.R., Haupt, J., and Nowak, R.: ‘Deterministic pilot sequences for sparse channel estimation in OFDM systems’. Proc. 17th Int. Conf. on Digital Signal Processing, (ICDSP), Corfu, Greece, July 2011, pp. 1 –7 5 Berger, C.R., Gomes, J., and Moura, J.M.F.: ‘Study of pilot designs for cyclic-prefix OFDM on time-varying and sparse underwater acoustic channels’. Proc. OCEANS, Santander, Spain, June 2011, pp. 1– 8 6 Qi, C., Wang, X., and Wu, L.: ‘Underwater acoustic channel estimation based on sparse recovery algorithms’, IET Signal Process., 2011, 5, (7), pp. 739– 747 7 Qi, C., and Wu, L.: ‘Optimized pilot placement for sparse channel estimation in OFDM systems’, IEEE Signal Process. Lett., 2011, 18, (12), pp. 749–752 8 Candes, E.J., and Tao, T.: ‘Near-optimal signal recovery from random projections: universal encoding strategies?’, IEEE Trans. Inf. Theory, 2006, 52, (12), pp. 5406–5425 9 Tropp, J.A.: ‘Greed is good: algorithmic results for sparse approximation’, IEEE Trans. Inf. Theory, 2004, 50, (10), pp. 2231– 2242
Fig. 2 Comparisons of channel estimation using different pilot subsets
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