Application of Compressed Sensing to DRM Channel Estimation

Transcription

1 Application of Compressed Sensing to DRM Channel Estimation Chenhao Qi and Lenan W School of Information Science and Engineering Sotheast University, Nanjing 10096, China {qch,wln}@se.ed.cn Abstract In order to redce the pilot nmber and improve the spectral efficiency, recently emerged compressed sensing (CS) techniqe is applied for digital broadcast channel estimation. According to the six channel profiles of the ETSI digital radio mondiale (DRM) standard, the sbspace prsit (SP) algorithm is employed for the delay spread and attenation estimation of each path in the case where the channel profile is identified and the mltipath nmber is known beforehand. The stop condition for SP is that the estimated sparsity eqals the mltipath nmber. For the case where the mltipath nmber is nknown, the orthogonal matching prsit (OMP) algorithm is employed for channel estimation, while the stop condition is that the estimation satisfies the level of the noise variance. Simlation reslts show that with the same nmber of pilots, CS algorithms with randomly placed pilots otperform traditional cbic-splineinterpolation-based least sqare (LS) channel estimation. SP is also demonstrated to be better than OMP when the mltipath nmber is known as apriori. I. Introdction Compressed sensing (CS) has recently emerged as a collection of principles and methodologies which enables efficient reconstrction of sparse signals from relatively few linear measrements [1], []. We roghly divide CS algorithms into two classes, inclding greedy algorithms and convex optimization algorithms. Greedy algorithms make a seqential locally optimal choice in an effort to determine a globally optimal soltion. They are matching prsit (MP) [3], orthogonal matching prsit (OMP) [4], sbspace prsit (SP) [5] and many other variants [6], [7], [8]. Compared to convex optimization algorithms, their comptational complexity is mch lower, which indicates they are more appropriate for practical applications. Recently, MP has been applied for pilot assisted channel estimation in orthogonal freqency division mltiplex (OFDM) systems [9], [10]. It is demonstrated that the pilot nmber can be sbstantially redced compared to the traditional interpolation based least sqare (LS) method while the performance of channel estimation is similar. Especially for the time-varying channel where channel estimation shold be freqently carried ot, CS algorithms are capable of saving a large nmber of pilots and improving the spectral efficiency. It is also beneficial for mlti-inpt mlti-otpt (MIMO) systems to employ CS algorithms since the pilots increase linearly with the nmber of transmit antennas. Althogh MP can find an approximated soltion from the overcomplete dictionary with asymptotic convergence, the shortcoming lies in the fact that it may select the same colmn several times and lower the efficiency. It is necessary to adapt more powerfl CS algorithms for relative applications. In this paper, we apply OMP and SP for the digital broadcast channel estimation. According to the ETSI digital radio mondiale (DRM) standard, six channel profiles are classified. For the case where the mltipath nmber is nknown, we apply OMP for channel estimation. For the case where channel profile is identified and the mltipath nmber is known beforehand, we apply SP for the delay spread and attenation estimation of each path. In both cases, we formlate the pilot assisted OFDM freqency domain channel estimation as a sparse recovery problem. We also compare their performance with traditional interpolation based LS method with differently placed pilots. The remainder of the paper is organized as follows. Section II formlates the DRM channel estimation. Section III and Section IV introdce the OMP and the SP algorithm. In Section V, the pilot placement and pilot nmber are discssed. Simlation reslts are presented in Section VI, and finally section VII concldes the paper. The notation sed in this paper is according to the convention. Symbols for matrices (pper case) and vectors (lower case) are in boldface. ( ) H,, 1, C, R, diag{ }, I L and CN denote conjgate transpose (Hermitian), absolte vale, l 1 -norm, l -norm, the set of complex nmber, the set of real nmber, diagonal matrix, identity matrix with the dimension L and complex Gassian distribtion, respectively. ˆφ denotes the estimate of the parameter of interest φ. II. Problem Formlation We model the channel implse response (CIR) of mltipath propagation as S h(τ, t) = α p (t)δ(τ τ p (t)) (1) p=1 where S, α p and τ p are the mltipath nmber, the channel coefficient and the delay spread for the p-th propagation path. According to the ETSI DRM standard [11], we have six channel profiles as listed in Tab.I. Channel No.1 is the pre AWGN channel, which is not common in practice. Channels No., No.4 and No.5 are two-path channels with increasing delay spread. Both channel No.3 and No.6 are for-path channels where eight parameters shold be estimated /11/$ IEEE

2 TABLE I Six channel profiles in the ETSI DRM standard Channel Path 1 Path Path 3 Path 4 No. α 1 τ 1 / ms α τ / ms α 3 τ 3 / ms α 4 τ 4 / ms With block fading channel assmption where the channel parameters are constant over each block and spposing perfect symbol synchronization, we model the eqivalent discrete CIR as S h(m) = α p δ((m τ p )T s ) () p=1 where T s is the sampling interval of the receiver. We notice that the elementary time period T of each OFDM symbol in the DRM standard is no more than 0.1 ms, so T s shold be less than 0.05 ms, which is very small compared to the maximm delay spread and reslts in a channel with relatively few nonzero taps. Spposing the nmber of channel taps to be L and S of them nonzero, we define it as an S -sparse channel. Considering an OFDM system, there are totally N sbcarriers, among which M sbcarriers are selected as pilots, with the positions represented by Q 1, Q,..., Q M (1 Q 1 < Q < < Q M N). We denote the transmitted pilots and the received pilots as X(Q 1 ), X(Q ),..., X(Q M ) and Y(Q 1 ), Y(Q ),..., Y(Q M ), respectively. We have y = Ah + η (3) where y = [Y(Q 1 ), Y(Q ),, Y(Q M )] T, h = [h(1), h(),, h(l)] T and η = [η(1),η(),,η(m)] T. η CN(0,σ I M ). And A = ZF (4) where 1 ω Q 1 ω Q 1 (L 1) F Np L = 1 1 ω Q ω Q (L 1) N ω Q M ω Q M (L 1) is a sbmatrix selected by the row index [Q 1, Q,..., Q M ] and thecolmnindex[0,1,..., L 1] from a standard N N Forier matrix. ω = e jπ/n. Z = diag{x(q 1 ), X(Q ),...,X(Q M )}. If the rows of A are more than its colmns, (3) can be solved by the traditional LS method. However, we are more interested in the nder-determined case where the rows of A is less than its colmns, which means the pilots are less than the nknown channel coefficients. Still we can se H(i) = Y(i) X(i), i = Q 1, Q,..., Q M (5) first to figre ot the channel transfer fnction (CTF) at the pilot sbcarriers and then make linear or cbic spline interpolations for the data sbcarriers. Obviosly this method will reslt in large deviations becase it doesn t se the sparse condition as apriori. Here we focs on low-complexity greedy CS algorithms and divide the sparse channel estimation problem into two cases. In the first case, the mltipath nmber is nknown, and we apply OMP for channel estimation. For the case where the channel profile is identified and the mltipath nmber is known beforehand, we apply SP for the delay spread and attenation estimation of each path. III. MP and OMP MP is a sort of algorithm that constrcts a sparse soltion by iteratively selecting dictionary elements best correlated with the residal part of the signal [3]. If the dictionary is a matrix, the objective of the constrction is to find a linear combination of matrix colmns closest to the signal. At each step, one colmn that best correlates with the crrent reside is added to crrent selection. Then it pdates the reside by projecting it onto the new selection. Here we briefly describe MP and OMP algorithm based on (3). First we generate a dictionary matrix D from A, A = DC (6) where D has the same dimension as A (D R N p L ) and each colmn of D is a nit vector (l -norm eqals one). C is a diagonal matrix with each diagonal element corresponding to the normized coefficient for each colmn of A. Letd i denote the i-th colmn of D and R k denote the reside at the k-th step. So the selected colmn index at k-thstepis l k = arg max i, R k > i {1,...,L} (7) where < d i, R k > represents the absolte vale of inner prodct between d i and R k. Now starting MP with an initial reside R 1 = y, the algorithm evolves by R k =< d lk, R k > d lk + R k+1 (8)

3 and replaces the reside R k with R k+1. Since R k+1 is orthogonal to d lk,wehave R k = < d l k, R k > + R k+1 (9) With increasing k (k = 1,,...), we minimize R k+1 till it satisfies the termination condition R k+1 σ (10) Spposing the iteration ends in K steps, the soltion is d = < d li, R i > d li (11) which is a linear combination of previos selected colmns. Finally, the estimation of h is ĥ MP = C 1 d = C 1 < d li, R i > d li (1) Althogh MP can rapidly find an approximation from overcomplete dictionary with asymptotic convergence, the shortcoming lies in the fact that it may select the same colmns several times and ths brings down the efficiency. Hence, OMP [1] is proposed with revision by sing reside s orthogonal component for the next iteration. Only the component that is orthogonal with the space spanned by the previos selected colmns is preserved. In most literatres, the set containing previos selected colmns is called the active set. Herewe denote the active set and its complementary set as I and I c, respectively. I I c = {1,...,L}. Unlike MP always selecting candidate colmn from {1,...,L} as in (7), OMP selects it only from I c, where the selected colmn index is l k = arg max < d i, R k > (13) i I c Then Gram-Schmidt orthogonalization is implemented to remove the component within the space spanned by I. < d lk, i > k = d lk i I i i (14) where { k } is an iteratively generated set that can be thoght as a base vector set of the space spanned by I. We initialize 1 to be d l1 and iteratively pdate the reside by R k = < R k, k > k k + R k+1 (15) In this way, OMP adds different colmns into the active set ntil the stoping condition (10) is satisfied. Spposing it ends in K steps (sally K L), the soltion is < R i, i > d = i i (16) And eventally the estimation of h is ĥ OMP = C 1 d = C 1 < R i, i > i i (17) Compared with MP, OMP is mch faster convergent. Even for the large dictionary matrix, the step is contable. It has already been demonstrated by [13], [4], [14] that in some circmstances OMP does scceed at finding the sparsest soltion. Bt the discssion is going on. OMP works in greedy way, which determines the final soltion is essentially sboptimal, not global optimal. Besides, OMP always adds a new colmn to the active set, bt never removes ot-dated colmns from the active set. When a selection error occrs, the iteration will contine till reaching the end withot correcting them adaptively. Althogh several revised versions of OMP have been proposed against these disadvantages [6], [7], [8], OMP is still one of the best candidates for practical applications becase of its reasonable tradeoff between the performance and complexity. IV. SP In the SP algorithm, the S colmns selection is iteratively refined from A ntil the stop condition is satisfied [5]. Althogh it is still developed based on the greedy rle, SP allows new colmns to enter into as well as to leave the active set. S colmns are simltaneosly selected rather than only one colmn as in MP and OMP. In this way, the sbspace spanned by S colmns is tracked down. Algorithm 1 Sbspace Prsit Algorithm Inpt: A, y, S Initialization: Normalize each colmn of matrix A with a diagonal coefficient matrix C so that A = D C Î = { S indices corresponding to S colmns of D on which y has the largest projections } R 1 = resid(y, DÎ) Iteration: k = 1,,... If R k = 0, qit the iteration; otherwise contine I = Î { S indices corresponding to S colmns of D on R k has the largest projections } Get D I and figre ot x = D I y I k = { S indices corresponding to S elements with the largest absolte vale in x } R k+1 = resid(y, D Ik ) If R k+1 > R k, qit the iteration otherwise, let Î = I k increase k by one and contine the iteration Otpt: One estimate vector ˆx is yielded with nonzero element indexed by Î satisfying ˆxÎ = D y Î The final reslt is ĥ sp = C 1 ˆx The SP algorithm solves the problem of (3) and works as follows. First, it begins with the same step as (6) to normalize each colmn of A. Then, it finds S colmns from D on which y has the S largest projections, and stores their indices into

4 a set Î. Meanwhile the reside R 1 is also obtained. The projection of y onto each colmn vector is defined as the absolte vale of their inner prodct which is similar to (7). Here we introdce the definition of the reside since it is different from MP and OMP. Definition 1: IfD H D is invertible, the reside of y on the matrix D is defined as where R = y DD y (18) D = (D H D) 1 D H (19) denotes the psedo inverse of D. We simply write it as R = resid(y, D). In the last step of initialization, we get an sbmatrix DÎ from D by index set Î and then figre ot the reside R 1 = resid(y, DÎ). After that we start the iteration. Another S indices corresponding to S colmns of D on which R k has the largest projections are selected and added to the previos S colmns set, forming I. The nmber of colmns in I may be less than S since it s possible to have overlap between two selections. Then we figre ot D I and a vector x = D I y, from which we choose S indices corresponding to S elements with the largest absolte vale. The active set I k at step k is obtained and a new reside R k+1 = resid(y, D Ik ) is also yielded. If R k+1 > R k (0) the iteration is terminated becase it indicates that the reside can t be smaller. Otherwise we replace Î with I k, increase k by one and contine the iteration. It s noticed that at the start point of every iteration, crrent reside R k is on focss. If R k is zero, we will also terminate the iteration. Finally one estimated vector ˆx is yielded. The nonzero locations of ˆx are indexed by Î and satisfy ˆxÎ = D y.the Î otpt is ĥ SP = C 1 ˆx (1) which is the estimated CIR for (3). V. Pilot Placement Recent advances in CS show that nder noiseless condition as η = 0in(3),h can be recovered from y with high accracy when A satisfies the restricted isometry property (RIP) [15]. Definition : A R m n satisfies RIP if (1 δ) h Ah (1 + δ) h () holds for all S -sparse vectors h R n ( h 0 S ). Defining δ s as the infimm of all possible δ that satisfies (), we say A satisfies RIP(S,δ s ). It can be easily obtained from () that 1 δ λ min λ max 1 + δ (3) where λ min and λ max denote the minimm and maximm eigenvales of A H A, respectively. Since () reqires all S - sparse vectors to be satisfied, it s difficlt to decide δ s and ths difficlt to verify (3). So far there s no method to test in polynomial time whether a given matrix satisfies RIP. Nevertheless, recent works by some mathematicians [15], [16] have shown that certain matrices satisfy RIP with high probability, i. e. random matrix from Forier ensemble, Gassian ensemble and binary ensemble. Concerning or channel estimation problem (3), we have to decide both the nmber and positions of pilots. For simplicity, we sppose all pilots have the same magnitde and phase, which means Z in (4) is an identity matrix and then A becomes a Forier sbmatrix. According to the RIP, we can randomly place the pilot symbols, which in essence is the random-row selection from standard Forier matrix. On the other hand, the nmber of pilots has to be at least twice of the sparsity S so as to exactly reconstrct h. And it s frther demonstrated by [17] that if the pilot nmber is N p = O ( (CtS log L)log(CtS log L)log S ) (4) where C and t (t > 1) are two parameters, the matrix A will satisfy RIP with probability at least 1 5e Ct. Therefore, in or application we will synchronosly generate psedorandom pilot placement in the same way for the transmitter and the receiver. Althogh it s well known that the best pilot placement for mobile OFDM systems is eqally spaced [18] and it is shown in [19] that the optimal pilot seqences are eqipowered, eqispaced and phase shift orthogonal sing LS channel estimation, we here employ CS algorithms instead of the LS estimator. According to the theoretical conclsion of RIP, we will se the random pilot placement. VI. Simlation reslts In or simlations, we set OFDM parameters according to the robstness mode B in the DRM standard as shown in Tab.II. The channel profile is set to be channel No.3 as described in Tab.I. We define the mean sqare error (MSE) as MSE{ĥ} = ĥ h h where ĥ is the estimate of h. TABLE II System parameters FFT length N FFT = 56 Used sbcarriers N = 06 Gard interval N g = 64 OFDM symbol period OFDM sample period T s = 6.7ms T = 83.3s Nmber of pilots M = 1 Nmber of channel mltipaths S = 4 CIR length L = 30 Modlation QPSK (5) Firstly, we sppose the mltipath nmber and the channel profile are nknown to the receiver. We compare the MSE

5 for OMP and LS. Eqal-spaced pilot placement and random pilot placement are employed for LS and OMP, respectively. In Fig.1, OMP with M = 1 is even better than cbic-splineinterpolation-based LS with M = 69 for SNR> 0dB, while their spectral efficiency is 94% and 66%, respectively. So OMP achieves more accrate channel estimation than LS while the pilot symbols are less sed OMP with M = 10 OMP with M = 1 SP with M = 10 SP with M = OMP with M = 1 OMP with M = 35 OMP with M = 69 LS cbic spline with M = 35 LS cbic spline with M = 69 MSE MSE SNR / db Fig.. MSE comparisons for OMP and SP SNR / db Fig. 1. MSE comparisons for OMP and LS Then we sppose the mltipath nmber is known as apriori to the receiver where we apply OMP and SP for the delay spread and attenation estimation of each path. When M = 10, SP is clearly sperior to OMP. When M = 1, SP is still a little better than OMP. The reason that SP performs better than OMP lies in the natre of each algorithm. At each iterative step, OMP always greedily selects one colmn vector while SP selects several colmns in batch. The possibility to correctly find one colmn with one selection is mch lower than with batch selection. Besides, once OMP selects one colmn into the active set, it never removes it, which means one error selection occrs, it will never be corrected in the later steps. VII. Conclsion In this paper we formlate the DRM channel estimation as a sparse recovery problem. It is shown that the CS algorithms are mch better than the traditional LS method. If the mltipath nmber is known beforehand, SP otperforms OMP. Ftre research will concentrate on the development of lower complexity CS algorithms with qantitative comparisons. References [1] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, vol. 5, no. 4, pp , Apr [] R. G. Baranik, Compressive sensing, IEEE Sig. Proc. Mag., vol. 4, no. 4, pp , Jly 007. [3] S. G. Mallat and Z. Zhang, Matching prsits with time-freqency dictionaries, IEEE Trans. Sig. Proc., vol. 41, no. 1, pp , Dec [4] J. A. Tropp and A. C. Gilbert, Signal recovery from random measrements via orthogonal matching prsit, IEEE Trans. Inf. Theory, vol. 53, no. 1, pp , Dec [5] W. Dai and O. Milenkovic, Sbspace prsit for compressive sensing signal reconstrction, IEEE Trans. Inf. Theory, vol. 55, no. 5, pp , May 009. [6] L. R. Neira and D. Lowe, Optimized orthogonal matching prsit approach, IEEE Sig. Proc. Lett., vol. 9, no. 4, pp , Apr. 00. [7] M. Andrle, L. R. Neira, and E. Sagianos, Backward-optimized orthogonal matching prsit approach, IEEE Sig. Proc. Lett., vol. 11, no. 9, pp , Sept [8] D. L. Donoho, Y. Tsaig, I. Drori, and J.-C. Starck, Sparse soltion of nderdetermined linear eqations by stagewise orthogonal matching prsit, Preprint, 006. [9] S. F. Cotter and B. D. Rao, Sparse channel estimation via matching prsit with application to eqalization, IEEE Trans. Commn., vol. 50, no. 3, pp , Mar. 00. [10] M. Sharp and A. Scaglione, Application of sparse signal recovery to pilot-assisted channel estimation, in Proc. ICASSP, LasVegas,NV, USA, Apr. 008, pp [11] ETSI, Digital radio mondiale (DRM) system specification, ETSI ES v..1, Oct [1] Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, Orthogonal matching prsit: Recrsive fnction approximation with applications to wavelet decomposition, in Proc. 7th Asilomar Conference on Signals, Systems and Compters, Los Alamitos, CA, USA, Nov. 1993, pp [13] J. A. Tropp, Greed is good: Algorithmic reslts for sparse approximation, IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 31 4, Oct [14] D. L. Donoho, M. Elad, and V. N. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inf. Theory, vol. 5, no. 1, pp. 6 18, Jan [15] E. J. Candes, J. Romberg, and T. Tao, Robst ncertainty principles: Exact signal reconstrction from highly incomplete freqency information, IEEE Trans. Inf. Theory, vol. 5, no., pp , Feb [16] E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, vol. 51, no. 1, pp , Dec [17] M. Rdelson and R. Vershynin, On sparse reconstrction from Forier and Gassian measrements, Commnications on Pre and Applied Mathematics, vol. 61, no. 8, pp , Nov [18] R. Negi and J. Cioffi, Pilot tone selection for channel estimation in a mobile OFDM system, IEEE Trans. Consm. Electron., vol. 44, no. 3, pp , Ag [19] I. Barhmi, G. Les, and M. Moonen, Optimal training design for MIMO OFDM systems in mobile wireless channels, IEEE Trans. Sig. Proc., vol. 51, no. 6, pp , Jne 003.