Sparse Multipath Channel Estimation Using Compressive Sampling Matching Pursuit Algorithm

Transcription

1 IEEE APWC parse Multpath Channel Estmaton Usng Compressve amplng Matchng Pursut Algorthm Guan Gu,, Qun Wan, We Peng and Fumyuk Adach. Dept. of Electrc Engneerng, Unversty of electrcal cence and echnology of Chna, Chengdu, 673, Chna. Dept. of Electrcal and Communcaton Engneerng, Graduate chool of Engneerng, ohoku Unversty, enda, , Japan Correspondence author: Abstract-Wdeband wreless channel s a tme dspersve channel and becomes strongly frequency-selectve. However, n most cases, the channel s composed of a few domnant taps and a large part of taps s approxmately zero or zero. o explot the sparsty of mult-path channel (MPC), two methods have been proposed. hey are, namely, greedy algorthm and convex program. Greedy algorthm s easy to be mplemented but not stable; on the other hand, the convex program method s stable but dffcult to be mplemented as practcal channel estmaton problems. In ths paper, we ntroduce a novel channel estmaton strategy usng compressve samplng matchng pursut () algorthm whch was proposed n []. hs algorthm wll combne the greedy algorthm wth the convex program method. he effectveness of the proposed algorthm wll be confrmed through comparsons wth the exstng methods. I. INRODUCION Coherent detecton n wdeband moble communcaton systems often requres accurate channel state nformaton at a recever. he study of channel estmaton for the purposes of channel equalzaton has a long hstory. In many studes, densely dstrbuted channel mpulse response was often assumed. Under ths assumpton, t s necessary to use a long tranng sequence. In addton, the lnear channel estmaton methods, such as least square () algorthm, always lead to bandwdth neffcency. It s an nterestng study to develop more bandwdth effcent method to acqure channel nformaton. Recently, the compressve sensng (C) has been developed as a new technque. It s regarded as an effcent sgnal acquston framework for sgnals characterzed as sparse or compressble n tme or frequency doman. One applcaton of the C technque s n channel estmaton. If the channel mpulse response follows sparse dstrbuton, we can apply the C technque. As a result, the tranng sequence length can be shortened compared wth the lnear estmaton methods. Recent measurements show that the sparse or approxmate sparse dstrbuton assumpton s reasonable [, 3]. In other words, the wreless channels n real propagaton envronments are characterzed as sparse or sparse clustered; these sparse or clustered channels are frequently termed as a sparse mult-path channel (MPC). An example of MPC mpulse response channel s shown n Fg.. Recently, the study on MPC has drawn a lot of attentons and concernng results can be found n lterature [4-6]. Correspondngly, sparse channel estmaton technque has also receved consderable nterest for ts advantages n hgh bt rate transmssons over multpath channel [7]. Channel Ampltude Domnant aps Length of MPC Fg.. An example of MPC. Channel length s 8. he domnant taps dstrbute unformly and the number of domnant taps s 8. It was necessary to state that we consder the real baseband channel model whle neglect the magnary part for smplfcaton. Explotng the sparse property of MPC, orthogonal matchng pursut () algorthm [8, 9] and convex program algorthm [] have been proposed. algorthm s fast and easy to be mplemented. However, the stablty of for sparse sgnal recovery has not been well understood yet. Donoho et al [] suggested that should be less stable than the convex program. ropp and Glbert [9] nvestgated the performance of algorthm by the measurement. It s reported n [9] that f the channel matrces satsfy some propertes, then algorthm can recover the sparse sgnals wth hgh probablty. However, the condtons on estmaton algorthm are more restrctve than the restrcted Isometry condton (RIC) []. Kuns and Rauhut [3] mathematcally proved that the frst teraton of can

2 parse Multpath Channel Estmaton usng Algorthm dentfy the maxmum domnant taps of channel wth the tranng sequences. However, because of the unavodable correlaton between columns of the tranng sequence, the nstablty of the algorthm easly leads to weak channel estmaton. Convex program method can resolve the nstablty of algorthm. Convex program algorthm, such as Dantzg elector () [4], s based on lnear programmng. he man advantage of convex program method s ts stablty and hgh estmaton accuracy. he convex problem method can work correctly as long as the RIC condtons are satsfed. However, ths method s computatonally complex and dffcult to be mplemented [5]. In ths paper, we wll ntroduce a novel MPC estmaton method usng compressve samplng matchng pursut () algorthm []. It has both the advantages of the greedy algorthm and the convex program. In other words, algorthm combnes low computatonal complexty and robustness on practcal channel estmaton. he study n [] focused on mathematcal descrpton on algorthm for sparse or approxmate sparse sgnal recovery problem. And the perfect channel state nformaton (CI) was assumed whle practcal channel estmaton was not consdered. In ths paper, we wll use the algorthm to deal wth the practcal channel estmaton problems. he rest of the paper s organzed as follows. parse multpath channel model s presented n ecton II. ecton III wll descrbe the exstng algorthm and propose a new MPC estmaton method by usng the algorthm. In secton IV, we wll compare the performance of the proposed method wth the exstng methods by smulatons. Fnally, conclusons are drawn n ecton V. II. MPC MODEL At frst, the symbols used n ths paper are descrbed as followng. he superscrpt stands for transposton. Bolded captal letters denote a matrx where bolded lowercase letters represent a vector. Notaton stands for the absolute value. Norm operator denotes vector norm,.e., the number of non-zero entres of the vector; denotes vector norm, whch s the sum of the absolute values of the vector entres. denotes norm. h and h ndcate estmate channel vector and actual channel vector, respectvely. We consder sngle-antenna wdeband propagaton systems, whch are often equvalent to frequency-selectve baseband N L channel model. An N-length tranng sequence X R ( N < L) wll be transmtted over a random statonary MPC. he equvalent baseband transmtted X and receved sgnals y = [ y y yn ] s gven by y[] t = x[ t τ][, h t τ] + z[] t () τ where τ s delay spread of multpath sgnal whch s characterzed by channel length L. Its matrx form can wrtten as Where y = Xh + z () x x x( N ) x( L ) x x x ( ) x N ( L ) X = x( N ) x( N ) x( N )( N ) x ( N )( L ) denotes the tranng sequence and read ts as row vector form X= [ x, x,..., x N ] ; h = represents the number of domnant channel taps of the MPC. uppose that there are domnant channel taps dstrbuted randomly over the channel, accordng to the MPC defnton, L ; z () t = [ z z zn ] s the addtve whte Gaussan nose (AWGN) wth zero mean and varance σ. III. COAMP ALGORIHM FOR MPC EIMAION In ths part, we wll ntroduce some propertes of compressed sensng (C) theory as a bass for the channel estmaton method. And then we wll show how to apply the algorthm to sparse channel estmaton. A. CREED ENING Consder the baseband channel model of (). If we want to guarantee accurate channel estmator, the tranng sequence X must satsfy two condtons. ) Restrcted Isomery property (RIP) []. he -RIC of a N L tranng sequence X, denoted byδ, s defned as the smallest value δ ( δ (,)) whch can satsfy the nequalty ( δ ) ( ) h Xh + δ h (3) for any MPC vector h. If () s satsfed, the tranng sequence X s sad to satsfy RIP of order and accurate channel estmator can be obtaned by usng C methods. From C perspectve, research on the RIP of the tranng sequence has two mportant purposes. Frst, RIP-based tranng sequence s a suffcent condton to robust probe sparse channel domnant taps. Furthermore, n the process of error performance analyss, RIC of tranng sequences play mportant role to mprove lower bound. ) Lower bound of length of tranng sequence X Due to the channel fadng and nose, how to determne the length of tranng sequence X s mportant n terms of both spectrum effcency and estmaton robustness. herefore, the length N of X must satsfy [6] ( ) 4 N C log L μx, (4) where C s a constant and μ X = L max, j X, j whch s known as the maxmum coherence between the th -column and j th -column of X B. MPC heorem. If N-length tranng sequence X satsfes the RIP and δ [], for any -sparse channel vector h, algorthm produces the channel estmator h % that satsfes h h % C max ε, h h % + z (5) { }

3 IEEE APWC for a gven parameter ε. And h % s a best -sparse approxmaton to h. Followng theorem, the prevously mentoned channel estmaton algorthm selects the maxmum tap of a sparse channel durng one teraton and the channel estmaton s carred out n an teratve way. Whle the proposed MPC wll select the entre domnant taps n each teraton and reduce the estmaton error teraton by teraton. Based on the channel model n (), the MPC method can be carred out n fve steps, as shown n Fg.. performance of other exstng algorthms such as,, and algorthms wll also be evaluated. In addton, the ME wth channel estmaton (known poston of domnant taps) s also evaluated as a reference. he smulaton condton s lsted n able. ab. mulaton condton Lnear method Estmaton methods Channel fadng C theory based method (ntroduced) Frequency-selectve fadng Channel length L 5 aps ampltude [.,.] U [.,.] No. of domnant taps 5 NR ranng sequence X db oepltz structure Length of X 5~45 Fg.. teps of MPC estmaton. he functon of step s to dentfy the postons of the domnant taps. hs step can be dvded nto two sub-steps. At frst, set P=Xr and choose maxmum domnant taps. he postons of selected domnant taps are denoted by Ω P. In the next, usng least square () method to calculate a channel estmator as h = arg mn y-xh, and select maxmum domnant taps from h. he postons of selected domnant taps n ths sub-step are denoted by Ω. he postons of domnant taps are merged by Ω=Ω P Ω n step. In step 3, calculate channel estmate h = X y Ω Ω on postons n Ω. In step 4, replace the non-domnant taps by zero. In step 5, update channel estmaton error, f the stoppng crteron { } : 4 h 4 h s satsfed, then output the channel estmaton result; otherwse repeat steps ~5. IV. IMULAION REUL AND DICUION In ths secton, the mean square error (ME) performance of the MPC estmaton method wll be evaluated by smulatons. For the purpose of comparson, the ME In the followng, we wll show the results of ME performance and computatonal complexty performance accordng to the CPU tme of laptop. A. Estmaton Error he ME s defned as M M m = ME = h-h % (6) It s obvous that smaller ME means more accurate channel estmaton and vce versa. he ME performance comparsons between the PMC estmaton method and the exstng estmaton methods are shown n Fg. 3 ~ 4. ME m Fg.3. ME of the overall taps at NR=dB. 3

4 parse Multpath Channel Estmaton usng Algorthm - computatonal complexty. Our smulatons are performance n MALAB 7 envronment usng a.4ghz Intel Core- processor wth GB of memory and under Mcrosoft XP 3 operatng system. ME Fg.4. ME of the domnant taps at NR=dB. Fg. 3 shows the result of (6) when all the taps of the channel are consdered and Fg. 4 shows the comparson result of (6) when only the domnant taps are consdered. It s found that when the tranng sequence length longer than 5, the method acheves better ME performance than the other exstng methods. hs means that the method can acheve the same ME performance by usng shorter tranng sequence. In other words, the method s more bandwdth effcent. Here, t was necessary to state that f the length of tranng sequence less than 5, the ME performance of s worse than whch was caused by followng reason: s a convex optmzaton algorthm and thus t converts to lnear program to resolve. Whle the s a support set estmaton whch apples hard threshold by selectng the largest domnant channel taps of a channel vector by applyng Least quare () on every teratve step. If the tranng sequence s very short and then estmate channel unstable on the range of hard threshold. o avod ths deteroratve ME performance on practcal, there have two potental schemes to mtgate. For one thng, we can relax the hard threshold at the cost of acceptable computatonal complexty. For other thng, we should guarantee lower bound of tranng sequence length so that robust estmaton of the. On the future work, adaptve threshold wll consder and further mprove channel estmator. o further study the ME performance, the cumulatve densty functon (CDF) by usng dfferent methods are compared and shown n Fg. 5 ~ 6. It can be clearly observed that the CDF curve of the method s very close to the lower bound ( channel estmator wth known poston of domnant taps) and much better than the other estmaton methods. B. Rough Estmaton of Computatonal Complexty o study the computatonal complexty of the ntroduced algorthm, we have evaluated the CPU tme n second to complete the channel estmaton for NR=dB. It s worth mentonng that although the CPU tme s not an exact measure of complexty, t can gve us a rough estmaton of CDF (%) CDF (%) CPU tme (second) ME Fg.5. ME CDF of the overall taps at NR=dB ME of domnant taps Fg.6. ME CDF of the domnant taps at NR=dB Fg.7. CPU tme for channel estmaton he comparson between,, and algorthms s shown n Fg. 7. It s seen that the computng tme of the algorthm s less than.5 seconds for both and, whle the computng tme of the

5 IEEE APWC algorthm s more than.3 seconds. It s shown that the method s also computatonally effcent. From the Fg. 7, we also fnd that take fewer computatonal tme than. Because of the MPC estmaton selects all domnant channel taps whle the MPC estmaton chooses maxmum channel tap on each teraton. V. CONCLUION AND FUURE WORK In ths paper, we have ntroduced a novel sparse channel estmaton method based on the C theory. he method has both advantages of the greedy algorthm and convex program algorthm. It has been shown that, when compared wth the exstng algorthms, our ntroduced method s both bandwdth and computatonally effcent. However, channel estmaton stll exst a potental mprovement gap. Because of the algorthm consdered the hard threshold to choose the set of domnant taps n Fg. EP-. On future work, we wll consder an adaptve channel estmaton method whch senses the random nose and other unexpected nterferences. ACKNOWLEDGEMEN hs work s supported n part by the Natonal Natural cence Foundaton of Chna under grant 67746, the Natonal Hgh echnology Research, Development Program of Chna (863 Program) under grant 8AAZ36 as well as n part by the Key Project of Chnese Mnstry of Educaton under grant 939, Chna cholarshp of Chna cholarshp Councl (CC) under grant No It s also supported n part by ohoku Unversty Global COE program "Global Educaton and Research Center for Earth and Planetary Dynamcs". REFERENCE [] D. Needell and J. A. ropp, ": Iteratve sgnal recovery from ncomplete and naccurate samples," Appled and Computatonal Harmonc Analyss, vol. 6(3), pp. 3-3, 8. [] Z. Yan, M. Herdn, A. M. ayeed, and E. Bonek, "Expermental study of MIMO channel statstcs and capacty va the vrtual channel representaton," ech. Rep., Unv. Wnsconsn-Madson, avalable meas.pdf., Feb. 7. [3] J. Kvnen, P. uvkunnas, L. Vuokko, and P. Vankanen, "Expermental nvestgatons of MIMO propagaton channels," Antennas and Propagaton ocety Internatonal ymposum, IEEE,. [4] W. F. chreber, "Advanced televson systems for terrestral broadcastng: ome problems and some proposed solutons," IEEE Proc., vol. 83, pp , Jun [5] R. teele, "Moble Rado Communcatons," IEEE Press, 99. [6] M. Kocc, D. Brady, and M. tojanovc, "parse equalzaton for real tme dgtal underwater acoustc communcatons," OCEAN'95, an Dego CA, pp. 47-4, Oct [7] C. Carbonell,. Vedantam, and U. Mtra, "parse channel estmaton wth zero tap detecton," IEEE ransactons on Wreless Communncatons, vol. 6(5), pp , May 7. [8] Z. G. Karabulut and A. Yongacoglu, "parse channel estmaton usng orthogonal matchng pursut algorthm," 4 IEEE 6th Vehcular echnology Conference, vol. 6(6), pp , 4. [9] J. A. ropp and A. C. Glbert, "gnal recovery from random measurements va orthogonal matchng pursut," IEEE ransacton on Informaton heory, vol. 53(), pp , 7. [] U. W. Bajwa, J. Haupt, G. Raz, and R. Nowak, "Compressed channel sensng," o appear n Proc. 4nd Annu. Conf. Informaton cences and ystems (CI'8), Mar. 9-, 8. [] D. L. Donoho, M. Elad, and V. N. emlyakov, "table recovery of sparse overcomplete representatons n the presence of nose," IEEE ransacton on Informaton heory, vol. 5(), pp. 6-8, Jan. 6. [] E. J. Candès, "he restrcted sometry property and ts mplcatons for compressed sensng," Compte Rendus de l'academe des cences, Pars, vol. ere I, 346, pp , 8. [3]. Kuns and H. Rauhut, "Random samplng of sparse trgonometrc polynomals II - orthogonal matchng pursut versus bass pursut," Found. Comput. Math., vol. 8, pp , Dec. 8. [4] E. Candès and. ao, "he Dantzg selector: tatstcal estmaton when p s much larger than n," Annals of tatstcs, vol. 35, pp , 6. [5] N. H. Nguyen and. D. ran, "he stablty of regularzed orthogonal matchng pursut algorthm," [6] E. Candès, J. Romberg, and. ao, "Robust uncertanty prncples: Exact sgnal reconstructon from hghly ncomplete frequency nformaton," IEEE ransacton on Informaton heory, vol. 5(), pp , Feb. 6. 5