Performance of Compressive Sensing Technique for Sparse Channel Estimation in Orthogonal Frequency Division Multiplexing Systems

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1 Performance of Compressve Sensng Technque for Sparse Channel Estmaton n Orthogonal Frequency Dvson Multplexng Systems P. Vmala 1 and G.Yamuna 1 Assstant Professor, Professor, Department of Electroncs and Communcaton Engneerng, Faculty of Engneerng and Technology, Annamala Unversty Annamala Nagar, Chdambaram, Taml Nadu. 1 vmalakathrau@gmal.com yamuna.sky@gmal.com Abstract Orthogonal Frequency Dvson Multplexng s a wdely adopted mult carrer modulaton n wreless communcaton systems due to ts effectve transmsson and effcent bandwdth utlzaton ablty. Wreless systems wth coherent data detecton requre the estmaton of channel at the recever. Commonly employed plot aded channel estmaton probes the channel wth known sequence called plots and process the output to estmate the channel wth lnear reconsructon technques lke LS and MMSE. Wreless channels encountered n practce exhbts sparse structure that are havng only a few domnant and many zeros coeffcents. A recent development n Compressed Sensng (CS) has encouraged the extensve search on the applcaton of sparse recovery algorthm to channel estmaton. CS provdes a constructve way to explot the channel sparsty whch reduces the number of plots and hence ncrease spectral effcency. Sparse channel estmaton performed usng sparse recovery algorthms provde better bt error rate compared wth tradtonal LS and MMSE technques. Further the qualty of CS based sparse recovery algorthms depend on coherence of plot structure therefore the plot structure sgnfcantly affects the performance. To prove the effcacy of sparse recovery algorthm over plot structure, performance of sparse recovery algorthm s evaluated for tradtonal combo and random plot patterns generated. The random plot wth reduced mutual coherence has acheved better performance usng sparse recovery algorthm. Keywords - Compressed sensng, Coherence of Plot Pattern, Orthogonal Frequency Dvson Multplexng, Sparse Channel Estmaton I. INTRODUCTION Orthogonal Frequency Dvson Multplexng (OFDM) has been wdely adopted due to ts hgh data rate, effcent spectral utlzaton and ablty to cope wth multpath fadng channel. Coherent recepton n dgtal wreless systems, need to obtan channel mpulse response accurately at the recever [1]. Estmaton of channel s a challengng problem n wreless communcaton systems. The two methods commonly employed to estmate the multpath channel at the recever are plot aded and blnd. The plot aded method estmates the channel by nsertng known sequence called plots nto some prefxed postons of subcarrers. The correspondng channel outputs are processed to estmate the channel parameters at plot subcarrer locatons and employ nterpolaton technques to estmate the channel at data subcarrers. Tradtonal plot aded methods, often comprsng of lnear Least Squares (LS) and Mnmum Mean-Square channel estmators (MMSE). It makes lower spectral effcency and power utlzaton wth buy back of quck response to the channel varaton []. On the other hand, the blnd method s based on the statstcs of channels and requres a large number of receved OFDM symbols to extract statstcal propertes. Furthermore ther performance s worse than the plot aded method. Therefore plot aded channel estmaton s wdely employed n modern wreless communcaton systems. Dependng upon the arrangement of plots among the subcarrers, the popular plot structures adopted n plot aded channel estmaton are block, combo and lattce type and among the plot structures combo type yelds better results by tradtonal channel estmaton methods [3]-[4]. The new standards such as 3GPP Long Term Evoluton (LTE) has been developed for hgher bandwdth demands n moble communcaton systems. These systems requre comparatvely large bandwdths, whch make t possble to resolve the ndvdual propagaton paths from transmtters to recevers. Ths result n a channel mpulse response wth only few domnant channel coeffcents and remanng coeffcents are zero or approxmately zero that s termed as sparse channel. Ttradtonal plot aded channel estmaton methods rely on lnear reconstructon methods and that are ncapable to explot the nherent sparsty of channels. The key dea of DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar

2 Compressve Sensng (CS) explots the sparsty of the channels whch mproves the spectral and energy effcency effectvely by reducng the requred number of plots [5]. Advances n the feld of CS have ganed a much nterest n sgnal processng and can be appled to sparse channel estmaton. We conduct a comparatve performance analyss for sparse channel estmaton usng tradtonal methods and CS reconstructon methods n terms of bt error rate usng combo plot structure. It s also analyzed that the effcency of CS methods are based on plot structure. The remander of the paper s organzed as follows. Secton provdes a descrpton about the compressed sensng technques. The secton 3 provdes about the OFDM system, frequency selectve channel model and system model for spare channel estmaton. The secton 4 descrbes conventonal and CS based channel estmaton methods. The Secton 5 presents the smulaton of channel estmaton and secton 6 descrbes concluson of the work. II. COMPRESSED SENSING THEORY The theory of CS extends the concept of tradtonal Nyqust samplng applcable only to band lmted sgnals to broader class of sgnals. CS theory states that certan sgnals can be recovered from far fewer samples and measurements than tradtonal methods [6]. CS reles on two prncples: sparsty and suffcent condton for recovery from sparse sgnals. n The measurement model of unknown sgnal vector x R usng matrx representaton s gven by y=ax (1) where y s measurement vector and A s an m n sensng matrx. Usng the tradtonal lnear measurement method, one needs m measurements such that m n. Theory of CS makes use of the fact that many natural sgnals have few non zeros entres or compressble to few non zeros and are referred as sparse. The sparsty expresses a key dea that the nformaton rate of a contnuous tme sgnal may be much smaller than ts suggested bandwdth. The sparse recovery from CS suggests that successful recovery of x s possble even when m s much smaller than n f x has few nonzero entres. A suffcent condton guaranteeng the perfect recovery of the sparse sgnal x from the model represented n equaton (1) s expressed n terms of Restrcted Isometry Property (RIP). A sensng matrx A satsfes the RIP of order K f there exsts a constant δ such that 1 x 1 Ax x () for any K sparse vector ( x K). The mnmum of all constants δ satsfyng the above condton s called 0 sometry constant δ K. But verfyng a sensng matrx A for RIP nvolves combnatoral computaton complexty, other than RIP the wdely used condton s mutual ncoherence [7]. However A satsfes the RIP ncurs tremendous computatonal complexty because t does not have proper lnear program. So an alternatve to RIP and smple to calculate the condton for A s the mutual coherence. The mutual coherence of a matrx A s the largest absolute nner product between any two columns a, a j of A. It can be wrtten as ( A ) max a, a (3) 1 jl and a smaller value of (A), wll lead to a more accurate recovery of x. Under above condtons of the matrx A, CS ndcates that as long as the unknown sgnal x s reasonably sparse, t s possble to recover x by ts reconstructon algorthms. III. SYSTEM AND CHANNEL MODEL OFDM s a multcarrer modulaton scheme that allows large number of closely spaced orthogonal sub carrer sgnals to carry data on parallel data streams. Each sub carrer s modulated wth conventonal modulaton scheme such as QAM or PSK at low symbol rate, mantanng total data rate smlar to conventonal sngle carrer modulaton schemes n the same band wdth. The modulated data are nserted wth plots and converted nto N parallel data streams n whch each sub-carrer conssts of ether data or plot symbol. After mplementng Inverse Fast Fourer Transform (IFFT), the th dscrete tme doman OFDM symbol can be expressed as x ( n) N 1 k0 X [ k] e jkn N j for n=0,1,,.,n-1 (4) where k represents the sub-carrer ndex. After addng cyclc prefx, the symbol x t s sent to a frequency selectve mult-path fadng channel and the receved symbol n dscrete doman can be represented as y t (n)=x t (n)*h(n)+w(n) (5) DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar

3 where w(n) s the Addtve Whte Gaussan Nose (AWGN) n tme doman and h(n) s the channel mpulse response. At the recever, after removng the cyclc prefx, the receved symbols are demultplexed usng a FFT block by 1 Y[ k] N N 1 n0 y( n) e j kn N Then parallel data forms are converted nto seral and demodulated. The channel s modeled as frequency selectve and can be expressed as L h( n) (( n ) T ) (7) 1 where (.) s a drac delta functon, L resolvable paths and each path has complex path gan and delay spread wth T s samplng nterval. The ever ncreasng demands of data rate, leads to decrease the samplng nterval T s compared to maxmum delay spread of the channel whch results n a channel wth few domnant coeffcents. The channel coeffcent vector h has only K domnant coeffcent, h s K sparse vector. CS technques explot the sparsty of such wreless channels [8]-[9]. Consder a plot aded N subcarrers OFDM system for sparse channel estmaton. N p subcarrers used for plot transmsson and N d = N-N p subcarrers used for data transmsson among N subcarrers. The plot postons n subcarrers are represented as [p 1,p,.,p Np ] where 1 p p... p N 1 Np. Accordng to CS, the estmaton of sparse channel at the recever can be modelled as [10] y A. h w (8) where, h=[h(1),h(),...,h(l)] T s channel mpulse response wth length L, w=[w(1),w(),...,w(n p )] T s an addtve whte Gaussan nose, y=[y(1),y(),...,y(n p )] T s a receved plot symbols among N p subcarrers of OFDM system. Sensng matrx A X. where X=dag{x(p 1 ), x(p ),., x(p Np )} s dagonal matrx of transmt plot symbols and FNp L F Np L s a dscrete Fourer sub-matrx constructed by selectng N p row denoted by plot locaton and L columns of full dscrete Fourer matrx. There s feasble soluton from the theory of CS, f channel vector h has only K domnant coeffcents such as K << L and rest of the coeffcents are zero. For ths case, the plots are kept less than the channel coeffcents (Np < L) whch sgnfcantly reduces plots and mproves the spectral effcency. Accordng to (3), f sensng matrx A holds mnmum coherence then, CS reconstructon algorthms yeld effcent reconstructon therefore; plot structure plays a major role n channel estmaton usng CS algorthms. IV. CHANNEL ESTIMATION TECHNIQUES The transmtted OFDM symbols over wreless communcaton channel are usually dstorted by the channel characterstcs. To compensate the channel effects at the recever, the coherent detectors requre knowledge of channel mpulse response. The model represented n (8) s lnear f N p > L. The Least-Square (LS) and Mnmum-Mean-Square-Error (MMSE) are popularly used tradtonal lnear technques for plot-aded channel estmaton n the OFDM systems at the recever. LS s a low complexty channel estmaton method however, wthout usng any knowledge of the statstcs of the channels. LS channel estmaton can be represented as Hˆ H 1 H 1 ( X X ) X Y X Y (9) LS Every component of the LS channel estmate at each subcarrer locatons k=0,1,,..,n-1 can be wrtten ˆ Y[ k] H LS [ k] (10) X[ k] Even though LS channel estmate suffers from a hgh MSE, t s wdely used for channel estmaton due to ts smplcty [11]. To mprove the LS channel estmaton s updated usng the weght matrx W defned as H ˆ ~ MMSE WX 1 Y WH (11) where W R H R, R ~ H H ~ s the auto correlaton matrx of H ~ and R ~ s the cross correlaton between HH orgnal channel vector and temporary channel estmate vector n frequency doman. These tradtonal plot aded channel estmaton reles on combo type plot arrangement snce t s senstve to frequency selectvty when comparng to the other arrangements. But these tradtonal technques are more or less relyng on lnear reconstructon strateges and because of ther dependence on lnear procedure; the number of plots used must be hgh. Regardless of choce of number of plots, the reconstructon error s also more f the 1 ~ ~ ~ H HH s (6) DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar

4 channel s sparse. The recently emerged CS explots the nherent sparsty of the wreless fadng channel and t depends on non lnear reconstructon procedure. CS based methods have two man advantages of reducng plot overhead and decreasng mean square error. If the knowledge of locatons of nonzero taps s known to the structural LS estmator then t gves best estmate. But the recever has no nformaton about the nonzero coeffcents locaton of h therefore; structural LS estmator cannot be realzed and CS based approach leads to optmal results. In plot aded channel estmaton, the plot subcarrers occupy a fracton of spectrum but they do not carry any nformaton and by reducng the number of plot subcarrers, the spectrum effcency can be ncreased. It s possble to estmate the channel coeffcents wth fewer numbers of plots usng CS based methods by consderng the sparsty of channel. The two man categores of CS based channel estmaton methods are l 1 mnmzaton and teratve greedy algorthms. The frst method s to mnmze the l 1 norm whch s based on lnear programmng technques. l 1 mnmzaton provdes accurate method for sparse sgnal recovery f sensng matrx A satsfes RIP. But the computatonal complexty of l 1 mnmzaton s hghly mpractcal. It leads to need of faster recovery algorthm that works n lnear tme. Recently proposed several low complexty teratve greedy algorthms rely on teratve approxmaton of the sgnals ether by dentfyng the support of the sgnal teratvely untl a convergence crteron s met or by fndng an mproved estmate of the sparse sgnal n each of the teraton that attempts to account for the msmatch to the measured data. Some greedy methods actually have the performance guarantees that are equal to those obtaned by convex optmzaton approaches. Two of the smplest and oldest greedy approaches are Orthogonal Matchng Pursut (OMP) and Iteratve Thresholdng. OMP s relable for reconstructng both sparse and near sparse sgnals [1]. OMP teratvely detect and estmate the locaton and value of the channel from the measurement y correctly wth hgh probablty. Iteratve method s usually faster than l 1 mnmzaton technques n the orders of magntude whle they may fall short of performance. Iteratve Thresholdng make correct measurements by soft or hard thersholdng from the nosy measurements of gven sparse sgnal. The number of teratons and problem set up at hand defnes the thresholdng functon. Sparse channel estmaton n OFDM system s smulated usng the teratve greedy OMP. Smplcty and compettve reconstructon performance s the sgnfcance of OMP algorthm. The smplest guarantee for OMP states that for exactly K-sparse h wll be recovered exactly n K teratons [13]-[14] f matrx A satsfyng ether the RIP or coherence. The basc dea of OMP algorthm to recover the sparse sgnal s to select a column from A at each teraton untl t reach the sparse degree K. OMP starts by selectng a column of A that s most correlated wth the measurement y. Ths step of the algorthm s repeated by correlatng the columns wth the sgnal resdual that s obtaned by subtractng the contrbuton of a partal estmate of the sgnal from the orgnal measurement vector. Algorthms contaned n ths category nclude Compressve Samplng Matchng Pursut (CoSaMP), Regularzed Orthogonal Matchng pursut (ROMP) and Stagewse Orthogonal Matchng pursut (StOMP). Among all these algorthm OMP s relable for reconstructng both sparse and near sparse sgnals. V. SIMULATION RESULTS Plot aded OFDM system s constructed usng MATLAB verson 7.1 tool wth parameters furnshed n Table I. Typcal sparse channel wth 5 domnant coeffcents over maxmum of 50 coeffcents s gven n Fg. 1. TABLE I. System Parameters Parameters Value Number of Total Subcarrers 56 Number of Plot Subcarrers 3 and 16 Number of Cyclc Prefx 64 Maxmum Number of Channel Coeffcents 50 Number of Nonzero Channel Coeffcents 5 Modulaton QPSK DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar

5 Magntude of coeffcent Channel coeffcent Fg. 1. Typcal Sparse Channel Sparse channel estmaton s performed wth tradtonal LS, MMSE and CS based OMP algorthm usng conventonally best combo type plots. Bt error rate performances of channel estmaton are evaluated for plot of length 16 and 3 for QPSK modulaton scheme and are presented n Fg. and 3. OMP algorthm acheves better performance when compared wth conventonal LS and MMSE algorthms, but bt error s only on the order of and n both the plots snce the coherence of combo plots s hgh. The performances of CS based methods are based on mutual coherence of the plot structure therefore a random plot structures are generated and the coherence of conventonal combo plot structure compared wth random plot structures and are presented n Fg. 4 as bar graph. The Fg. 5 shows the performance comparson for channel estmaton wth combo and random plot structure of length 16 nterms of bt error rate. The random plot structure has acheved mnmum bt error rate rangng about 10-3 for sgnal to nose rato of 30 db when compared wth conventonal combo plot because of reduced coherence. CS algorthms provde better reconstructon performance for sparse channel f mutual coherence of the plot structure s mnmum LS CE MMSE CE omp CE Bt Error Rate SNR n db Fg.. Bt Error Rate Vs Sgnal to Nose Rato for LS, MMSE and OMP Channel Estmatons wth 16 Plots DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar

6 LS CE MMSE CE omp CE Bt Error Rate SNR n db Fg. 3. Bt Error Rate Vs Sgnal to Nose Rato for LS, MMSE and OMP Channel Estmatons wth 3 Plots Fg. 4. Mutual Coherence Value for Combo and Random Plot Patters Combo Plot RandomPlot1 RandomPlot BER SNR n db Fg. 5. Bt Error Rate Vs Sgnal to Nose Rato usng OMP Channel Estmaton usng Combo and Random Plot Patterns wth 16 plots DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar

7 VI. CONCLUSION AND FUTURE WORK Tradtonal and CS channel estmaton technques have been appled for plot aded sparse channel estmaton n OFDM system. CS algorthm provdes better BER performance than tradtonal algorthms lke LS and MMSE algorthms. By explotng the sparse nature of the wreless channels, CS technque reduces the plot over head also. The sgnfcant mprovement n the performance of CS algorthm can be obtaned by plot structure f t acheves the smaller value of mutual coherence. Random plot generaton s not approprate for practcal systems and therefore selecton of plot structure that acheves mnmum value of mutual coherence would be promsng work for future technologes. REFERENCES [1] Y. L, Plot-Symbol-Aded Channel Estmaton for OFDM n Wreless Systems, IEEE Transactons on Vehcular Technology, vol. 49, no.4, pp , Aug [] M. Hseh and C. We, Channel estmaton for OFDM Systems Based on Comb-Type Plot Arrangement n Frequency Selectve Fadng Channels, IEEE Transactons on Consumer Electroncs, vol. 44, no.1, pp. 17-5, Feb. 00. [3] S. Coler, M. Ergen, A. Pur and A. Baha, Channel Estmaton Technques Based on Plot Arrangement n OFDM Systems, IEEE Transactons on Broadcastng, vol. 48, no. 3, pp. 3 9,Oct. 00. [4] Yong Soo Cho, Jeakwonkn, Won Young Yang and Chung G. Kang, MIMO OFDM Wreless Communcaton wth MATLAB, IEEE Press, John Wley & Sons Pvt. Ltd, 010. [5] Davd L. Donoho, Compressed Sensng, IEEE Transactons on Informaton Theory, vol.5, no. 4, pp , Apr., 006. [6] E.J.Candes, The Restrcted Isometry Property and ts Implcatons for Compressed Sensng, Comptes Rendus Mathematque, vol.346, no.9-10, pp , May 008. [7] R. Baranuk, M. Davenport, R. Devore, and M. Wakn, A Smple Proof of the Restrcted Isometry Property for Random Matrces, Constructve Approxmaton, vol. 8, no. 3, pp , Dec.008. [8] Georg Taubock and Franz Hlawatsch, A Compressed Sensng Technque for OFDM Channel Estmaton n Moble Envronments: Explotng Channel Sparsty for Reducng Plots, Proceedngs of IEEE Internatonal Conference on Acoustcs, Speech and Sgnal Processng Las Vegas, NV, March Apr [9] Waheed U. Bajwa, New Informaton Processng Theory and Methods for Explotng Sparsty n Wreless Systems, Ph.D. Dssertaton, Unversty of Wsconsn, Madson, WI, 009. [10] Chenhao Q and Lenan Wu, A Hybrd Compressed Sensng Algorthm for Sparse Channel Estmaton n MIMO OFDM Systems, Proceedngs of IEEE Internatonal Conference on Acoustcs, Speech and Sgnal Processng, Prague,Czech Republc, May 011, pp [11] Tan-Mng Ma, Yu-Song Sh, and Yng-Guan Wang, A Low Complexty MMSE for OFDM Systems over Frequency-Selectve Fadng Channels, IEEE Communcatons Letters, vol. 16, no. 3, pp , Mar. 01. [1] Joel A. Tropp and Anna C. Glbert, Sgnal Recovery from Random Measurements Va Orthogonal Matchng Pursut, IEEE Transactons on Informaton Theory, vol. 53, no. 1, pp , Dec [13] Arvnd Ganesh, Zhan Zhou and Y Ma, Separaton of a Subspace Sgnal: Algorthms and Condtons, Proceedngs of IEEE Internatonal Conference on Acoustcs Speech Sgnal Processng, Tape, Tawan, Apr. 009, pp [14] Jan Wang and Byonghyo Shm, On the Recovery Lmt of Sparse Sgnals usng Orthogonal Matchng Pursut, IEEE Transactons on Sgnal Processng, vol. 60, no.9, pp , Sept. 01. AUTHOR PROFILE P. Vmala receved her Bachelor Degree n Electroncs and Communcaton Engneerng from Jayaram College of Engneerng, Trchy, Taml Nadu, Inda n 000 and Masters Degree n Process Control and Instrumentaton Engneerng from Annamala Unversty, Chdambaram, Tamlnadu, Inda n 010. She s currently workng as Assstant Professor n the Department of Electroncs and Communcaton Engneerng, Faculty of Engneerng and Technology, Annamala Unversty. Her current research areas are Dgtal Sgnal Processng, Sparse Sgnal Processng, Dgtal Communcaton and Wreless Communcaton. G. Yamuna receved her Bachelor Degree n Electroncs and Communcaton Engneerng from Natonal Insttute of Technology (NIT), Trchy, Taml Nadu, Inda n She receved her Master Degree n Power System from Annamala Unversty n the year She receved her Ph.D. Degree n Electrcal Engneerng from Annamala Unversty n 010. She has publshed many techncal papers n natonal and nternatonal conferences and journals. Currently, she s workng as a Professor n the Department of Electroncs and Communcaton Engneerng, Faculty of Engneerng and Technology, Annamala Unversty, Taml Nadu, Inda. She s an Assocate Edtor and Revewer for several Natonal and Internatonal Journals. Her research area of nterest ncludes Sgnal Processng, Image Processng, Wreless Communcaton Systems, Antenna Desgn and Informaton Securty. DOI: /jet/017/v91/ Vol 9 No 1 Feb-Mar