APPROXIMATE FULL DIVERSITY LINEAR EQUALIZERS FOR DOUBLY SELECTIVE CHANNELS

Transcription

1 17th Europea Sigal Processig Coferece (EUSIPCO 2009 Glasgow, Scotlad, August 24-28, 2009 APPROXIMATE FULL DIVERSITY LINEAR EQUALIZERS FOR DOUBLY SELECTIVE CHANNELS Shakti Prasad Sheoy, Irfa Ghauri, ad Dirk T.M. Slock EURECOM, Dept. of Mobile Commuicatios, 2229 Route des Crêtes, Sophia Atipolis Cedex, Frace Ifieo Techologies Frace SAS, GAIA, 2600 Route des Crêtes, Sophia Atipolis Cedex, Frace ABSTRACT It is kow that liear miimum mea square error zeroforcig (MMSE-ZF equalizatio ca achieve full joit multipath-doppler diversity offered by doubly selective chaels [1] whe appropriate precodig is applied at the trasmitter. However, brute force implemetatio MMSE- ZF equalizer ivolves a matrix iversio operatio that results i sigificat computatioal burde. I order to alleviate this problem, first a iterative implemetatio of MMSE- ZF equalizer based o polyomial expasio (PE approximatio is proposed. The, the structure of a matrix ivolved i this approximatio is exploited to reduce the computatioal complexity of the PE approximatio. Simulatio results are provided to show that while this approach reduces the computatioal complexity compared to the brute-force implemetatio of the MMSE-ZF equalizer, it does ot effect the diversity order. 1. INTRODUCTION Practical wireless commuicatio chaels are proe to sigal fadig due to the presece of multiple sigal paths (time dispersive chael, time-varyig ature of the chael (frequecy dispersive chael or both (time-frequecy dispersive or the so called, doubly dispersive chael. However, its is possible for the receiver to employ equalizatio techiques that optimally exploit the iheret diversity i these chaels as a coveiet couter-measure agaist fadig. For istace, the frequecy selectivity of time dispersive chaels provide multipath diversity due to the presece of multiple idepedetly fadig compoets i the chael. I block trasmissio systems, whe the chael coherece time is shorter tha the trasmit block legth, temporal variatios of the chael provides Doppler diversity [2] which ca be exploited by the receiver. Doubly selective chaels offer joit multipath-doppler diversity that ca be haressed by suitable equalizatio techiques ad proper precodig. I [3], the authors used the Complex-Expoetial Basis Expasio Model (CE-BEM [4] with Q+1 basis fuctios to model the doubly selective chael of memory L ad showed that by employig liear precoded block trasmissio, the maximum diversity i the chael is upper bouded by (Q + 1(L + 1 ad ca be achieved whe maximumlikelihood equalizatio (MLE is used at the receiver. However, MLE icurs a huge computatioal complexity therefore the diversity order achieved by liear equalizatio (LE which is a low-complexity albeit sub-optimal alterative to optimal maximum-likelihood equalizatio (MLE was ivestigated i [5] [6] [1]. It is ow kow that liear miimum mea squared error zero forcig (MMSE-ZF receivers achieve maximal diversity offered by time/frequecy ad doubly selective chaels. Recogizig the fact that the structure of MMSE-ZF receivers ca be exploited i order to reduce the complexity of these full-diversity achievig receivers, low complexity liear equalizers were proposed i [7] ad [8] for frequecy selective chaels. Sice the computatioal complexity i the MMSE-ZF receiver is predomiatly due to the matrix iversio ivolved i buildig the equalizer, we propose a reduced-complexity implemetatio of the MMSE-ZF receiver based o polyomial expasio (PE approximatio. PE approximatio is ot a ew idea, it was first proposed i [9] where, i order to avoid explicit matrix iversio, the Cayley Hamilto theorem [10] was applied to express the matrix iverse as a fiite sum of weighted matrix polyomials. The weights themselves were chose to optimize a desired performace metric at the output of the equalizer. I this paper, we adopt a approach where the complexity reductio is achieved by first approximatig the MMSE-ZF receiver usig PE approximatio a the exploitig the structure of a matrix ivolved i the PE approximatio. I the sequel, we will explai our approach i more detail ad show that a decrease i the computatioal effort is achieved with o loss o the diversity order of the resultig approximated equalizer. 2. SIGNAL MODEL I Fig. 1 we show the block diagram of the trasmissio model. At the trasmitter, complex data symbols s[i] are s[i] Parser s[k] x[k] y[k] Θ h i,l Equalizer Figure 1: Block diagram of trasmissio model. first parsed ito N-legth blocks.the -th symbol i the k-th block is give by [s[k]] = s[kn + ] with [0, 1,..., N 1]. Each block s[k] is precoded by a M N matrix Θ where M N ad the resultat block x[k] is trasmitted over the block fadig chael. We cosider a chael memory of order L. It is well kow that the temporal variatio of the chael taps i doubly selective chaels with a fiite Doppler spread ca be captured by fiite Fourier bases. We therefore use CE-BEM [4] with Q + 1 basis fuctios to model the time variatio of each tap i a block duratio. The basis coefficiets remai costat for the block duratio but are allowed to vary with every block. The time-varyig chael for each block trasmissio is thus completely described by the Q + 1 Fourier bases ad (Q + 1(L + 1 coefficiets. ŝ[i] EURASIP,

2 I geeral Q is chose such that Q = 2 f max MT s where 1/T s is the samplig frequecy ad f max is the Doppler spread of the chael. The coefficiets themselves are assumed to be zero-mea complex i.i.d Gaussia radom variables. Usig i as the discrete time (sample idex, we ca represet the l-th tap of the chael i the k-th block h i,l = h q (k, le j2πfqi, l [0, L], f q = (q Q/2/M. The correspodig receive sigal is formed by collectig M samples at the receiver to form [y(km + 0, y(km + 1,...,y(kM + M 1] T. Whe M L, this block trasmissio system ca be represeted i matrix-vector otatio as [3] H ds [k; 0]Θs[k] + H ds [k; 1]Θs[k 1] + v[k], (1 where v[k] is a AWGN vector whose etries have zeromea ad variace σv 2 ad is defied i the same way as y[k]. H ds [k; 0] ad H ds [k; 1] are M M matrices whose etries are give by [H ds [k; t]] r,s = h (km+r,tm+r s with t [0, 1], r, s [0,..., M 1]. Defiig D[f q ] as a diagoal matrix whose diagoal etries are give by [D[f q ]] m,m = e j2πfqm, m [0, 1,..., M 1], ad further defiig [H q [k; t]] r,s = h q (k, tm + r s as Toeplitz matrices formed of BEM coefficiets, it is straightforward to represet Eq. (1 as 1 t=0 D[f q ]H q [k; t]θs[k t] + v[k], (2 3. PRECODED TRANSMISSION IN DOUBLY SELECTIVE CHANNELS The precodig matrix Θ cosidered here is give by Θ = F H P+Q T 1 T 2. where represets the Kroecker product of matrices, F P+Q is a (P + Q-poit DFT matrix, T 1 = [I P 0 P Q ] T, T 2 = [I K 0 K L ] T. P ad K are chose such that M = (P + Q(K+L ad N = PK. This precoder was proposed i [3] ad was show to eable diversity order of (Q + 1(L + 1 for ML receivers i doubly selective chaels. Whe this precoder is applied at the trasmitter, the received sigal ca be represeted as (F H P+Q I K+L H[k]s[k] + v[k], (3 where H[k] is give by (see Sec. 7 for derivatio. (J P+Q [q]t 1 (D K+L [f q ] H q [k; 0]T 2. (4 I the followig we will drop the block idex k i the iterest of simplifyig otatios. For this scheme, it was show i [1] that MMSE-ZF equalizatio collects full diversity offered by the chael. I other words, the slope of the outage probability curve of the MMSE-ZF receiver i the high-snr regime is (Q+1(L+1. This beig the case, it is of iterest to explore the possibility of lowerig the complexity of such a receiver sice brute-force implemetatio of the MMSE- ZF receiver ivolves the computatio of the matrix iverse (H H H 1 which is computatioally expesive. 4. POLYNOMIAL EXPANSION APPROXIMATION FOR LE IN DOUBLY SELECTIVE CHANNELS I order to reduce the above metioed computatioal cost, we propose here a reduced complexity implemetatio of the MMSE-ZF equalizer for doubly selective chaels. We start with a alterative represetatio of the received sigal i Eq. (3 y = H tv Θs + v, where H tv represets the chael matrix i the time-domai ad ca i tur be represeted as the sum of two matrices H tv = H κ + H ν, H κ = (D P+Q [f q (K + L] e j ωq H q, H ν = (D P+Q [f q (K + L] (D K+L [f q ] e j ωq I K+L H q. ω q = ω q (K + L 1/2. Represetig the received sigal i this form allows us to iteratively estimate the trasmit symbol vector s. The symbol estimate after the m-th iteratio is give by ŝ (m = (H κ Θ (y H ν Θŝ (m 1. (5 where the superscript represets the Moore-Perose pseudo-iverse. From Eq. (5, we ca derive the sigal to iterferece oise ratio (SINR expressio for the -th symbol of the symbol estimate ŝ (m as ρ[g s G H s SINR = ], ρḡḡ H + [G v G H ], (6 v, where ρ is the sigal to oise ratio (SNR ad ḡ is the -th row of G s without the elemet [G], ad G s = I + ( 1 m ((H κ Θ H ν Θ m+1, G v = ( ( 1 k ((H κ Θ H ν Θ k (H κ Θ. Alteratively, it is possible to evisage a polyomial expasio approximatio for the MMSE-ZF receiver that miimizes the mea squared error at the receiver. I this case, the symbol vector estimate after m iteratios is give by ŝ (m = Λ k R k z. (7 where R = (H κ Θ H ν Θ, z = (H κ Θ y, ad the diagoal scale factor matrices Λ k of order N are estimated by pluggig i the expressio for ŝ (m i Eq. (7 i the LMMSE criterio Λ opt k = arg mi E s ŝ (m 2. (8 Λ k :k {0,1,m} Note that Eq. (5 correspods to the special case of Eq. (8 where the diagoal elemets of all Λ k are uity. Aother special case of Eq. (8 where the diagoal matrices Λ k are 334

3 reduced to scalar weightig coefficiets λ k are addressed before (for istace i [9]. Let λ,k = [Λ k ], ad λ = [λ,0,, λ,m ] the Eq. (8 ca be solved by fidig the optimum λ opt separately for each trasmit symbol {0, 1,, N 1} i the symbol vector s as λ opt = argmi E s[] λ q[] 2. (9 λ q[] = [w 0 [] w 1 [] w m []] T ad w m [] are elemets of w m = R m z. Oce the N vectors correspodig to λ opt are obtaied the diagoal matrices Λ k are formed ad substituted i Eq. (7 to get the symbol estimate. The SINR at the output of this equalizer is give by SINR MMSE PE = ρ[g s G H s ], ρḡḡ H + [G v G H v ],, (10 where ḡ is ow the -th row of G s without the elemet [G], ad G s = G v = Λ k R k (I + (H κ Θ H ν Θ, Λ k R k (H κ Θ. Sice both Eq. (7 ad Eq. (5 require the calculatio of the pseudo-iverse (H κ Θ we ow focus our attetio to reducig the complexity of the matrix iversio that eeds to be performed i order to obtai (H κ Θ. Notice that H κ Θ ca be factored as show i (12 where we replace the blockcirculat-with-toeplitz-blocks (BCTB matrix i (11 with a block-circulat-with-circulat-blocks matrix (BCCB. i.e., H BCCB = (J P+Q [q] e j ωq H c q. where H c q is a circulat matrix whose first colum is the same as the first colum of H q, This allows us to take advatage of the fact that H BCCB is diagoalizable as D = (F H P+Q FH K+L H BCCB(F P+Q F K+L, where Θ N,F = N(Θ H, ad N(. deotes the ull space of a matrix. The solutio to Eq. (14 allows us to compute (DΘ F as Θ F D 1 [I D H Θ N,F (Θ H N,F D 1 D H Θ N,F 1 Θ H N,F D 1 ] (15 which ivolves iversio of a matrix of dimesio M N i place of iversio of matrix of dimesio M i the bruteforce approach. Moreover, Θ is oly depedet o the precodig matrix hece it ca also be precomputed ad used F across blocks. 4.1 Optimizatio of trasmissio parameters The reductio i computatioal complexity of (H κ Θ usig Eq. (15 is related to the redudacy i the trasmit block (i.e., M N. Oe ca therefore optimize the trasmissio parametersp, Q, K, L to reduce the computatioal complexity of overall PE approximatio of the MMSE-ZF equalizer. Such a optimizatio has to take ito accout the iterrelatios of the trasmissio parameters. For a fixed f max ad T s, we kow that L is fixed but Q icreases proportioally to M due to the fact that α = Q/M is fixed due to the relatio Q = 2 f max MT s. This also implies that the available diversity at the receiver icreases with M 5. NUMERICAL RESULTS I this sectio we provide simulatio results to show that the approximatio for the MMSE-ZF equalizer proposed i the paper does ot etail ay loss i terms of the diversity order collected by the receiver. It is well accepted that the slope of the outage probability curve i the high SNR regime provides a accurate estimate of the diversity order of a receiver. The outage probability curve was plotted by computig the post-equalizatio SINR (SNR for the case of brute for MMSE-ZF equalizer for the symbol idex correspodig to N/2 sice it suffers the maximum iter-symboliterferece withi the block. Mote-Carlo simulatios were carried out for a fixed trasmissio rate for differet SNR poits. For the brute-force implemetatio of the MMSE-ZF receiver, the post-equalizatio SNR for a arbitrary symbol idex i the symbol block s is give by SNR = ρ [(H H H 1 ],, Now pluggig this ito (12 we have where H represets the equivalet doubly selective chael after precodig ad ρ is the SNR. For the polyomial expasio equalizers, the post-equalizatio SINR was computed H κ Θ = (I P+Q F K+L DΘ F. as give by Eq. (6 ad Eq. (10. Whe the post-equalizatio where Θ F = (I P+Q F H Θ which i tur leads us to K+L SINR was below the SNR required to support the fixed trasmissio rate, the chael was declared to be i outage. (H κ Θ = (DΘ F (I P+Q F K+L H. (13 The slope of the outage probability curve thus obtaied provides a estimate of the diversity order. I additio to The problem of computig (H κ Θ is thus reduced to the this, we compare the slope of the receiver to that of the problem of computig (DΘ F. This ca be accomplished matched filter boud (MFB which is kow to collect all by formulatig the problem of computig the pseudo-iverse the available diversity i the chael. as that of fidig the N (M N matrix Ξ that correspods Fig. 2 illustrates the evolutio of the diversity order slope to the solutio of the miimizatio problem [7] achieved agaist the order of approximatio i the polyomial expasio equalizer i Eq. (5. It is see that the slope arg mi Tr{(Θ Ξ FD 1 +ΞΘ N,F D 1 H (Θ flattes out uderstadably at lower order approximatios FD 1 +ΞΘ N,F D 1 } due to large approximatio errors but starts to stabilize at (14 about secod order approximatio of the equalizer. 335

4 H κ Θ = (F H I P+Q K+L (J P+Q [q] e j ωq H q (T 1 T 2, (11 H κ Θ = (F H P+Q I K+L (J P+Q [q] e j ωq H c q(t 1 T 2. (12 0 th order approximatio 1 st order approximatio d order approximatio 3 rd order approximatio PE order 1 MMSE PE order 1 PE order 2 MMSE PE order 2 PE order 3 MMSE PE order Figure 2: Evolutio of diversity order for differet iteratios. Figure 4: Compariso of performace of the two PE approximatios. miimizes the MSE at the receiver Eq. (7 is show i Fig. 4. We see a sigificat ehacemet i performace for the first order approximatio whe compared to the PE approximatio i Eq. (5. However, for higher orders the the differece i their performace appears egligible MFB Brute force MMSE ZF 1 st order approximatio 2 d order approximatio Figure 3: Diversity order of LE approximated by PE. Fig. 3 shows the compariso of the diversity order of brute-force implemetatio of the MMSE-ZF equalizer for doubly selective chaels for the case of Q = 2 ad L = 1. Observe that the slope of the outage probability curves for both the implemetatios are the same. The polyomial expasio equalizer has a SNR offset whe compared to the brute force implemetatio which is to be expected sice the equalizer is a approximatio of the MMSE-ZF receiver however, it succeeds i collectig full diversity offered by the doubly selective chael at relatively low order of approximatio. The performace of PE approximatio that 6. CONCLUSIONS I this cotributio we propose two approximatios for the MMSE-ZF equalizer for doubly selective chaels. The proposed equalizers are based o polyomial expasio approximatio ad lead to a decrease i the computatioal complexity of the equalizer whe compared with the brute-force implemetatio. Simulatio results show that such a approximatio does ot icur a pealty i terms of the diversity order achieved by the approximated equalizers. 7. APPENDIX Due to the presece of the zero-paddig matrix T 2, it ca be easily show that the iter-block-iterferece compoet i the received sigal is zero, i.e., H q [k; 1]Θs[k 1] = 0. As a result, the received block i Eq. (1 ca ow be represeted as D[f q ]H q [k; 0]Θs[k] + v[k], (A1 Usig stadard Kroecker product idetities, oe ca show that H q [k; 0]Θ = F H T P+Q 1 H q [k; 0]T 2, (A2 336

5 where H q [k; 0] is a K + L K + L Toeplitz matrix formed by the first K + L rows ad colums of H q [k; 0]. Eq. (A1 ca the be re-writte as Note that D[f q ] (F HP+Q T 1 H q [k; 0]T 2 s[k]+v[k], (A3 D[f q ] = D P+Q [f q (K + L] D K+L [f q ], (A4 The above equatio represets D[f q ] as Kroecker product of time-variatio over two scales. D P+Q [f q (K + L] is a diagoal matrix of size P + Q that represets time-variatio at a coarse scale (complex-expoetials sampled at sub-samplig iterval of (K + LT s ad D K+L [f q ] is a diagoal matrix of size K +L that represets the time-variatio over a fier grid correspodig to the samplig period T s. Usig Eq. (A4 ad stadard matrix idetities, we ca decompose the received sigal as i Eq. (A6 where J[q] = J (q Q/2 ad J is a circulat matrix with [0, 1, 0 1 P+Q 2 ] T as the first colum. Sice the matrix (F H P+Q I K+L has o effect o the diversity of the doubly selective chael, for the aalysis of the diversity order of MMSE-ZF receiver, the effective chael matrix ca be represeted as (J P+Q [q]t 1 (D K+L [f q ] H q [k; 0]T 2, (A7 Fig. 5 provides a illustratio of the structure of the equivalet chael matrix due to precodig. Here H q represets the product matrix D K+L [f q ] H q [k; 0] for ease of illustratio. I particular, it is a block-toeplitz matrix with costituet blocks which are i tur formed by the product of a diagoal matrix D K+L [f q ] ad a Toeplitz matrix formed by the correspodig BEM coefficiets of the q-th basis fuctio. REFERENCES [1] S. P. Sheoy, I. Ghauri, ad D. T. M. Slock, Diversity order of liear equalizers for doubly selective chaels, i SPAWC 2009, 10th IEEE Iteratioal Workshop o Sigal Processig Advaces i Wireless Commuicatios, Jue 21-24, 2009, Perugia, Italy, Ju [2] A. Sayeed ad B. Aazhag, Joit multipath-doppler diversity i mobile wireless commuicatios, Commuicatios, IEEE Trasactios o, vol. 47, o. 1, pp , Ja [3] X. Ma ad G. Giaakis, Maximum-diversity trasmissios over doubly selective wireless chaels, Iformatio Theory, IEEE Trasactios o, vol. 49, o. 7, pp , July (P+Q (K+L H Q H Q H Q 01 K 0 0 = D K+L[f 0] H 0[k; 0] P K 0 Figure 5: Equivalet chael matrix for doubly selective chael. [4] G. Giaakis ad C. Tepedelelioglu, Basis expasio models ad diversity techiques for blid idetificatio ad equalizatio of time-varyig chaels, Proceedigs of the IEEE, vol. 86, o. 10, pp , Oct [5] C. Tepedelelioglu, Maximum multipath diversity with liear equalizatio i precoded OFDM systems, Iformatio Theory. IEEE Trasactios o, vol. 50, o. 1, pp , Ja [6] S. P. Sheoy, I. Ghauri, ad D. T. M. Slock, Diversity order of liear equalizers for block trasmissio i fadig chaels, i Asilomar 2008, Coferece o Sigals, Systems, ad Computers, October 26-29, 2008, Asilomar, USA, Oct [7] C. Tepedelelioglu ad Q. Ma, O the performace of liear equalizers for block trasmissio systems, Global Telecommuicatios Coferece, GLOBECOM 05. IEEE, vol. 6, pp. 5 pp., Nov.-2 Dec [8] S. Sheoy, F. Negro, I. Ghauri, ad D. Slock, Low- Complexity Liear Equalizatio for Block Trasmissio i Multipath Chaels, i Wireless Commuicatios ad Networkig Coferece, WCNC IEEE, April 2009, pp [9] S. Moshavi, E. G. Katerakis, ad D. L. Schillig, Multistage liear receivers for DS-CDMA systems, Iteratioal Joural of Wireless Iformatio Networks, vol. 3, pp. 1 17, October [10] G. H. Golub ad C. F. V. Loa, Matrix Computatios. The Johs Hopkis Uiversity Press, K+L ( (D P+Q [f q (K + L]F H P+QT 1 (D K+L [f q ] H q [k; 0]T 2 s[k] + v[k], (F H P+Q I K+L ( (J P+Q [q]t 1 (D K+L [f q ] H q [k; 0]T 2 s[k] + v[k], (A5 (A6 337