FREQUENCY-DOMAIN TURBO EQUALIZATION FOR SINGLE CARRIER MOBILE BROADBAND SYSTEMS. Liang Dong and Yao Zhao

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1 FREQUENCY-DOMAIN TURBO EQUALIZATION FOR SINGLE CARRIER MOBILE BROADBAND SYSTEMS Liang Dong and Yao Zhao Department of Eectrica and Computer Engineering Western Michigan University Kaamazoo, MI ABSTRACT This paper investigates mobie broadband communications, which undergo time-varying radio channes with arge mutipath deay spread. Space-time bock transmission and singe carrier moduation are adopted at the transmitter to combat the inter-symbo interference resuted from the mutipath deay. The singe carrier modems with frequency-domain equaization have simiar performance and compexity as orthogona frequency-division mutipexing, yet ess sensitive to radio frequency impairments. At the receiver, we propose the Turbo equaization that consists of a minimum mean-square error equaizer in the frequency domain and a channe decoder. In order to cope with the time-varying characteristics of the mobie channe, the received data is partitioned into bocks for the inear equaizer, and then combined as a decoding frame for the decoder. Simuation resuts show good performance of the proposed scheme with feasibe computationa compexity. I. INTRODUCTION Singe carrier (SC) moduation and orthogona frequency-division mutipexing (OFDM) are two major techniques to combat the inter-symbo interference (ISI) characterizing the dispersive channes of wireess broadband systems. Much work has been done to compare these two approaches, 2, 3. Athough OFDM has aready been appied in many practica appications, SC moduation is gaining popuarity. SC moduation uses a singe carrier, instead of many sub-carriers in OFDM. Therefore, the peak-to-average ratio of the transmitted power for SC-moduated signa is smaer. This means a smaer inear range of power ampification to support a given average power, thus an inexpensive power ampifier compared with the OFDM system. When combined with frequency-domain equaization, the SC approach deivers performance simiar to OFDM, with essentiay the same overa compexity 4. When mutipe antennas are used at the transmitter, the space-time bock transmission can be used aong with SC moduation to improve system performance over frequency-seective channes 5. Diversity transmission using Aamouti s space-time bock-codes (STBC) 6 has been proposed in severa wireess standards due to its many attractive features. It achieves fu spatia diversity at fu transmission rate for two transmit antennas, without requiring channe state information at the transmitter. At the receiver, the maximum ikeihood decoding for Aamouti s STBC requires ony simpe inear processing. In this paper, we foow 5 to use the Aamouti-ike space-time bock transmission over broadband channes. The bocks are extended with cycic prefix () and transmitted with SC. Convoutiona code is adopted as the forward error contro method. The receiver impements Turbo equaization. The Turbo equaization, first proposed in 7, borrowed the idea of Turbo decoding to detect iterativey the origina information bits impaired by ISI. The outer code is usuay a convoutiona code. The ISI channe, equivaent to the inner code, is considered as a rate one convoutiona code in rea Gaois fied 8. The extrinsic information transfer (EXIT) chart is used as a theoretica too for performance anaysis. Our proposed Turbo equaizer is composed of a inear minimum mean-square error (MMSE) equaizer in the frequency domain and a soft-input soft-output (SISO) decoder. MMSE spacefrequency equaization for SC mutipe-input mutipeoutput (MIMO) systems over frequency-seective channes is proposed in 9, but it can not be impemented iterativey to form Turbo equaization because it assumes fixed statistics about the transmitted information. To reaize the Turbo equaization, we proposed a frequency-domain MMSE equaizer based on dynamic a priori information 0 to cooperate with the SISO outer decoder. The a priori information of the MMSE equaizer is updated by the SISO decoder at each iteration. of 7

2 TX b c {s(2k), s(2k+)} Channe Encoder Moduator Space-Time Bock Transmission TX2 Fig.. Transmitter baseband bock diagram: channe encoding and space-time bock transmission. Moreover, in order to cope with fast channe variation, the symbos are partitioned into short bocks for space-time bock transmission. The received data are partitioned into bocks before converted to the frequency domain and fed into the inear MMSE equaizer. The equaized symbos are brought back to the time domain, and combined as a ong decoding frame for the deintereaver and the decoder. With itte variation of the channe during the transmission of two bocks, the independence of the frequency tones is amost preserved. The sma crosstak between the frequency tones can be further deat with by the frequencydomain equaization. This paper is organized as foows. Section II describes the system mode of the singe carrier spacetime transmission and reception. Section III detais the receiver structure and scheme of the frequency-domain Turbo equaization. Numerica resuts through simuations are presented in Section IV, which demonstrate an improved ink. Finay, concusions are drawn in Section V. II. SYSTEM MODEL A. Space-Time Transmission Fig. depicts the bock diagram of a wireess transmitter with two antennas. The binary bits b are encoded using a convoutiona code. The coded bits c are randomy intereaved in order that the infuence of error bursts is reduced at the input of the decoder at the receiver. The intereaved coded bits are then moduated and mapped to symbos and grouped into bocks of ength N. A pair of two bocks are transmitted through the two antennas using Aamouti s space-time bock transmission scheme. The bock ength N can be adjusted to adapt to the time-varying rate of the mobie channe. We choose N to be short enough such that the channe can be regarded as time-invariant during at east the transmission of two bocks. The transmission uses a singe carrier frequency. In this paper, we assume a coherent symbo-spaced receiver frontend with perfect symbo timing and describe the system with an equivaent discrete-time baseband mode. Let s(2k) and s(2k + ) denote two consecutive symbo bocks with normaized power as s(2k) = s(2kn), s(2kn + ),, s(2kn + N ) T () s(2k + ) = s((2k + )N), s((2k + )N + ),, s((2k + )N + N ) T (2) These two symbo bocks are transmitted through the two antennas in the foowing form anaogous to the Aamouti space-time code, s(2k) Ps (2k + ) s(2k + ) Ps (2k) time space (3) where P is a permutation matrix that is drawn from a set of permutation matrices {P (n) } N n=0. Each P(n) performs a reverse cycic shift, such that when it is appied to a N vector s = s(0),..., s(n ) T, the p-th entry of P (n) s is s((n p + n) mod N). For exampe P (0) s = s(n ), s(n 2),..., s(0) T (4) P () s = s(0), s(n ), s(n 2),..., s() T (5) Suppose that the transmit fiter, the broadband ISI channe, and the receive fiter can be represented by a discrete-time inear fiter with finite-ength impuse response (FIR) of ength L. The FIR fiter can be determined by the sequence h µ = h µ (0),..., h µ (L ) T, where µ =, 2 indicates the transmit antenna. For mutipe transmitter-receiver antenna pairs with different channe memory, L is the ongest fiter ength. As shown in Fig. 2, a cycic prefix () of ength L is added to each transmitted bock. Therefore, the inter-bock interference can be eiminated by removing the at the receiver. It shoud be noted that with the ength of the determined by the channe fiter ength L, the shorter the origina bock ength N, the more overhead is there in the transmitted frame. 2 of 7

3 matrix P, has the foowing property TX s(2k) _ P s*(2k+) PH (j) µ P = H (j)h µ (0) TX 2 Fig. 2. s(2k+) P s*(2k) Bock 2k Bock 2k+ Transmitted sequence through two antennas B. Space-Time Reception When the receiver synchronizes to the symbo bocks, the received data in two consecutive bocks are given by x(2k) = H (2k) s(2k) + H (2k) 2 s(2k + ) +n(2k) (6) x(2k + ) = H (2k+) Ps (2k + ) +H (2k+) 2 Ps (2k) + n(2k + ) (7) where n is the zero-mean Gaussian noise vector with covariance matrix σni. 2 With the remova of at the receiver, the channe H (j) µ,where j = 2k, 2k + denotes the j-th symbo bock, can be represented by an N N circuant matrix as µ (0) 0 µ () µ () µ (0) µ (2) µ (L 2) µ (L 3) µ (L ) H (j) µ = µ (L ) µ (L 2) 0 0 µ (L ) µ (0) (8) Here, we assume that the channe is invariant during a bock transmission. Hence, H (j) µ has an eigendecomposition as H (j) µ = F H Λ (j) µ F (9) where F is the orthonorma discrete Fourier transform (DFT) matrix whose (k, )th entry is F k, = N e j(2π/n)k (k, = 0,..., N ). Λ µ (j) is a diagona matrix whose (k, k)th entry is given by the kth DFT coefficient of the first coumn of H µ (j). In addition, the circuant matrix, when operating with the permutation As depicted in Fig. 3, the receiver divides the symbo bocks to generate x(2k) and Px (2k + ), which are then passed through FFT modues to be converted to the frequency domain. The outputs are given by Fx(2k) FPx = (2k + ) X(2k) S(2k) S(2k + ) Λ (2k) Λ (2k) 2 Λ (2k+) 2 Λ (2k+) } {{ } Λ (2k) + Fn(2k) } FPn (2k + ) {{ } W(2k) () where S(j) = Fs(j), j = 2k, 2k +. The fitered noise W(2k) remains white with covariance matrix σni. 2 The FFT output X(2k) is eft mutipied by Λ (2k)H, which is a process caed space-time inear combining. The combiner output is passed through the frequency-domain MMSE equaizer to obtain the estimates Ŝ(2k) and Ŝ(2k + ). The estimates are brought back to the time domain via IFFT modues. Finay, the coded bits are detected and decoded to the information bits. The receiver detais wi be expained in the next section. III. FREQUENCY-DOMAIN TURBO EQUALIZATION A. Receiver Structure As shown in Fig. 3, the receiver consists of a spacetime inear combiner, foowed by the Turbo equaizer. The Turbo equaizer has two stages, the frequencydomain inear MMSE equaizer and the SISO channe decoder. The two stages are separated by a deintereaver (a bock abeed π ) and an intereaver (a bock abeed π ). It is assumed that perfect channe estimation is avaiabe at the receiver, that is, Λ (2k) is known to the receiver. With the assumption that the channe is invariant during at east the transmission time of one symbo bock, Λ (j) µ (µ =, 2) is diagona. With the assumption that the channe is invariant during at east the transmission time of two symbo bocks, Λ (2k) is diagona. However, when the mobie communication ink undergoes fast time-varying channes, both Λ (j) µ and Λ (2k) have non-zero off-diagona eements. That is, the mutipe frequency tones at the receiver experience crosstak. Frequency crosstak is due to the time-seectivity of the 3 of 7

4 RX {x(2k), x(2k+)} S/P x(2k) Px*(2k+) FFT FFT Space-Time Linear Combiner MMSE Equaizer S(2k) S(2k+) IFFT IFFT LLR Mixer Le(s) - Channe Decoder L(b dec) L(c dec) Fig. 3. Receiver baseband bock diagram: space-time combining and Turbo equaization. mobie channes. A vector inear equaizer wi be used to estimate S(2k) and S(2k + ) together. The reception sti benefits from the space-time bock transmission and the space-time inear combining, because the off-diagona eements are comparativey sma. For simpicity, the bock index (2k) is dropped from now on, i.e. X = X(2k), Λ = Λ (2k), W = W(2k). Let S(2k) S =, thus S(2k + ) S = F 0 0 F Foowing (), we have s(2k) s(2k + ) Λ H X = Λ H ΛS + Λ H W W = Fs. (2) (3) where the fitered noise W is coored with covariance matrix as shown in (4) at the bottom of the next page. Binary phase shift keying (BPSK) moduation is considered in this paper. The a priori og-ikeihood ratio (LLR) of the symbo bit is fed back to the MMSE equaizer, and the equaizer outputs the a posteriori LLR of the symbo bit. The a priori LLR is given by L(s(i)) = n The a posteriori LLR is given by L(s(i) X) = n P s(i) = + P s(i) = ps(i) = + X ps(i) = X (5) (6) Using Bayes rue, it can be written as px s(i) = + L(s(i) X) = n px s(i) = L e(s(i)) P s(i) = + P s(i) = L(s(i)) + n where L e (s(i)) is the extrinsic information. (7) The extrinsic information of bocks of ength N at the output of the MMSE equaizer is cacuated and combined into a decoding frame. The decoding frame is deintereaved to generate the a priori LLR of the coded bit for the channe decoder. The decoder outputs an update of the LLR of the coded bit and the information bit. The extrinsic information of the coded bits based on the code constraints is intereaved and fed back to the inear MMSE equaizer as the a priori information in the next iteration. The a posteriori LLR of the information bit is used to make hard decision at the ast iteration. Note that the extrinsic information from the MMSE equaizer and from the decoder are statisticay independent at the first iteration, but become more and more correated in the subsequent iterations. Consequenty, the improvement through iteration wi diminish graduay. 4 of 7

5 B. Frequency-Domain Linear MMSE Equaizer The frequency-domain estimation Ŝ at the output of the inear MMSE equaizer is given by Ŝ = C XX C XS H (X E(X)) + E(S) = (ΛC SS Λ H + σni) 2 ΛC SS H (X ΛE(S)) +E(S) (8) where C XX, C XS and C SS are covariance matrices, defined as C xy Cov(x, y) = E(xy H ) E(x)E(y H ). According to (2), E(S) = FE(s) with the ith eement E(s(i)) depending on the a priori LLR L(s(i)) as E(s(i)) = k P s(i) = k k {+, } ( ) P s(i) = + = tanh n 2 P s(i) = ( ) L(s(i)) = tanh 2 (9) Due to the independence of the intereaved symbos {s(i)} and the BPSK moduation, the covariance matrix C SS can be cacuated as C SS = FC ss F H = Fdiag ( ( E(s) 2 ) ) F H (20) where the ith eement of E(s) 2 is E(s(i)) 2, given in (9). The a priori LLR L(s(i)) is updated in each iteration, so the MMSE estimator must be recacuated. Once the symbo vaues in the frequency domain are estimated, the time-domain vaues can be obtained by performing the inverse FFT as ŝ = F H Ŝ (2) At the first Turbo iteration, it is assumed that s(i) is equay ikey + or which yieds E(S) = FE(s) = 0 and C SS = I. The MMSE estimator as shown in (8) can be simpified as Ŝ = (ΛΛ H + σ 2 ni) Λ H X (22) Assume that the probabiity density function pŝ(i) s(i) = k, k = ± is Gaussian with mean µ i,k = E{ŝ(i) s(i) = k} and variance σ 2 i,k = cov{ŝ(i), ŝ(i) s(i) = k}. The a posteriori LLR L(s(i) ŝ(i)), which is an approximation of L(s(i) X), can be expressed as pŝ(i) s(i) = + L(s(i) ŝ(i)) = n pŝ(i) s(i) = L e(s(i)) P s(i) = + P s(i) = L(s(i)) (23) + n The extrinsic information L e (s(i)) can be cacuated as L e (s(i)) = n exp( ŝ(i) µ i,+ 2 /σ 2 i,+ ) exp( ŝ(i) µ i, 2 /σ 2 i, ) = ŝ(i) µ i, 2 σ 2 i, ŝ(i) µ i,+ 2 σ 2 i,+ (24) Let A = (ΛC SS Λ H + σ 2 ni) ΛC SS H. From (8) and (2), we have ŝ(i) = u i F H A(ΛS ΛE(S) + W) + E(s(i)) (25) where u i = 0,..., 0,, 0,..., 0. It can be derived ith that µ i,k = u i F H AΛ Fd (i,k) + E(s(i)) (26) ( ) ) σi,k 2 = u i ( F H A Λ FC ss s(i)=k F H Λ H + σ 2 n I A H H F u i where and d (i,k) = 0,..., 0, k E(s(i)), 0,..., 0 T ith C ss s(i)=k = Cov(s, s s(i) = k) = C ss + diag(0,..., 0, 2E(s(i))(E(s(i)) k), ith 0,..., 0). (27) At the first Turbo iteration, (26) and (27) can be simpified as µ i,k = ku i F H H AΛ Fu i (28) σi,k 2 = u i ( F H A ( ΛΛ H + σ 2 n I ) ) A H F u H i (29) C W = Λ (2k)H Λ (2k) + Λ (2k+) 2 Λ (2k+)H 2 Λ (2k)H Λ (2k) 2 Λ (2k+) 2 Λ (2k+)H Λ (2k)H 2 Λ (2k) Λ (2k+) Λ (2k+)H 2 Λ (2k)H 2 Λ (2k) 2 + Λ (2k+) Λ (2k+)H σ 2 n (4) 5 of 7

6 C. SISO Channe Decoder The bock ength N is chosen reativey sma such that the channe can be assumed time-invariant during the transmission of two bocks. Aso, the short bock ength eases the computation of the MMSE equaizer. The a priori LLRs of the coded bits of mutipe bocks are combined as a decoding frame. The SISO channe decoder takes in the entire frame and outputs the updated LLRs of the coded bits L(c(i) decoding), as we as the LLRs of the information bits L(b(i) decoding) based upon the code constraints. We use the og maximum a posteriori probabiity (Log-MAP) decoding agorithm to cacuate the a posteriori LLRs of the coded bits and the information bits. The a posteriori LLRs of the coded bits can be written as 2 L(c(i) decoding) P c(i) = + decoding = n P c(i) = decoding (s = n,s) Σ + ps = s, s + = s ps = s, s + = s (s,s) Σ (30) whie the a posteriori LLRs of the information bits can be written as L(b(i) decoding) P b(i) = + decoding = n P b(i) = decoding (s = n,s) U + ps = s, s + = s ps = s, s + = s (s,s) U (3) where Σ + and Σ are the sets of a state pairs s = s and s + = s that correspond to the coded bit c(i) = + and c(i) =, respectivey. U + and U are the sets of a state pairs s = s and s + = s that correspond to the information bit b(i) = + and b(i) =, respectivey. IV. SIMULATIONS The performance of the proposed scheme is evauated through simuations. As mentioned in Sections II-A and III-C, in order to cope with the time-varying characteristics of the mobie channe, we partition a ong decoding frame into mutipe bocks of ength N for the space-time transmitter and the equaizer. The equaized bocks are combined as input to the channe decoder. This partition can aso improve the computationa efficiency. Specificay in our simuations, MI at output (input) of MMSE Equaizer (Decoder) MI at output (input) of Decoder (MMSE Equaizer) Fig. 4. EXIT chart, E b /N 0 = 0.8 db. 2 6 BPSK-moduated symbos are partitioned into 256 pairs of bocks (tota 52 bocks). Each bock, that contains N = 28 symbos, is processed through the inear frequency-domain MMSE equaizer and the outputs construct a 2 6 -symbo frame to be fed into the deintereaver and the decoder. Convoutiona codes and the Log-MAP decoding agorithm are used. Fig. 4 shows the EXIT chart at E b /N 0 = 0.8 db, where E b is the energy of the information bit. The coding gain and the antenna gain are considered. The EXIT chart tracks the mutua information between an information bit and its a posteriori extrinsic LLR at the outputs of the equaizer and the decoder. Fig. 5 depicts the BER performance versus E b /N 0 at mutipe iterations. As shown in these figures, the mutua information increases with each iteration, which indicates more accurate decoding resuts. Consequenty, the BER becomes smaer with each iteration. The simuated system has two transmit antennas and one receive antenna, and is abeed as 2. The mutiantenna channe is expoited through the Aamouti-ike space-time bock transmission. The proposed scheme is designed to hande ISI encountered in broadband systems. Specificay in our simuation, the ength of the channe impuse response is L = 6 symbo periods. For comparison, we aso consider the case of the frequency-seective channe with a singe transmit antenna, abeed ISI channe, and the case of the frequency-fat channe with two transmit antennas, abeed 2 non-isi channe. The comparison of the BER performance at the fourth Turbo iteration is shown in Fig. 6. It is reveaed that the performance of our pro- 6 of 7

7 BER 0 3 BER Iteration Iteration 2 Iteration 3 Iteration 4 Iteration E /N (db) b x (ISI Channe) x (ISI Channe) 2 x (Non ISI Channe) E /N (db) b 0 Fig. 5. Bit error rate at different iterations. Fig. 6. Bit error rate comparison at the 4th iteration. posed scheme over the 2 ISI channe is about.2 db away from that of the 2 non-isi channe at a BER of 0 3. Over ISI channes, better performance is achieved in the 2 case than in the case. This resuts from the diversity gain obtained through space-time bock transmission. V. CONCLUSION A frequency-domain Turbo equaization scheme is proposed for space-time bock transmission over SC broadband channes. The equaization is impemented through the a priori information based inear MMSE detector, whose transformation matrix is updated in each iteration according to the feedback from the SISO decoder. Simuation resuts show that the proposed scheme is promising to combat the ISI. Better performance with two transmit antennas can be achieved due to the diversity gain resuted from the space-time bock transmission. Furthermore, by partitioning the ong decoding frame into short bocks, the proposed approach is abe to dea with time-varying mobie channes, given that the channe varies sighty during the transmission of two bocks. References N. Wang and S. D. Bostein, Comparison of -based singe carrier and OFDM with power aocation, IEEE Trans. Commun., vo. 53, no. 3, pp , Mar Z. Wang, X. Ma, and G. B. Giannakis, OFDM or singecarrier bock transmissions? IEEE Trans. Commun., vo. 52, no. 3, pp , Mar N. Benvenuto and S. Tomasin, On the comparison between OFDM and singe carrier moduation with a DFE using a frequency-domain feedforward fiter, IEEE Trans. Commun., vo. 50, no. 6, pp , June D. Faconer, S. L. Ariyavisitaku, A. Benyamin-Seeyar, and B. Eidson, Frequency domain equaization for singe-carrier broadband wireess systems, IEEE Commun. Mag., vo. 40, no. 4, pp , Apr W. M. Younis, A. H. Sayed, and N. A-Dhahir, Efficient adaptive receivers for joint equaization and interference canceation in mutiuser space-time bock-coded systems, IEEE Trans. Signa Processing, vo. 5, no., pp , Nov S. M. Aamouti, A simpe transmit diversity technique for wireess communications, IEEE J. Seect. Areas Commun., vo. 6, no. 8, pp , Oct C. Douiard, M. Jezeque, C. Berrou, A. Picart, P. Didier, and A. Gavieux, Iterative correction of intersymbo interference: Turbo equaization, Eur. Trans. Teecommun., vo. 6, no. 5, pp , Sept R. Koetter, A. C. Singer, and M. Tücher, Turbo equaization, IEEE Signa Processing Mag., vo. 2, no., pp , Jan X. Zhu and R. D. Murch, Layered space-frequency equaization in a singe-carrier MIMO system for frequency-seective channes, IEEE Trans. Wireess Commun., vo. 3, no. 3, pp , May M. Tücher, A. Singer, and R. Koetter, Minimum mean squared error equaization using a priori information, IEEE Trans. Signa Processing, vo. 50, no. 3, pp , Mar S. Zhou and G. B. Giannakis, Singe-carrier space-time bock-coded transmissions over frequency-seective fading channes, IEEE Trans. Inform. Theory, vo. 49, no., pp , Jan X. Wang and H. Poor, Iterative (turbo) soft interference canceation and decoding for coded CDMA, IEEE Trans. Commun., vo. 47, pp , Juy of 7

University of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): /GLOCOM.2003.

University of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): /GLOCOM.2003. Coon, J., Siew, J., Beach, MA., Nix, AR., Armour, SMD., & McGeehan, JP. (3). A comparison of MIMO-OFDM and MIMO-SCFDE in WLAN environments. In Goba Teecommunications Conference, 3 (Gobecom 3) (Vo. 6, pp.