Training sequence optimization for frequency selective channels with MAP equalization

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1 532 ISCCSP 2008, Malta, March 2008 raining sequence otimization for frequency selective channels with MAP equalization Imed Hadj Kacem, Noura Sellami Laboratoire LEI ENIS, Route Sokra km 35 BP 3038 Sfax, unisia Aline Roumy IRISA-INRIA Camus de Beaulieu Rennes Cedex, France Inbar Fijalkow Laboratoire EIS, UMR 8051 ENSEA-UCP-CNRS, 6 av du Ponceau Cergy-Pontoise, France Abstract In this aer, we address the roblem of otimization of the training sequence length for frequency selective channels when a Maximum a osteriori MAP equalizer is used he otimal length of the training sequence is found by maximizing an effective signal-to-noise ratio SNR and an effective channel caacity of the training-based transmission scheme We study these roblems of otimization when the training and data owers are equal and when they are allowed to be different When the owers can be different, we give the otimal ower allocation I INRODUCION Equalization is used to combat intersymbol interference on frequency selective channels he otimal equalizer [1] to be used is based on Maximum a osteriori MAP detection It makes decision on a symbol-by-symbol basis and is otimum since it minimizes the bit error robability when the channel is known by the receiver In ractice, the channel imulse resonse is estimated by sending known training symbols When the length of the training sequence increases, the variance of the channel estimation error decreases, but the information throughut decreases as well hus, a trade-off has to be found Several methods have been roosed to design the otimal training sequence length he solution resented in [2] and [3] is based on maximizing a lower bound of the caacity of the training-based scheme resectively for a single-inut single-outut SISO frequency selective channel and for a multile-inut multile-outut MIMO flat fading channel Another aroach tries to find the otimal sequence that minimizes the Mean Square Error MSE of the channel estimator for different systems [4]-[7] All these works do not take into account the receiver used In this aer, we consider the articular case where a MAP equalizer is used for a transmission over a SISO frequency selective channel We introduce simle exressions of the effective Signal-to-Noise Ratio SNR and the effective channel caacity for the training-based scheme We find the training sequence lengths maximizing these quantities when the training and data owers are equal When the owers can be different, we also give the otimal ower allocation he aer is organized as follows In Section II, we describe the transmission system model Section III studies the otimization of the training sequence length when the training and data owers are equal In Section IV, we find the otimal ower allocation and the otimal training sequence length when the training and data owers are allowed to be different hroughout this aer scalars and matrices are lower and uer case resectively and vectors are underlined lower case he oerator denotes the transosition, and I m is the m m identity matrix x, x and x are resectively the greatest integer lower than x, the smallest integer greater than x and the absolute value of x II RANSMISSION SYSEM MODEL We consider a data transmission system over a frequency selective channel he inut information bit sequence is maed to the symbol alhabet A For simlicity, we will consider only the BPSK modulation A = { 1, 1} We assume that transmissions are organized into bursts of symbols he channel is suosed to be invariant during the transmission he received baseband signal samled at the symbol rate at time k is L 1 x k = h l s k l + n k 1 l=0 where L is the channel memory and s k, for 1 L k 1, are the transmitted symbols In this exression, n k are modeled as indeendent random variables of real white Gaussian noise with normal robability density function N 0, σ 2 where N α, σ 2 denotes a Gaussian distribution with mean α and variance σ 2 he term h l is the l th ta gain of the channel, which is assumed to be real valued he channel is estimated by using a training sequence of length 2L 1 We assume that this sequence has ideal autocorrelation and crosscorrelation roerties Let s = s L,, s 1 L be the vector of training symbols and h = h 0,, h L 1 the vector of channel tas he least square channel estimate ĥ = ĥ0,, ĥl 1 is given by [8]: ĥ = H L s H L s 1 HL s x 2 where H L s is the L+1 L Hankel matrix having the first column s L,, s 0 and the last row s0,, s 1 L and x = x 0,, x L is the outut of the channel corresonding to the training sequence /08/$2500 c 2008 IEEE

2 ISCCSP 2008, Malta, March Hence, we obtain 1 ĥ h N 0, σ 2 H L s H L s 3 σ = N 0, 2 L+1σ I 2 L where σ 2 is the transmit ower during the training hase =10 =27 =100 III OPIMIZAION OF HE RAINING SEQUENCE LENGH FOR EQUAL POWERS We consider at the receiver a MAP equalizer using the BCJR algorithm [1] We first assume that the transmit owers during the training and data transmission hases are equal to the unit We will be interested in the otimization of the training sequence length by maximizing an effective SNR and an effective channel caacity that we will define A Maximization of the effective SNR When the channel is estimated, the Bit Error Rate BER at the outut of the MAP equalizer can be aroximated at high SNR as [9] P Q d min 1 + 2σ L L where d min is the channel minimum distance [10] Hence, the equivalent signal to noise ratio at the outut of the MAP equalizer using the channel estimate is given by SNR eq, ĥ = d2 min 1 + L L Increasing the training sequence length leads to an imrovement of the channel estimate quality but also to a loss in data throughut hus, in order to take account this loss, we define an effective SNR at the outut of the equalizer as SNR eff,eq, ĥ = = SNR eq, ĥ d 2 1 min L 6 4σ L+1 Our goal is to maximize SNR eff,eq, ĥ under the constraints and r 0, where r 0 = 2L 1 Let x R + 1 and fx = d2 min x 1 + x L+1 L Notice that SNR eff,eq, ĥ = f Let f x be the second derivative of fx Since f x < 0, for x R +, the function f is concave hus, it has a unique maximum reached for x R +, such as: x = 1 + L + L 7 We consider the two ossible cases: - If x < r 0 < 4L 1 then the length of the training sequence maximizing SNR eff,eq, ĥ is equal to r 0 - If r 0 x 4L 1 then = r 1 where r 1 = arg max x { x, x } fx We can summarize the revious results as follows: = r 1 r r 0 8 BER SNR eff Fig 1 MAP equalizer BER erformance versus SNR eff for different values of the length of the training sequence { x if x 0; where x + = 0 elsewhere When 4L 1, the otimum value of SNR eff,eq, ĥ can be aroximated by SNR eff,eq,ĥ d2 min 1 + L L + L Simulation results In our simulations, we consider Channel3 with imulse resonse 05; 071; 05 Figure 1 shows the BER erformance of the MAP equalizer when the channel is estimated, for different values of the length of the training sequence = 10, 27 and 100 with resect to SNR eff = SNR, where SNR is the signal-to-noise ratio at the inut of the MAP equalizer We set the number of symbols er burst to 256 According to the revious analytical results, the otimal length of the training sequence is = 27 his is confirmed by the simulations since they show that the equalizer resents its best erformance, at high SNR, when = B Maximization of the effective channel caacity In the case of channel estimation, the channel caacity of the training-based scheme using the MAP equalizer is given by C = log + SNR eq, ĥ 10 In order to take into account the loss in channel caacity due to the ilot symbols, we define an effective channel caacity as C eff = 1 2 log1 + SNR eq, ĥ 11 We define gx = x log 1 + d2 min x L+1 x+1 for x R +, then C eff = g Since g x < 0, for x R +, g is concave Hence, it has an unique maximum reached for y R + As g 0g < 0, then according to the

3 534 ISCCSP 2008, Malta, March 2008 theorem of intermediate values y ]0, [ hus, two cases are considered: - If g r 0 0 then = r 0 - If g r 0 > 0 then y ]r 0, [ and = argmax y { y, y }gy Hence, = r 2 r r 0 12 where r 2 = arg max y { y, y } gy IV JOIN OPIMIZAION OF HE RAINING SEQUENCE LENGH AND POWER ALLOCAION We assume now that the training and data owers are allowed to be different hus, the ilot symbols are transmitted with a ower σ 2 and the d = data symbols are transmitted with a ower σd 2 A Maximization of the effective signal-to-noise ratio When the channel is estimated by the least square estimator and ilot and data symbols have different ower levels, the exression of the BER at the outut of the MAP equalizer is given by P = Q d 2 min 4 σd 2 σ Lσ 2 d L + 1σ his result can be roved by using the same roof as in [9] while taking into account the ilot and data owers he equivalent signal-to-noise ratio at the outut of the MAP equalizer becomes SNR eq, ĥ = d2 min 4 σd 2 σ Lσ 2 d L + 1σ In this case, we define the effective signal-to-noise ratio SNR eff,eq, ĥ as: SNR eff,eq, ĥ = d 2 min σ 2 d Lσ 2 1 d 4 σ L + 1σ 2 15 Now, consider the following otimization roblem: max SNR eff,eq, ĥ, σ, 2 d, σd 2 st σ 2 + σd 2 d = σt d = σ, 2 σd 2 0 r 0 where σ 2 t is the total transmit energy er burst We denote the fraction of the total transmit energy used in the data transmission hase as σ 2 d d = ασ 2 t, 0 < α < 1 17 hus, the effective SNR can be written as SNR eff,eq, ĥ = ασ2 t d 2 1 min α L 1+ 1 α L+1 18 he roblem 16 is equivalent to max SNR eff,eq, ĥ, α st r 0 0 < α < 1 19 Proosition 1 he otimal training sequence length and the otimal ilot symbol ower maximizing the effective SNR under the constraints of 16 are given by = r 3 r r 0 σ 2 = 1 α σ2 t 20 where r3 = arg max x { x, x } f 1 x, α x, f 1 x, α = ασ 1, 2 t d2 min αxl 4σ α x x L+1 x = L, α x = Ax AxLx Ax Lx and Ax = xx L + 1 he roof of Proosition 1 is given in the Aendix he ower of data symbols maximizing SNR eff,eq, b h is then given by σd 2 = α σt 2 21 he otimum value of the effective signal-to-noise can be aroximated by SNR eff,eq, b d2 min σ2 t A 1 α 2 h 22 L Simulation results Figure 2 shows the BER at the outut of the MAP equalizer with resect to SNR eff for Channel3, = 256 and σt 2 = 1 We consider the scenarios given in able I According to 20, the theoretical values of the otimal length of the training sequence and the otimal ilot symbol ower are resectively = 23 and σ 2 = 118 Simulations in Figure 2 confirm that the MAP equalizer best erformance are achieved when = and σ 2 = σ 2 Scenario σ 2 S1 σ 2 S2 50 σ 2 S3 10 σ 2 S4 075σ 2 S5 15σ 2 ABLE I SCENARIOS CONSIDERED IN FIGURE 2 B Maximization of the effective channel caacity In the following, we define C eff, the effective channel caacity as: C eff = 1 log 1 + SNR 2 eq, ĥ, α 23

4 ISCCSP 2008, Malta, March S1 S2 S3 S4 S * =13 =20 = BER C eff SNR eff Fig 2 BER versus SNR eff : MAP equalizer erformance for different values of and σ 2 for Channel3, = 256 and σ2 t = Fig 3 Effective channel caacity versus σ 2 for different values of the length of the training sequence, L = 3, = 256, σt 2 = 02 and σ2 = 1 σ 2 We want to solve the otimization roblem given by: max C eff, α = 1 2 log 1 + d2 min ασ 2 t L+11 α L+11 α+lα st r 0 0 < α < 1 Let x, α R + and g 1 x,α = x 2 log 1+ d2 min ασ2 t x L+11 α xx L+11 α+lαx 02 Hence, C eff, α = g 1, α Proosition 2 he length of the training sequence and the training ower maximizing the effective channel caacity are given by = arg max y { y, y } g 1 y, α y σ 2 = 1 α σ2 t 25 C eff ρ=12db ρ=10db ρ=8db ρ=6db Fig 4 Effective channel caacity versus for L = 3, = 256 and σt 2 = 1 for different values of ρ where α x = Ax AxLx Ax Lx, Ax = xx L + 1, and y = arg min r0 x g 1 x x, α x he roof of Proosition 2 is given in the Aendix Simulation results Figure 3 shows the effective channel caacity as a function of the ilot symbol ower for L = 3, = 256, σt 2 = 02 and σ 2 = 1 for different values of By using 25, we have = 13 and σ 2 = 046 his is confirmed by the simulations since the effective channel caacity is maximized for these values Figure 4 shows the effective channel caacity as a function of for L = 3, = 256 and σt 2 = 1 for different values of ρ = 1/σ 2 We notice that the length of the training sequence maximizing the effective channel caacity grows when the noise variance at the inut of the MAP equalizer increases able II gives the values of and σ 2 obtained by solving 24 for different values of σt 2 We suose that the total length of the burst is = 256, the channel memory length L = 3 and ρ = 8dB When σt 2 increases, decreases and σ 2 increases V CONCLUSION In this aer, we consider the roblem of otimizing the training sequence length when a MAP equalizer is used We study two cases: the case where the training and data owers are equal and the case where they are allowed to be different We define an effective signal-to-noise ratio and an effective channel caacity for the training-based transmission scheme We find the training sequence lengths maximizing these quantities in the case of equal owers When the owers

5 536 ISCCSP 2008, Malta, March 2008 AND σ2 σt 2 σ ABLE II MAXIMIZING C eff VERSUS σt 2 FOR = 256, L = 3 AND ρ = 8dB are allowed to be different, we also give the otimal ower allocation A Proof of Proosition 1 VI APPENDIX Let {x, α} [L, + [ R + and f 1 x, α = ασ2 t d 2 1 min αxl 1+ 1 α x x L+1 hus, SNR eff,eq, ĥ = f 1, α Now, consider the following relaxed otimization roblem: max f 1 x, α st r 0 x 0 < α < 1 26 Since, 2 f 1 α < 0, f 2 1 is concave with resect to α, for a fixed value of x hus, when x is fixed, f 1 has a unique maximum achieved for α = α x such as α x = Ax AxLx Ax Lx 27 where Ax = xx L+1 Notice that 0 < α x < 1 Hence, by using the combinatorial otimization [11], the solution of 26 is:, α = arg max f 1m, α m, α 28 r 0 m where m is an integer Let x [L, + [ and F 1 x = f 1 x, α x = σ2 t d 2 min Ax AxLx + Lx 29 As F 1 x < 0, for x [L, + [, F 1 is concave hus, F 1 has a unique maximum reached for x = x [L, + [ By calculating the derivative of F 1, we can show that it is maximized for x = L Suose that r2 0 L 1, then x r 0 Let 1, α 1 be the solution of 19 Suose that 1 / { x, x } Hence, there are the two ossibilities: 1 < x = x 1 or 1 > x = x 2 As 2 f 1 x x, α < 0, f 2 1 is concave with resect to x for a fixed value of α - If 1 < x 1 < x 2, we have f 1 1, α 1 f 1 x 1, α 1 f f 1 x 2, α 1 Hence, 1x 2,α 1 f 1 1,α 1 x 2 1 < f1x2,α1 f1x1,α1 x 2 x 1 On the other hand, f 1 x, α is concave with resect to x for a fixed value of α hus, f 1x 2,α 1 f 1x 1,α 1 x 2 x 1 f 1x 2,α 1 f 1 1,α 1 x 2 1 which is imossible - If 1 > x 2 > x, f 1 x, α f 1 x 2, α f 1 1, α f Hence, 1 1,α f 1 x,α 1 x < f 1 1,α f 1 x 2,α 1 x 2 On the other hand, f 1 x, α is concave in x for a given value of f α hus, 1 1,α f 1x,α 1 x f11,α f 1x 2,α 1 x 2 which is imossible hus, the training sequence length maximizing the effective signal-to-noise ratio is r3 = arg max f 1 x, α x x { x, x } If < r2 0 L 1, then x < r 0 As F 1 is concave, = r 0 In conclusion, B Proof of Proosition 2 = r 3 r r 0 30 he effective channel caacity can be written as C eff, α = g 1, α where g 1 x, α = 1 x 2 log 1 + d2 min ασ 2 t xx L+11 α x xx L+11 α+lαx, for {x, α} [r 0, + [ R + Since, 2 g 1 α < 0, g 2 1 is concave with resect to α for a fixed value of x hus, when x is fixed, g 1 has a unique maximum reached for α = α x such as α x = Ax AxLx Ax Lx 31 where Ax = xx L + 1 Note that 0 < α x < 1 Hence, the solution of 24 is [11]:, α = arg max g 1m, α m, α 32 r 0 m where m is an integer Let x [r 0, + [ and G 1 x = g 1 x, α x = x log 1 + σ2 t d2 min x L+1 4σ 2 AxLx+Lx 33 As G 1x < 0, for x [r 0, + [, G 1 is concave Besides, G 1 < 0 hus, we consider the two ossible cases: - If G 1r 0 0 G 1r 0 = min r0 x G 1x g1 x r 0, α r 0 = min r0 x g1 x x, α x, then = r 0 and α = α r 0 - If G 1r 0 > 0, then according to the theorem of intermediate values, there exists a unique y ]r 0, [ such as g1 x y, α y = 0 Hence, = arg max y { y, y } G 1 y and α = α

6 ISCCSP 2008, Malta, March REFERENCES [1] L R Bahl, J Cocke, F Jelinek, and J Raviv, Otimal decoding of linear codes for minimizing symbol error rate, IEEE rans Inf heory, vol I-32, 1 100, March 1974 [2] B Hassibi and B M Hochwald, How much training is needed in multile-antenna wireless links?, IEEE rans on Inf heory, vol 49, no 4, , Aril 2003 [3] H Vikalo, B Hassibi, B Hochwald, and Kailath, On the caacity of frequencey-selective channels in training-based transmission schemes, IEEE rans on Signal Processing, vol 52, no 9, , Setember 2004 [4] I Barhumi, G Leus, and M Moonen, Otimal training design for MIMO OFDM systems in mobile wireless channels, IEEE rans on Signal Processing, vol 51, no 6, , June 2003 [5] -L ung, K Yao, and RE Hudson, Channel estimation and adative ower allocation for erformance and caacity imrovement of multileantenna ofdm systems, IEEE Worksho on Signal Processing Advances in Wireless Communications SPAWC 01,, no 2, March 2001 [6] F Wong and B Park, raining sequence otimization in MIMO systems with colored interference, IEEE rans on communications, vol 52, no 11, , November 2004 [7] S Buzzi, M Los, and S Sardellitti, Performance of iterative data detection and channel estimation for single-antenna and multile-antennas wireless communications, IEEE rans on Vehicular echnology, vol 53, no 4, , July 2004 [8] S Crozier, D Falconer, and S Mahmoud, Least sum of squared errors LSSE channel estimation, IEE Proceedings, vol 138, , August 1991 [9] N Sellami, A Roumy, and I Fijalkow, he imact of both a riori information and channel estimation errors on the MAP equalizer erformance, IEEE rans on Signal Processing, vol 54, no 7, , July 2006 [10] GD Forney, Maximum-likelihood sequence estimation for digital sequences in the resence of intersymbol interference, IEEE rans on Inf heory, vol 18, , May 1972 [11] HP Christos and K Steiglitz, Combinatorial otimization: algorithms and comlexity, Prentice Hall, New Jersey, USA, 1982

TO combat the effects of intersymbol interference, an equalizer

TO combat the effects of intersymbol interference, an equalizer 2716 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 7, JULY 2006 The Impact of Both a Priori Information and Channel Estimation Errors on the MAP Equalizer Performance Noura Sellami, Aline Roumy,