Performance analysis of the MAP equalizer within an iterative receiver including a channel estimator

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1 Performance analysis of the MAP eqalizer within an iterative receiver inclding a channel estimator Nora Sellami ISECS Rote Menzel Chaer m 0.5 B.P 868, 308 Sfax, Tnisia Aline Romy IRISA-INRIA Camps de Bealie 3504 Rennes Cedex, France Inbar Fijalow Laboratoire ETIS, UMR 805 ENSEA-UCP-CNRS, 6 av. d Poncea 9504 Cergy-Pontoise, France Abstract In this paper, we consider an iterative receiver composed of a Maximm A Posteriori MAP eqalizer and a MAP decoder. Dring the iterations, the eqalizer and the decoder exchange extrinsic information and se them as a priori in order to improve their performance. We consider here an iterative receiver inclding a channel estimator. We propose to stdy analytically the impact of both the a priori information and the channel estimation errors on the eqalizer performance. We show that it is eqivalent to a shift in terms of signal-to-noise ratio SNR and we provide an analytical expression. Simlation reslts show that the analytical expression we give approximate qite well the eqalizer performance. I. INTRODUCTION The optimal receiver for a freqency selective coded channel performs joint eqalization and decoding which maes its complexity generally prohibitive. A soltion achieving a good complexity/performance trade-off is to se an iterative receiver composed of a soft-inpt soft-otpt SISO eqalizer and a SISO decoder []. The basic idea behind iterative processing is to exchange extrinsic information among the eqalizer and the decoder in order to achieve sccessively refined performance. The optimal SISO algorithm, in the sense of imm bit error rate, to be sed for eqalization and decoding is the symbol MAP algorithm []. Hence, in this paper, we consider an iterative receiver composed of a MAP eqalizer and a MAP decoder. We propose to stdy analytically the impact on the eqalizer performance of both the channel estimation errors and the a priori information provided by the channel decoder. To do that, we follow the approach of [3] and [4] which stdied the impact of channel estimation errors on the eqalizer performance. In [3], Gorohov stdied the impact of channel estimation errors on the performance of the Viterbi eqalizer and showed that it is eqivalent to a loss in SNR and evalated this loss. In [4], we have extended the stdy to a List-type MAP eqalizer prefiltered by the whitened matched filter, in the case of mltiple-inpt mltiple-otpt MIMO systems. In [5], we assmed that the channel is perfectly estimated by the receiver and we showed that the se of the a priori information by the MAP eqalizer is eqivalent to a gain in SNR. In this paper, we consider a more realistic scheme where the channel estimation is not perfect. In this case, we will show that the se of both the a priori information and the channel estimate by the MAP eqalizer is eqivalent to a shift in SNR and we will give a closed form of this shift. This wor is a first step in the stdy of the convergence analysis of iterative receivers. Most analyses are based on extrinsic information transfer EXIT charts [6]. These analyses se generally simlations since it is difficlt to stdy analytically the performance of a MAP eqalizer having a large nmber of states. Actally, analytical stdies based on the EXIT fnction have been performed when the trellis has only two states [7]. The contribtion of or paper is to give an analytical stdy of the MAP eqalizer performance when the nmber of states is greater than two. Throghot this paper scalars and matrices are lower and pper case respectively and vectors are nderlined lower case.. T denotes the transposition and I m is the m m identity matrix. II. SYSTEM MODEL We consider a coded data transmission system over a freqency selective channel depicted in Figre. The inpt information bit seqence is first encoded with a convoltional encoder. The otpt of the encoder is interleaved, mapped to the symbol alphabet A. For simplicity, we will consider only the BPSK modlation A = {+, }. We assme that transmissions are organized into brsts of T symbols. The channel is spposed to be invariant dring one brst. The received baseband signal sampled at the symbol rate at time is L x = h l s l + n l=0 where L is the channel memory. In this expression, n are modeled as independent samples of a real white Gassian noise with normal probability density fnction pdf N 0, σ where N α, σ denotes a Gassian distribtion with mean α and variance σ. The term h l is the l th tap gain of the channel, which is assmed to be real valed. Let s = s T,..., s L T be the L + T -long vector of coded symbols and n = n T,..., n 0 T be the T -long noise vector. The otpt of the channel is the T -long vector x = x T,..., x 0 T defined as x = τhs + n

2 where τh is a T T + L Toeplitz matrix with its first row being h 0, h,..., h L, 0,..., 0 and its first colmn h 0, 0,..., 0 T. c s Encoder Interleaver Modlation Channel + n x x Channel estimate Eqalizer L e s Deinterleaver Ls Fig.. Interleaver L d s Transmitter strctre Decoder Data estimate Fig.. Transmitter strctre When the channel is nown and no a priori information is provided to the eqalizer, the data estimate according to the seqence MAP criterion eqivalently the maximm lielihood ML criterion since there is no a priori is given by ŝ MAP = arg x τh : A T +L. 3 Now, we consider a particlar error event characterized by its length m [8]. Ths, we sppose that there exists an interval of size m sch that all the symbols of ŝ are different from the corresponding symbols of s while the preceding symbol and the following one are the same for s and ŝ. Define s m and ŝ m to be the vectors of symbols corresponding to this interval and the vector of errors e m = ŝ m s m. A sbevent E m of the error event is that ŝ m is better than s m in the sense of the ML metric E m : x m τ m hŝ m x m τ m hs m 4 where x m is the sbvector of x and τ m h is the bloc of τh corresponding to the error interval. The probability P E m of E m is given by [8]: εm P E m = Q σ where ε m = τ m he m and Qα = π α exp y dy. Let Σ m be the set of all possible error events of length m. Then, the probability, P Σ m, that any error event is of length m is bonded by the sm of the probabilities of the sbevents E m 5 P Σ m E m P E m. 6 Let d be the channel imm distance [8]. Becase of the exponential decrease of the Gassian distribtion fnction, the overall probability of error P Σ m P Σ m will be doated at high SNR by the term involving the imm vale d of ε m. Ths P Σ Q d σ. 7 Or goal is to find an approximation of P Σ when the eqalizer is integrated into an iterative receiver inclding a channel estimator. III. ITERATIVE RECEIVER As shown in Figre, the receiver consists of two soft-inpt soft-otpt SISO processors, the eqalizer and the decoder. We consider only the MAP approach for both eqalization and decoding, sing the BCJR algorithm []. The MAP eqalizer comptes the a posteriori probabilities APPs on the coded bits, P s = s x, h, s A, L T, sing the received vector x and the channel estimation vector h= h0,, h T L, and otpts the log-lielihood ratios LLRs [6]: L e s = L s x, h L s = log P s = + x, h P s = x, h log P s = + 8 P s = which are the a posteriori LLRs L s x, h s the a priori LLRs L s. These a priori LLRs are provided by the decoder. At the first receiver iteration, L s = 0 since no a priori information is available. The LLRs L e s are then deinterleaved and provided to the decoder as inpt information, in order to refine its reliability. The MAP decoder comptes the APPs P s = s r, r = L e s L,, L e s T T, and otpts the LLRs L d s = log P s = + r P s = r log P s = + P s =. These LLRs are then interleaved and provided to the eqalizer as a priori, L s, at the next iteration. After some iterations, hard decisions are taen on the information bits by the decoder. IV. PERFORMANCE ANALYSIS Now, we want to evalate the impact of both the a priori information and the channel estimation errors on the MAP eqalizer performance. The stdy will be done here for the eqalizer sing the seqence MAP criterion. It holds for the symbol MAP eqalizer sing the BCJR algorithm [] since the two eqalizers have almost the same performance as shown in [9, page 84]. Proposition: Sppose we are given a freqency selective channel with a memory of L and an additive white Gassian noise AWGN with noise variance σ. Assme that the otpts of an AWGN channel with noise variance σa are also available as observations corresponding here to the a priori observations. Sppose that the channel is estimated by sing a perfect training seqence of length T 0, having ideal

3 atocorrelation properties. The estimates h l of the tap gains h l, for 0 l L, are ths modeled as h l = h l + σ e l, where l are independent Gassian random variables with zero mean and variance and σ e = σ T0.Then, at high SNR, the MAP eqalizer sing the a priori information and the channel estimate is eqivalent to the MAP eqalizer having no a priori information and a perfect channel nowledge bt with an eqivalent signal-to-noise ratio SNR ˆ est = SNR + 8µ d + Lr + 8µ d where SNR is the tre signal-to-noise ratio, µ = σ σ a and r = σe σ. Remar: The representation of the a priori information as the otpts of an AWGN channel is a good approximation of the decoder otpts. Actally, it has been shown in [][0][6] that it is eqivalent to have at the eqalizer inpt a set of observations 9 z = s + w 0 where w N 0, σa. Ths, the LLRs L s fed bac from the decoder can be modeled as independent and identically distribted i.i.d samples from a random variable with the conditional pdf N s σ a, 4 σ a for some σ a [0][6]. Proof: The proof is divided into three parts. First, the probability of an error sbevent of length m, P E m, is derived and then pper bonded. Finally, the overall probability of error, P Σ, is calclated in order to find an approximation of the eqivalent SNR. Proof-part: P E m Let h= h 0,, h L T be the vector of tre channel parameters and h= h0,, h T L be its estimate. Taing into accont the a priori information and the channel estimation errors, the a posteriori probability of the seqence s is given by ps x, z, h x τ hs exp σ exp z s σa where z= z T,..., z L T and τ h is a T T + L Toeplitz matrix with its first row being h0, h,..., h L, 0,..., 0 and its first colmn T Replacing h0, 0,..., 0. The data estimate according to the seqence MAP criterion is then given by x σ ŝ MAP = arg τ h + σa z : A T +L A sbevent E m of the error event of length m is that ŝ m is better than s m in the sense of the seqence MAP metric E m : x m τ m hŝ m + σ σ a z m ŝ m 3 σ x m τ m hs m + σa z m s m where z m is the sbvector of z and τ m h is the bloc of τ h corresponding to the error interval. Let µ = σ σ a, y= x T, x T,, x 0, µz T,, µz L T, T M = τh T, µi T +L a L T + L matrix and b= n T, n T,, n 0, µw T,, µw L T. Using and 0, we can write y = Ms + b. 4 Using h instead of h, the data estimate according to the seqence MAP criterion is given by, y ŝ MAP = arg M : A T +L 5 where M = τ h T, µit +L T. Hence, 3 is eqivalent to E m : y m M m ŝ m ym M m s m 6 where y m is the m+l sbvector of y corresponding to the error interval and M m = τ m h T, µim T. Let e m = ŝ m s m, then 6 is eqivalent to the event M m e m e T M m m y T m M m s m. 7 Let M m h = M m M m and b m = y m M m s m, then we obtain M m e m e T M m m T M m h s m + e T M m m T b m. 8 Using the assmptions given in [3], we obtain thatτ m he m εm + ξ T0 where ξ T0 tends in probability to 0, hence M m e m = τm he m + 4mµ ε m + 4mµ. 9 M m e m by εm +4mµ and M m e m by E m = M m e m in 8, leads to ε m + 4mµ E T m b m ET m M m h s m 0 Since the lower bloc of M m h is eqal to zero, we have

4 E T m M m h s m = ε T m M m h s m. We sppose that h = h h N 0, C, C being the covariance matrix of h. Defining C m s = H L s m CH L s m T, H L s m being the Hanel matrix sch as H L s m h = M m h s m, we obtain where χ s N 0, s with ε m + 4mµ χ s s = 4σ E m + 4ε T m C msε m = 4σ E m + + 4mµ ε m ε T m C msε m σ ε m 3 Hence, the probability of the error event P E m is given by P E m =Q E m + σ =Q ε m + 4mµ σ ε m + 4mµ ε m + + 4mµ ε m ε T m C msε m σ ε m / ε T m C msε m σ ε m We sppose here that a perfect training seqence of length T 0 is sed and then h l = h l + σ e l, where l are modeled as independent complex Gassian random variables with zero mean and variance and σ e = σ T0. Ths, C m s LσeI m, and ε T m C msε m Lσ e ε m. This leads to Ẽm P E m = Q σ ε m + 4mµ = Q + σ where r = σe σ. Lr + 4mµ ε m / / P Σ Q d σ 4 5 Proof-part: pper bond for P E m : In order to find an approximation of P Σ, the overall probability of error, we now want to find a lower bond for the qantity Ẽm. Actally, at high SNR, this term will doate the sm of the probabilities of the error events becase of the exponential decrease of the Gassian distribtion fnction. Let s consider first the case of perfect channel nowledge. By definition, ε m d. Moreover, we have m. Ths, a lower bond for Ẽm = ε m + 4mµ is given by bondµ = d + 8µ. 6 In [5], we showed that this bond is tight. When the channel is not perfectly estimated, the probability of the error event can be rewritten as P E m = Q ε m + 4mµ. 7 σ ε m Lr + ε m + 4mµ We assme here that L << T 0 which is eqivalent to Lr <<. In this case, ε m Lr is negligible compared to ε m. When the error seqence allowing to attain the imm distance is of length m =, we can consider that the qantity obtained by calclating Ẽm taing m = and ε m = d is a lower bond for Ẽ m. Otherwise, an exhastive search has to be done to find the lower bond. Proof-part3: P Σ As in the case withot a priori and perfect channel nowledge, at high SNR, the overall probability of error P Σ can be approximated by + 8µ d + + 8µ d / Lr. 8 Ths, the expression of the error probability given in 8 can be seen as the one given in 7 with an eqivalent signal-tonoise ratio SNR ˆ est = SNR + 8µ d + Lr + 8µ d V. SIMULATION RESULTS. 9 In or simlations, we consider the following channels: Channel3: 0.5; 0.7; 0.5 Channel5: 0.9; 0.50; 0.58; 0.50; 0.9 The modlation sed is the BPSK. The transmissions are organized into brsts of 5 symbols. Figres 3 and 4 show the Bit Error Rate crves with respect to the SNR, for and for r = σe σ = 0.3. different vales of the ratio µ = σ σ a This vale of the ratio r corresponds to an ideal training seqence of length T 0 =. Each crve is obtained while the ratio µ is ept constant. The solid lines indicate the receiver performance obtained by simlations. The dotted lines are obtained by shifting the crve corresponding to the case with no a priori and with a perfect channel nowledge µ = 0, r = 0 by the vales of the SNR shift: 0 log 0 + 8µ 0 log d 0 + Lr + 8µ d. Table. shows the vales of the imm error distance d and the imm distance inpt error seqence for the channels of interest [].

5 Channel3 Channel5 d Error seqence,, T able. For Channel3, we notice that the theoretical crves dotted lines approximate well the. However, for larger µ µ = 0.83, the approximation becomes slightly erroneos. Moreover, Figre 4 shows that the approximation is better for Channel5. We can conclde that the approximation holds in general for σ < σ a and that it is better for Channel5. Now, we propose to test the reliability of the term de to the channel estimation errors in 9. In Figre 5, the solid crves indicate the performance of the eqalizer fed by the a priori when the channel is estimated r = 0.3, for different vales of µ. The dotted crves are obtained by shifting the crves given by simlations for different vales of µ and r = 0 by the vales of the SNR loss: 0 log 0 + Lr + 8µ d. We notice that the dotted crves approximate well the solid ones independently of the vales of µ. Hence, the gap observed between the theoretical crves and the crves obtained by simlations in figres 3 and 4 is de to the term corresponding to the effect of the a priori and not to the term corresponding to the channel estimation errors in µ= Fig. 3. Comparison of the eqalizer performance and the theoretical performance when r = σe = 0.3, for different vales of µ, for Channel3. σ Fig. 4. Comparison of the eqalizer performance and the theoretical performance when r = σe = 0.3, for different vales of µ, for Channel5. σ µ= Fig. 5. The solid crves indicate the performance of the eqalizer within the iterative receiver when the channel is estimated r = 0.3. The dotted crves are obtained by shifting the crves given by simlations for different vales of µ and r = 0 by the vales of the SNR loss. VI. CONCLUSION In this paper, we considered an iterative receiver composed of a MAP eqalizer and a MAP decoder. We proposed to stdy analytically the impact of both the a priori information and the channel estimation errors on the eqalizer performance. We gave an approximation of the error probability which allows s to find an expression of the shift in terms of the SNR de to the se of the a priori information and the channel estimate. Simlation reslts showed that this expression gives a qite good approximation especially for long channels. This wor is a first step in the stdy of the convergence analysis of iterative receivers. REFERENCES [] C.Doillard, M.Jézéqel, C.Berro, A.Picart, P.Didier, and A.Glaviex, Iterative correction of intersymbol interference: trbo-eqalization, Eropean Trans. Telecommn., vol. 6, no. 5, pp , 995. [] L.R.Bahl, J.Coce, F.Jeline, and J.Raviv, Optimal decoding of linear codes for imizing symbol error rate, IEEE Trans. Inf. Theory, vol. IT-3, pp.84-87, March 974. [3] A.Gorohov, On the performance of the Viterbi eqalizer in the presence of channel estimation errors, IEEE Signal Process. Letters, vol. 5, no., pp. 3-34, December 998. [4] N.Sellami, I.Fijalow, and S.Perrea, Performance analysis of a Listtype eqalizer over estimated MIMO freqency selective channels, EUSIPCO 0, Tolose, France, September 00. [5] N.Sellami, A.Romy, and I.Fijalow, On the analysis of the MAP eqalizer performance within an iterative receiver, Signal Process. Advances in Wireless Comm. SPAWC 04, Lisbon, Portgal, Jly 004. [6] M.Tüchler, R.Koetter, and A.Singer, Trbo eqalization: principles and new reslts, IEEE Trans. on Comm., vol. 50, no. 5, pp , May 00. [7] A.Romy, S.Gemghar, G.Caire, and S.Verd, Design methods for irreglar repeat accmlate codes, IEEE Trans. on Inf. Theory, to appear Agst 004. [8] G.D.Forney, Jr., Maximm-lielihood seqence estimation for digital seqences in the presence of intersymbol interference, IEEE Trans. Inf. Theory, vol. 8, pp , May 97. [9] S. Benedetto and E. Biglieri, Principles of Digital Transmission with Wireless Applications, NewYor: Klwer/Plenm, 999. [0] S.Ten Brin, Convergence of iterative decoding, IEEE Electronic Letters, vol. 35, pp , May 999. [] J.Hagenaer, E.Offer, and L. Pape, Iterative decoding of binary bloc and convoltional codes, IEEE Trans. on Inf. Theory, vol. 4, Isse, pp , March 996. [] W.Ser, K.Tan, and K.Ho, A new method for detering nnown worst-case channels for maximm-lielihood seqence estimation, IEEE Trans. on Comm., vol. 46, no., pp , Febrary 998.