Semi-blind equalization for high speed GMSK-based mobile communications

Transcription

1 1 Semi-blind equalization for high speed GMSK-based mobile communications Raphaël Gallego, Member, IEEE, Florence Alberge, Member, IEEE, Pierre Duhamel, Fellow, IEEE, and Alexandre Rouxel, Member, IEEE Abstract This paper is concerned with semi-blind equalization techniques applied to bursty GMSK-based mobile communication systems. In bursty time-division multiple access mobile communication systems, the standards specify a maximum relative speed between transmitter and receiver at a given carrier frequency. However, in practical situations the maximum Doppler spread resulting from these maximum relative speed and the frequency desynchronization between the mobile and the base station can greatly exceed the specified conditions. So in the worst cases the practical rate of channel variation along the burst can be much more important than planned in the standards. In this paper, new semi-blind algorithms are proposed to provide reliable bursty communication through a multipath fast fading channel. The underlying methods belong to the Conditional Maximum Likelihood family, and can be implemented as adaptive algorithms. In high speed scenarios, we show that the proposed solutions outperform the Viterbi algorithm used in classical implementations, while providing similar performances in a fixed setting. Index Terms Semi-blind equalization, SIMO model, deterministic maximum likelihood, mobile communications, short burst communications, GSM/GPRS system, Viterbi algorithm, fast fading channels. I. INTRODUCTION In mobile communications systems (MCS), signals are transmitted through multipath mixed-phase time-variant channels. Mobile multipath channels have non-desirable impact on the transmitted signal, including inter-symbol interference (ISI) and channel variations. To deal with these effects, equalization techniques are usually applied to the received signal. In bursty MCS, such as Global System for Mobile (GSM), data are split in finite sequences or bursts so that the time-variant channels can be considered nearly constant during each transmitted burst. Stateof-the-art bursty MCS equalizers make use of a training sequence (TS), located in the middle of the burst (a R. Gallego and A. Rouxel are with the Advanced Baseband Department, Wavecom S.A., 3 Esplanade du Foncet, 944 Issy-les-Moulineaux Cedex, France ( rgallego@ieee.org; alexandre.rouxel@ieee.org) F. Alberge and P. Duhamel are with the Signal Processing Department, LSS-Supélec, 3 rue Joliot-Curie, 9119 Gif-sur-Yvette Cedex, France ( florence.alberge@lss.supelec.fr; pierre.duhamel@lss.supelec.fr)

2 midamble ), to estimate the channel parameters in the receiver. These channel parameters are then used to recover the transmitted data. Nevertheless, in situations involving fast fading channels, TS-based equalization has limitations since the information about the channel parameters that can be obtained from the TS is essentially non-reliable. Indeed, the channel value at the end of the burst can be quite different from the estimate provided by the training midamble. For example, a GSM cellular phone moving at 300 Km/h (for a frequency carrier of 1800 MHz) results in a Doppler shift of 500 Hz which leads to a coherence time of nearly 0.8 ms (corresponding to more than 00 symbols in GSM). Taking into account that a typical GSM half-burst contains approximately 60 symbols, this implies non-negligible channel variations during the equalization process. Blind non-cooperative equalization methods have been applied to bursty MCS in order to obtain satisfactory performances without using any training symbols [1], [], but they lack robustness to channel order overestimation, among others [3]. Semi-blind equalization techniques are attractive as they take advantage of both training-based and non-cooperative approaches [4], [5]. In blind equalization, whereas the use of High Order Statistics (HOS) is necessary in Single-Input-Single-Output (SISO) systems, both Higher and Second Order Statistics (SOS) approaches can be considered in the case of SIMO (Single-Input-Multiple-Output) systems. HOS approaches were first used for blind equalization [6], [7], [8], [9], [10], but they usually need a large number of samples compared to SOS-based methods, so their applicability in fast fading environments is limited because statistics cannot be assumed stable, even along the (short) burst. L. Tong et al introduce in [11] the use of SOS in blind equalization/estimation through to the extraction of spatial or temporal diversity, leading to SIMO models (see also [1], [13]). Different SOS methods have been studied in literature, like subspace methods [14], [15] or maximum likelihood methods [16], [17], [18]. In general, Maximum Likelihood (ML) methods are attractive due to their good near-asymptotic performance and relative simplicity [19]. Among these, Statistical ML (SML) approaches [18] achieve the best performance though at the cost of local minima in the minimization problem. Conversely, the Deterministic ML (DML) approaches [16], [17] lead to poor performances at low SNR since the method does not make use of any knowledge of the source. This lack of performance prevents the use of these methods in typical mobile communications scenarios. Conditional ML (CML) methods have also been proposed [0], [1]. These methods are a tradeoff between SML and DML approaches, since they use only part of the available prior information. Slock et al proposed in [0] a Gaussian ML (GML) approach in order to regularize the criterion; however, the prior information introduced (a Gaussian distribution for the transmitted symbols) may seem unrealistic. Another CML approach has been proposed in [1] by Alberge et al, with a convenient usage of the prior information. In fact, the classical SML and DML approaches can be viewed as special cases of this CML approach which leads to a good tradeoff between the asymptotic performance when the process is initialized close to the global minimum, and the number of local minima. In what follows, we focus on bursty MCS based on GSM using the Gaussian Minimum Shift Keying (GMSK) modulation, also known as the nd Generation (G) for mobile communication systems. The same modulation is

3 3 also used in GSM supporting General Packet Radio Service (GPRS), also referred as the.5g, and in the Enhanced GPRS (EGPRS) system (Modulation and Coding Scheme 1 to 4), also referred as the.75g []. In order to obtain a SIMO model (where CML approaches can be used), diversity should be available at the receiver. The use of extra antennae leading to spatial diversity increases the cost of the Radio Frequency front end. This solution can be prohibitive in mobile terminals. Oversampling can be used to gain temporal diversity, but its efficiency requires an excess bandwidth [3], and the GMSK used in GSM has small excess bandwidth [4]. However, the used GMSK can be approximated as a linear modulation involving a second order SIMO model. This result is obtained using the Laurent s linear decomposition [5] applied to a GMSK modulation with a symbol duration by bandwidth product (BT) equal to 0.3, as defined in the GSM standard and its evolutions []. This model is commonly used in the literature (e.g., [6], [7]) and will be adopted in this paper 1. The 3rd Generation Partnership Project (3GPP) specifies the speed (depending on the carrier frequency) between the transmitter and the receiver under which the system should guarantee a certain level of performance [8]. However, the standards do not require specific performances for speeds higher than 50 Km/h for 900 MHz carriers (i.e., a maximum Doppler spread of approximately 00 Hz). Classical Viterbi-based receivers still provide a reliable performance at these speeds with the GSM bursts length [9], [30] but an increasing number of situations with higher-speed applications (like mobile Internet in high-speed trains running at more than 300 Km/h) is now considered. In these cases, the Viterbi algorithm performance is not sufficient to fulfill the minimum specified requirements [8]. This paper deals with semi-blind equalization when applied to GMSK-based mobile communications in multipath fast fading channel conditions. Our work focuses on a family of the CML approaches presented in [31]. Our contribution is two-fold. First, block algorithms are proposed for blind equalization (CMLBA), and are then adapted to semi-blind context. We introduce here a new CML criterion (named CML-P in the paper), based on Theorem 1 of Section III, which states that the conditioning of given set of small sub-matrices is better than the one used for the whole burst. The new algorithm derived (CMLBAP) improves performance w.r.t. the previous CMLBA and is computationally more efficient. The performance achieved by the algorithm based on this new criterion (CMLBAP) is shown to be similar to that of the Viterbi algorithm in the block implementation, for the targeted range of SNR s in the fixed setting, and better than Viterbi under time-varying channel conditions [3]. Second, the CML-P criterion is extended to an adaptive scenario. Starting from the adaptive algorithms (CMLAA) analyzed in [33] based on CML criterion, a new adaptive algorithm (CMLAAi) is proposed which achieves performance similar to those of CMLBAP in a fixed setting and, in the same time, is well adapted to time-varying channel conditions. Simulations show the improvement given by the new algorithm w.r.t. the Viterbi one in a high speed environment (300 Km/h for 1800 MHz carrier) [34]. 1 In the EDGE (Enhanced Data Rates for GSM Evolution), packet (EGPRS) and circuit switched (ECSD) systems, which both use a 8-PSK modulation for high data rate modes, this linear decomposition is used to derived the 8-PSK emission filter from the GMSK modulation in order to obtain optimally closed spectrum for all emitted signals [4] and then to minimize the effect of this evolution on the cellular network plan.

4 4 The paper is organized as follows. The background as well as the common definitions and notations used in this paper are given in Section II. Our contribution to block algorithms is detailed in Section III. The case of adaptive algorithms is analyzed in Section IV. Simulation results and performance comparison are given in Section V. Finally, conclusion and final remarks are given in Section VI. II. BACKGROUND AND DEFINITIONS A. Diversity extraction It has been shown in [5] that any binary phase modulation with constant amplitude, as the GMSK, can be expressed as a sum of amplitude modulated pulses. Furthermore, it was also shown that a good approximation is achieved by keeping only the first term of this decomposition. In GMSK-based communications as defined in the GSM standard [4] a linear approximation is used. Given that the source is differentially encoded to avoid the zero crossing (as specified in the standards) and applying this approximation to the GMSK emitted signals defined in the GSM system, the following model is given. Denoting s(n) = ±1 the transmitted binary data and b(n) the noise samples, the received signal, before de-rotation (operation that consists in shifting each complex-valued symbol by 90 degrees), can be written as: x(n) = k h(k) s(n k) j n k + b(n), (1) h = h0 h p g () where h 0 denotes the first order filter from the GMSK linear decomposition, h p stands for the physical propagation channel, and g is used for the reception filter. As shown in [7], the main pulse h 0 contains approximately 99,5 per cent of the total GMSK pulse energy. Once a de-rotation operation is performed, the received signal reads: x(n) j n x(n) = k h(k) s(n k) j k b(n) j n = k h(k) s(n k) + b(n), (3) with h(k) h(k) j k and b(n) b(n) j n, (4) where h(k) is the global channel impulse response after de-rotation. Taking into account that h(k) is complex and s(n) a real BPSK signal, the de-rotated signal x(n) sampled at the baud rate can be seen as the output of a twosubchannel system (one for the real part, one for the imaginary part). This approach allows the use of algorithms that are usually applied in a SIMO (Single-Input-Multiple-Output) context [7]. B. SIMO Model. Definitions and Notations Let N denote the number of received symbols, M the channel order (thus, M + 1 is the length of the channel impulse response), and L the diversity order (L = in our case). Define:

5 5 The transmitted symbols sequence of size (M + N) 1, s M+N [ s( M),..., s(0),..., s(n 1) ] T The (sub-)channel impulse response of size (M + 1) 1, h c n [ h c n (M),..., hc n (0)] T, c = {1,..., L} and its SIMO equivalent, of size L (M + 1) 1, h n [ [ h 1 ] T [ ] n... h L T ] T n Finally, the received symbols sequence of size L N 1, X N [ x 1 (0),.., x L (0),..., x 1 (N 1),.., x L (N 1) ] T (n defines the time index) Now assume that the channel is not time-varying during a given burst of length N, that is, h n = h, n = 0...N 1. In this (block) approach, the received signal can be written as follows: X N = T N, M+N (h) s M+N + B N (5) where B N is the vector of size L (N) 1 stacking the received noise samples and T N, M+N (h) is a Sylvester matrix of size (L N) (M + N) defined as: T N, M+N (h) = C. Conditional Maximum Likelihood (CML) approach h 1 (M)... h 1 (0) h (M)... h (0) h 1 (M)... h 1 (0) h (M)... h (0). (6) 1) Maximum Likelihood approach: The criterion developed in our work belongs to the family of Maximum Likelihood (ML) criteria. ML approaches are often used because of their good performances when using a limited number of symbols, which is the case of short bursty mobile communications. Other approaches need a larger number of symbols to achieve sufficient performances, even if they can have a lower computational cost [35]. If no a priori statistics are assumed for the source, then both symbols and channel are to be estimated by the receiver, leading to a Deterministic ML (DML) approach [16], [17]. The DML criterion, which is minimized over both the channel and symbols, reads: JB DML ( ) h,sm+n = TN, M+N (h) s M+N X N. (7) The channel identifiability conditions for the SOS-SIMO case can be found in Chapter 7 of [3]

6 6 If full knowledge of the source statistics is assumed (i.e. equal probability among a given alphabet), then a SML approach [18] can be considered. This knowledge has the ability of improving the performances, at the cost of an increase in the number of local minima. The SML criterion is expressed as: JB SML ( ) h,sm+n = TN, M+N (h) s M+N X N, (8) with s M+N { 1, +1}. ) The CML criterion: from DML to SML: The criterion used in our work intends to achieve a good tradeoff between the existence of local minima and performances. Starting from DML approach, the idea consists in introducing new hypothesis that will allow to improve the performances while controlling the local minima problem. In DML approach, both symbols and channel are unknown variables and the criterion derived from the joint minimization is non-convex, leading to the existence of local minima, even if the criterion is convex w.r.t. each variable (s or h) separately. Besides, if the addition of new hypothesis keeps this convexity unchanged, then we may expect that no sizeable number of new local minima will be added to the global criteria. Actually, the stability of the global minimum is of crucial importance for the recursive procedure that will be implemented later. In our CML approach symbols are considered as random variables with a given probability density function (pdf), different from the real one, but reflecting some a priori knowledge of the symbols. This class of methods has been first introduced in [0] to regularize the criterion, in where a GML (Gaussian ML) approach is analyzed (see also [36]). In our case, a truncated exponential function is used, allowing to keep the above mentioned convexity: where Z is a normalization factor. p(s(n)) = 0, if s(n) > 1 p(s(n)) = 1 Z eks (n), if s(n) 1 p(s M+N ) = Π n p(s(n)) (9) As we can observe, when k 0, the pdf tends to a constant function on the interval [ 1, +1]. On the other hand, when k, the criterion corresponds to an SML approach, since symbols are constrained to be { 1, +1}. The choice of k follows from a tradeoff between the quantity of a priori information used and the respect of the convexity property [33]. Considering that the noise in (5) is AWGN (Additive White Gaussian Noise) and with variance equal to σ, the likelihood function conditioned on both the channel and the symbols is expressed as: where K is a normalization factor. p(x N h,s M+N ) = ( = K exp 1 ) σ X N T N, M+N (h)s M+N Taking into account the prior information used for the symbols given by (9), the following likelihood function (10)

7 7 for the symbols estimation can be derived: p(x N,s M+N h) = p(x N h,s M+N ) p(s M+N ) = K exp ( X N T N, M+N (h)s M+N + σ k s M+N ), (11) for s M+N 1, where K is a normalization factor. Thereby, the CML criterion can be expressed as follows: JB CML ( ) h,sm+n = TN, M+N (h) s M+N X N γ sm+n, sm+n B M+N (1) with B K = {s R K ; 1 s +1, 1 i K} and γ = kσ. (13) Thus, the γ parameter controls the amount of a priori information taken into account. The upper bound for which the criterion remains convex w.r.t. each variable separately is obtained by forcing the second derivative of (1) w.r.t. the symbols to be zero [33]. This results in: where λ min (A) denotes the minimum eigenvalue of A. γ = λ min { TN, M+N (h) H T N, M+N (h) } (14) III. BLOCK APPROACH Among the M+N symbols contained into s M+N, let N k (resp. N u ) denote the number of known (resp. unknown) symbols at the receiver. Thus, the total symbol sequence can be written as: [ [snk] T ] ] T T ŝ M+N = [ŝnu. (15) Actually, in a mobile radiocommunications scenario, the known symbols used by a semi-blind algorithm are those of the training sequence 3. A. CML Block Algorithm (CMLBA) The block algorithm can be derived from the CML criterion above and adapted for the semi-blind context. A new notation is introduced in the criterion in order to take into account the known symbols knowledge. It is defined as the known interference directly subtracted from the received symbols and described by the following equations: If N0 (s N1 ;h) = V N0, N 1 (h) s N1, (16) If N0 (s N1,...,s NP ;h) = If N0 (s N1 ;h) If N0 (s NK ;h). (17) 3 Because of the equivalence of the backward/forward equalization processes in the GSM System, in this paper we assume that we work always with the forward half-burst, so that the midamble can be seen as a training sequence located at the beginning of the half-burst

8 8 The matrix V N0, N 1 (h) denotes the matrix that results when keeping only the N 1 columns of a Sylvester T N0, N T (h). The context clearly indicates which N 1 columns correspond to a given matrix V N0, N 1 (h). Thus, the semi-blind criterion can be expressed as: JB CML ( ) h,snu = TN, M+N (h) s M+N X N γ snu = [ V N, Nk (h) V N, Nu (h) ] s N k X N γ snu s Nu = VN, Nu (h) s Nu ( X N V N, Nk (h) s Nk ) γ snu = VN, Nu (h) s Nu ( X N If N (s Nk ;h) ) γ snu (18) with γ = λ min { VN, Nu (h) H V N, Nu (h) }. (19) In [37], another form for the equation (5) is used for the CML approach, that is equivalent to the classical one: T N, M+N (h) s M+N = U N, M+1 (s M+N ) h (0) where U N, M+1 (s M+N ) is a Symbols matrix of size (L N) (L (M + 1)) defined as: I L sm (n) T I L sm (n 1) T U N, M+1 (s M+N (n)) =. I L sm (n N + 1) T where I L denotes the identity matrix of size L L and where stands for the Kronecker product. That allows to rewrite the criterion in an equivalent form but well-suited to be minimized by the variable h: JB CML ( ) h,sm+n = UN, M+1 (s M+N ) h X N γ sm+n. (1) Thus, the block algorithm consists in the separate minimization with respect to s and h variables, so an iterative procedure is deduced as follows. Starting from an initial estimate of the channel ĥ(0), we can write: for k = 1:ite B end ŝ (k) N u = arg min J CML(ĥ(k 1) ) B,s Nu s Nu B Nu () [ ŝ (k) M+N = [snk ] T [ŝ(k) ] ] T T N u (3) ĥ (k) = argmin h J CML B ( h,ŝ (k) M+N ) (4) The inequality constraint for symbols minimization (13) is resolved by a relaxation method (see [38] for details).

9 9 The number of (block) iterations (ite B ) performed is 0, which has been found to be a good compromise between performance and complexity according to our results. B. The Partitioning Procedure As already explained on the CML criterion, the γ parameter given by (19) plays a very important role, because it controls the amount of prior information taken into account. Thereby, our interest is to have λ min as large as possible (and, so, the largest γ). Theorem 1 below shows that a segmentation of () into a set of equations increases λ min. Theorem 1: Let T 1 and T be two sub-matrices of a matrix T such that T = [T 1 T ] then where λ T min λ T min λt1 min and λ T min λt min resp. λtp min stands for the smallest eigenvalue of TH T resp. T H p T p, p=1,. Proof: Let N resp. N p denote the number of columns of T resp T p. λ T min = arg min v C N v H T H Tv v H v Let v = [ v H 1 0H N ] H where v1 C N1 and 0 N C N, then vh T H Tv v H v v C N. In particular, The same line of arguments leads to: λ T min vh T H T v v H v λ T min arg min v 1 C N 1 = vh 1 TH 1 T 1v 1 v H 1 v 1 v H 1 TH 1 T 1v 1 v H 1 v 1 λ T min λt min. v 1 C N1. = λ T1 min. This method is easily extended to more partitions. In this case, the minimum eigenvalues corresponding to the associated sub-matrices increase, as well as the amount of a priori information that can be taken into account while keeping a convex criterion. Of course, the number of columns in each sub-matrix should remain larger than the channel order M for the sub-systems to be solvable. So, the optimal number of columns is M + 1. Thus, the partitioning procedure consists in the (vertical) division of the global filtering matrix T N, M+N (h). Apart from the known symbols, the rest of the matrix which is related to the unknown symbols is partitioned in P small sub-matrices of M + 1 columns. This procedure not only allows to take profit from the λ min condition, but also permits to realize P small optimizations (by the relaxation method) instead of a large one. So the benefit of the partitioning procedure is two fold: increase the performances of the block algorithm (because of a better choices of λ min ) and reduce the computational cost of the whole algorithm

10 10 C. CML Block Algorithm - Partitions (CMLBAP) In order to illustrate the advantages of the partitioning procedure, we derive a new algorithm named CMLBAP (P for Partitions), which employs the criterion detailed below: JBP CML ( ) h,snp = TN, M+N (h) s M+N X N γ snp = VN, Nu (h) s Nu ( X N If N (s Nk ;h) ) γ snp = [ V N, N1 (h)... V N, NP (h) ] [ s N1.. s NP ] ( X N If N (s Nk ;h) ) γ s Np = V N, Np (h) s Np ( X N If N (s Nk ;h) If N (s N1, s Np 1, s Np+1, s NP ;h) ) γ snp = VN, Np (h) s Np ( X N If N (s Nk ;h) If c N(s Np ;h) ) γ snp, (5) with γ = λ min { VN, Np (h) H V N, Np (h) } (6) and If c N (s N p ) = If N (s N1, s Np 1, s Np+1, s NP ;h). (7) Index p denotes the partition we deal with. The V Nu, N p (h) matrix corresponds to the associated channel matrix of the p th partition. P is calculated from the division of the total number of symbols to be estimated (N u ) by the size of these small matrices (N p = M + 1). If the calculated P is not an integer, the size of the last partition (N P ) is modified for convenience to fit an integer number of partitions into the N u symbols. Finally, If c N (s N p ) is introduced to simplify notation, and is defined as the complementary interference for the partition p, that is, the subtracted interference to the partition we deal with, considering the symbols of the rest of the partitions as known. Behavior of the algorithm is similar to the CMLBA one, but the minimization of variable s is now realized by P minimizations corresponding to the P small matrices V Nu, N p (h). Thus, considering a given block iteration k, the equations of the new algorithm (for the symbols minimization) can be written as follows: for p = 1:P ŝ (k) N p = arg min J CML(ĥ(k 1) ) BP,s Np s Np B Np = arg min s Np B Np VN, Np (ĥ(k 1) ) s Np ( X N Îf N (s N k ;ĥ(k 1) ) ˆ If c N (s N p ;ĥ(k 1) ) ) ˆγ snp, (8) with ˆγ = λ min { VN, Np (ĥ(k 1) ) H V N, Np (ĥ(k 1) ) } (9) ŝ (k) N u = [[ ŝ (k) N 1 ]T... [ŝ(k) N P ]T ] T (30)

11 11 end The minimization w.r.t. the channel remains unchanged (4). IV. ADAPTIVE APPROACH Until now, we have implicitly supposed that the vehicle speed w.r.t. the base station was zero or practically zero. This implies that the true channel impulse response is (practically) constant along the burst. When acting in real conditions, e.g., when communicating from high speed trains, the true channel response may vary rapidly. In extreme situations, the last channel response located at the end of the burst (ĥend ) can be totally different from the one situated near the training sequence (ĥini ). Thus, whereas in the block approach the implicit model of the received signal involves a fixed channel h, the underlying model of the adaptive approach is a true time varying model, involving various filters H [] along the burst where i is the time index. as a consequence, different channel estimates ĥ are computed along the burst in the adaptive approach. The channel and symbols estimates are updated upon the arrival of a new symbol. At a given time i, the estimated symbols sequence of size N k + i read: [ ŝ N k +i = [snk ] T [ŝ ] ] T T i. (31) Once all the unknown symbols contained in the burst have been processed by the receiver (i = N u ), the above estimated sequence corresponds to the global estimated symbols sequence as in the block approach (15): [ ŝ (Nu) [snk] T [ŝ(n N k +N u = u) ] ] T T N u = ŝm+n. (3) A. CML Adaptive Algorithm (CMLAA) The adaptive procedure is derived as follows. At each time i, only one symbol and channel minimization is performed, that is, a unique block iteration is done. In this symbols minimization, only the Q + 1 more recent symbols of ŝ M+N are updated. Let us introduce the following notations. Let su ˆ Q+1 denote the sequence containing the Q + 1 updated symbols at time i, whereas sp ˆ M denotes the previous symbols (the past interference) used in the estimation of su ˆ. Thus, the estimated symbols sequence reads, during the steady state: [ ŝ N k +i = [snk ] T [ŝ(i 1) T [ (i 1)] T [ ] T i M Q 1] sp ˆ M su ˆ Q+1 This behavior is depicted in Fig.1, where we consider a sliding window moving along the burst. For the channel minimization, because of only a given number of symbols are updated, a new recursive procedure is derived; in such procedure, the channel estimate ĥ is updated from the result of the previous minimization, ĥ (i 1). Details on this recursive procedure are given in the Appendices. ] T

12 1../../figures/eps/gsm_burst_sliding_window.eps Fig. 1. The burst in adaptive approach: sliding window The computational cost of the algorithm compared to its block equivalent is considerably reduced. This cost depends directly on to the number of symbols updated at each time i. In practice, a good tradeoff is achieved for Q = M + 1 (i.e., the length of the channel impulse response). The recursive criterion is detailed below: JR CML ( h,snk +i) = T N k M+i, N k +i (h) s N k +i X γ snk +i, (33) with N k M+i γ = λ min { T N k M+i, N k +i (h)h T N k M+i, N k +i (h)}. (34) Taking into account that we update only the Q + 1 symbols of su Q+1, the minimization of T N k M+i Q 1, N k +i Q 1 (h) s X X N k M+i Q 1 Q+1 N k +i Q 1 + T Q+1, M+Q+1 γ su Q+1 (h) s M+Q+1 (35) it is equivalent to the minimization of T Q+1, M+Q+1 (h) s M+Q+1 X Q+1 γ su Q+1. (36)

13 13 Therefore, the next criterion is considered: JR CML ( ) h,su Q+1 = = T Q+1, M+Q+1 (h) s M+Q+1 X Q+1 = [ V p Q+1, M (h) V u Q+1, Q+1 (h)] X Q+1 γ su Q+1 = V u Q+1, Q+1 (h) su Q+1 ( X Q+1 V p Q+1, M (h) sp M = V u Q+1, Q+1 (h) su Q+1 γ su Q+1 sp M su Q+1 ) γ su Q+1 ( X Q+1 If Q+1 (sp M ;h)) γ su Q+1, (37) with γ = λ min { V u Q+1, Q+1 (h)h V u Q+1, Q+1 (h)}. (38) Of course, the above criterion (33) can be expressed by the next equivalent form used for channel minimization thanks to (1): JR CML ( h,snk +i) = U N k M+i, M+1 (s N k +i ) h X N k M+i γ snk +i. (39) As we can observe, this recursive approach is now well suited to be applied to time-varying channel. A forgetting factor λ is introduced to follow the variations of the channel impulse response. The value of this parameter depends on the speed of this variation and must be adjusted consequently. The adaptive criterion is as follows: JA CML ( h,su Q+1, λ ) = 1/ ( Λ Q+1 T Q+1, M+Q+1 (h) s M+Q+1 X ) Q+1 γ su Q+1 = Λ 1/ Q+1 V u Q+1, Q+1 (h) su Q+1 Λ 1/ (X Q+1 Q+1 If Q+1 (sp M ;h)) γ su Q+1, (40) with and γ = λ min { V u Q+1, Q+1 (h)h V u Q+1, Q+1 (h)} (41) If Q+1 (sp M ;h) = V p Q+1, M (h) sp M (4) where Λ K is, in our case (L = ), a diagonal matrix diag {λ K 1, λ K 1,..., λ, λ, 1, 1}. The equations for the adaptive procedure that uses the above criterion are detailed in Appendix A. As it is shown, a given estimate ĥ is updated from the previous estimate ĥ(i 1), because only a finite number of symbols has been updated. That is why we name this procedure as a prls procedure (for pseudo-rls), because it is a Recursive Least Squares (RLS)-like procedure (with L = ) where, instead of updating one symbol at each time index i, a complete window of Q + 1 symbols is updated [31].

14 14 The advantage of this procedure w.r.t. the RLS procedure is that it allows to reduce the possible errors of estimation, since every symbol is estimated Q + 1 times. The disadvantage is an increase of the computational cost in the symbols minimization (at each iteration i, an array on Q + 1 symbols is updated in prls, whereas only one symbol is updated in a typical RLS procedure). The equations of the prls procedure for the calculus of the channel impulse response estimates are detailed in [31]. The simulations given in Section V. V-B show the improvement in terms of Raw Bit Error Rate (RBER) reached by the adaptive algorithms when comparing to the block ones in the case of time-varying channels (v > 0 km/h). However, in a fixed setting (v = 0 km/h), a degradation w.r.t. the new CMLBAP is observed. This can be explained by the fact that the above presented adaptive algorithm is deduced from the CMLBA one [33]. Thus, our objective is to reduce the degradation of the CMLAA w.r.t. the CMLBAP in a fixed setting. B. CML Adaptive Algorithm - iterative (CMLAAi) From the comparison of CMLBAP and CMLAA criteria shown in (5) and (40), one can observe an important difference in the symbols minimization. Whereas in CMLBAP all the interference from the estimated symbols is taken into account for each partition s Np, only the past interference is used in CMLAA for the a given updated window of symbols su Q+1, as observed in (7) and (4) respectively. Actually, both past and future interferences are used in CMLBA, if a time line was considered in the block approach. Thus, in this section, we propose to include this future interference subtraction term in the adaptive algorithm: the future interference is subtracted to the window su Q+1 in order to improve performance of CMLAA. Moreover, the fact of including this new term in the criterion of the adaptive procedure allows to iterate (like in the block case) the whole procedure in order to improve the estimation of this new subtracted interference. The improvements w.r.t. the adaptive procedure are two fold: a better conditioning of the associated channel matrices (leading to better performance) and the capacity to improve the estimation of the subtracted interferences when iterating the whole adaptive procedure. Denoting the symbols corresponding to this future interference sf m, being m equal to the channel order M for

15 15 convenience, the new adaptive criterion can be expressed as: JAI CML ( h,su Q+1, λ ) = = Λ D 1/( Q+1, m T Q+1+m, M+Q+1+m (h)s M+Q+1+m X ) Q+1+m γ (i,k) su Q+1 = Λ D Q+1, m1/ ([ V p Q+1+m, M sp (i,k) M su Q+1 sf (i,k 1) m X (h) V u Q+1+m, Q+1 (h) V f Q+1+m, m (h)] Q+1+m ) γ (i,k) su Q+1 = Λ D 1/ Q+1, m V u Q+1+m, Q+1 (h) su Q+1 Λ D Q+1, m 1/ (X Q+1+m If(i,k) (i,k) (i,k 1) Q+1+m (sp M,sf m ;h) ) γ (i,k) su Q+1, (43) with γ (i,k) = = λ min { V u Q+1+m, Q+1 (h)h Λ D Q+1, mv u Q+1+m, Q+1 (h)} (44) and If (i,k) (i,k) (i,k 1) Q+1+m (sp M,sf m ;h) = = V p Q+1+m, M (h) sp M + V f Q+1+m, m (h) sf m. (45) Because of the introduction of the future interference, a new definition has been introduced for the matrices containing the forgetting factor λ. Thus, Λ D is also a diagonal matrix like Λ but the weighting by λ is done in both directions (past/future): [ ] ΛK1 Λ D K 1, K = λ λ... The equations of the new procedure are detailed in Appendix B. λ K λ K. (46) Fast convergence of the CMLAAi can be achieved by minimizing the criterion (43) but now in both (temporal) directions, that is, in the direct (forward) as shown above and in the reverse direction (backward). First, we introduce

16 16 the following new definitions: s R N+M [ s R (n), n = 0... N 1... N + M 1 ] T, h R c [ n h Rc ] T, n (k), k = M... 0, c = {1,..., L} h R n [ [ h R 1]T [ n... h RL]T ] T, n X R N c [ x R c (n), n = 0... N 1, c = {1,..., L} ] T, X R N [ x R 1 (0),.., x R L (0),..., x R 1 (N 1),.., x L (N 1) ] T. Taking in account these definitions, we can write that: s R (n) s(n 1 n), h R c n (k) h c n (M k), x R c (n) x c (N 1 n). Thus, the filtering equation given in (5) can be written now in the time reverse form as: X R N = T N, N+M (h R ) s R N+M. (47) Therefore, with the above definitions, the same form for the filtering equation is kept, so the same equations for the direct sense are used for the reverse sense. Since for a given time index n, the same criterion is minimized along the iterations k, both direct and reverse iterations converge to the same minimum. Performance of the new adaptive algorithm (CMLAAi) w.r.t. the non-iterative adaptive algorithm CMLAA in terms of RBER is shown by the simulation experiments in Section V. V-B. As expected, an improvement can be observed when comparing the CMLAA to the CMLAAi with only one iteration. On the other hand, no noticeable performance gain is observed after more than 4-5 iterations. For convenience, we will take ite A = 4 (two forward and backward) iterations as default parameter for the simulations in this paper. Indeed, when comparing the new adaptive algorithm with iterations to the block algorithm with partitions in terms of computational cost, we can observe that the complexity is roughly the same when taking into account the number of symbols minimizations per relaxation executed by each one. For the block case, considering the optimal number of partitions, we have a total of P ite B = 1 0 = 40 minimizations and, in the adaptive case, N u ite A = 60 4 = 40 minimizations, too. (This calculus is based on the fact that the minimizations performed by the relaxation method represent more than 90 % of the total computational cost of the algorithms.) V. SIMULATION RESULTS In order to compare the theoretical performances of the new algorithms w.r.t. the previous ones, a simplified model of the GSM system is considered. Simulation conditions are based on the model presented in (1). Instead of considering exact GMSK signal, the linear approximation is used [6], [7]. The GMSK signal is substituted by a BPSK signal followed by a rotation operation. Furthermore, neither transmission or reception filters are taken

17 17 into account. This avoids the performance degradation due to the channel truncation (as in a real GSM conditions) so the asymptotical performances of the algorithms can be tested. In this model, the simulated channel impulse responses {h n, n}, as defined in (), are considered as normalized Gaussian random variables. The length of all impulse responses is supposed to be known. Actually, only channel variation and the noise limits the performance of these algorithms. A. Simulation Conditions The normal GSM burst is divided in two parts (see Fig.), as the standard [39] specifies, each one composed of N k +N u symbols. The known symbols of the TS located at the beginning of the semi-burst are used for initializing all the algorithms. Indeed, a channel estimate (h ini ) is derived from a straightforward Least Square Estimation (LSE) of this known sequence [40].../../figures/eps/gsm_burst_simplified_model.eps Fig.. The simplified GSM model Two half bursts made of N u = 61 random binary data (BPSK) are generated. Like in GSM systems, a CAZAC (Constant Amplitude and Zero Auto-Correlation) sequence [41], [4] of N k = 6 known symbols is used for the TS. After a rotation operation, the whole burst is then transmitted over a randomly generated channel (having L = sub-channels) of length M + 1 = 5. Once a de-rotation operation (see Section II. II-A) is performed by the receiver, equalization algorithms are applied. Soft decision symbols are then sliced and compared with the transmitted ones. The RBER is used in order to evaluate the performance of the presented algorithms. The block diagram of the simulation chain is shown in Fig.3.

18 18../../figures/eps/gsm_chain_simplified_model.eps Fig. 3. The simplified GSM model B. Performance Comparison First, a performance comparison of the semi-blind block algorithms is given (Fig.4). Static channels (v = 0 km/h) are used in this simulation. Classic Viterbi (Block) Algorithm (VTBA) is considered as a reference for comparison. We compare the performance of the following algorithms: CMLBA o proposed in [33] using ĥini from the TS to initialize the algorithm (this approach does not take into account the interference generated by the training sequence), CMLBA proposed in Section III. III-A, CMLBAP proposed in Section III. III-C which uses the partitioning procedure. First, comparing CMLBA o with CMLBA, we observe the gain (more than 1.5 db for a RBER of 10 ) obtained when taken into account the past interference generated by the known symbols of the training sequence. Second, the partitioning procedure applied in CMLBAP provides a noticeable gain of performance w.r.t. the CMLBA, despite its reduced complexity: for a RBER of 10 the gain is approx. 0.5 db, and for a RBER of 10 3 this gain is improved up to 0.75 db. Furthermore, no degradation is observed when comparing the CMLBAP to the VTBA (Viterbi Block Algorithm) in the typical targeted range (SNR under 10 db). Indeed, a little improvement is achieved by CMLBAP w.r.t. to VTBA in low SNR ratios (a gain of 0.5 db for a RBER of 10 ). This is due to the fact that the Viterbi-based algorithm only uses the information given by the known symbols of the TS, whereas the semi-blind CMLBAP algorithm uses both known and unknown symbols. In the next simulation (Fig.5), the semi-blind adaptive algorithm CMLAA given in Section IV. IV-A is compared to the block ones (CMLBA and CMLBAP) in a time-varying environment. As before, the VTBA is used as a reference for comparison. The terminal speed used for the simulated channel is v = 300 Km/h (f c = 1800 MHz),

19 19../../figures/eps/simplified_model_fixed_channel_cmlbaocmlbacmlbapvtba[black].eps Fig. 4. CMLBAP vs. CMLBA / CMLBA o (Static Channel) higher than the standard requirements [8] (v = 130 Km/h (f c = 1800 MHz), but more appropriate for some actual situations (high speed trains). As expected, performance of all algorithms is limited by channel variation, which leads to an error floor. CMLbased algorithms outperform in all cases the VTBA. The reason is that, at this speed, the channel estimated from the training sequence is accurate only near the TS. CML-based algorithms estimate the channel several times, where the Viterbi-based one just uses the estimate given by the TS. As we can also observe, CMLAA outperforms CMLBA but not CMLBAP in this time-varying situation. However, none of these algorithms can attain the required target RBER for the system to be working correctly. In order to evaluate the possible degradation of the adaptive version (CMLAA) w.r.t. the block algorithms, a new simulation is given in Fig.6, but now in a fixed setting. An important degradation of CMLAA w.r.t. CMLBA/CMLBAP is observed when a static channel is employed:

20 0../../figures/eps/simplified_model_variant_channel_vtbacmlbacmlbapcmlaa[black].eps Fig. 5. CMLAA vs. CMLBAP / CMLBA (Time-Varying Channel) approx. 1 db for a RBER of 10. This is due to the fact that the CMLAA is derived from the CMLBA, which does not use the partitioning procedure. Simulation given in Fig.7 show the enhancement provided by the new adaptive algorithm (CMLAAi, presented in Section IV. IV-B) in a fixed setting due to the use of the all available interference, as in the CMLBAP. No noticeable degradation of CMLAAi w.r.t. the block one (CMLBAP) is now observed. Finally, a new simulation (Fig.8) is performed in order to evaluate the improvement of CMLAAi versions w.r.t. the CMLAA algorithm in a time-varying environment. Same channel variation conditions than before were used (v = 300 Km/h for a f c = 1800 MHz. As we can observe, the new CMLAAi (based on CMLBAP) outperforms CMLAA (based on CMLBA). Whereas the CMLAAi has an error floor (in terms of RBER) of.5 10, the CMLAA (resp. CMLBAP) only achieves an error floor of 4 10 (resp ). The difference is even more considerable when comparing to the VTBA,

21 1../../figures/eps/simplified_model_fixed_channel_cmlaacmlbacmlbap[black].eps Fig. 6. CMLAA vs. CMLBAP / CMLBA (Static Channel) which has an error floor higher than VI. CONCLUSION New algorithms for semi-blind equalization have been presented in this paper. Based on the CML approach [1], a partitioning procedure is developed and a new criterion (CML-P) is derived. This new criterion can be viewed as a customization of the use of the available a priori information for the symbols estimation. Instead of considering the same prior information for all the symbols, as previous CML approach does, CML-P customizes the amount of information taken into account for each sub-set (partition) of the estimated symbols and takes advantage of better conditioned channel matrix to add more a a priori information on the source statistics. A new (block) algorithm (CMLBAP) has been derived, which outperforms (in terms of RBER) the previous CML-based one (CMLBA) [1] according to our simulation results, despite its reduced complexity. Furthermore, in a fixed setting, it is shown to have performances similar to those of the optimal ML Viterbi algorithm. In

22 ../../figures/eps/simplified_model_fixed_channel_cmlaacmlaaitrcmlbap[black].eps Fig. 7. CMLAAi vs. CMLAA / CMLBAP (Static Channel) time-varying channel conditions, the CMLBAP performs better than the Viterbi algorithm. The same concepts have been used to develop an adaptive version of such algorithm (CMLAAi) which leads to an improved tracking capacity. This new algorithm outperforms the Viterbi one (lower error floor is achieved) in high speed environment (v = 300 Km/h for f c = 1800 MHz), according to our simulations experiments. Future research will be focused on the introduction of an a priori information on the channel estimation variation. For instance the introduction of a realistic dynamic model for the time varying channel is under investigation, like the use of an hyper-model with the known dynamics of a time-varying channel [43], [44], in order to improve the ratio complexity/performance as in the case of symbols estimation. An analytical performance analysis of the proposed techniques will also be considered, as well as a study of the complexity of these techniques when compared to the Viterbi algorithm.

23 3../../figures/eps/simplified_model_variant_channel_vtbacmlbapcmlaacmlaaitr[black].eps Fig. 8. CMLAAi vs. CMLAA / CMLBAP (Time-Varying Channel) ACKNOWLEDGMENT The authors would like to thank the digital signal processing department of Wavecom S.A. for supporting this research, and Serguei Burykh for his help in proofreading the manuscript. APPENDIX I CMLAA EQUATIONS The CML Adaptive Algorithm (CMLAA) is given below:

24 4 for i = 1:N u su ˆ Q+1 = arg min J CML(ĥ(i 1) A,su Q+1, λ ) su Q+1 B Q+1 = arg min Λ 1/ Q+1 T u Q+1, Q+1 (ĥ(i 1) ) su Q+1 su Q+1 B Q+1 with ˆ γ = Λ 1/ (X Q+1 Q+1 Îf(i 1) Q+1 (sp M ;ĥ(i 1) ) ) ˆ γ su Q+1, (48) = λ min { T u Q+1, Q+1 (ĥ(i 1) ) H Λ Q+1 T u Q+1, Q+1 (ĥ(i 1) ) } (49) ĥ = argmin h JA CML ( h,ŝ M+Q+1, λ) = ĥ(i 1) + [ R ] 1 A {U Q+1, M+1 (ŝ M+Q+1 )H [ ] Λ Q+1 X Q+1 U Q+1, M+1 (ŝ M+Q+1 )ĥ(i 1) [ λ U (i 1) Q, M+1 (ŝ(i 1) M+Q )H Λ Q X (i 1) Q ]} U (i 1) Q, M+1 (ŝ(i 1) M+Q )ĥ(i 1) (50) end where: [ ] 1 R A = A 1 A + A 1 A U(i 1) Q, M+1 (ŝ(i 1) M+Q )H [ ΛQ λ U(i 1) Q, M+1 (ŝ(i 1) M+Q )A 1 A ] 1 U(i 1) Q, M+1 (ŝ(i 1) M+Q )H U (i 1) Q, M+1 (ŝ(i 1) M+Q )A 1 A, (51) A 1 A = [ λr (i 1) A ] 1 [ λr (i 1) U Q+1, M+1 (ŝ M+Q+1 )H A ] 1 [ Λ Q U Q+1, M+1 (ŝ M+Q+1 )[ λr (i 1) ] A 1 ] 1 U Q+1, M+1 (ŝ M+Q+1 )H U Q+1, M+1 (ŝ M+Q+1 )[ λr (i 1) ] 1. A (5)

25 5 APPENDIX II CMLAAI EQUATIONS The iterative CML Adaptive Algorithm is given below: for k = 1:ite A for i = 1:N u su ˆ (i,k) Q+1 = arg min J CML(ĥ(i 1,k) AI,su Q+1, λ ) su Q+1 B Q+1 = arg min Λ D Q+1, m T su Q+1 B Q+1 ŝ (i,k) M+Q+1+m X Q+1+m Q+1+m, M+Q+1+m (ĥ(i 1,k) ) γ (i,k) ˆ su Q+1 = arg min Λ D 1/ Q+1, m T u Q+1+m, Q+1 (ĥ(i 1,k) ) su Q+1 B Q+1 with su Q+1 Λ D 1/ ( Q+1, m X Q+1+m Îf(i 1,k) ˆ γ (i,k) = Q+1+m ( ˆ (i,k) sp M, sf ˆ (i,k) m )) γ (i,k) ˆ su Q+1, (53) λ min { T u Q+1+m, Q+1 (ĥ(i 1,k) ) H Λ D Q+1, mt u Q+1+m, Q+1 (ĥ(i 1,k) ) } (54) and (i 1,k) ÎfQ+1+m( sp ˆ (i,k) M, ˆ m ) = sf (i,k) = T p Q+1+m, M (ĥ(i 1,k) ) sp ˆ (i,k) M + T f Q+1+m, m (ĥ(i 1,k 1) ) sf ˆ (i,k 1) m (55) ĥ (i,k) = arg min h JAI CML ( (i,k) h,ŝ M+Q+1+m, λ) = ĥ(i 1,k) + [ R ] 1 A {U Q+1+m, M+1 (ŝ(i,k) [ Λ D Q+1, m X Q+1+m U Q+1+m, M+1 (ŝ(i,k) M+Q+1+m )H M+Q+1+m ) ĥ(i 1) ] λ U (i 1) Q+m, M+1 (ŝ(i 1,k) M+Q+m )H Λ D Q, m ]} U (i 1) Q+m, M+1 (ŝ(i 1,k) M+Q+m )ĥ(i 1,k) [ X (i 1) Q+m (56) end end REFERENCES [1] D. Boss, T. Petermann, and K.-D. Kammeyer, Impact of blind versus non-blind channel estimation on the BER performance of GSM receivers, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics (HOS 97), pp. 6 66, July 1997.

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