Adaptive Blind Equalizer for HF Channels

Adaptive Blind Equalizer for HF Channels Miroshnikova. Department of Radio engineering Moscow echnical University of Communications and Informatic Moscow, Russia miroshnikova@ieee.org Abstract In this paper, the problem of blind adaptive equalization of HF channel is considered using a state space approach. HF channel is multipath propagation channel due to multiple reflections on ionospheric layers, which causing intersymbol interference (ISI) at the receiver output. he problem of ISI suppression is referred to as deconvolution or channel equalization. At present, a family of methods for equalizing and estimating the channel based on the criterion of minimum mean-square error (MMSE) has become widespread in engineering practice. hese methods assume the use of a training or pilot symbols known by the receiver to estimate the channel and "train" the equalizer. However, in order to increase the bandwidth efficiency blind deconvolution or channel equalization methods have been developing recently more actively. he essence of these methods is the task of equalization and estimation of the channel from sensors outputs without any a priory knowledge of the original signals. he classical criterion for blind equalization is the constant modulus (CM) criterion, which is an extension of the odart algorithms family. However, this method has slow convergence, and is not applicable to all modulation methods used in the HF systems. he paper considers a learning algorithm based on information theory and natural gradient is used to solve the optimization problem. o effectively use the algorithm in a changing channel environment, it is suggested to use an additional optimization of the algorithm step size. he channel model is state-space description of a dynamic system. he main advantage of state-space model is that is flexible and can be used for internal description of system. Based on developed state-space model and measurement models, an adaptive optimum-size blind equalization algorithm is proposed to track the HF channel variation in time. Proposed algorithm is compared to CMA and classical stochastic gradient descent algorithms for blind deconvolution. In numerical simulations, it is observed that the proposed approach can track the channel variations with good performance. During computer simulations under good, moderate and poor HF ionospheric channel conditions, it is observed, that proposed adaptive equalization algorithm with adaptive step-size for blind deconvolution provides reliable equalization error and can track the variation of the channel in time with high accuracy Keywords blind deconvolution; adaptive equalization; HF communication; natural gradient; I. IRODUCIO One of the most common criteria used for channel estimation and blind equalization is minimum mean square error (MMSE). he main algorithms built on the criteria of minimum MSE are least squares algorithm [7], recurrent least squares method (RLS) [] and Kalman filter [2]. hese algorithms are widely used in practice due to their low computational complexity and the possibility of their use in equalizers with the direct adjustment of coefficients []. he use of the above algorithms assumes the presence of test (training) sequence in transmitted signal for the channel estimation and "training" of the equalizer. However, in order to increase the transmission speed (in particular, in HF communication channels), in recent years, methods for blind deconvolution and equalization have been actively developing [,3-5]. he objective of these methods is signal separation and equalization without a priori knowledge of channel and parameters of useful signals, i.e. without using the test sequence. he possibility of increasing transmission speed, and versatility of blind methods made them an attractive for research in modern communication systems. However, these methods have several significant drawbacks, such as: high computational complexity, the requirement of a larger array of samples for processing on the reception, in comparison with the classical methods, in most cases, slow convergence and inability to track fast variations of the channel parameters. here are several methods of blind equalization. he first works in this field are the works of Sato and odart. he classical criterion for the construction of blind equalizers is the criterion of a constant modulus (CM) [0], which is an extension of the odart algorithm family. owdays, there are two main classes of blind deconvolution algorithms: first group is based on information theory (minimization of mutual information or maximization of entropy) and second is based on high order cumulants estimation. Busgang algorithm and the natural gradient algorithm are representatives of the first category. Busgang algorithm is iterative algorithms and uses a classical method of stochastic gradient descend to optimize the objective function depending on the output signal of the equalizer. hese algorithms are simple and easy to implement, but it can converge to incorrect solutions. he natural gradient algorithm has been developed by Amari, with the aim of overcoming the disadvantages of Busgang algorithm [9]. Studies have shown [0], the algorithm of natural gradient can significantly improve the efficiency of blind equalizers in comparison with the classical gradient methods. In the case of second group, higher-order statistics of samples from the output of the antenna array (AR) is used to compute the coefficients of the blind equalizer. he main drawback of this approach is that it requires a pre-procedure of "whitening" the data before maximization of the objective function, which may require substantial computing resources. 978--5386-786-/7/$3.00 207 IEEE

II. BLID DECOVOLUIO MODEL Consider the AR of P elements, which receives a linear mixture of source signals s i. In this case, we assume that signals in the mixture are statistically independent and P o solve the problem of blind equalization, we represent the vector of samples at the output of the AP in the form: x( n) = H( p) s( n p) () x n x x2 x P = - is a vector of output samples from antenna array at time lag n Where ( ) [,,, ] ( ) [,,, ] s n s s2 s = vector of samples of the transmitted signals at time lag n H ( p) P mixing (channel) matrix at time lag p. In the z-domain, the mixing matrix can be represented as: H z H p z = p (2) For the case of a multipath channel, the element hi f matrix H can be written as: M j2 im im hi aimα ime πϑ τ m= = (3) M- number of paths for i-th channel a im the direction vector of path m for the i-th signal, which depends on the angle of arrival. αim m -th path gain for the i-th signal ϑim ppler shift of path m for the i-th signal signal τ im path m delay for i-th signal hus, in the simulation, parameters of the ionospheric channels, according to the recommendations IU-R, can be considered. he task of blind equalization is estimation of signal sources using only the samples x(n). In general, for blind deconvolution problem, this task is reduced to finding the inverse filter, the output of which can be represented as: y( n) = W( p) x( n p) (4) ( ) [,,, ] y n y y2 y = ector of samples at the output of the separation system W( p) separation matrix at time lag p.. In z-domain separation matrix can be represented as a filter with transfer function: W z W p z = p (5) If the channel model is causal FIR filter, it is possible to write the matrix W(z) in the form: L = p (6) p= 0 W z W p z lobal mixing function is defined as: z = W z H z (7) ( ) III. SAE-SPACE CHAEL MODEL Consider a communication channel in the form of a dynamical system using state space approach. With this approach, mixing model consist of two equations: state and measurement. State equation can be written as: ( n ) A ( n) Bs( n) ( n) ε + = ε + + ϑ (8) Measurement equation can be written as: ( ) ε ϑ( ) x n = C n + Ds n + n (9) ε ( n) L state vector of dimension s(n) - vector of source signals samples A is L L transition matrix, B is a matrix L, D is a matrix of dimension (P ), a C matrix of P d dimension, and an -noise matrix. he separating system can be represented as: ε n+ = Aε n + Bx n + Lϑ n (0) y ( n) Cε ( n) Dx( n) = + () hen the transfer function of the mixing system can be expressed as: H z = C zi A B+ D (2) he required transfer function of the separation system can be expressed as: W z = C zi A B+ D (3) If the channel is modeled as a non-recursive LDS (FIR filter of order L), the signal at the output of the separation system can be written: L p (4) p= 0 ( ) = ( ) y k W x k p hen the matrices C and D can be written as: C = [ WW 2 WL ] (5) D = W 0 (6) ( n) = x ( n, ), x ( n L) ε (7)

In other words, the state vector is determined by the samples at the output of the antenna array. Both matrices A and B in this case are deterministic. he output signal y (n) will be an estimate of the source signals in the following sense: Λ y n = W z H z s n = P z s n (8), Where P is a permutation matrix Λ ( z) a diagonal matrix with elements Δ λiz ere λ nonzero scaling constant, Δ a non-negative integer hus, source signals can be fully recoverable by the separation system having a transfer function W( z) if the global mixing matrix satisfies [8]: ( ) Λ( ) z = W z H z = P z (9) IV. ALORIHM OULIE Evaluation of W(z) in the case of modelling the channel as a FIR filter is reduced to the evaluation of the matrices C and D. Using Kulbach-Leibher divergence as an optimization parameter, the objective function takes the form [8]: J ( yw, ) = logdet( DD ) log( fi( yi) ) (20) 2 ( ) fi yi approximation of the true probability density function of the distribution of estimates of the signal sources. f y he selected non-linear function In practice, instead of i ( i) φ( y) s used, which depends on assumptions about the distribution of the signal source. he second term of the expression (20) makes the signals in the mixture maximally statistically independent. he direct minimization of this function does not restore the signals, but only makes the signals at the output of the separation scheme mutually independent. he signal estimates y ( n ) obtained at the output are scaled, rearranged and delayed versions of the original signals [0] A. atural gadient approach If the channel is modeled as a non-recursive LDS (FIR filter of order L), then the signal at the output of the AP can be written as expression () hen, using the natural gradient algorithm [8], the iterative algorithm takes the form: W n+ = μ I φ y n y n W n (2) ( ) ( ( )) ( ) i= μ ( ) p( ) 0 0 p ( ) φ ( ) W n+ = y n x n W n (22), where p =,2,..., L onlinear activation function. Its choice depends on the distribution function of the source signals. B. radient algorithm with Adaptive step-size Simulation results for the algorithm (4) for typical ionospheric channels shows that gradient algorithms for blind deconvolution and equalization are often doesn t converge (especially for the "perturbed" HF channels). For better adaptation of the algorithm to changing parameters of the HF channel, it was decided to use the step - size adaptation. he steady-state error given by the algorithm As can be seen from the figure, in order to achieve a value of <0.2, the classical gradient algorithm will require about 8000 thousand iterations, in the case of the natural gradient algorithm of the order of 2000 iterations, with an optimal step of about 900 iterations. Figure 2 shows the dependence of the error rate (BER) on the ratio For systems using an equalizer with tuning by training sequence and blind equalizer. As an equalizer using a training sequence, an equalizer with feedback by decision is chosen, with the adaptation of the LMS algorithm and the RLS algorithm. decreases with the step size decreasing, the speed of convergence increases for step sizes smaller than half of maximum value required for stable algorithm operation. hus, the aim is to increase the step -size to a value sufficient for its stable operation in the initial iterations of the algorithm when the parameters are far from their optimal values and to reduce step size, to reduce steady-state error, when approaching the optimal values [7]. Application of gradient method [9] to the step size in (4) gives the following expression for the step size adaptation: As a result, we get: ( k) = ( k ) ( k) μ μ γ ( ) ( k ) J W μ (23) P y ( n ) ϕ ( y( n ) ) μ( n) = μ( n ) + γ ( k) + + μ ( n ) + μ( n ) ( y ( n ) ϕ( y( n ))) y ( n ) ϕ( y( n ) ) + y ( n ) y( n )) ϕ ( y( n ) ) ϕ( y( n ) )) (24) In order to guarantee the stability of the system during the adaptation process, it is required to limit the range of possible step size values by a range [ δ, μmax] where δ a small value, in order to avoid the sudden stopping of the algorithm, μmax the upper bound of the step to ensure the stability of the algorithm. One of the drawbacks of this approach is that you need to choose the "step for the step". Modeling has shown that the choice of magnitude does not have a significant effect on the operability of the algorithm. From this point of view, the choice of μmax s more important.

V. SIMULAIO RESULS he measure of performance for blind equalization is the value of the intersymbol interference Misi = + j p pij i p pij (24) l= max p, j pij j= max p,i pij As can be seen from formula (24), Misi = 0 only if (z) satisfies condition (7) Simulation conducted using a simulation model of the ionospheric channel according to IU-R F. 487 recommendation. Channel in middle latitudes for the good, moderate and poor HF ionospheric channel conditions was considered. Antenna array with two elements (P=2) and two source signals, one with 8-psk modulation and other with QAM-6 modulation (=2) were considered. he nonlinear activation function in (4),was selected tanh(y). Figure shows the Misi performance for the algorithm of the classical gradient, natural gradient and the natural gradient with adaptive step-size. Fig.. Characteristic of the change for the algorithms of the classical gradient (solid line), natural gradient (dashed line) and natural gradient with the optimal pitch (dash-dot line). As can be seen from the figure, to achieve the values Misi <0.2 the classical gradient algorithm will require about 8,000 iterations, in the case of the natural gradient algorithm it will require 2000 iterations, using the adaptive step size about 900 iterations. Figure 2 shows BER performance for systems using the equalizer setting on the training sequence and blind equalizers. he MIL-SD-0B system, using DFE equalizer with LMS algorithm and RLS algorithm was considered as referenced system. As can be seen from the comparison graphs shown in figure 2, the system with classical equalizers are better in BER performance, however, the system with blind equalization allow to realize high transmission speeds by saving the channel resource. References [] A.A. Kuchumov, V.S. Priputin, Application of methods of blind signal separation to improve the noise immunity of radio monitoring complex in decameter wavelength, Proc. Of X-th Industry Scientific and echnical Conference " Information Society echnologies ", pp. 42-43, Moscow, 206. [ in. [2] V.. Kartashevsky, Demodulation in channel with memory with the Kalman filter, «Radiotekhnika» (Radioengineering), no3, 994. [ in [3].E. Miroshnikova. Performance analysis of blind source separation algorithms, Radiotekhnika (Radioengineering), no3, pp 37-42, 206. [ in [4].E. Miroshnikova, Adaptive filtering for blind identification "Synchronization, forming and processing systems in the infocommunications" (SynchroInfo), vol 7, o, pp. 42-25, 206 [ in. [5].E. Miroshnikova, Analysis of blind identification algorithms for HF channel identification, H&ES Research, vol.8, o3, pp. 30-34, 204 [in [6] V.S.Priputin, Blind signal separation based on second order statistics for spatially polarization signal processing, -comm: elecommunications and ransport, vol.6, o6,, pp. 36-39, 204 [in [7] K. A Mayyas,. Aboulnasr, Robust variable step-size LMS-type algorithm : Analysys and simulation, IEEE rans. Signal Processing, vol. 45, pp 63-639, May 997 [8] A. Cichocki, L Chang., atural gradient approach to blind deconvolution of dynamical systems, Proc.of second International Workshop on ICA and BSS ICA-2000,, pp 27-32, 9-22 June 2000, Helsinki, Finland. [9] S.C. Douglas eneralized gradient step sizes for stochastic gradient adaptive filters, Proc.Int. Conf. Acoustic, Speech, Sigmal Processing, Detroit, MI,vol.2, pp. 396-399,May 995 [0] D.. odard, Self-recovering equalization and carrier tracking in twodimensional data communication systems, IEEE ransactions on Communications, vol.28, pp.867 875, 980 [] R.J. Ober, Balanced canonical forms, Identification, Adaptation, learning, ato ASI Series, Springer,996, pp 20-83. Fig. 2. Dependencies of the error rate (BER) on the ratio for systems using equalizer with feedback by decision, with adaptation of LMS algorithm (dashed line) and RLS algorithm (dot-dash line), and a blind equalizer with step adaptation (solid line).