 Research Article
 Open Access
A Gaussian Mixture Approach to Blind Equalization of BlockOriented Wireless Communications
 Frederic Lehmann (EURASIPMember)^{1}Email author
https://doi.org/10.1155/2010/340417
© Frederic Lehmann. 2010
 Received: 6 October 2009
 Accepted: 30 June 2010
 Published: 18 July 2010
Abstract
We consider blind equalization for block transmissions over the frequency selective Rayleigh fading channel. In the absence of pilot symbols, the receiver must be able to perform joint equalization and blind channel identification. Relying on a mixed discretecontinuous statespace representation of the communication system, we introduce a blind Bayesian equalization algorithm based on a Gaussian mixture parameterization of the a posteriori probability density function (pdf) of the transmitted data and the channel. The performances of the proposed algorithm are compared with existing blind equalization techniques using numerical simulations for quasistatic and timevarying frequency selective wireless channels.
Keywords
 Channel Estimation
 Minimum Mean Square Error
 Pilot Symbol
 Constant Modulus Algorithm
 Blind Equalization
1. Introduction
Blind equalization has attracted considerable attention in the communication literature over the last three decades. The main advantage of blind transmissions is that they avoid the need for the transmission of training symbols and hence leave more communication resources for data.
The pioneering blind equalizers proposed by Sato [1] and Godard [2] use lowcomplexity finite impulse response filters. However, these methods suffer from local and slow convergence and may fail on illconditioned or timevarying channels.
Other authors have proposed symbolbysymbol softinput softoutput (SISO) equalizers based on trellis search algorithms. Two such approaches have been proposed so far to achieve SISO equalization in a blind or semiblind context. The first approach relies on fixed lag smoothing [3] and was further simplified in [4] by allowing pruning and decision feedback techniques. The second approach uses fixed interval smoothing [5, 6]. The aforementioned methods employ a trellis description of the intersymbol interference (ISI) [7], where each discrete ISI state has its associated channel estimate. Another fixed interval method, based on expectationmaximization channel identification, has appeared recently [8], but this technique is restricted to static channels.
In this paper, we will consider fixed interval smoothing, which is adapted to blockoriented communications. After modeling the ISI and the unknown channel at the receiver side, we obtain a combined statespace formulation of our communication system. Specifically, the ISI is modeled as a discrete state space with memory, while the (potentially timevarying) channel is modeled as an autoregressive (AR) process.
Main Contributions and Related Work
The main technical contribution of this work is the introduction of a blind equalization technique based on Gaussian mixtures. A major problem in blind equalization is that multiple modes arise in the a posteriori channel pdf, which originate from the phase ambiguities inherent to digital modulations. Assume that an equalizer is able to represent only a single mode, as is usually the case, it is likely that the wrong mode is retained during a fading event or due to the occasional occurrence of high observation noise. In such a situation, a classical equalizer is not able to recover the correct phase determination in a blind mode, since no pilot symbol is available. Therefore, all the subsequent decisions in the frame will be erroneous with high probability. The main feature of the proposed algorithm is that the multimodality of the channel a posteriori pdf is explicitly taken into account thanks to a Gaussian mixture parameterization. We derive the corresponding SISO smoothing algorithm suitable to solve our blind equalization problem.
Note that the idea of Gaussian mixture processing has been presented in [9] in the context of MIMO decoding and that the results in this paper have been partially presented in [10]. Also the idea of exploiting Gaussian mixtures for blind equalization appeared previously in a different form [11].
Organization
Section 2 describes the adopted system model. Section 3 introduces the Gaussian mixturebased blind equalization technique. Finally, in Section 4, the performances of the proposed technique are assessed through numerical simulations and compared with existing blind equalization techniques.
Notations
Throughout the paper, bold letters indicate vectors and matrices. will denote a complex Gaussian distribution of the variable , with mean and covariance matrix . denotes the identity matrix, while denotes the allzero matrix. The symbol denotes the Kronecker product. The operator will denote the determinant of a matrix.
2. System Model
where is a complex white Gaussian noise sample with singlesided power spectral density .
It is well known that can be represented graphically by a trellis diagram containing states [7].
3. Blind SISO Equalization Using a Gaussian Mixture Approach
In this section, we derive a fixedinterval smoother by propagating a mixture of Gaussians per ISI state forward and backward in the ISI trellis. Consequently, the ISI state and the channel state will be jointly estimated. Finally, the desired a posteriori probabilities for the bits are obtained by a simple marginalization step.
3.1. Forward Filtering
In (16), each discrete state is associated with a mixture of Gaussians, where is a design parameter of choice.
Theorem 1.
Proof.
In the above expression, we easily recognize the integral as the wellknown prediction step of Kalman filtering [14]. Moreover, the multiplication by is the correction step of Kalman filtering. Therefore, can be written as (17).
3.2. Complexity Reduction Algorithm (CRA)
3.3. Backward Filtering
where the discrete summation extends over the states , for which a valid transition exists.
Theorem 2.
The proof is obtained by injecting (26) into (25) and using the same arguments as in the demonstration of Theorem 1.
3.4. Smoothing
Theorem 3.
where denotes the dimension of the continuous valued state variable.
Proof.
After straightforward algebraic manipulations on the product of two Gaussian densities, the desired result (32) is obtained.
3.5. Complexity Evaluation
It is well known that the complexity of one recursion of the Kalman filter is [16], where denotes the dimension of the continuousvalued state estimate. However, in our forward and backward filters, the complexity of one recursion of the Kalman filter reduces to due to the block diagonal form of and the fact that the matrix inversion reduces to a division by a scalar. Thus, the overall complexity per information bit of the forward and backward filter is . The complexity per information bit of the smoothing pass can be evaluated as , due to the matrix inversions.
4. Numerical Results
4.1. Comparison with Existing Methods
We consider a memory Rayleigh fading channel simulated with the method introduced in [17]. The standard deviations of the resulting three complex processes are set at . The block size is bits. As illustrated in Figure 1, a tail of length bits is used, which enables the proposed algorithm to start with the correct initial and final ISI state when processing each frame. This is necessary to remove the phase ambiguity inherent to BPSK modulation. We assume that each data block is affected by an independent channel realization. denotes the average energy per bit.
We compare the bit error rate (BER) of our method with two kinds of blind equalizers. The first kind of blind equalizers consists of Baudrate linear filters optimized with the constant modulus algorithm (CMA) [2] or with the first cost function (FCF) introduced in [18]. These equalizers are iterated times back and forth on each data block, in order to aleviate the slow convergence problem [19]. There is also the issue of the ambiguities inherent to these blind equalizers. Differential encoding of the transmitted data is used to solve the phase ambiguity problem. Also, a length known preamble is used to resolve the delay ambiguity. The lengths of the CMA and FCF equalizers were optimized by hand, to and coefficients, respectively. The second kind of blind equalizer is the persurvivor processing (PSP) algorithm [20], which is similar in spirit to the proposed method, since it is a trellisbased algorithm operating on the conventional state ISI trellis [7] and using Kalman filtering for channel estimation. However, since the path pruning strategy employs the Viterbi algorithm [7], it is not an SISO method.
4.2. Performances on a Realistic Channel Model
SalehValenzuela model parameters.
Parameter and units  Notation and numerical value 

Intercluster arrival rate (1/s) 

Intercluster decay constant (s) 

Intracluster arrival rate (1/s) 

Intracluster decay constant (s) 

5. Conclusions
In this paper, we introduced a new blind receiver for softoutput equalization. The proposed method, adapted from an earlier work on fixedinterval blind MIMO demodulation, is well suited for block transmissions. In essence, the algorithm propagates Gaussian mixtures forward and backward in the conventional ISI trellis, in order to perform joint ISI state and channel estimation. Numerical simulations showed that the proposed method outperforms several wellknown blind equalization schemes and works well even in fastfading scenarios.
Future extensions of this work include the application of blind Gaussian mixturebased equalization to Rician fading and higherorder modulations. Since the proposed algorithm is soft output in nature, its application to turbo equalization will also be investigated.
Declarations
Acknowledgment
The author wishes to thank the editor and the anonymous reviewers, whose constructive comments were very helpful in improving the presentation of this paper.
Authors’ Affiliations
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