A Gaussian Mixture Approach to Blind Equalization of Block-Oriented Wireless Communications
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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 discrete-continuous state-space 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 quasi-static and time-varying frequency selective wireless channels.
KeywordsChannel Estimation Minimum Mean Square Error Pilot Symbol Constant Modulus Algorithm Blind Equalization
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  and Godard  use low-complexity finite impulse response filters. However, these methods suffer from local and slow convergence and may fail on ill-conditioned or time-varying channels.
Other authors have proposed symbol-by-symbol soft-input soft-output (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  and was further simplified in  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) , where each discrete ISI state has its associated channel estimate. Another fixed interval method, based on expectation-maximization channel identification, has appeared recently , but this technique is restricted to static channels.
In this paper, we will consider fixed interval smoothing, which is adapted to block-oriented communications. After modeling the ISI and the unknown channel at the receiver side, we obtain a combined state-space formulation of our communication system. Specifically, the ISI is modeled as a discrete state space with memory, while the (potentially time-varying) 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  in the context of MIMO decoding and that the results in this paper have been partially presented in . Also the idea of exploiting Gaussian mixtures for blind equalization appeared previously in a different form .
Section 2 describes the adopted system model. Section 3 introduces the Gaussian mixture-based 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.
Throughout the paper, bold letters indicate vectors and matrices. Open image in new window will denote a complex Gaussian distribution of the variable Open image in new window , with mean Open image in new window and covariance matrix Open image in new window . Open image in new window denotes the Open image in new window identity matrix, while Open image in new window denotes the Open image in new window all-zero matrix. The symbol Open image in new window denotes the Kronecker product. The operator Open image in new window will denote the determinant of a matrix.
2. System Model
3. Blind SISO Equalization Using a Gaussian Mixture Approach
In this section, we derive a fixed-interval smoother by propagating a mixture of Open image in new window Gaussians per ISI state forward and backward in the ISI trellis. Consequently, the ISI state Open image in new window and the channel state Open image in new window will be jointly estimated. Finally, the desired a posteriori probabilities for the bits Open image in new window are obtained by a simple marginalization step.
3.1. Forward Filtering
In the above expression, we easily recognize the integral as the well-known prediction step of Kalman filtering . Moreover, the multiplication by Open image in new window is the correction step of Kalman filtering. Therefore, Open image in new window can be written as (17).
3.2. Complexity Reduction Algorithm (CRA)
3.3. Backward Filtering
The proof is obtained by injecting (26) into (25) and using the same arguments as in the demonstration of Theorem 1.
where Open image in new window denotes the dimension of the continuous valued state variable.
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 Open image in new window , where Open image in new window denotes the dimension of the continuous-valued state estimate. However, in our forward and backward filters, the complexity of one recursion of the Kalman filter reduces to Open image in new window due to the block diagonal form of Open image in new window 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 Open image in new window . The complexity per information bit of the smoothing pass can be evaluated as Open image in new window , due to the matrix inversions.
4. Numerical Results
4.1. Comparison with Existing Methods
We consider a memory- Open image in new window Rayleigh fading channel simulated with the method introduced in . The standard deviations of the resulting three complex processes Open image in new window are set at Open image in new window . The block size is Open image in new window bits. As illustrated in Figure 1, a tail of length Open image in new window 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 Open image in new window phase ambiguity inherent to BPSK modulation. We assume that each data block is affected by an independent channel realization. Open image in new window 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 Baud-rate linear filters optimized with the constant modulus algorithm (CMA)  or with the first cost function (FCF) introduced in . These equalizers are iterated Open image in new window times back and forth on each data block, in order to aleviate the slow convergence problem . 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- Open image in new window known preamble is used to resolve the delay ambiguity. The lengths of the CMA and FCF equalizers were optimized by hand, to Open image in new window and Open image in new window coefficients, respectively. The second kind of blind equalizer is the per-survivor processing (PSP) algorithm , which is similar in spirit to the proposed method, since it is a trellis-based algorithm operating on the conventional Open image in new window -state ISI trellis  and using Kalman filtering for channel estimation. However, since the path pruning strategy employs the Viterbi algorithm , it is not an SISO method.
4.2. Performances on a Realistic Channel Model
Saleh-Valenzuela model parameters.
In this paper, we introduced a new blind receiver for soft-output equalization. The proposed method, adapted from an earlier work on fixed-interval 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 well-known blind equalization schemes and works well even in fast-fading scenarios.
Future extensions of this work include the application of blind Gaussian mixture-based equalization to Rician fading and higher-order modulations. Since the proposed algorithm is soft output in nature, its application to turbo equalization will also be investigated.
The author wishes to thank the editor and the anonymous reviewers, whose constructive comments were very helpful in improving the presentation of this paper.
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