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A Gaussian Mixture Approach to Blind Equalization of Block-Oriented Wireless Communications

Open Access
Research Article
Part of the following topical collections:
  1. Advanced Equalization Techniques for Wireless Communications

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 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.

Keywords

Channel Estimation Minimum Mean Square Error Pilot Symbol Constant Modulus Algorithm Blind Equalization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 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 [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 expectation-maximization 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 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 [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 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.

Notations

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

The transmitted data are organized in GSM-like bursts containing Open image in new window bits, as illustrated in Figure 1. For ease of exposition, we consider binary phase shift keying (BPSK) modulation, so that the bit is transmitted at instant Open image in new window , Open image in new window . The tail is a short all-one vector of length equal to the channel memory, used to set the final ISI state of the current data burst to a known value. At the same time, this also sets the initial ISI state of the next data burst to the same known value. Since blind equalization is of interest, no additional pilot symbols are inserted in the data stream.
Figure 1

Data format.

We assume a discrete Rayleigh fading channel of memory Open image in new window as depicted in Figure 2. The elements of the impulse response Open image in new window are modeled as independent zero-mean complex Gaussian random variables. For a static channel, the channel state is defined as Open image in new window and evolves according to the trivial dynamical system
Figure 2

Channel model.

For a time-varying channel, let Open image in new window and Open image in new window denote the maximum Doppler shift and the symbol duration, respectively. The channel autocorrelation can be modeled as follows [12]:
where Open image in new window is the zero-order Bessel function of first kind. A good approximation of the channel statistics is obtained with an order two autoregressive process [13] of the following form:
where Open image in new window and Open image in new window . The driving noise term Open image in new window is chosen as a zero-mean white complex Gaussian process of variance
so as to normalize the channel coefficients to unit variance [14]. Consequently, the channel state, in the case of a time-varying wireless channel, is defined as Open image in new window and evolves according to the dynamical system
where state transition matrix is given by
and the process noise vector is given by
The received complex noisy observation at instant Open image in new window has the following form:

where Open image in new window is a complex white Gaussian noise sample with single-sided power spectral density Open image in new window .

We define the ISI state Open image in new window as the decimal form of the binary subsequence Open image in new window , which can take Open image in new window discrete values. Let Open image in new window denote the ISI state transition function defined by the following relation:

It is well known that Open image in new window can be represented graphically by a trellis diagram containing Open image in new window states [7].

For a time-varying channel, it is now clear from (6) and (9) that our communication system can be described as a mixed discrete-continuous state space of the form Open image in new window , whose dynamics are given by
The second equation in (11) can be interpreted as a linear dynamical system with state transition matrix given by (7) and zero-mean Gaussian process noise, with covariance matrix
The third equation in (11) can be interpreted as a linear observation process, where the observation matrix Open image in new window has the form
From (1) and (9), a slightly different state-space representation is obtained for the static channel as a special case of (11) by choosing Open image in new window , Open image in new window and

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

A recursive expression of Open image in new window , where Open image in new window is obtained by noting that
where the discrete summation extends over the states Open image in new window , for which a valid transition Open image in new window exists. In general, the multiplications and integration in (15) cannot be expressed in closed form, therefore, we introduce the following Gaussian mixture parameterization at instant Open image in new window

In (16), each discrete state Open image in new window is associated with a mixture of Open image in new window Gaussians, where Open image in new window is a design parameter of choice.

Theorem 1.

A closed form expression of Open image in new window is obtained as follows:
where the means Open image in new window and covariance matrices Open image in new window associated with the state transition Open image in new window are obtained from the following recursions:
and the weights Open image in new window are given by

Proof.

Injecting (16) into (15), one obtains

In the above expression, we easily recognize the integral as the well-known prediction step of Kalman filtering [14]. 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).

Figure 3 illustrates how the Gaussian mixture Open image in new window is computed on a Open image in new window -state trellis. The components of the Gaussian mixtures Open image in new window and Open image in new window undergo a Kalman prediction and correction given the hypothesized data symbol on the valid trellis branch Open image in new window and Open image in new window , respectively. The resulting Gaussian mixture Open image in new window is obtained as a weighted sum of the resulting mixtures.
Figure 3

Example of forward propagation of Gaussian mixtures (with Open image in new window ) on a Open image in new window -state trellis.

3.2. Complexity Reduction Algorithm (CRA)

A problem with (17) is that each discrete state Open image in new window is now associated with a mixture of more than Open image in new window Gaussians. This means that the number of terms in the Gaussian mixture will grow with time. In order to keep the computational complexity constant for each time instant, we need to approximate the exact expression given by (17) as follows:
so that again Open image in new window Gaussians with weight Open image in new window , mean Open image in new window and covariance Open image in new window , Open image in new window are associated with each state Open image in new window , as in (16). We do this by applying the CRA proposed in [15]. Assume that Open image in new window (resp. Open image in new window ) is a multivariate Gaussian, whose weight, mean, and covariance are given by Open image in new window , Open image in new window , and Open image in new window (resp. Open image in new window , Open image in new window , Open image in new window ). In [15], a practical measure of similarity between the two densities is given by
where Open image in new window denotes the Kullback-Leibler distance. Then, pairs of similar Gaussians with minimal Open image in new window are repeatedly merged until Open image in new window Gaussians remain using the following approximation:

3.3. Backward Filtering

Let Open image in new window denote the total number of available observations and Open image in new window . A time-reversed version of the forward filter in Section 3.1 can also be derived. We seek a recursive expression of the likelihood Open image in new window , propagated backward in time. We obtain the following recursion:

where the discrete summation extends over the states Open image in new window , for which a valid transition Open image in new window exists.

Theorem 2.

Assume that the following Gaussian mixture parameterization:
is available at instant Open image in new window , a closed form expression of Open image in new window is obtained as
where the means Open image in new window and covariance matrices Open image in new window associated with the state transition Open image in new window are obtained from the following recursions:
and the weights Open image in new window are given by

The proof is obtained by injecting (26) into (25) and using the same arguments as in the demonstration of Theorem 1.

Figure 4 illustrates how the Gaussian mixture Open image in new window is computed on a Open image in new window -state trellis. The components of the Gaussian mixtures Open image in new window and Open image in new window undergo a Kalman correction and backward prediction given the hypothesized data symbol on the valid trellis branch Open image in new window and Open image in new window , respectively. The resulting Gaussian mixture Open image in new window is obtained as a weighted sum of the resulting mixtures.
Figure 4

Example of backward propagation of Gaussian mixtures (with Open image in new window ) on a Open image in new window -state trellis.

Again, we need to apply the CRA of Section 3.2. Complexity reduction algorithm to (27), so that Open image in new window admits the desired form

3.4. Smoothing

A two-filter smoothing formula is obtained as follows:

Theorem 3.

Using the Gaussian mixture approximations for the forward and the backward filter introduced in Sections 3.1 and 3.3, respectively, a closed form expression of Open image in new window is obtained as follows:
where the covariances associated to transition Open image in new window are
and the means associated to transition Open image in new window are
for Open image in new window . The expression of the weights is given by
The coefficient Open image in new window has the following form:

where Open image in new window denotes the dimension of the continuous valued state variable.

Proof.

In (31), the term Open image in new window has been calculated as (21) and the integral, also appearing in (25), has already been computed as
Therefore, (31) can be rewritten as follows:

After straightforward algebraic manipulations on the product of two Gaussian densities, the desired result (32) is obtained.

Figure 5 illustrates how the Gaussian mixture Open image in new window is computed on a Open image in new window -state trellis. The components of the Gaussian mixtures Open image in new window undergo a Kalman correction and backward prediction given the hypothesized data symbol on the valid trellis branch Open image in new window . The resulting Gaussian mixture is multiplied with the Gaussian mixture Open image in new window computed in the forward pass and by the scalar Open image in new window , so as to obtain Open image in new window .
Figure 5

Illustration of the computation of smoothed Gaussian mixtures (with Open image in new window ) on a Open image in new window -state trellis.

Since we are interested in soft-output equalization, we must compute smoothed bit-by-bit marginal probabilities. Let Open image in new window be the set of state transitions Open image in new window such that the information bit Open image in new window , with Open image in new window . Taking (32) at instant Open image in new window and marginalizing out the vector Open image in new window , we obtain
The hard decision can then be written as follows:
Similarly, the a posteriori pdf of the channel vector is obtained as a Gaussian mixture by marginalizing out all possible ISI state transitions
Under the minimum mean square error (MMSE) criterion, the forward filtered channel vector estimated at instant Open image in new window is obtained by marginalizing out the ISI state variable
Similarly, the MMSE smoothed estimate of the channel vector at instant Open image in new window is obtained by marginalizing out all possible ISI state transitions

3.5. Complexity Evaluation

It is well known that the complexity of one recursion of the Kalman filter is Open image in new window [16], 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 [17]. 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) [2] or with the first cost function (FCF) introduced in [18]. 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 [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- 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 [20], 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 [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.

Figure 6 illustrates the BER of the different equalizers as a function of Open image in new window for a static Rayleigh fading channel ( Open image in new window ). The CMA and FCF equalizers reach an error floor due to residual ISI. The same phenonenon is observed for the PSP algorithm and is due to the misacquisition problem analysed in [21]. Our Gaussian mixture smoother in the degenerate case, where the channel estimation is performed with only Open image in new window Gaussian per discrete ISI state, is clearly outperformed by the proposed algorithm with Open image in new window , which attains performances close to perfect channel state information (CSI). Note that our Gaussian mixture smoother with Open image in new window is essentially the same as [3], but adapted to block transmissions, since fixed-lag smoothing has been replaced by fixed-interval smoothing.
Figure 6

BER performances of blind equalization at Open image in new window .

Figure 7 illustrates the channel mean square error (MSE). The forward filtered and the smoothed channel estimates are computed according to (42) and (43), respectively. We note that smoothing provides a dramatic improvement over forward filtering alone both for Open image in new window and Open image in new window . We interpret this result by the fact that the smoothing pass, by exploiting the knowledge of future observations, is able to correct errors and phase ambiguities due to fading events or occasionally high noise in the forward pass. As a reference, the channel MSE attained by a genie-aided Kalman smoother (with known symbols) is also shown.
Figure 7

Channel MSE of blind Gaussian mixture-based equalization at Open image in new window .

We also study a fast-fading scenario with Open image in new window , in order to study the robustness of our algorithm against a large Doppler spread. Figures 8 and 9 illustrate the BER and the channel MSE, respectively. Considering the BER performances, we observe that the CMA, FCF, and PSP equalizers as well as the Gaussian mixture smoother with Open image in new window reach an error floor, while the performances of the Gaussian mixture smoother with Open image in new window are still satisfactory. Again, the channel MSE after the smoothing pass is much smaller than after forward filtering alone.
Figure 8

BER performances of blind equalization at Open image in new window .

Figure 9

Channel MSE of blind Gaussian mixture-based equalization at Open image in new window .

4.2. Performances on a Realistic Channel Model

A realistic model for quasi-static indoor multipath propagation is given by the Saleh-Valenzuela channel model [22], whose impulse response is given by
The Open image in new window represent the propagation delays of the rays, which arrive in clusters. These clusters and the rays within each cluster form Poisson arrival processes, with different but fixed rates. The Open image in new window are independent complex zero-mean Gaussian coefficients whose variances decay exponentially with the cluster and ray delay. In our simulations, the parameters of the Saleh-Valenzuela channel model are given in Table 1. Therefore, assuming rectangular transmit and receive pulse shaping as well as perfect symbol and frame synchronization, we obtain the equivalent discrete channel model described in Figure 2 with
Here, the equivalent discrete channel model is truncated to Open image in new window , thus, all existing rays with propagation delay Open image in new window will generate unmodeled ISI. Using the fact that the Open image in new window are statistically independent, we obtain
In general, the taps of the discrete-time channel model are correlated by the transmit and receive pulse shapes and the correlation depends on the power delay profile (PDP), which is of course unknown to the receiver. Therefore, the standard hypothesis of statistical independence of the channel coefficients Open image in new window in Section 2, corresponds to neglecting the correlations in (46).
Table 1

Saleh-Valenzuela model parameters.

Parameter and units

Notation and numerical value

Intercluster arrival rate (1/s)

Open image in new window

Intercluster decay constant (s)

Open image in new window

Intracluster arrival rate (1/s)

Open image in new window

Intracluster decay constant (s)

Open image in new window

In Figure 10, we test the Gaussian mixture smoother with Open image in new window , for a Saleh-Valenzuela channel model with parameters given in Table 1. We observe that the bit error rate (BER) and the frame error rate (FER) exhibit an error floor. This floor can be interpreted as the result of ignoring the correlation between the channel coefficients and to unmodeled ISI due to the fact that the equivalent discrete channel model is truncated to a memory- Open image in new window channel.
Figure 10

BER and FER performances of blind SISO equalization on a quasi-static Saleh-Valenzuela channel model.

5. Conclusions

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.

Notes

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.

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Copyright information

© Frederic Lehmann. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Institut TELECOM, TELECOM SudParis, Department CITIUMR-CNRS 5157Evry CedexFrance

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