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.
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. will denote a complex Gaussian distribution of the variable , with mean and covariance matrix . denotes the identity matrix, while denotes the all-zero matrix. The symbol denotes the Kronecker product. The operator will denote the determinant of a matrix.
2. System Model
The transmitted data are organized in GSM-like bursts containing 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 , . 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.
We assume a discrete Rayleigh fading channel of memory as depicted in Figure 2. The elements of the impulse response are modeled as independent zero-mean complex Gaussian random variables. For a static channel, the channel state is defined as and evolves according to the trivial dynamical system
For a time-varying channel, let and denote the maximum Doppler shift and the symbol duration, respectively. The channel autocorrelation can be modeled as follows :
where is the zero-order Bessel function of first kind. A good approximation of the channel statistics is obtained with an order two autoregressive process  of the following form:
where and . The driving noise term is chosen as a zero-mean white complex Gaussian process of variance
so as to normalize the channel coefficients to unit variance . Consequently, the channel state, in the case of a time-varying wireless channel, is defined as 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 has the following form:
where is a complex white Gaussian noise sample with single-sided power spectral density .
We define the ISI state as the decimal form of the binary subsequence , which can take discrete values. Let denote the ISI state transition function defined by the following relation:
It is well known that can be represented graphically by a trellis diagram containing states .
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 , 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 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 , and
3. Blind SISO Equalization Using a Gaussian Mixture Approach
In this section, we derive a fixed-interval 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
A recursive expression of , where is obtained by noting that
where the discrete summation extends over the states , for which a valid transition 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
In (16), each discrete state is associated with a mixture of Gaussians, where is a design parameter of choice.
A closed form expression of is obtained as follows:
where the means and covariance matrices associated with the state transition are obtained from the following recursions:
and the weights are given by
Injecting (16) into (15), one obtains
In the above expression, we easily recognize the integral as the well-known prediction step of Kalman filtering . Moreover, the multiplication by is the correction step of Kalman filtering. Therefore, can be written as (17).
Figure 3 illustrates how the Gaussian mixture is computed on a -state trellis. The components of the Gaussian mixtures and undergo a Kalman prediction and correction given the hypothesized data symbol on the valid trellis branch and , respectively. The resulting Gaussian mixture is obtained as a weighted sum of the resulting mixtures.
3.2. Complexity Reduction Algorithm (CRA)
A problem with (17) is that each discrete state is now associated with a mixture of more than 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 Gaussians with weight , mean and covariance , are associated with each state , as in (16). We do this by applying the CRA proposed in . Assume that (resp. ) is a multivariate Gaussian, whose weight, mean, and covariance are given by , , and (resp. , , ). In , a practical measure of similarity between the two densities is given by
where denotes the Kullback-Leibler distance. Then, pairs of similar Gaussians with minimal are repeatedly merged until Gaussians remain using the following approximation:
3.3. Backward Filtering
Let denote the total number of available observations and . A time-reversed version of the forward filter in Section 3.1 can also be derived. We seek a recursive expression of the likelihood , propagated backward in time. We obtain the following recursion:
where the discrete summation extends over the states , for which a valid transition exists.
Assume that the following Gaussian mixture parameterization:
is available at instant , a closed form expression of is obtained as
where the means and covariance matrices associated with the state transition are obtained from the following recursions:
and the weights 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 is computed on a -state trellis. The components of the Gaussian mixtures and undergo a Kalman correction and backward prediction given the hypothesized data symbol on the valid trellis branch and , respectively. The resulting Gaussian mixture is obtained as a weighted sum of the resulting mixtures.
Again, we need to apply the CRA of Section 3.2. Complexity reduction algorithm to (27), so that admits the desired form
A two-filter smoothing formula is obtained as follows:
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 is obtained as follows:
where the covariances associated to transition are
and the means associated to transition are
for . The expression of the weights is given by
The coefficient has the following form:
where denotes the dimension of the continuous valued state variable.
In (31), the term 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 is computed on a -state trellis. The components of the Gaussian mixtures undergo a Kalman correction and backward prediction given the hypothesized data symbol on the valid trellis branch . The resulting Gaussian mixture is multiplied with the Gaussian mixture computed in the forward pass and by the scalar , so as to obtain .
Since we are interested in soft-output equalization, we must compute smoothed bit-by-bit marginal probabilities. Let be the set of state transitions such that the information bit , with . Taking (32) at instant and marginalizing out the vector , 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 is obtained by marginalizing out the ISI state variable
Similarly, the MMSE smoothed estimate of the channel vector at instant 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 , where 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 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 . 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 Baud-rate linear filters optimized with the constant modulus algorithm (CMA)  or with the first cost function (FCF) introduced in . These equalizers are iterated 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- 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 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 -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.
Figure 6 illustrates the BER of the different equalizers as a function of for a static Rayleigh fading channel (). 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 . Our Gaussian mixture smoother in the degenerate case, where the channel estimation is performed with only Gaussian per discrete ISI state, is clearly outperformed by the proposed algorithm with , which attains performances close to perfect channel state information (CSI). Note that our Gaussian mixture smoother with is essentially the same as , but adapted to block transmissions, since fixed-lag smoothing has been replaced by fixed-interval smoothing.
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 and . 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.
We also study a fast-fading scenario with , 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 reach an error floor, while the performances of the Gaussian mixture smoother with are still satisfactory. Again, the channel MSE after the smoothing pass is much smaller than after forward filtering alone.
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 , whose impulse response is given by
The 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 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 , thus, all existing rays with propagation delay will generate unmodeled ISI. Using the fact that the 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 in Section 2, corresponds to neglecting the correlations in (46).
In Figure 10, we test the Gaussian mixture smoother with , 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- channel.
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.
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Lehmann (EURASIPMember), F. A Gaussian Mixture Approach to Blind Equalization of Block-Oriented Wireless Communications.
EURASIP J. Adv. Signal Process.2010, 340417 (2010). https://doi.org/10.1155/2010/340417