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

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. 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.

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

(1)

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 [12]:

(2)

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 [13] of the following form:

(3)

by letting

(4)

where and . The driving noise term is chosen as a zero-mean white complex Gaussian process of variance

(5)

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 and evolves according to the dynamical system

(6)

where state transition matrix is given by

(7)

and the process noise vector is given by

(8)

The received complex noisy observation at instant has the following form:

(9)

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:

(10)

It is well known that can be represented graphically by a trellis diagram containing 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 , whose dynamics are given by

(11)

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

(12)

The third equation in (11) can be interpreted as a linear observation process, where the observation matrix has the form

(13)

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

(14)

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

(15)

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

(16)

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

Theorem 1.

A closed form expression of is obtained as follows:

(17)

where the means and covariance matrices associated with the state transition are obtained from the following recursions:

(18)

and the weights are given by

(19)

Proof.

Injecting (16) into (15), one obtains

(20)

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

(21)

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 [15]. Assume that (resp. ) is a multivariate Gaussian, whose weight, mean, and covariance are given by , , and (resp. , , ). In [15], a practical measure of similarity between the two densities is given by

(22)

where denotes the Kullback-Leibler distance. Then, pairs of similar Gaussians with minimal are repeatedly merged until Gaussians remain using the following approximation:

(23)

where

(24)

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:

(25)

where the discrete summation extends over the states , for which a valid transition exists.

Theorem 2.

Assume that the following Gaussian mixture parameterization:

(26)

is available at instant , a closed form expression of is obtained as

(27)

where the means and covariance matrices associated with the state transition are obtained from the following recursions:

(28)

and the weights are given by

(29)

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

(30)

3.4. Smoothing

A two-filter smoothing formula is obtained as follows:

(31)

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 is obtained as follows:

(32)

where the covariances associated to transition are

(33)

and the means associated to transition are

(34)

for . The expression of the weights is given by

(35)

The coefficient has the following form:

(36)

where denotes the dimension of the continuous valued state variable.

Proof.

In (31), the term has been calculated as (21) and the integral, also appearing in (25), has already been computed as

(37)

Therefore, (31) can be rewritten as follows:

(38)

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

(39)

The hard decision can then be written as follows:

(40)

Similarly, the a posteriori pdf of the channel vector is obtained as a Gaussian mixture by marginalizing out all possible ISI state transitions

(41)

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

(42)

Similarly, the MMSE smoothed estimate of the channel vector at instant is obtained by marginalizing out all possible ISI state transitions

(43)

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.