RADIAL BASIS FUNCTION NETWORKS AS EQUALIZERS

The Radial Basis Function Network (RBFN) [4] is a two layer processing structure as shown in fig.2. The hidden layer consists of an array of computing nodes. Each node contains a parameter vector called centre and the unit calculates a squared distance between the centre and the network input vector. The squared distance is then divided by a parameter called width and the result is passed through a non-linear function. The second layer is a linear combiner with a set of connection weights. The overall response of the RBF network is a mapping f_r,

$$f_r({\bf y})=\sum_{i=1}^{n}w_i \phi(\vert\vert{\bf y- c_i}\vert\vert^2/\rho_i)$$ (2)

where n is the number of computing nodes, c_i are the RBF centres, $\rho_i$ are the widths of the nodes, $\phi(.)$ is the basis function and w_i are the weights. A different type of approach is also proposed in [7].

Comparing the network response with the optimal Bayesian equalizer solution it has been shown [4] that both have an identical structure. The RBF network is therefore an ideal structure for implementing the optimal equalizer. Given channel, co-channel and the noise statistics, it is known exactly how to specify all the parameters of the RBF network. The number of hidden nodes n is equal to number of noise free observation states and the RBF centres are placed at these states. The non-linear function $\phi$ is chosen as an exponential function $\phi(y)=exp(-y)$ because it is a bounded and localized function. All the widths have the same value $\rho = 2\sigma_e^2$, which is twice as large as the noise variance. Each hidden node than implements a component conditional density function and the weights are fixed corresponding to $\alpha$ or $-\alpha$, where $\alpha$ is some small constant. The RBF network then realizes precisely the optimal equalizer.

The equalizer decision function in (2) provides a localized behavior. When the equalizer input is far from all the channel states the equalizer may fail to provide a proper decision function. To overcome this we propose a modification of (2) equation can be modified in the normalized form to provide non-localized behavior providing the right decision to all input vectors. The normalized equation would be

$$f_r({\bf y})=\frac{\sum_{i=1}^{n}w_i\phi(\vert\vert{\bf y-c_i... ...{\sum_{i=1}^{n}\phi(\vert\vert{\bf y-c_i}\vert\vert^2/\rho_i)}$$ (3)

In practice the signals would be complex in nature so (3) has to be modified to accommodate complex signals, thus we can write this as

$$f_r({\bf y})=\frac{\sum_{i=1}^{n}(w_{R_i}+iw_{I_i})\phi(\vert... ...{\sum_{i=1}^{n}\phi(\vert\vert{\bf y-c_i}\vert\vert^2/\rho_i)}$$ (4)

where

$$\vert\vert{\bf y-c_i}\vert\vert^2=({\bf z-c_i)}^H({\bf z-c_i)}.$$ (5)

The representation of a complex RBF is as shown in fig.3 [8][9]. From the figure we can easily conclude that a CRBF network (CRBFN) with N complex inputs and a complex output can be viewed alternatively as a real RBFN with 2N real inputs and two real outputs. In this regard the CRBFN is a straightforward extension of the real RBFN.

The estimation of the decision function needs in (2) and (3) needs the channel estimation for the evaluation of the equalizer decision function. The channel state estimation needs the channel information which in most cases is not available. Under these circumstances the channel states can be estimated during the training period. This can be achieved with the help of any adaptive algorithm like LMS or RLS, but this technique fails for non-linear channels. Algorithms like Orthogonal Least Squares (OLS) can also be used for this purpose [5], but they are not practical for on-line purpose. The channel states can also be calculated according to some clustering algorithm, like the k-means algorithm. The advantage of this is that there is no necessity of the channel model being known, and this model works well for practical channels. For complex signals, the clustering algorithm alone is not sufficient, one needs an adaptive mechanism, which will train it simultaneously, and so better output could be obtained. For this purpose we have used a combination of clustering algorithm and stochastic gradient (SG) algorithm. The SG algorithm provides an effective means to overcome poor network initialization and resultant performance degradation, which can be especially problematic for networks with localized basis functions. In this paper we have considered a combination of clustering and SG algorithm, for the training of CRBFN, and we have applied it in equalization of complex channels with QAM signals. We have carried out extensive simulations that justify the use of CRBFN's use for channel equalization.

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