RADIAL BASIS FUNCTION NETWORK AS
EQUALIZERS

The Radial Basis Function Network (RBFN) [4] is a two layer processing structure as shown in Figure 1. 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,

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

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].

  

Figure 1: Schematic of RBF network.

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 processing means to implement 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 (1) provides a localized behaviour. This 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

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

The estimation of the decision function needs in (1) and (2) 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 suffers failure which arises due to non-linearity. The channel states can also be calculated directly with the help of some clustering algorithm. The training time may be large in this but the convergence due to this blind clustering procedure is guaranteed.

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