TRAINING PROCEDURE FOR RBF NETWORKS

The training of the radial basis function is basically the positioning of centres at desired positions or desired states.Although, the desired signal states are initially unknown, they are the cluster means, in the observation states. Learning can therefore be achieved efficiently using a supervised or decision directed clustering algorithm and this has been investigated in [6]. When co-channel interference is also present then a simple supervised clustering is not sufficient, so there has to be a two stage procedure and this needs that the supervised clustering procedure is followed by unsupervised clustering procedure. The supervised clustering procedure is

$$if~~(d_0(k) == state~i) \{$$

$${ c_i(k) =state\_counter_i * c\_i(k-1) + y(k);}$$

$${ state\_counter_i += 1;}$$

$${ c_i(k) /= state\_counter_i;}$$

$$\}$$ (6)

Because of the underlying data structure, a rapid convergence of this supervised clustering procedure is guaranteed, and the algorithm is very simple and robust. The supervised/decision directed clustering algorithm infers which desired signal state appears. Suppose that the ith desired signal state occurs. The noise-free observation state that actually appears must be within the ith group of the l noise free observation state states. The unsupervised algorithm then searches through the this group to find the state that actually appears and to adjust the corresponding RBF accordingly. The unsupervised clustering computes squared distances between the centres and the data vector y(k), selects a minimum squared distance and moves the corresponding centre closer to y(k). if the ith desired signal state appears at k, the computational procedure of the unsupervised clustering algorithm is given as

$$a_j(k) = \vert\vert y(k) - c_{i_j}(k-1)\vert\vert^2, 1 \leq j \leq l ,$$ (7)

$$j^*=arg[min\{a_j(k), i \leq j \leq l \}],$$ (8)

c_ij(k) = c_ij(k-1)+g_c(y(k)-c_ij(k-1)). (9)

where g_c is the learning rate for the centres.

It should be emphasized that the above combined clustering algorithm is all that is required to adapt an RBF network in the case of equiprobable symbols. If the assumption of equiprobable events is violated, then the weights of the RBF network can be adapted using the following LMS type algorithm.

$$a_i(k)=\vert\vert y(k)-c_i(k-1)\vert\vert^2,~ ~ 1\leq i\leq n,$$ (10)

$$\phi_i(k)=exp(-a_i(k)/\rho),~ ~ 1\leq i\leq n,$$ (11)

$$\epsilon(k)=d(k)-\sum_{1=1}^{n}w_i(k-1)\phi_i(k),$$ (12)

$$w_i(k)=w_i(k-1)+g_w\epsilon(k)\phi_i(k),~ ~ 1\leq i\leq n$$ (13)

where c_i is the centre of RBF, w_i is the weight of RBF, $\phi$ is the basis function, g_c is the learning rate for centres, g_w is the learning rate for weights, $\rho$ is the width of RBF, $\epsilon$ is the error in the approximation.

For complex signals, the stochastic gradient (SG) algorithm gives better results. The SG algorithm does not guarantee convergence to globally optimum network parameters, however it does appear to converge to reasonable solutions in practice. The method can be used as a single-stage learning algorithm if training data are only sequentially available or as the second stage method of two stage algorithm where centers are spread parameters and are predetermined by a method such as the OLS or clustering technique. The advantage of SG algorithm is that all the free network parameters are adapted simultaneously, usually yielding improved overall solutions. Also, the algorithm is well-suited for on-line adaptive signal processing. The SG algorithm for CRBFN is as follows.

$$w_{i,n+1}=w_{i,n}+\mu_we_n\phi_i({\bf y}_n)$$ (14)

$$\sigma_{i,n+1}=\sigma_{i,n}+\mu_\sigma\phi_i({\bf y}_n)[w_{R_... ...\vert\vert{\bf y}_n-{\bf c}_{i,n}\vert\vert^2}{\sigma_{i,n}^3}$$ (15)

$${\bf c}_{i,n+1}={\bf c}_{i,n}+\mu_c\phi_i({\bf y}_n)\left[ \f... ...n}}Im\{e_n\}Im\{{\bf y_n-c}_{i,n}\}} {\sigma_{i,n}^2} \right]$$ (16)

where w are the weights, e is the error, $\sigma$ is the width, $\mu_w , \mu_c ,\mu_\sigma$ are the learning rates of weights, centres, and width respectively

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