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Hebbian-Based Maximum Eigenfilter
For the matched filter considered in Example 2, the eigenvalue 1 and associated eigenvector q1 are respectively defined by
Consider a random vector X, a realization of which is denoted by the sample vector x. Let
where the vector s, representing the signal component, is fixed with a Euclidean norm of one.The random vector V, representing the additive noise component, has zero mean and covariance matrix σ2I. The correlation matrix of X is given by
The largest eigenvalue of the correlation matrix R is therefore
The associated eigenvector q1 is equal to s. It is readily shown that this solution satisfies the eigenvalue problem
Hence, for the situation described in this example, the self-organized linear neuron (upon convergence to its stable condition) acts as a matched filter in the sense that its impulse response (represented by the synaptic weights) is matched to the signal component s.