Discrete-time models of complex nonlinear processes, whether physical, biological, or economical, are usually under the form of systems of coupled difference equations. In analyzing such systems, one of the rst tasks is to nd a state-space description of the process-that is, a set of state variables and the associated state equations. We present a methodology for nding a set of state variables and a canonical representation of a class of systems described by a set of recurrent discrete-time, time-invariant equations. In the eld of neural networks, this is of special importance since the application of standard training algorithms requires the network to be in a canonical form. Several illustrative examples are presented.
We present an optical implementation of an improved version of the Kohonen map neural network applied to the recognition of handwritten digits taken from a postal code database. Improvements result from the introduction of supervision during the learning stage, a technique that also simplifies the map layer labeling. The experimental implementation is based on a frequency-multiplexed raster computer-generated hologram used to realize the required N(4) interconnection. The setup is shown to be equivalent to a 64-channel correlator. Computer simulations are used to study various detection and classification procedures. The results of the optical experiments, obtained with binary phase computer-generated holograms, are presented and shown to be in excellent agreement with the simulations.
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