Generalization of the linear matched filter concept to nonlinear matched filters

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Abstract
A nonlinear matched filter based image correlator is investigated. The linear matched filter is expressed as a bandpass function containing the amplitude and phase of the Fourier transform of the reference signal. The bandpass filter function is then applied to a kth law nonlinear device to produce the nonlinear matched filter function. Analytical expressions for the nonlinear matched filter are provided. The effects of the nonlinear transfer characteristics on the correlation signals at the output plane are investigated. The correlation signals are determined in terms of the nonlinear characteristics used to transform the filter. We show that the nonlinear filter results in a sum of infinite harmonic terms. Each harmonic term is envelope modulated due to the nonlinear characteristics of the device, and phase modulated by m times the phase modulation of the linear filter function. The correct phase information of the filter is recovered for the first-order harmonic of the series. The envelope of each harmonic term is proportional to the kth power of the Fourier transform magnitude of the reference signal. We show that various types of filter such as the continuous phase-only filters can be produced simply by varying the severity of the nonlinearity. Nonlinear filters provide higher correlation peak intensity and a better defined correlation spot.