Theory of the binary phase-only filters for optical pattern recognition

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In this paper, we presentan analysis of the binary phase-only filter (BPOF).1,2 The linear filter function is expressed as the real part of the conventional matched filter, which contains the amplitude and phase of the Fourier transform of the reference signal. The linear filter function is then applied to a binary nonlinear device to produce the BPOF function. Our paper provides analytical expressions for the correlation signals produced by the BPOF. An expression for the BPOF can be obtained by an approach similar to that used in the analysis of nonlinearly transformed filters by the transform method.3 We have investigated the effects of the binary nonlinear transformation on the correlation signals at the output plane. We show that the BPOF results in an infinite sum of harmonic terms. Each harmonic term is phase-modulated by m times the phase modulation of the linear filter function. The amplitude modulation of the harmonic terms is removed in the binary nonlinear transformation. We show that for the first-order harmonic term, the nonlinear transformation preserves the phase of the matched filter. The higher-order terms are added to the first-order term and may be considered as degradation of the filter. The intensity of the higher order terms is less than that of the first-order signal. One component of the first-order term is identical to the continuous phase-only filter. However, the firstorder phase-only filter is degraded by the higher-order intermodulation terms. This explains why the performance of the binary phase-only filter is similar to that of the continuous phase-only filter.