Generalized joint fractional fourier transform correlators: a compact approach

Fractional correlation was introduced recently. We generalize the architecture of a joint (Fourier) transform correlator (JTC) to achieve the joint fractional (Fourier) transform correlator (JFrTC) such that fractional correlation can be obtained. Here the Fourier transform in the JTC is replaced by the fractional Fourier transform, and four different JFrTC architectures can be implemented. The mathematical derivations for these JFrTC architectures are given, together with the simulation verifications. The JFrTC can provide a correlation signal similar to a delta function but with a small discrimination ratio, such that it is insensitive to additive noise. In a conventional JTC the distance between the two desired correlation signals at the output plane is fixed and depends on the distance between the input and the reference signals. However, with a given fractional order and an additional phase mask the separation distance between the two correlation signals at the output plane of a JFrTC can be larger or smaller than that of a JTC. This property is useful for the applications of real-time target tracking. Unlike in a previous approach [Appl. Opt. 36, 7402 (1997)], we need only two fractional Fourier transformations instead of three to achieve fractional correlation.