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2-7. Optical correlator using a holographic filter. 3 PATTERN RECOGNITION VIA MATCHED FILTERING A. 3-1) 2 where S(v) is the signal spectrum, |iV(v)| is the noise spectral density, and k is a constant. For the optical counterpart of this matched filter, one can simply replace the temporal variables t and v by their spatial equivalents (x, y) and (M, V), respectively. 3-2) 54 SILVERIO P. ALMEIDA AND GUY INDEBETOUW where H(u, υ) = & [h(x9 y)~] and G*(u, v) = & [g*( - x, - y)~]. In the case where the noise spectral density can be assumed constant, the filter transfer function and impulse response become H(u, v) = k'G*(u, v) and h(x, y) = k'g*( - x, - y).

10-8) (0, elsewhere. and use it as our first estimate of the spectrum. Thus let F^u, v) = G(u, v). Its inverse FT gives / x (x, y). Since we know that the function is spatially bounded, we form the correction 0i (x, y) = / i (x, y) rect(x/L) reet{y/W). 10-9) With Gx(x, y) = ^2\j9i(x> y)~\> w e f ° r m the function $F(u,v), F2(u,v) = < IGxfav), {u,v)e@, elsewhere. 10-10) 1. THEORY OF THE OPTICAL FOURIER TRANSFORM 39 This correction incorporates the a priori knowledge we have from observing the spectrum over the passband of the system.

For values of u not too near the origin, the second integral can be omitted in relation to the first. In that case var[^V(w)] = \S(u)\2K *1/K (1 - K\x\) sine2 xLdx^— l/K Because S(u) ~ (^w(u)), K> \S{u)\2. 7-19) which is to be contrasted with Eq. 7-9), which holds for the unsmoothed case. The smoothing filter has reduced the variance of the measurement by the factor K/L. As stated earlier, a similar result is given in the literature for the Gaussian process. In this simple calculation we ignored the effect of W(u) on S(u) for the sake of convenience.

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