the filter tunes through the power spectrum at the source. High 
period waves are received first followed by low period waves. 
The band width is constant for a constant Die For a larger fixed 
X, aS time increases the filter tunes through the power spectrum 
at the source more slowly and its band width is narrower. 
A square cornered sharp cutoff filter is physically impossible. 
It is, however, a relatively simple step to extend the forecast 
diagram to a Fresnel Filter. The procedure is to return to the 
methods of Chapter 5 and solve the problem given by 
7(O,t) = cos(Sze + W( Hans) 
if O<t.,<D, and by 7(o,t) = 0 otherwise. The transformation 
given by t,, = t' + D/2 would break the function down into an odd 
and an even part about t' = 0. Formulas similar to those in Chap- 
ter 5 would result except that the original simplifications in the 
derivation permitted by the use of the whole number of waves and 
the oddness (in the sense of not even) of the function would not 
be available. Note that the third step in equation (5.2) shows 
that the Fourier spectrum will be continuous at w = 2m/Ton41° 
The result would be functions similar to equations (5.15), (5.16) 
and (5.17), and it would be possible to show that a modified form 
of Ge + Ho would give the square of the modulation envelope in the 
given wave train. The arbitrary phase would be in the trigonometric 
term. 
Finally, the Fresnel Filter would be obtained as given by 
equation (7.50). In the filter, emphasis is placed on the varia- 
tion with » for a fixed x and t. If x is small, and ifmw = gt p/ 2x 
then the range of integration is from a large positive number to 
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