Page 312 - Introduction to Information Optics

Exercises 29 /
+
5.8 Show the two wavelet components h~(x) and h (x) are solutions of the
two equations

- .j \ 9
0 — lt)~
2 + 2 +
-d h (x) + V h (x) = ——^-y.
2
' 77r(l - if)
The right-hand side of the two equations represents the elementary
courant at the origin, which is generated by the converging wavelet h~(x)
+
and then generates the diverging wavelet h (x), which is, in fact the
Huygens wavelet.
5.9 Write the Wigner distribution function of a summation of two signals,
f(x) + g(x), and show the cross-talk term, which indicates the bilinear
property of the transform.
5.10 Show that the Fourier transform of a circular harmonic function of an
image is equal to the circular harmonic function of the Fourier transform
of the image.
5.11 Show that the fractional Fourier transform can be optically implemented
by using a segment of certain length of the gradient index fiber lens.
5.12 Prove that Mellin transforms can be implemented by using Fourier
transforms after some geometric transformation.
5.13 Show that the phase mask must satisfy Eq. (5.47) with the saddle-point
method in the implementation of the mapping of the coordinate system
in Eq. (5.45).
5.14 Find the phase-mask amplitude transmittance for logarithmic polar trans-
formation.