Page 311 - Introduction to Information Optics

296 5. Transformation with Optics

5.2 Prove a spherical wavefront of a radius R, described as

in the x-y plane in the paraxial condition.
5.3 When using the local frequency concept to analyze the nonstationary
signal, the windowed Fourier transform, or Gabor transform, is widely
used.
(a) Find the mathematical definition of the Gabor transform.
(b) Prove the inverse Gabor transform for recovering the original signal.
(c) Compare the Gaussian window used in the Gabor transform with the
wavelet window.
5.4 The wavelet transform is mainly used for detecting irregularities in the
signal. The wavelet transform of a regular signal can be all zero. When a
signal is regular, its (n + 1) order derivative and the derivatives of orders
higher than (n + 1) are equal to zero. What property should the wavelet
have in order to ensure the wavelet transform coefficients of this signal
are all equal to zero ?
5.5 Prove that any electromagnetic waves, which are solutions of the Maxwell
equations, should be limited in the light-cone

2
2
2
P = pi - c \p\ = o.
5.6 According to the expression in Eq. (5.15) of the electromagnetic wavelets
in the 4D space-time, write down expressions for:
(a) The reference electromagnetic wavelet (mother wavelet).
(b) The family of wavelets with dilations and scaling in 4D space-time.
5.7 Show that the electromagnetic wavelets given by Eq. (5.15) can be
decomposed into

+
h(x) =h~(x) +h (x),

where x = x(r, t) and

, ,
h~(x) = 2 4
2n c \r/c\
1
+
h (x) = 3
27rV|r/c|[|r/c|-t-0 '
+
h~(x) converges to the origin and h (x) diverges from the origin.