Page 285 - Introduction to Information Optics

270 5. Transformation with Optics

Thus, we have shown that the optical diffraction is a wavelet transform of
a monochromatic optical field with electromagnetic wavelets. We have decom-
posed a monochromatic optical field into electromagnetic wavelet basis and
shown that the complex amplitude of the field is reconstructed by the inverse
wavelet transform, which is equivalent to the Huygens-Fresnel formula.
Hence, in the case of monochromatic field, electromagnetic wavelets are the
monochromatic spherical wavelets proposed by Huygens, and wavelet decom-
position gives the expression of the Huygens principle.

5.6. WIGNER DISTRIBUTION FUNCTION

5.6.1. DEFINITION

The Wigner distribution function is a mapping of a signal from the
space-coordinate system to the space-frequency joint coordinates system. The
space-frequency joint representation is useful for analysis of nonstationary and
transient signals, as discussed in Sec. 5.4.2. The Wigner distribution of a
function f(x) is defined in the space domain as

W f(x, ft>) = / x + / * x- exp(-Mx') dx',
I / (x + | j /*
»/ \ '*""' /

which is the Fourier transform of the product of a signal f(x/2) dilated by a
factor of 2 and the inverted signal /*( — x/2) also dilated by a factor of 2. Both
the dilated signal and its inverse are shifted to the left and right, respectively,
by x along the x-axis. By the definition in Eq. (5.22) the Wigner distribution
function is a nonlinear transform and is a second-order or bilinear transfor-
mation. The Wigner distribution of a ID signal is a 2D function of the
spatial-frequency ca and the spatial shift x in the space domain.
The Wigner distribution function can be also defined in the frequency
domain and is expressed as

W f(o), x) = -L F ((a + y j F* (u) - y j Qxp(jxco') dv, (5.23)
«/ \ / \ ** /

where F(co) is the Fourier transform of f(x). From Eqs. (5.22) and (5.23) one
can see that the definitions of the Wigner distribution functions in the space
and frequency domains are symmetrical.