Page 334 - Phase Space Optics Fundamentals and Applications

Sampling and Phase Space 315

insight when we bestow it with many of the WDF properties. For ex-
ample, a signal with a finite bandwidth might have a PSD shown in
Fig. 10.1.
In Fig. 10.1a we show the PSD of a signal that has a finite bandwidth.
It is well known that in the strict mathematical sense any signal that
has such a property must, as a consequence, have infinite spatial sup-
port. However, in many practical cases we assume that the signal
has an approximately finite support in both domains, and the PSD of
such a signal is shown in Fig. 10.1b. The subject of finite support is ad-
dressed in greater detail in Sec. 10.3. The PSD that is bounded in both
x and k is much easier to use for illustrating some of the properties of
the WDF. For example, the application of the FT to our signal brings
◦
about a 90 rotation of the WDF and therefore the PSD as shown in
Fig. 10.1c. The Fresnel transform causes a horizontal shearing of the

k k k

x x x

(a) (b) (c)

k k k

x x x

(d) (e) (f)

FIGURE 10.1 The PSD of a signal (a) having finite bandwidth and therefore
an infinite spatial support, (b) having approximately finite spatial support
and finite bandwidth, (c) which is a Fourier transform of signal represented
in part b,(d) which is a Fresnel transform of the signal represented in part b,
(e) which is a signal represented in part b after being multiplied by a chirp
function, and (f) which is a linear canonical transform of the signal
represented in part b.