Page 331 - Phase Space Optics Fundamentals and Applications

312 Chapter Ten

cases it is even useful not to apply some WDF properties (in partic-
ular, the property of bilinearity) to the PSD in order to simplify our
understanding. Such omissions must be done with care and with good
reason. In Sec. 10.2 we discuss the PSD in greater detail, and then we
demonstrate the application of the PSD in understanding sampling
theory and in simulating optical systems.
The central theme of this chapter is to show that the WDF and the
PSD are useful tools in understanding the sampling of signals with an
LCT of finite support. This subject can be further complicated if we
consider signals which are sampled, then transformed by a LCT, and
then sampled again. The topic is of considerable interest because it
is central to the numerical implementation (or simulation) of optical
processes. 34–42,53–88 The volume of publications on the subject in the
last 10 years highlights its relevance to contemporary optics as does
the industrial application of these algorithms in today’s optoelectronic
world. The double sampling considerably complicates matters, and it
forces us to consider sampling criteria of a signal in two transforma-
tion domains sequentially. The first sampling operation considerably
affects the second, and vice versa; i.e., the second sample must be
considered as also shaping the first sampling operation. In this case a
new type of aliasing can be encountered which is discussed for the first
time in this chapter. Again we find that the most intuitive approach
to this subject is through the PSD.
This chapter is broken down as follows. In Sec. 10.2 we discuss some
initial concepts that are utilized in the following sections. In Sec. 10.3
we review how a signal can have a finite support in some LCT do-
main. In Sec. 10.4 we discuss how Nyquist sampling and generalized
sampling may be discussed both qualitatively and quantitatively in
an elegant fashion using the WDF and PSD. In Sec. 10.5 we progress
to discuss sampling of a signal in two domains for the purposes of
simulating a quadratic-phase system, and finally in Sec. 10.6 we offer
a brief conclusion.

10.2 Notation and Some Initial Concepts
10.2.1 The Wigner Distribution Function
and Properties
The WDF is a time-frequency distribution and is mathematically defined
in terms of this spatial (x) distribution as follows

∞

{u(x)}(x, k) = u x − u ∗ x + exp(− j2 k ) d = W uu (x, k)
∗
2 2
−∞
(10.1)