Page 329 - Phase Space Optics Fundamentals and Applications

310 Chapter Ten

theorems, but also it paves the way for rich new theorems and algo-
rithm designs. This is the subject of this chapter.
Until recently, sampling of electromagnetic signals was performed
primarilyusing thetheoremsdevisedinthe first halfofthelastcentury
1
3
2
4
by Nyquist, Shannon Whittaker, and Kotelnikov. We note that the
5
sampling theory of diffraction patterns was introduced by Francia
in 1955. Modern sampling theory has evolved far beyond Nyquist-
Shannon sampling, and a thorough account of contemporary sam-
pling theory and discrete signal processing can be found in Refs. 6
and 7. In this chapter we focus primarily on Shannon sampling (and
a recently generalized derivative of it) and its simplified description
using phase space. In this chapter we limit the scope of the discussion
to classical sampling. The rules of classical sampling and interpola-
tion may be summarized as follows: It is assumed that the continu-
ous signal is band-limited in frequency. Sampling this signal with a
sampling rate at least as fast as twice the maximum frequency (the
Nyquist rate) allows for the continuous signal to be “interpolated”
from the discrete sampled values by applying the Shannon interpola-
tion formula. This can be explained using the Fourier transform. 8–11
Sampling creates an infinite number of copies of the signal’s Fourier
transform, all adjacent along the frequency axis, with a separation
equal to the inverse of the sampling interval. If the sampling rate is
high enough, the copies will not overlap with one another due to
their assumed finite support. The continuous signal may be interpo-
lated by isolating one of these copies, achieved by multiplying by an
appropriate rect function, which amounts to convolving the sampled
signal with a sinc function. This convolution is known as Shannon
interpolation. 2
Recently there has been considerable interest 12–25 in the literature
on the subject of sampling certain optical signals at rates below the
Nyquist rate and still managing to interpolate the continuous signal.
To the best of our knowledge, the first demonstration of this was by
12
Gori in 1981 in which he investigated the sampling of Fresnel diffrac-
tion patterns. This collective body of work 12–25 has demonstrated that
the requirement of imposing the property of band-limitedness in the
Fourier domain is too strict a requisite for interpolation to be achiev-
able. It is sufficient that the signal be bounded in any one of an in-
finite set of domains which are output domains of the linear canon-
ical transform (LCT). 13,19–21,23−33 The Fourier transform 8−11 and the
Fresnel transform 9,12,14,18 are special cases of the LCT. If the signal is
bounded within some finite support in such a domain, then the sig-
nal can be sampled at a rate, proportional to the finite LCT support
width. TheNyquistsampling rate, whichisproportionaltothe Fourier
support width (bandwidth), is merely a special case of this more gen-
eralized sampling theorem. Importantly, the phase-space investiga-
tion of these concepts that follows unearths an interesting insight that