Page 341 - Phase Space Optics Fundamentals and Applications

322 Chapter Ten

theory. The Shannon sampling theorem assumes that a signal has this
property.Atheoremthatappearsinmanytextbooksstatesthatasignal
and its Fourier transform cannot both have compact support (Ref. 9, p.
26; Ref. 89). We will refer to this as the Fourier transform compact support
theorem. This theorem is a corollary of the Paley-Wiener theorem. 90−91
However, for numerical work, we must assume that a signal may
be approximately represented using a finite number of samples. This
is achieved in the discrete Fourier transform by pretending that the
signal is periodic in both space and frequency. Inevitably, this will
result in some error referred to as aliasing.

10.3.2 Band-limitedness and the LCT
The FT is a special case of the LCT. Given that a signal and its FT
cannot both have compact support, it is natural to ask, Can a signal
and its LCT ever have compact support, and if so, when? The question
of compact support and band-limitedness is important in relation to
the LCT 92 as the development of a generalized sampling theory 93 is
one of the topics we address in this chapter. We now present theorems
that describe how the LCT preserves, destroys, or transforms com-
pact support or band-limitedness. The proofs of these theorems are
omitted, but can be found in Ref. 92.
We first consider the case of a LCT with none of the ABCD pa-
rameters equal to zero. Such a LCT is entirely destructive of compact
support and band-limitedness. Given a quadratic-phase system (QPS)
characterized by an ABCD matrix with no elements equal to zero, and
given an input waveform u(x) that either has compact support or

is band-limited, the output waveform L M {u(x)}(x ) neither is band-
limited nor has compact support. This case is represented by the first
two lines in Table 10.1. Equivalently, if the ABCD matrix has no en-

tries equal to zero and the output waveform L M {u(x)}(x ) has compact
support or is band-limited, then the input waveform u(x) neither is
band-limited nor has compact support. This case is represented by
lines 3 and 4 in Table 10.1. A number of other cases are also shown in
Table 10.1. For example, if C = 0, as it does in the case of the Fresnel
transform (A = D = 1,B = z, C = 0), the property of finite band-
width will be conserved through the transform. Similarly the property
of infinite bandwidth will also be conserved. For a Fourier transform
A = D = 0,B = 1, and C =−1. Since A = 0, a property of finite band-
width will not be preserved through the Fourier transform; rather, it
will produce a property of finite support. If D = 0, the reverse also
holds true.
At this point the reader might reasonably ask, What will the PSD
of signal with a finite support w in an LCT domain (with parameters
ABCD) look like? As a simple illustration we show such a signal in
Fig. 10.6a. The signal will have a local bandwidth which is equal to b.
We show two lines, parallel to the boundary support of the signal,