Page 344 - Phase Space Optics Fundamentals and Applications

Sampling and Phase Space 325

∞ ∞
2 2
E = |u(x)| dx = |U(k)| dk (10.24)
−∞ −∞
The dual equality in Eq. (10.24) follows from Rayleigh’s theorem.
Earlier in Fig. 10.1b we showed the PSD of a signal u(x) in which the
signal energy lies within a rectangular area. In the next section we
discuss sampling and interpolation. As a prelude we note that the
signal u(x) is completely determined if it is sampled equidistantly in
x with sample space x such that the Nyquist criterion is satisfied.
Therefore the number of samples N required to completely describe
u(x)is N = d/ x ≥ db.Clearly,forthemostefficientuniformsampling
x = 1/b and N = db, the space-bandwidth product (SBP) of the
signal. In general, signals may have an irregularly bounded WDF,
and one such case is shown in Fig. 10.1d.

10.4 Sampling a Signal
In the last section we discussed signals that had the properties of
compact support and band-limitedness and the effect that different
types of LCT would have on these two properties. It is well known
that a signal that has the property of band-limitedness can be sam-
pled and interpolated exactly from these samples. This is true only
if the signal has been sampled at a rate greater than or equal to the
Nyquist rate, which is determined by the bandwidth of the signal.
In this section we review this sampling theorem, using phase-space
diagrams. As shown in the last section, often the LCT of a band-
limited signal is no longer band-limited. Therefore, if one were to
rigorously follow the laws of Nyquist and Shannon, one would ar-
rive at the conclusion that such a signal could not be sampled and
interpolated from these samples. Recent work suggests otherwise. It
has been shown that a more general sampling theorem must be em-
ployed for signals of this type. As we show in this section, by far the
simplest way that one can deduce this generalized sampling theorem
is to again employ phase-space diagrams. In fact we shall see that
we need only heuristically apply some of the rules we have learned
thus far on the PSD in order to fully derive the generalized sampling
theorem in the briefest of fashions. We begin with a discussion of
Nyquist.

10.4.1 Nyquist-Shannon Sampling
The comb function was discussed in Sec. 10.2.5.1. We begin with a
signal u(x) with finite bandwidth equal to B with the PSD shown