Page 340 - Phase Space Optics Fundamentals and Applications

Sampling and Phase Space 321

k

x

k

x
(a) (b)

FIGURE 10.5 The rect function: (a) Wigner distribution function of rect(k)
and (b) the phase-space diagram of that function.

of the signal samples in the space domain with a sinc function. Since
this process is so central to sampling, it is important that we define
the WDF of a rect function in the frequency domain, defined as

1 ∀|k|≤ 1/2T
rect(Tk) = (10.16)
0 ∀|k| > 1/2T
Using Eq. (10.16) as input to the WDF integral defined in terms of
the Fourier transform [see Eq. (10.2)] results in the following WDF for
the rect function.

2x
{rect(Tk)}(x, k) = 2T(1 − 2T|k|) sinc (1 − 2T|k|) (10.17)
T
In Fig. 10.5a we illustrate this function. We see that it is bounded
in the frequency domain. We note that integrating this function along
the k axis results in sinc(x/T) while integrating along the x axis results
in rect(Tk). The PSD for this function is shown in Fig. 10.5b.

10.3 Finite Supports
10.3.1 Band-limitedness in Fourier Domain
If a signal u(x) is zero-valued outside of some finite range, that is,
u(x) = 0 for |x| < D, it is said to have compact support. If the FT of a
signal has compact support, i.e., if U(k) for |k| < B, where U(k) is the
FT of u(x), we say that u(x)is band-limited. Such a signal has a PSD
shown in Fig. 10.1. We furthermore refer to u(x) as having bandwidth
B. This concept of band-limitedness is very important in sampling