Page 347 - Phase Space Optics Fundamentals and Applications

328 Chapter Ten

10.4.2 Generalized Sampling
We now consider signals that have finite support in some LCT domain
with parameters ABCD. In the case of these signals it is possible to
derive a rigorous sampling theory that can be interpreted and derived
in the simplest possible way by employing phase-space diagrams. We
take the signal u(x) with PSD shown in Fig. 10.6a. We define this signal
to have a finite support w in some LCT domain with parameters ABCD
and to have local bandwidth b. From the third row of Table 10.1 we
know that u(x) must have an infinite bandwidth and an infinite spatial
support. Therefore Nyquist-Shannon sampling cannot be applied in
the conventional sense. If this signal can be sampled at some rate
T and reconstructed from these samples u(nT), we must derive a
new sampling criterion and a new interpolation formula. In fact we
already have derived both. We refer the reader to Sec. 10.2.5.1 where
2
we discussed the sheared comb function exp( j x ) T (x). The PSD
for this function is given in Fig. 10.4, where = 1/ tan(− ). The WDF
of this sheared-sampled signal is given by

2 k
u T (x) exp( j x ) (x, k) = {u(x)}(x, k) ∗
2
T (x) exp( j x ) (x, k) (10.29)
If we match up the values of , then the convolution along k of
the PSD of u(x) shown in Fig. 10.6 and the sheared comb function
that result will be an infinite number of replicas of a band-limited
signal with bandwidth b similar to that shown in Fig. 10.7b.From
Eq. (10.21) matching up the values of , this implies that for the comb
function = 1/ tan(− ) = D/B. When b = 1/T, we get the PSD
shown in Fig. 10.7b, where we can see that the replicas are just shy of
overlapping one another. To avoid aliasing, we must have T ≤ 1/b.
From Eq. (10.21) this forces the following condition: The signal may be
reconstructed by multiplying the resultant signal by Trect(Tk) in the
Fourier domain. This step in the reconstruction is identical to Shannon
interpolation described by Eq. (10.28).
2 2 x
{u(x) exp( j x )}(x, k) = {u T (x) exp( j x )}(x, k) ∗
{rect(Tk)}(x, k) (10.30)
The final part of the reconstruction is accomplished by multiply-
ing by the conjugate of the original shear. The overall reconstruction
algorithm is given by

D 2 D 1 x
2
u(x) = exp − j x u T (x) exp j x ∗ sinc
B B T T
∞
D , 2 D x − nT
2
= exp − j x u(nT) exp j (nT) sinc
B B T
n=−∞
(10.31)