Page 342 - Phase Space Optics Fundamentals and Applications

Sampling and Phase Space 323

Input Signal LCT Output Signal
Finite Finite Finite Finite
Support Bandwidth A B C D Support Bandwidth
√
X = 0 = 0 = 0 = 0 X X
√
X = 0 = 0 = 0 = 0 X X
√
X X = 0 = 0 = 0 = 0 X
√
X X = 0 = 0 = 0 = 0 X
√ √
X = 0 X
X = 0 X
√ √
X = 0 X
X = 0 X
√ √
X = 0 X
X = 0 X
√ √
X = 0 X
X = 0 X

Note: If a signal is to have a finite support in some domain, it must belong to one
of the above possible sets.
TABLE 10.1 Properties of Finite Support and Band-limitedness for Linear
Canonically Transformed Signals

defined as follows:

x + k tan = b tan
2
(10.18)
x + k tan =−b tan
2
After we apply the LCT, there will occur a simple coordinate trans-
formation to the PSD shown in Fig. 10.6b. This coordinate shift is given
by Eqs. (10.8) and (10.9). In this domain the signal clearly has a finite
support in x equal to w. Clearly our two lines have now become
w
x =−
2 (10.19)
w
x =
2
After applying the coordinate shift to Eq. (10.19), we get

(A+ C tan )x + (B + D tan )k = b tan
2

(A+ C tan )x + (B + D tan )k = b tan (10.20)
2