Page 132 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 113

where = p /2. From this definition, it is easy to see that
2
RW f (x , ) =|F 2 / (x )| (4.19)
so that the RWT can be also interpreted as a two-dimensional repre-
sentation of all the FrFTs of the original function.
Another interesting relationship can be established between the
RWT and the AF associated with the input signal. For our input signal
the AF is defined as

A{ f (x), ,x }= A f ( ,x )

+∞
x x

= f x + f ∗ x − exp(−i2 x) dx (4.20)
2 2
−∞
which can be understood as the two-dimensional FT of the WDF, i.e.,
+∞ +∞

F 2D {W f (x, ), ,x }= W f (x, ) exp [−i2 ( x + x )] dx d
−∞ −∞

= A f ( , −x ) (4.21)
There is a well-known relationship between the two-dimensional FT
of a function and the one-dimensional Fourier transformation of its
projections. This link is established through the central slice theorem,
which states that the values of the one-dimensional FT of a projection
at an angle give a central profile—or slice—of the two-dimensional
FT of the original signal at the same angle. If we apply this theorem
to the WDF, it is straightforward to show that

F{RW f (x , ), }= A f ( cos , − sin ) (4.22)
i.e., the one-dimensional FT of the RWT for a fixed projection angle
provides a central profile of the AF A f ( ,x ) along a straight line

forming an angle − with the axis. These relationships together
with other links between representations in the phase space are sum-
marized in Fig. 4.2.
To conclude this section, we consider the relationship between the
RWTofaninputone-dimensionalsignal f (x) andtheRWTofthesame
signal but after passing through a first-order optical system. In this
case, the input signal undergoes a canonical transformation defined
through four real parameters (a, b, c, d) or, equivalently, by a 2×2 real
matrix

a b
M = (4.23)
c d