Page 128 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 109

which also can be defined in terms of the Fourier transform (FT) of
the original signal

+∞

F{ f (x), }= F( ) = f (x) exp (−i2 x) dx (4.2)
−∞
as

+∞

W f (x, ) = F + F ∗ − exp (i2 x) d (4.3)
2 2
−∞
It is interesting to remember that any WDF can be inverted to recover,
up to a phase constant, the original signal or, equivalently, its Fourier
transform. The corresponding inversion formulas are 3

+∞

1 x + x

f (x) = W f , exp [i2 (x − x )] d (4.4)
f (x ) 2

∗
−∞
+∞

1 +

F( ) = W f x, exp [−i2 ( − )x] dx (4.5)
F ( ) 2

∗
−∞
Note that these equations state the uniqueness of the relationship be-
tween the signal and the corresponding WDF (except for a phase con-
stant). It is straightforward to deduce from these formulas that the
integration of the WDF on the spatial or spatial-frequency coordinate
leads to the modulus square of the signal or its Fourier transform,
respectively, i.e.,
+∞

2
| f (x)| = W f (x, ) d (4.6)
−∞
+∞

2
|F( )| = W f (x, ) dx (4.7)
−∞
These integrals, or marginals, can be viewed as the projection of the
function W f (x, ) in phase space along straight lines parallel to the
axis [in Eq. (4.6)] or to the x axis [in Eq. (4.7)]. These cases are particular
ones of all possible projections along straight lines of a given function
in phase space. In fact, for any function of (at least) two coordinates,