Page 130 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 111

and therefore Eq. (4.8) can be reformulated as
⎧
+∞

⎪
⎪ x x
⎪
⎪ g x, − dx for = 0,
⎪
⎪ sin tan 2
⎪
⎪
⎪ −∞
⎪
⎪
⎪ +∞
⎪
⎨
R g (x , ) = g (x ,y) dy for = 0 (4.11)
⎪
⎪
⎪ −∞
⎪
⎪
⎪ +∞
⎪
⎪
⎪
⎪
⎪ g (x, x ) dx for =
⎪
⎪ 2
⎩
−∞
Thus, when we consider as projected function W f (x, ), we can define
the generalized marginals as the Radon transform of this WDF, namely,
+∞

(x , ) =
R{W f (x, ),x , }= R W f W f (x, ) d
−∞
+∞

= W f (x cos − sin ,x sin + cos ) d
−∞
⎧
+∞

⎪
⎪ x x
⎪
⎪ W f x, − dx for = 0,
⎪
⎪ sin tan 2
⎪
⎪
⎪ −∞
⎪
⎪
⎪ +∞
⎪
⎨
= W f (x , ) d for = 0
⎪
⎪
⎪ −∞
⎪
⎪
⎪ +∞
⎪
⎪
⎪
⎪
⎪ W f (x, x ) dx for =
⎪
⎪ 2
⎩
−∞
(4.12)
where, in the last expression, we have explicitly considered the equa-
tions for the integration lines in the projection. In terms of the original
signal, this transform is called its Radon-Wigner transform. It is easy
to show that
+∞ +∞

x
(x , ) = RW f (x , ) = f x cos − sin +
R W f
2
−∞ −∞

x
× f ∗ x cos − sin −
2
× exp[−i2 (x sin + cos )x ] dx d (4.13)