Page 131 - Phase Space Optics Fundamentals and Applications

112 Chapter Four

By performing a proper change in the integration variables, the fol-
lowing more compact expression can be obtained

RW f (x , )
⎧ 2
+∞
⎪ x 2 x x
⎪ 1 f (x) exp i exp −i2
⎪ dx for = 0,
⎪ sin tan sin 2
⎨
−∞
= 2
⎪| f (x 0 = x)| for = 0
⎪
⎪
⎪
⎩ 2
F x /2 = for =
2
(4.14)
From this equation it is clear that
RW f (x , ) ≥ 0 (4.15)
This is a very interesting property, since the WDF cannot be positive
in whole phase space (except for the particular case of a Gaussian
signal). Note also that from Eq. (4.14) a symmetry condition can be
stated, namely,
RW f (x , − ) = RW f (−x , ) (4.16)
∗
so that for real signals, that is, f (x) = f (x)∀x ∈ R, one finds
∗
RW f (x , − ) = RW f (−x , ) (4.17)

and, therefore, for this kind of signal the reduced domain ∈ [0, )
in the Radon transform is clearly redundant. In this case, the range
∈ [0, /2] contains in fact all the necessary values for a full definition
of the RWT.
Equation (4.14) also allows one to link the RWT with another inte-
gral transform defined directly from the original signal, namely, the
fractional Fourier transform (FrFT). This transformation, often con-
sidered a generalization of the classic Fourier transform, is given by
p
F { f (x), }
⎧
⎪ exp[i( + /tan )] +∞
2

⎪ √ f (x)
⎪
⎪ i sin
⎪
⎪ −∞
⎨ 2
× exp i x exp −i2 x dx for = 0,
= F p ( ) = tan sin 2
⎪
⎪ f ( ) for = 0
⎪
⎪
⎪
⎪
⎩
F( ) for =
2
(4.18)