Page 54 - Introduction to Information Optics

1.5. Signal Analysis 39

1.5.5 WIGNER DISTRIBUTION

There is another form of time-frequency signal representation, similar to
ambiguity function, defined by

f i2 vt f 2nv>t
W(r, v) = u(t}u*(x -t)e~ * dt = U*(v')U(v - v>'' dv', (1.127)

which is known as the Wigner distribution function. Instead of using the
correlation operator [i.e., u(t)u*(t + T)], Wigner used the convolution operator
[i.e., M(£)M*(T — ?)] for his transformation. The physical properties of the
Wigner distribution function (WDF) can be shown as

2
W(t, v)dv = \u(t)\ , (1.128)

2
W(T,v)dT; = \U(v)\ , (1.J29)

W(t,v)dtdv = l, (1.130)

in which the WDF has been normalized to unity for simplicity.
One of the interesting features of the WDF is the inversion. Apart from the
association with a constant phase factor, the transformation is unique, as can
be shown,

W(t, v)e~ i4nvt dt = U(2v)U*(Q), (1.131)

(r, v)e i4nvt dv = w(2t)M*(0). (1.132)

If the constant phase factors u*(0) or U*(0) happen to be very small or even
zero, then we can reformulate the transformations by using the maximum value
2
2
of |tt(t)| or |C/(t)| , which occurs at r max, or v max , respectively, as given by,
W(T, v)exp[-47r(v - v max)T]Jr = U(2v - v max)l/*(v max), (1.133)

W(T, V) eXp[/47lv(T - T max)] dv = U(2l - T ma>*(T max). (1.134)