njit-etd2003-081 - New Jersey Institute of Technology

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3.6.3 The Choi-Williams (Exponential) Distribution
Choi and Williams presented a new approach where they address the main draw back of
the Wigner distribution (cross terms)[40]. They used a generalized ambiguity function
[40] and chose an exponential kernel, that is:
one obtains:
Substituting equation (3.53) in equation (3.36) and integrating with respect to v
where pew (t, f),z(t),z * (t) are the Choi-Williams distribution, an analytical signal and
complex conjugate of the analytical signal. The ability to suppress the cross terms
comes by controlling a .
In Figure 3.3, three cases were presented, where each contains two sine waves
with frequencies of 100 and 400 hertz. The Choi Williams distribution was performed
for the three cases but with different values for a . Note, in case "a", a is 10000, which
makes the kernel equal to one, and the Wigner distribution is obtained. In case "b" and
"c", a is 50 and 1 respectively which produces a kernel, which is peaked, near the origin
in the v, r plane and hence offers better cross term suppression.
Hence one can control the relative suppression of the cross terms by reducing the
value of a . The Choi Williams distribution satisfies many of the desirable properties, as
described below:
1. The Choi Williams distribution is real. To prove this, one can replace the v, r with
— v,—r respectively in to the kernel function and perform the manipulation: