njit-etd2003-081 - New Jersey Institute of Technology

where ρw (t, f), z(t), and z * (t) are the Wigner distribution, an analytic signal and the
complex conjugate of the analytic signal respectively. From equation (3.36) one see that
for each particular time one are adding up pieces made from the product of the signal at
a past time multiplied by the signal at an equal future time. The Wigner distribution
satisfies many properties, which are described as follows:
The WD is a real valued function that is:
Since the kernel of the WD is one for any value of v and r, then the complex
conjugate of the kernel is always one. That is;
This is the constraint of the kernel for the distribution to be real.
The WD satisfies the time and frequency shift properties as long as the kernel of
the WD is not a function of time and frequency that is;
where z(t),s(t),ρz(t, f) and ρs(t,f) are the signal, the shifted signal, the distribution of
signal and the distribution of the shifted signal respectively. To prove this property, one