Language Reference

TFPWV (x, window <, overlap> <, fftlen> <, center> <, nthreads> <, hilbert_tsf> ) ;

This function is supported by the IML procedure and the iml action.

The TFPWV function computes the pseudo-Wigner-Ville (PWV) transform of the input vector x. The function returns a three-column matrix that has rows, where

Here series_length is the dimension of the argument x, and window_length is the dimension of the argument window.

The three columns of the resulting output are as follows:

The input arguments are as follows:

The pseudo-Wigner-Ville distribution is a modification of the Wigner-Ville distribution. The Wigner-Ville distribution of a continuous time series is obtained by computing the Fourier transform of for fixed t as varies. So the Wigner-Ville distribution of a continuous, possibly complex-valued, time series is given by :

It can be shown that the is real even when takes complex values.

The pseudo-Wigner-Ville distribution is obtained by adding an additional term in the defining integral. The pseudo-Wigner-Ville distribution of a continuous time series is given by

where is a window function.

The Wigner distribution of discrete time series is defined as follows (Claasen and Mecklenbräuker 1980b, 1980a; Debnath 2002):

From the preceding formula, it follows that is the discrete-time Fourier transform of and provides the basis for the computation here.

Given an input time series , the computation can be considered to be the evaluation of a function , which measures the value of the pseudo-Wigner-Ville distribution at time n and frequency for different values of n and f. Now can be defined: given a possibly complex-valued time series and a window of odd length , where window=(, define

The preceding summation is performed with the following convention: any term in the summation for which both and do not lie between 0 and series_length – 1 is replaced with 0.

If and , the output of the PWV function is determined by the evaluation of the complex value for and for
. If u denotes the real part of , then a row of output consists of the triplet of values as f varies over and n varies over .

The pseudo-Wigner-Ville distribution has some undesirable properties. It displays annoying artifacts for multicomponent time series (Cohen 1995), but you can get better results by replacing the input with the analytic signal that corresponds to the input (Boashash 1988). For this reason, the TFPWV function provides the hilbert_tsf option, which replaces the original series with its analytic signal before computation, and the center option, which removes the mean from the series so that an overall mean effect does not show up in the output. Before any computation, the input series is first transformed depending on the value of the hilbert_tsf and center parameters. First the value of the center parameter is checked; if it is 1, then the input series is replaced by the centered series that is obtained by subtracting the series mean from each term of the series. If the value of the hilbert_tsf parameter is also 1, this possibly centered series is replaced by the analytic signal that corresponds to the centered input.

The following statements demonstrate the TFPWV function:

x = 1:10;
window_len=3; overlap=0; fftlen=4;
window = tfwindow(window_len);              /* Hanning window */
output = tfpwv(x, window, overlap, fftlen);
print output[colname={"f" "t" "pwv"}];