Modified Wigner distribution function

The Wigner distribution (WD) was first proposed for corrections to classical statistical mechanics in 1932 by Eugene Wigner. The Wigner distribution, or Wigner-Vile Distribution (WVD) for analytic signals, also has applications in time frequency analysis. Compared to the short-time Fourier transform the Wigner distribution gives better auto term localisation compared to the smeared out STFT. However when applied to a signal with multi frequency components cross terms appear due to its quadratic nature. In 1994 L.Stankovic proposed a novel technique, now mostly referred to as S-method, resulting in the reduction or removal of cross terms.

Mathematical definition

The concept of the S-method is a combination between the STFT and the Pseudo Wigner Distribution (PWD), the windowed version of the WD.

• Wigner distribution

$W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau$

• Pseudo Wigner distribution

$W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau /2)w^{*}(-\tau /2)x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau$

• S-method

$SM(t,f)=\int _{-\infty }^{\infty }P(\theta )Y(t,f+\theta /2)Y^{*}(t,f-\theta /2)d\theta$

$where\ Y(t,f)=\int _{-\infty }^{\infty }w(\tau )x(t+\tau )e^{-j2\pi f\tau }d\tau \ Is\ the\ STFT\$

$P(\theta )$ Is a windowing function in the frequency domain resulting in the cross term removal.

See also

• Time-frequency representation
• short-time Fourier transform
• Gabor transform
• Wigner distribution function

References

• L. Stankovic, “A Method for Time-Frequency Signal Analysis”, IEEE Trans. on Signal Processing, vol. 42, no. 1, Jan. 1994