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–Ville 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

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

$\text{where }Y(t,f) = \int_{-\infty}^\infty w(\tau) x(t+\tau) e^{-j2\pi f\tau} \, d\tau\text{ is the STFT}.$

P(θ) 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