Modified Wigner distribution function

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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 refered to as S-method, resulting in the reduction or removal of cross terms.

[edit] 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(θ) Is a windowing function in the frequency domain resulting in the cross term removal.

[edit] See Also


[edit] References

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