Time-frequency smoothing interpretation The time-windowing function $h$ introduced in (4.18) or (4.19) attenuates interference components that oscillate in the frequency direction. To suppress interference terms oscillating in the time direction, we must smooth in that direction with a low-pass function $G$. The resulting distributions $$\tilde{P}_x^k(t,\nu)=\int_{-\infty}^{+\infty} G(u)\ \frac{\mu_k(u)}{\sqrt{\lambda_k(u)\lambda_k(-u)}}\ T_x(t,\lambda_k(u)\nu;\Psi)\hspace*{2cm}$$ $$\times T_x^*(t,\lambda_k(-u)\nu;\Psi)\ du,$$ (4.20) are called the smoothed pseudo affine Wigner distributions. It is important to notice that, like the (smoothed) pseudo Wigner-Ville case with the localization on linear chirps, (smoothed) pseudo affine Wigner distributions are no longer localized on power-law group delays. Nevertheless, as $Q$ (the quality factor of the wavelet $\Psi$) tends towards infinity and $G(u)$ to the all-pass function, this localization property is asymptotically recovered since $\tilde{P}_x^k$ converges to $P_x^k$. Besides, since (4.20) can be implemented efficiently, this convergence property provides us with an efficient-implementation approximation of any affine Wigner distribution (by considering the corresponding pseudo affine Wigner distribution with a large $Q$). Expression (4.20) is used in the function tfrspaw.m which computes these (smoothed) pseudo affine Wigner distributions.
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