Derivation Recall from section 4.1.1 that we obtained the pseudo Wigner-Ville distribution by introducing a window function into the Wigner-Ville distribution. An analogous windowing operated on the affine Wigner distributions (4.17) leads to the pseudo affine Wigner distributions. But in contrast to the pseudo Wigner-Ville case, this windowing must be frequency-dependent, to ensure that the resulting time-scale distribution remains scale-covariant. As a result, the smoothing in frequency is constant-Q, rather than constant-bandwidth as in the pseudo Wigner-Ville distribution. The general expression of this new class of distributions, expressed in the time-domain, writes: $$\tilde{P}_x^k(t,\nu)=\nu\ \int_{-\infty}^{+\infty} \mu_k(u)\ \le... ...h[\nu \lambda_k(u) (\tau-t)]\ e^{-j2\pi\lambda_k(u)\nu (\tau-t)}\ d\tau\right]$$ $$\times\left[\int_{-\infty}^{+\infty} x(\tau_p)\ h[\nu \lambda_k(-u) (\tau_p-t)]\ e^{-j2\pi\lambda_k(-u)\nu (\tau_p-t)}\ d\tau_p\right]^* du\ \$$ (4.18) where $h$ is the time-windowing function. By analogy with the pseudo Wigner-Ville distributions, we call these distributions the pseudo affine Wigner distributions. An efficient online implementation can be obtained if we reorder (4.18) to yield $$\tilde{P}_x^k(t,\nu)=\int_{-\infty}^{+\infty} {\mu_k(u)\over\sqrt... ...mbda_k(-u)}}\ T_x(t,\lambda_k(u)\nu;\Psi)\ T_x^*(t,\lambda_k(-u)\nu;\Psi)\ du,$$ (4.19) where $T_x(t,\nu;\Psi)$ is the continuous wavelet transform (see section 3.2.1), and $\Psi(\tau)=h(\tau)\ e^{j2\pi \tau}$ is a bandpass wavelet function.
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