Defining the wavelet bispectrum (Q2659734)

As discussed in the paper, bispectral analysis is an effective signal processing tool for investigating interactions between oscillations. For an admissible wavelet function \(\psi\) such that \(\hat{\psi}\ge 0\), \(\hat{\psi}\) vanishes on \((-\infty,0]\) and \(\int_0^\infty \frac{\hat{\psi}(r)}{r}dr<\infty\), the continuous wavelet transform of any real-valued signal \(x\in L^2(\mathbb{R})\) is defined to be \[ W_{x}(f,t)=W_{\psi,\kappa,x}(f,t)=\frac{f}{\kappa} \int_{\mathcal{R}} x(\tau) \overline{\psi\left(\frac{(\tau-t)f}{\kappa}\right)} d\tau, \] where \(f>0\) is the frequency, \(t\in \mathbb{R}\), and \(\kappa\) is a parameter. The original wavelet bispectrum defined in [\textit{B. V. van Milligen} et al., ``Nonlinear phenomena and intermittency in plasma turbulence'', Phys. Rev. Lett. 74, No. 3, 395--398 (1995), \url{doi:10.1103/PhysRevLett.74.395}; ``Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas, 2, No. 8, 3017--3032 (1995), \url{doi:10.1063/1.871199}] is given by \[ B^I_{xyz}(s_1,s_2):=\int_T^{T+\Delta I} W_x(s_1,t) W_y(s_2, t) \overline{ W_z((s_1^{-1}+s_2^{-1})^{-1},t)} dt, \] where \(x\), \(y\), \(z\) are finite energy signals and \(I=[T, T+\Delta]\) is a time interval. The paper argued in Section 1 that the original wavelet bispectrum is not quantitative of its bispectral content of an area of scale-scale space and is merely qualitative. Therefore, the authors introduced another definition of a wavelet bispectrum in Section 3.1 that for \(\lambda\in [0,1]\), \[ D_\psi(\lambda):=\int_{-\infty}^\infty \int_{-\infty}^\infty \mathring{\psi}(e^{r_1})\mathring{\psi}(e^{r_2}) \mathring{\psi}(\lambda e^{r_1}+(1-\lambda)e^{r_2}) dr_1 dr_2, \] where \(\mathring{\psi}(f):=\hat{\psi}(\frac{1}{f})\) and \(\hat{\psi}(f):=\int_{-\infty}^{\infty} \psi(\tau) e^{-2\pi i f \tau} d\tau\) is the Fourier transform of \(\psi\) with \(f\) being the frequency variable. The authors observed in identity (44) that \[ \frac{1}{8} A_1 A_2 A_3 e^{i\theta}= \frac{1}{D_\psi(\lambda)} \int_{\mathbb{R}^2} W_{\psi,\kappa,x}(e^{\zeta_1},t) W_{\psi,\kappa,y}(e^{\zeta_2},t) \overline{W_{\psi,\kappa,z}(e^{\zeta_1}+e^{\zeta_2},t)} d(\zeta_1,\zeta_2), \] where \(x(t)=A_1 \cos(2\pi \lambda \nu t+\phi_1), y(t)=A_2\cos(2\pi (1-\lambda)\nu t+\phi_2)\) and \(z(t)=A_3\cos(2\pi \nu t+\phi_1+\phi_2-\theta)\) are signals with constant frequencies. Then the authors defined in Definition~4 the logarithmic-frequency wavelet bispectral density function \(b_{\psi,\kappa,xyz}: (0,\infty)^2\times \mathbb{R}\rightarrow \mathbb{C}\) in Eq. (45) by \[ b_{\psi,\kappa,xyz}(f_1,f_2,t)=D_\psi\left( \frac{f_1}{f_1+f_2}\right)^{-1} W_{\psi,\kappa,x}(f_1,t) W_{\psi,\kappa,y}(f_2,t)\overline{W_{\psi,\kappa,z}(f_1+f_2,t)}, \] for finite energy signals \(x\), \(y\), \(z\). The new wavelet bispectrum is its associated measure given in Eq. (47) as follows: \[ \int_A \frac{b_{\psi,\kappa,xyz}(f_1,f_2,t)}{f_1f_2} d(f_1,f_2,t), \] where \(A\subseteq \mathbb{R}^3\) is a measurable set. Then the authors justified the new definition of wavelet bispectrum in Theorem 8 by showing that the new definition of wavelet bispectrum matches the traditional bispectrum of sums of sinusoids, in the limit as the frequency resolution tends to infinity. The authors also provided a few numerical examples to illustrate the advantages of the newly defined wavelet bispectrum in Section 5 by using a particular admissible wavelet \(\psi\) given by \(\hat{\psi}_\sigma(r):=e^{-2(\pi \sigma \log r)^2}, r>0\) with a parameter \(\sigma>0\).