Draft:Causal wavelet

• Comment: Fails WP:GNG, requires significant coverage in multiple independent secondary sources. Dan arndt (talk) 06:50, 5 January 2023 (UTC)

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In mathematics, a causal wavelet is a continuous wavelet used in the real time continuous wavelet transform. The causal mother wavelet is defined as:

$h(t)=\psi (t)I_{t\geq 0}=e^{-{\frac {t^{2}}{2}}}e^{j2\pi f_{0}t}I_{t\geq 0}$

where $\psi (t)$ is a Morlet wavelet.

Hence, the Fourier transform of causal mother wavelet

$|H(\omega )|^{2}={\frac {4}{4(\omega -2\pi f_{0})+1}}$

and satifies the Admissibility Criterion $\int _{0}^{\infty }{\frac {|H(\omega )|^{2}}{\omega }}\,d\omega <\infty$, then the causal wavelet transform is reversible. Furthermore, we can observe that $|H(\omega )|$ reach the maximum value at $\omega =2\pi f_{0}$. Therefore, when the $f_{0}$ is high, the convolution of causal wavelet is a high pass filter and vice versa. While we usually chose $f_{0}={\frac {5}{2\pi }}$ for a Morlet wavelet, hence we have

$h(t)=e^{-{\frac {t^{2}}{2}}}e^{j5t}I_{t\geq 0}$ and the real form $h(t)=e^{-{\frac {t^{2}}{2}}}\cos {5t}I_{t\geq 0}$.

Moreover, we define the causal wavelet transform as

$W_{h}[x(t)](a,b)={\frac {1}{\sqrt {b}}}\int _{-\infty }^{\infty }x(t){\overline {h({\frac {t-a}{b}})}}\,dt=\int _{-\infty }^{\infty }x(t){\overline {h_{(a,b)}(t)}}\,dt$

where $h_{(a,b)}(t)={\frac {1}{\sqrt {b}}}h({\frac {t-a}{b}})$ is called the daughter wavelet of the causal wavelet.

Simulation

Simulation on causal signal

Consider a causal signal $x(t)$, which there is a value $t_{0}$ such that $x(t)=0,\forall t<t_{0}$, for example

$x(t)=\sin {\frac {5(t-512)}{16}}\cos {\frac {-(t-512)}{32}}I_{t\geq 512}$

and we define the causal mother wavelet as $h(t)=e^{-{\frac {t^{2}}{2}}}\cos {5t}I_{t\geq 0}$.

Moreover, we define the inner product in $L^{2}(\mathbb {R} )$

$\langle f(t),g(t)\rangle =\int _{-\infty }^{\infty }f(t)g(t)\,dt$, for $f(t),g(t)\in L^{2}(\mathbb {R} )$.

Then, we can define the Signal-to-noise ratio (SNR) w.r.t. the wavalet $\psi (t)$ as

$SNR_{\psi }(x(t))=\mathop {\max } _{a,b}{\frac {\int _{-\infty }^{\infty }|x(t)|^{2}\,dt}{\int _{-\infty }^{\infty }|x(t)-\psi _{(a,b)}(t)|^{2}\,dt}}=\mathop {\max } _{a,b}{\frac {\langle x(t),x(t)\rangle }{\langle x(t),x(t)\rangle +\langle \psi _{(a,b)}(t),\psi _{(a,b)}(t)\rangle -2\langle x(t),\psi _{(a,b)}(t)\rangle }}$.

We can see that the causal wavelet $h(t)$ is always better than the Morlet wavelet $g(t)$ for the SNR of the causal signal $x(t)$.

Since $\mathop {\arg \max } _{a,b}{\frac {\langle x(t),x(t)\rangle }{\langle x(t),x(t)\rangle +\langle \psi _{(a,b)}(t),\psi _{(a,b)}(t)\rangle -2\langle x(t),\psi _{(a,b)}(t)\rangle }}=\mathop {\arg \max } _{a,b}\{2\langle x(t),\psi _{(a,b)}(t)\rangle -\langle \psi _{(a,b)}(t),\psi _{(a,b)}(t)\rangle \}$

and $\langle h_{(a,b)}(t),h_{(a,b)}(t)\rangle <\langle g_{(a,b)}(t),g_{(a,b)}(t)\rangle$ for the same $a,b$, so as $\langle x(t),h_{(a,b)}(t)\rangle =\langle x(t),g_{(a,b)}(t)\rangle$ for the same $a,b$,

we get $SNR_{h}(x(t))>SNR_{g}(x(t))$.