- #1
divB
- 87
- 0
Hi,
I shall show the following:
[tex]
(f*g) \star (f*g) = (f\star f)*(g\star g)
[/tex]
where [itex]*[/itex] denotes convolution and [itex]\star[/itex] cross-correlation. Writing this in terms of integral & regrouping:
[tex]
\int_{\phi} \left(\int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1\right) \cdot \left(\int_{\tau_2} f(\tau_2) g(t+\phi-\tau_2) d\tau_2\right) d\phi \\
= \int_{\tau_2} \int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1 \int_{\phi} f(\tau_2) g(t+\phi-\tau_2) d\tau_2 d\phi
[/tex]
But now I am stuck. How should I bring both f into one integral? Both are functions of a differerent variable and [itex]\int f(x)dx \cdot \int f(x)dx \neq \int f(x)\cdot f(x) dx[/itex]...
Thanks for any pointer...
I shall show the following:
[tex]
(f*g) \star (f*g) = (f\star f)*(g\star g)
[/tex]
where [itex]*[/itex] denotes convolution and [itex]\star[/itex] cross-correlation. Writing this in terms of integral & regrouping:
[tex]
\int_{\phi} \left(\int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1\right) \cdot \left(\int_{\tau_2} f(\tau_2) g(t+\phi-\tau_2) d\tau_2\right) d\phi \\
= \int_{\tau_2} \int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1 \int_{\phi} f(\tau_2) g(t+\phi-\tau_2) d\tau_2 d\phi
[/tex]
But now I am stuck. How should I bring both f into one integral? Both are functions of a differerent variable and [itex]\int f(x)dx \cdot \int f(x)dx \neq \int f(x)\cdot f(x) dx[/itex]...
Thanks for any pointer...