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Sorting multiple integrals

  1. Feb 6, 2013 #1
    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...
     
  2. jcsd
  3. Feb 6, 2013 #2

    haruspex

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    I don't think the argument to g is right.
     
  4. Feb 6, 2013 #3
    Hi, through you very much. But why? t-tau2 is from the convolution and (t+phi) because it is the 2nd operand for the x-correlation...
     
  5. Feb 6, 2013 #4

    haruspex

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    Actually, the left hand integral in the same line was also wrong. Both must mention phi:
    ##h(t) = (f*g)(t) = \int_{\tau=0}^t f(\tau)g(t-\tau).d\tau##
    ##(h*h)(t) = \int_\phi h(\phi)h(t-\phi).d\phi = \int_{\phi=0}^t \left(\int_{\tau=0}^t f(\tau)g(\phi-\tau).d\tau\right)\left(\int_{\tau=0}^{t-\phi} f(\tau)g(t-\phi-\tau).d\tau\right).d\phi ##
     
  6. Feb 6, 2013 #5

    Mute

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    Your second line should be ##(h \star h)(t)##, as it's a cross-correlation, so it's a plus ##\phi## in the integral arguments rather than a minus ##\phi##, and the limits are the real line:

    $$(h \star h)(t) = \int_{-\infty}^\infty d\phi~ h(\phi)h(t+\phi) = \int_{\phi=-\infty}^\infty d\phi~ \left(\int_{\tau_1=0}^t d\tau_1~f(\tau_1)g(\phi-\tau_1)\right)\left(\int_{\tau_2=0}^{t+\phi} d\tau_2 f(\tau_2)g(t+\phi-\tau_2)\right). $$
     
  7. Feb 6, 2013 #6

    haruspex

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    OK, I missed the distinction in the signs, thanks. So the error in the OP is that ##f\left(t-\tau_1\right)## should have been ##f\left(\phi-\tau_1\right)##
     
    Last edited: Feb 6, 2013
  8. Feb 7, 2013 #7
    Hi, thank you both of you! That's indeed a mistake.

    However, I just don't see it, maybe I miss the forest for the trees (aehm, integrals ...)

    In the end, I need to group both "g" and both "f" together into one integral.
    However f has distinct variables (tau1 and tau2).
    The g has even 4 (!) different variables: t, phi, tau1, tau2

    I tried substitution, but I just can't get rid of them ...
     
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