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Math Help - Cross Correlation and Convolution

  1. #1
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    Cross Correlation and Convolution

    I'm trying to follow a simple textbook explanation of cross correlation:

    Where \star indicates cross correlation, \otimes indicates convolution, and f^* would be the complex conjugate of f:

    f \star g = f^*(-t) \otimes g = \int_{-\infty}^\infty f^*(-\tau)g(t - \tau) \, d\tau

    Substituting \tau' = -\tau, we have:

    = \int_{-\infty}^\infty f^*(\tau')g(t + \tau') \, (-d\tau')

    Then, the next step is where I'm confused. How do you get from the last step to:

    = \int_{-\infty}^\infty f^*(\tau)g(t + \tau) \, d\tau
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  2. #2
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    Re: Cross Correlation and Convolution

    There's an error in the previous step. Making the variable substition \tau' = -\tau will also invert the sign of the limits, so switching them around will cancel the minus sign.
    Thanks from VinceW
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  3. #3
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    Re: Cross Correlation and Convolution

    Quote Originally Posted by Ridley View Post
    There's an error in the previous step. Making the variable substition \tau' = -\tau will also invert the sign of the limits, so switching them around will cancel the minus sign.
    Of course! Thanks!
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