# Thread: Cross Correlation and Convolution

1. ## Cross Correlation and Convolution

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

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

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

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

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

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

$\displaystyle = \int_{-\infty}^\infty f^*(\tau)g(t + \tau) \, d\tau$

2. ## Re: Cross Correlation and Convolution

There's an error in the previous step. Making the variable substition $\displaystyle \tau' = -\tau$ will also invert the sign of the limits, so switching them around will cancel the minus sign.

3. ## Re: Cross Correlation and Convolution

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