We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following:
Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$
Correlation: $(f(t)\star g(t))(\tau) = \int_{-\infty}^\infty{f^\ast(t)g(\tau+t)dt}$
In the english wikipedia and in other sources I found that the following relationship should hold:
$(f(t)\star g(t))(\tau) = (f^\ast(-t)\ast g(t))(\tau)$
Is this correct? If so, how can i prove this? Usually, i would try substitution, but how to change the $g(\tau+t)$ to $g(\tau-t)$?
Best Answer
I figured out the answer while writing down the question. Here it is:
$(f^\ast(-t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f^\ast(-t)g(\tau-t)dt} = -\int_{\infty}^{-\infty}{f^\ast(t)g(\tau+t)dt} = \int_{-\infty}^{\infty}{f^\ast(t)g(\tau+t)dt} = (f(t)\star g(t))(\tau)$
In the second step, the substitution $t\to -t$ took place.