What's the difference between convolution and crosscorrelation?

What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing convolution?

The meanings of convolution and cross-correlation are quite different.

To see what happens in a simple setting, consider some independent integer valued random variables $X$ and $Y$ with respective distributions $p=(p_n)_n$ and $q=(q_n)_n$. Then:

• The convolution $p\ast q$ is the distribution $s=(s_n)_n$ defined by $$s_n=\sum\limits_kp_kq_{n-k}=P[X+Y=n]$$ for every $n$. Thus, $p\ast q$ is the distribution of $X+Y$.

• The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by $$c_n=\sum\limits_kp_kq_{n+k}=P[Y-X=n]$$ for every $n$. Thus, $p\circ q$ is the distribution of $Y-X$.

To sum up, considered at the level of distributions of independent random variables, the composition $\ast$ acts as an addition while the composition $\circ$ acts as a difference.