Writing convolution in other terms

Question

Could I get some help with this question. I just started convolution and I'm still not clear on a lot of things

Answer 1

Use the change of variables, let $x=-\tau-b$, then we have $\tau=-x-b$ and $dx=-d\tau$. Hence

$ \begin{eqnarray*} \nu(t)&=&\int_{-\infty}^\infty\chi(x) h(-x-b+at)\ dx\\ &=&y(at-b). \end{eqnarray*}$