The Time scaling properties of convolution states that if we time scale both the inputs of the convolutions by , the convolution result we get is also time scaled by (and multiplied by ).

In other word, if , then

Proof

First we apply the definition of convolution

If we put in instead of as input, we get

Then we can substitute in (). We have two cases for and .

When , when , and

When , when , and

Combine the results, we get that in both cases: