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I am having trouble calculating a convolution of functions composition.

Given $x(t)$ and $h(t)$, compute the convolution of $2x(2t) \star h(2t) $

I tried the following, but I'm not sure if it's correct:

Set $g(t) \stackrel{\tiny{\text{def}}}{=} 2x$, $s \stackrel{\tiny{\text{def}}}{=} 2t$. So now we need to calculate $g(s) \star h(s)$:

$\displaystyle { g(s) \star h(s) = \int_{-\infty}^{+\infty}g(\tau)h(s-\tau)d\tau \underset {s=2t}{=} \int_{-\infty}^{+\infty}g(\tau)h(2t-\tau)d\tau }$

Please advise. Any help will be appreciated.

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1 Answer 1

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Consider : $$F(t)=x(t) \star h(t) \longrightarrow F(\omega)= x(\omega)h(\omega)$$ $$G(t)=x(2t) \star h(2t) \longrightarrow \dfrac{1}{2}x( \dfrac{\omega}{2} )\dfrac{1}{2}h( \dfrac{\omega}{2})$$ $$\dfrac{1}{2}x( \dfrac{\omega}{2} )\dfrac{1}{2}h( \dfrac{\omega}{2})= \dfrac{1}{4} F(\dfrac{\omega}{2} ) $$ $$\dfrac{1}{2}F(2t) \longleftarrow \dfrac{1}{4} F(\dfrac{\omega}{2} ) $$ $$G(t)= \dfrac{1}{2}F(2t) $$

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