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I know that the definition of convolution is the following:

$$ f(t) * g(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \mathrm d \tau $$

Then, which is the correct one between the two:

$$ f(t) * g(-t) = \int_{-\infty}^{\infty} f(\tau) g(\tau + t) \mathrm d \tau \qquad (1) $$

$$ f(t) * g(-t) = \int_{-\infty}^{\infty} f(\tau) g(\tau - t) \mathrm d \tau \qquad (2) $$

I need the explanation, too.

Naetmul
  • 282

1 Answers1

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Let $h(t)=g(-t)$. Then the convolution becomes $$\int_{-\infty}^\infty f(\tau)h(t-\tau)d\tau=\int_{-\infty}^\infty f(\tau)g(\tau-t)d\tau$$ This is because $h(t)=g(-t)\implies h(t-\tau)=h(t')=g(-t')=g(\tau-t)$

Samrat Mukhopadhyay
  • 16,277