<Dd> f ⋅ g = F − 1 (F (f) ∗ F (g)) (\ displaystyle f \ cdot g = (\ mathcal (F)) ^ (- 1) (\ big \ () (\ mathcal (F)) \ (f \) * (\ mathcal (F)) \ (g \) (\ big \))) </Dd> <P> By similar arguments, it can be shown that the discrete convolution of sequences x (\ displaystyle x) and y (\ displaystyle y) is given by: </P> <Dl> <Dd> x ∗ y = D T F T − 1 (D T F T (x) ⋅ D T F T (y)), (\ displaystyle x * y = \ scriptstyle (\ rm (DTFT)) ^ (- 1) \ displaystyle (\ big () \ scriptstyle (\ rm (DTFT)) \ displaystyle \ (x \) \ cdot \ \ scriptstyle (\ rm (DTFT)) \ displaystyle \ (y \) (\ big)),) </Dd> </Dl> <Dd> x ∗ y = D T F T − 1 (D T F T (x) ⋅ D T F T (y)), (\ displaystyle x * y = \ scriptstyle (\ rm (DTFT)) ^ (- 1) \ displaystyle (\ big () \ scriptstyle (\ rm (DTFT)) \ displaystyle \ (x \) \ cdot \ \ scriptstyle (\ rm (DTFT)) \ displaystyle \ (y \) (\ big)),) </Dd>

Fourier transform of a product is a convolution