Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms. Let f {displaystyle f} and g {displaystyle g} be two functions with convolution f ∗ g {displaystyle f*g} . (Note that the asterisk denotes convolution in this context, not standard multiplication. The tensor product symbol ⊗ {displaystyle otimes } is sometimes used instead.) In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Versions of the convolution theorem are true for various Fourier-related transforms. Let f {displaystyle f} and g {displaystyle g} be two functions with convolution f ∗ g {displaystyle f*g} . (Note that the asterisk denotes convolution in this context, not standard multiplication. The tensor product symbol ⊗ {displaystyle otimes } is sometimes used instead.) If F {displaystyle {mathcal {F}}} denotes the Fourier transform operator, then F { f } {displaystyle {mathcal {F}}{f}} and F { g } {displaystyle {mathcal {F}}{g}} are the Fourier transforms of f {displaystyle f} and g {displaystyle g} , respectively. Then where ⋅ {displaystyle cdot } denotes point-wise multiplication. It also works the other way around: By applying the inverse Fourier transform F − 1 {displaystyle {mathcal {F}}^{-1}} , we can write: