Fourier transform property: Convolution

Statement: if $g_1(t)\leftrightarrow G_1(f)$ and $g_2(t)\leftrightarrow G_2(f)$, then

$$g_1(t)*g_2(t)\leftrightarrow G_1(f)\,G_2(f),\qquad g_1(t)\,g_2(t)\leftrightarrow G_1(f)*G_2(f)$$

where $g_1(t)*g_2(t)=\int_{-\infty}^{\infty}g_1(\tau)g_2(t-\tau)d\tau$.

Explanation: convolution in the time domain becomes simple multiplication in the frequency domain (time-convolution theorem), and multiplication in time becomes convolution in frequency (multiplication theorem).

Significance:

• The output of an LTI system $y(t)=x(t)*h(t)$ is found easily as $Y(f)=X(f)H(f)$; filtering is just multiplying spectra.
• Multiplication of a message by a carrier or by a sampling pulse train corresponds to convolving the spectra, explaining modulation and sampling spectra.