# Convolution and The Fourier Transform.

Cuthbert Nyack
The convolution can be obtained in the time domain via the convolution integral and in the frequency domain by taking the inverse Fourier Transform of the product of the spectra of the 2 functions.

The applet below compares the convolution obtained in the time and frequency domains. Fn changes the function, a and b changes the shape of the function and wr changes the range of the integration used for the inverse Fourier transform.

eg below compares the convolution obtained via the time and frequency domains when Fn = 1. Convolution can sometimes be used to simplify the process of finding the Fourier Transform (FT) of a pulse in cases where the pulse can be written as the product of 2 functions and the FT of both is known. According to the convolution theorem the FT of the product function is the convolution of the spectra of the 2 multiplying functions. Consider the cosine pulse shown in magenta above. This is the product of a cosine (shown in dashed red) and a triangular pulse(shown in dashed blue). The spectrum of a cosine is 2 delta functions at ±(cosine frequency) and the spectrum of the triangular pulse is the sinc2 function centered at the origin. The spectrum of the cosine pulse is the convolution of the 2 spectra. The plot above shows the spectrum of the cosine in dashed red, that of the triangular pulse in dashed blue and the spectrum of the cosine pulse in magenta which is obtained by convolving the 2 dashed ones.