Linear Filtering Based on the Discrete Fourier Transform

The DFT provides an efficient way to calculate the time-domain convolution of two signals.
One of the most important applications of the Discrete Fourier Transform (DFT) is calculating the time-domain convolution of signals. This can be achieved by multiplying the DFT representation of the two signals and then calculating the inverse DFT of the result.

You may doubt the efficiency of this method because we are replacing the convolution operation with two DFTs, one multiplication, and an inverse DFT operation. However, since there are efficient methods of calculating the DFT, collectively called the Fast Fourier Transform (FFT), we can gain significant computational saving by using the DFT to perform the time-domain convolution.”