Convolution is the operation “performed” by digital filters. We can say that the filter convolves an input signal with its impulse response to produce an output .
Convolution is described mathematically as:
(4)
Convolution can be interpreted as a sample-by-sample multiplication and sum of the signal and a time ”flipped” and delayed version of the signal .
We can derive the time/frequency transform pair for the convolution process as follows:
We have just derived the very important result that convolution in the time domain corresponds to multiplication in the frequency domain:
(5)
For long signal lengths ( or so), it is much faster to transform signals with the FFT (and IFFT) and to perform the convolution as a frequency-domain multiplication.