The DFT followed by the inverse DFT results in an exact reconstruction of the original signal. This provides an easy way to implement a digital filter, as illustrated in figures 26 and 27.
Figure 26: The overlap and add method for linear filtering
Figure 27: overlap and add method in the time domain
A FIR linear filter would apply a constant multiplicative weighting in the frequency domain. For filter lengths greater than about 60 samples it is faster to use the FFT than direct convolution. Note that zero padding of the input is needed to ensure that there are no wrap-around effects.
This framework also allows for more general filtering where the filter may be time dependent or dependent on the analysed signal.