4.1 Discrete Fourier series for discrete-time periodic signals4.2 Discrete Fourier transform for discrete-time aperiodic signals4.3 The inverse discrete Fourier transform and its computation4.4 Properties of the DFT4.5 The fast Fourier transform4.6 Practical implementation issues4.7 Computation of convolution using DFT4.8 Frequency ranges of some natural and man-made signals