Relation of the DFT to Fourier Series
We now show that the DFT of a sampled signal (of length ), is proportional to the Fourier series coefficients of the continuous periodic signal obtained by repeating and interpolating . More precisely, the DFT of the samples comprising one period equals times the Fourier series coefficients. To avoid aliasing upon sampling, the continuous-time signal must be bandlimited to less than half the sampling rate (see Appendix D); this implies that at most complex harmonic components can be nonzero in the original continuous-time signal.
If is bandlimited to , it can be sampled at intervals of seconds without aliasing (see §D.2). One way to sample a signal inside an integral expression such as Eq.(B.5) is to multiply it by a continuous-time impulse train
where is the continuous-time impulse signal defined in Eq.(B.3).
We wish to find the continuous-time Fourier series of the sampled periodic signal . Thus, we replace in Eq.(B.5) by
If the sampling interval is chosen so that it divides the signal period , then the number of samples under the integral is an integer , and we obtain
where . Thus, for all at which the bandlimited periodic signal has a nonzero harmonic. When is odd, can be nonzero for , while for even, the maximum nonzero harmonic-number range is .
Duration and Bandwidth as Second Moments
The Continuous-Time Impulse