## Fourier Series (FS) and Relation to DFT

In continuous time, a periodic signal , with period seconds,B.2 may be expanded into a Fourier series with coefficients given by

 (B.5)

where is the th harmonic frequency (rad/sec). The generally complex value is called the th Fourier series coefficient. The normalization by is optional, but often included to make the Fourier series coefficients independent of the fundamental frequency , and thereby depend only on the shape of one period of the time waveform.

### 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

 (B.6)

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

By the sifting property of delta functions (Eq.(B.4)), the Fourier series of isB.3

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 .

In summary,

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