## 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 .

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Differentiation Theorem
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Fourier Transform (FT) and Inverse