## 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

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*

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

^{B.3}

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Fourier Transform (FT) and Inverse