## 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}

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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Differentiation Theorem

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Fourier Transform (FT) and Inverse