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

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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University