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






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


We wish to find the continuous-time Fourier series of the
sampled periodic signal . Thus, we replace
in
Eq.
(B.5) by




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,
Next Section:
Differentiation Theorem
Previous Section:
Fourier Transform (FT) and Inverse