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
![$ \delta(t)$](http://www.dsprelated.com/josimages_new/mdft/img1713.png)
![$ \,$](http://www.dsprelated.com/josimages_new/mdft/img131.png)
We wish to find the continuous-time Fourier series of the
sampled periodic signal . Thus, we replace
in
Eq.
(B.5) by
![$\displaystyle x_s(t) \isdef x(t)\cdot \Psi_T(t).
$](http://www.dsprelated.com/josimages_new/mdft/img1714.png)
![$ \,$](http://www.dsprelated.com/josimages_new/mdft/img131.png)
![$ x_s$](http://www.dsprelated.com/josimages_new/mdft/img1715.png)
![\begin{eqnarray*}
X_s(\omega_k) = \frac{1}{P} \int_0^P x_s(t) e^{-j\omega_k t} d...
...1}{P} \sum_{n=0}^{\lceil P/T\rceil-1} x(nT) e^{-j\omega_k nT} T.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1716.png)
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
![\begin{eqnarray*}
X_s(\omega_k)
&=& \frac{T}{P} \sum_{n=0}^{N-1} x(nT) e^{-j\o...
...{1}{N}\hbox{\sc DFT}_{N,k}(x_p),
\quad k=0,\pm 1, \pm 2, \dots
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1718.png)
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:
Duration and Bandwidth as Second Moments
Previous Section:
The Continuous-Time Impulse