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
 |
(B.6) |
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
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
![$ x_p\isdef [x(0),x(T),\dots,x((N-1)T)]$](http://www.dsprelated.com/josimages_new/mdft/img1719.png)
. Thus,

for all

at which the bandlimited
periodic signal

has a nonzero
harmonic. When

is odd,

can be nonzero for
![$ k\in[-(N-1)/2,(N-1)/2]$](http://www.dsprelated.com/josimages_new/mdft/img1721.png)
, while for

even, the maximum nonzero
harmonic-number range is
![$ k\in[-N/2+1,N/2-1]$](http://www.dsprelated.com/josimages_new/mdft/img1722.png)
.
In summary,
Next Section: Duration and Bandwidth as Second MomentsPrevious Section: The Continuous-Time Impulse