Let

denote any continuous-time
signal having a
continuous Fourier transform

Let
denote the samples of

at uniform intervals of

seconds.
Then

can be exactly reconstructed from its samples

if

for all

.
D.3
Proof: From the continuous-time
aliasing theorem (§
D.2), we
have that the discrete-time
spectrum

can be written in
terms of the continuous-time
spectrum

as
where

is the ``digital frequency'' variable.
If

for all

, then the above
infinite sum reduces to one term, the

term, and we have
At this point, we can see that the
spectrum of the sampled signal

coincides with the nonzero spectrum of the continuous-time
signal

. In other words, the
DTFT of

is equal to the
FT of

between plus and minus half the
sampling rate, and the FT
is zero outside that range. This makes it clear that spectral
information is preserved, so it should now be possible to go from the
samples back to the continuous waveform without error, which we now
pursue.
To reconstruct

from its samples

, we may simply take
the inverse Fourier transform of the zero-extended DTFT, because
By expanding

as the DTFT of the samples

, the
formula for reconstructing

as a superposition of weighted
sinc
functions is obtained (depicted in Fig.
D.1):
where we defined
or
The ``sinc function'' is defined with

in its argument so that it
has zero crossings on the nonzero integers, and its peak magnitude is
1. Figure
D.2 illustrates the appearance of the sinc function.
We have shown that when

is bandlimited to less than half the
sampling rate, the IFT of the zero-extended DTFT of its samples

gives back the original continuous-time signal

.
This completes the proof of the
sampling theorem.

Conversely, if

can be reconstructed from its samples

, it must be true that

is bandlimited to
![$ [-f_s/2,f_s/2]$](http://www.dsprelated.com/josimages_new/mdft/img1818.png)
, since a sampled signal only supports frequencies up
to

(see §
D.4 below). While a real digital signal

may have energy at half the sampling rate (frequency

),
the phase is constrained to be either 0 or

there, which is why
this frequency had to be excluded from the sampling theorem.
A one-line summary of the essence of the sampling-theorem proof is
where

.
The sampling theorem is easier to show when applied to
sampling-rate
conversion in discrete-time,
i.e., when simple
downsampling of a
discrete time signal is being used to reduce the sampling rate by an
integer factor. In analogy with the continuous-time
aliasing theorem
of §
D.2, the
downsampling theorem (§
7.4.11)
states that downsampling a digital signal by an integer factor

produces a digital signal whose spectrum can be calculated by
partitioning the original spectrum into

equal blocks and then
summing (aliasing) those blocks. If only one of the blocks is
nonzero, then the original signal at the higher sampling rate is
exactly recoverable.
Next Section: Appendix: Frequencies Representable
by a Geometric SequencePrevious Section: Aliasing of Sampled Signals