Proof of Shannon's sampling theorem
In order to recover the signal
from it's samples exactly, it is necessary to sample
at a rate greater than twice it's highest frequency component.
Introduction
As mentioned
earlier ,
sampling is the necessary fundament when we want to apply digital signalprocessing on analog signals.
Here we present the proof of the sampling theorem.
The proof is divided in two. First we find an expression for the spectrum of the signal resulting fromsampling the original signal
.
Next we show that the signal
can be recovered from the samples.
Often it is easier using the frequency domain when carrying out a proof,and this is also the case here.
Key points in the proof
- We find an
equation for the spectrum of the sampled signal
- We find a
simple method to reconstruct the original signal
- The sampled signal has a periodic spectrum...
- ...and the period is
Proof part 1 - spectral considerations
By sampling
every
second we obtain
.
The inverse fourier transform of this
time discrete signal is
For convenience we express the equation in terms of the real angular
frequency
using
.We then obtain
The inverse fourier transform of a continuous signal is
From this equation we find an expression for
To account for the difference in region of integration we split the integration in
into subintervals of length
and then take the sum over the resulting integrals to obtain the complete area.
Then we change the integration variable, setting
We obtain the final form by observing that
,
reinserting
and multiplying by
To make
for all values of
, the integrands in
and
have to agreee, that is
This is a central result. We see that the digital spectrum consists of a sum of shifted versions of
the original, analog spectrum. Observe the periodicity!
We can also express this relation in terms of the digital angular frequency
This concludes the first part of the proof. Now we want to find a reconstruction formula, so
that we can recover
from
.
Proof part ii - signal reconstruction
For a
bandlimited signal the inverse fourier transform is
In the interval we are integrating we have:
. Substituting this relation into
we get
Using the
DTFT relation for
we have
Interchanging integration and summation (under the assumption of convergence) leads to
Finally we perform the integration and arrive at the important reconstruction formula
(Thanks to R.Loos for pointing out an error in the proof.)
Summary
Go to
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Introduction
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Illustrations
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Matlab Example
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Hold operation
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Aliasing applet
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System view
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Exercises
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