Aliasing of Sampled Signals
This section quantifies aliasing in the general case. This result is then used in the proof of the sampling theorem in the next section.
It is well known that when a continuous-time signal contains energy at
a frequency higher than half the sampling rate , sampling
at
samples per second causes that energy to alias to a
lower frequency. If we write the original frequency as
, then the new aliased frequency is
,
for
. This phenomenon is also called ``folding'',
since
is a ``mirror image'' of
about
. As we will
see, however, this is not a complete description of aliasing, as it
only applies to real signals. For general (complex) signals, it is
better to regard the aliasing due to sampling as a summation over all
spectral ``blocks'' of width
.
Continuous-Time Aliasing Theorem
Let denote any continuous-time signal having a Fourier
Transform (FT)









![$\displaystyle X_d(e^{j\theta}) = \frac{1}{T} \sum_{m=-\infty}^\infty X\left[j\left(\frac{\theta}{T}
+ m\frac{2\pi}{T}\right)\right].
$](http://www.dsprelated.com/josimages_new/mdft/img1788.png)

![$ [-\pi/T,\pi/T]$](http://www.dsprelated.com/josimages_new/mdft/img1790.png)
Proof:
Writing as an inverse FT gives




The inverse FT can be broken up into a sum of finite integrals, each of length
, as follows:

Let us now sample this representation for at
to obtain
, and we have

since and
are integers.
Normalizing frequency as
yields
![$\displaystyle x_d(n) = \frac{1}{2\pi}\int_{-\pi}{\pi} e^{j\theta^\prime n}
\f...
...t(\frac{\theta^\prime }{T}
+ m\frac{2\pi}{T}\right) \right] d\theta^\prime .
$](http://www.dsprelated.com/josimages_new/mdft/img1800.png)



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Sampling Theorem
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Introduction to Sampling