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

.
Let

denote any continuous-time
signal having a
Fourier
Transform (FT)
Let
denote the samples of

at uniform intervals of

seconds,
and denote its
Discrete-Time Fourier Transform (
DTFT) by
Then the
spectrum 
of the sampled signal

is related to the
spectrum 
of the original continuous-time signal

by
The terms in the above sum for

are called
aliasing
terms. They are said to
alias into the
base band
![$ [-\pi/T,\pi/T]$](http://www.dsprelated.com/josimages_new/mdft/img1790.png)
. Note that the summation of a
spectrum with
aliasing components involves addition of
complex numbers; therefore,
aliasing components can be removed only if both their
amplitude
and phase are known.
Proof:
Writing

as an inverse FT gives
Writing

as an inverse DTFT gives
where

denotes the normalized discrete-time
frequency variable.
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
Since this is formally the inverse DTFT of

written in terms of

,
the result follows.
Next Section: Sampling TheoremPrevious Section: Introduction to Sampling