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.
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