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*
. 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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