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.