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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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