NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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Aliasing of Sampled Continuous-Time Signals
This section quantifies aliasing in the general case. This
result is then used in the proof of
Shannon's 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 
, then 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 fundamental
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 .
Theorem.
(Continuous-Time
Aliasing Theorem) 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 amplitudeand
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
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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