NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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A natural question to investigate is what frequencies
are possible. The angular step of the point 
along the unit circle in the complex
plane is 
. Since 
, an angular step larger than 
,
say 
is indistinguishable from the angular step
.
Therefore, we must restrict the angular step 
to a length
range in order to avoid ambiguity.
Recall that we need support for both positive and
negative frequenciessince equal magnitudes of each must be
summed to produce realsinusoids,
as indicated by the identities
The length
range which is symmetric about zero is
which, since 
, corresponds to the frequency
range
However, there is a problem with the point at 
: Both
and 
correspond to the same point 
in
the
-plane.
Technically, this can be viewed as
aliasing of the frequency on
top of 
, or
vice versa. The practical impact is that it is not possible in
general to reconstruct a sinusoid from its samples at this
frequency. For an obvious example, consider the sinusoid at half
the
sampling-rate sampled on its zero-crossings: 
. We cannot be expected to reconstruct a
nonzero signal from a string of zeros! For the signal 
, on the other hand, we sample the positive and
negative peaks, and everything looks looks fine. In general, we
either do not know the amplitude,
or we do not know phase of a sinusoid sampled at exactly twice its
frequency, and if we hit the zero crossings, we lose it altogether.
In view of the foregoing, we may define the valid range of
''digital frequencies'' to be
While you might have expected the open interval 
, we are free to give the point
either the largest positive or largest negative representable
frequency. Here, we chose the largest
negative frequency since it corresponds to the assignment of
numbers in
two's complement arithmetic, which is often used to implement
phase registers in sinusoidal oscillators. Since there is no
corresponding positive-frequency component, samples at
must be interpreted as samples of clockwise circular motion around
the unit circle at two points. Such signals are any alternating
sequence of the form
, where 
can be be complex. The amplitude at is
then defined as 
, and the phase is 
.
We have seen that uniformly spaced samples can represent
frequencies
only in the range 
, where 
denotes the sampling rate. Frequencies outside this range yield
sampled sinusoids indistinguishable from frequencies inside the
range.
Suppose we henceforth agree to sample at higher than
twice the highest frequency in our continuous-time signal. This is
normally ensured in practice by lowpass-filtering
the input signal to remove all signal
energy at
and above. Such a filter is called ananti-aliasing filter,
and it is a standard first stage in anAnalog-to-Digital (A/D)
Converter (ADC). Nowadays, ADCs are normally implemented within
a single integrated circuit chip, such as a CODEC (for
''coder/decoder'') or ''multimedia chip''.
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