Appendix: Frequencies Representable
by a Geometric Sequence

Consider

, with

. Then we can write

in polar form as
where

,

, and

are
real numbers.
Forming a geometric sequence based on

yields the sequence
where

. Thus, successive integer powers of

produce a
sampled complex sinusoid with unit amplitude, and
zero phase. Defining the
sampling interval as

in seconds
provides that

is the
radian frequency in radians per
second.
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

is indistinguishable from
the angular step

. Therefore, we must restrict the
angular step

to a length

range in order to avoid
ambiguity.
Recall from §
4.3.3 that we need support for both positive
and
negative frequencies since equal magnitudes of each must be summed
to produce real
sinusoids, 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 sequence of zeros! For the signal

, on the other hand, we sample
the positive and negative peaks, and everything 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 one 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 per revolution. Such signals appear as an
alternating sequence of the form

, where

can 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 an
anti-aliasing filter, and it is a standard first stage in an
Analog-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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