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

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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