Consider , with . Then we can write in polar form as
Forming a geometric sequence based on yields the sequence
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.
The length range which is symmetric about zero is
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''.
Informal Derivation of Taylor Series