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Appendix: Frequencies Representable

by a Geometric Sequence

Consider , with . Then we can write in polar form as

*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

*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

*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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Informal Derivation of Taylor Series

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Sampling Theorem