### Sampled Sinusoids

In discrete-time audio processing, such as we normally do on a computer,
we work with *samples* of continuous-time signals. Let
denote the sampling rate in Hz. For audio, we typically have
kHz, since the audio band nominally extends to kHz. For compact
discs (CDs), kHz,
while for digital audio tape (DAT), kHz.

Let
denote the *sampling interval* in seconds. Then to
convert from continuous to discrete time, we replace by , where
is an integer interpreted as the *sample number*.

The sampled generalized complex sinusoid is then

Thus, the sampled case consists of a sampled complex sinusoid multiplied by a sampled exponential envelope .

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