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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Generalized Complex Sinusoids