Sampled Sinusoids

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Sampled Sinusoids

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Sampled Sinusoids

In discrete-time audio processing, such as we must do on a computer, we work with samples of continuous-time signals. Let $f_s$ denote thesampling rate in Hz. For audio, we typically have $f_s>40$ kHz, since the audio band nominally extends to $20$ kHz. For compact discs (CDs), $f_s = 44.1$ kHz (or very close to that--I once saw Sony device using a sampling rate of $44,025$ Hz), while for digital audio tape (DAT), $f_s = 48$ kHz.

Let $T\ denote the sampling period in seconds. Then to convert from continuous to discrete time, we replace $t$ by $nT$, where $n$is an integer interpreted as the sample number.

The sampled generalized complex sinusoid (which includes all other cases) is then

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