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Sampled Sinusoids

In discrete-time audio processing, such as we normally do on a computer, we work with samples of continuous-time signals. Let $ f_s$ denote the sampling 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, while for digital audio tape (DAT), $ f_s= 48$ kHz.

Let $ T\isdef 1/f_s$ denote the sampling interval 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 is then

y(nT) &\isdef & \left.{\cal A}\,e^{st}\right\vert _{t=nT}\\
&=& {\cal A}e^{s n T} = {\cal A}\left[e^{sT}\right]^n \\
&\isdef & A e^{j\phi} e^{(\sigma+j\omega) nT} \\
&=& A e^{\sigma nT} \left[\cos(\omega nT + \phi) + j\sin(\omega nT + \phi)\right] \\
&=& A \left[e^{\sigma T}\right]^n
\left[\cos(\omega nT + \phi) + j\sin(\omega nT + \phi)\right].

Thus, the sampled case consists of a sampled complex sinusoid multiplied by a sampled exponential envelope $ \left[e^{\sigma
T}\right]^n = e^{-nT/\tau}$ .

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``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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University