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

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

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