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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Sinusoids and Exponentials
       Complex Sinusoids
          Powers of z

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Powers of z

Choose any two complex numbers $ z_0$ and $ z_1$, and form the sequence

$\displaystyle x(n) \isdef z_0 z_1^n, \quad n=0,1,2,3,\ldots\,. \protect$ (4.10)

What are the properties of this signal? Writing the complex numbers as

\begin{eqnarray*} z_0 &=& A e^{j\phi} \\ z_1 &=& e^{sT} = e^{(\sigma + j\omega)T}, \end{eqnarray*}

we see that the signal $ x(n)$ is always a discrete-time generalized (exponentially enveloped) complex sinusoid:

$\displaystyle x(n) = A e^{\sigma n T} e^{j(\omega n T + \phi)} $

Figure 4.17 shows a plot of a generalized (exponentially decaying, $ \sigma<0$) complex sinusoid versus time.

Figure 4.17: Exponentially decaying complex sinusoid and projections.
\includegraphics[scale=0.8]{eps/circledecaying}

Note that the left projection (onto the $ z$ plane) is a decaying spiral, the lower projection (real-part vs. time) is an exponentially decaying cosine, and the upper projection (imaginary-part vs. time) is an exponentially enveloped sine wave.

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Next: Phasor and Carrier Components of Sinusoids
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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