Powers of $z$

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

$$x(n) \isdef z_0 z_1^n, \quad n=0,1,2,3,\ldots\,. \protect$$ (4.1)

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:

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

Figure 4.9 shows a plot of a generalized (exponentially decaying, $\sigma<0$) complex sinusoid versus time.

Figure: Exponentially decaying complex sinusoid and its projections.

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.