Phasor and Carrier Components of Complex Sinusoids

If we restrict $z_1$ in Eq. (4.1) to have unit modulus, then $\sigma=0$ and we obtain a discrete-time complex sinusoid.

$$x(n) \isdef z_0 z_1^n = \left(Ae^{j\phi}\right) e^{j\omega n T} = A e^{j(\omega n T+\phi)}, \quad n=0,1,2,3,\ldots \protect$$ (4.2)

where we have defined

$$\begin{eqnarray*} z_0 &\isdef & Ae^{j\phi}, \quad \hbox{and}\\ z_1 &\isdef & e^{j\omega T}. \end{eqnarray*}$$