Phasor

It is common terminology to call $z_0 = Ae^{j\phi}$ the complex sinusoid's phasor (or ``phase vector''), and $z_1^n = e^{j\omega n T}$ its carrier wave.

For a real sinusoid,

$$x_r(n) \isdef A \cos(\omega n T+\phi),$$

the phasor is again defined as $z_0 = Ae^{j\phi}$ and the carrier is $z_1^n = e^{j\omega n T}$ . However, in this case, the real sinusoid is recovered from its complex-sinusoid counterpart by taking the real part:

$$x_r(n) =$$   re$$\left\{z_0z_1^n\right\}$$

The phasor magnitude $\left\vert z_0\right\vert=A$ is the amplitude of the sinusoid. The phasor angle $\angle{z_0}=\phi$ is the phase of the sinusoid.

When working with complex sinusoids, as in Eq.(4.11), the phasor representation $Ae^{j\phi}$ of a sinusoid can be thought of as simply the complex amplitude of the sinusoid. I.e., it is the complex constant that multiplies the carrier term $e^{j\omega nT}$ .