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

For a real sinusoid,

$\displaystyle 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:

$\displaystyle x_r(n) =$   re$\displaystyle \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}$ .

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2018-02-13 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University