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Complex Amplitude

Note that the amplitude $ A$ and phase $ \phi$ can be viewed as the magnitude and angle of a single complex number

$\displaystyle {\cal A}\isdefs A e^{j\phi} $

which is naturally thought of as the complex amplitude of the complex sinusoid defined by the left-hand side of either Eq.$ \,$(1.9) or Eq.$ \,$(1.10). The complex amplitude is the same whether we are talking about the continuous-time sinusoid $ A e^{j(\omega t+\phi)}$ or the discrete-time sinusoid $ A e^{j(\omega nT+\phi)}$.

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Complex Sinusoids