Complex Amplitude

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

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

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 $A e^{j(\omega nT+\phi)}$.