Complex Amplitude or Phasor

• When two complex numbers are multiplied, their magnitudes multiply and their angles add:
  $$r_1e^{j\theta_1}r_2e^{j\theta_2} = r_1r_2e^{j(\theta_1 + \theta_2)}.$$

• If the complex number $X = Ae^{j\phi}$ is multiplied by the complex exponential signal $e^{j\omega_0 t}$, we obtain
  $$x(t)=Xe^{j\omega_0 t}= Ae^{j\phi}e^{j\omega_0 t} = Ae^{j(\omega_0t + \phi)}.$$

• The complex number $X$ is referred to as the complex amplitude, a polar representation of the amplitude and the initial phase of the complex exponential signal.

• The complex amplitude is also called a phasor as it can be represented graphically as a vector in the complex plane.