Complex Exponential Signals

The complex exponential signal (or complex sinusoid) is defined as

$$x(t) = Ae^{j(\omega_0t + \phi)}.$$

It may be expressed in Cartesian form using Euler's formula:

$$x(t) = Ae^{j(\omega_0t + \phi)}$$

$$= A\cos(\omega_0t + \phi) + jA\sin(\omega_0t + \phi).$$

As with the real sinusoid,

• A is the amplitude given by $\vert x(t)\vert$
  

  $$\vert x(t)\vert \triangleq \sqrt{\mbox{re}^2\{x(t)\}+\mbox{im}^2\{x(t)\}}$$

  $$\equiv \sqrt{A^2[\cos^2(\omega t+\phi) + \sin^2(\omega t+\phi)]}$$

  $$\equiv A \mbox{for all $t$}$$

  $$( \cos^2(\omega t+\phi) + \sin^2(\omega t+\phi)=1).$$

  
  
• $\phi$ is the initial phase
• $\omega_0$ is the frequency in rad/sec
• $\omega_0t + \phi$ is the instantaneous phase, also denoted $\arg x(t)$.