Real and Complex Exponential Signals

How does the Complex Exponential Signal compare to the real sinusoid?

• As seen from Euler's formula, the sinusoid given by $A\cos(\omega_0t + \phi)$ is the real part of the complex exponential signal. That is,
  $$A\cos(\omega_0t + \phi) =$$   re$$\{Ae^{j(\omega_0t+\phi)}\}.$$

• Recall that sinusoids can be represented by the sum of in-phase and phase-quadrature components.
  

  $$A\cos(\omega_0t + \phi) = \{Ae^{j(\omega_0 t + \phi)} \}$$

  $$= \{Ae^{j(\phi + \omega_0 t)} \}$$

  $$= A \{e^{j\phi} e^{j\omega_0 t} \}$$

  $$= A \{\left(\cos\phi + j\sin\phi\right)\left(\cos(\omega_0 t) + j\sin(\omega_0 t)\right) \}$$

  $$= A \{\cos\phi\cos(\omega_0 t) - \sin\phi\sin(\omega_0 t)$$

  $$\qquad + j(\cos\phi\sin(\omega_0t) + \sin\phi\cos(\omega_0t)) \}$$

  $$= A\cos\phi\cos(\omega_0 t) - A\sin\phi\sin(\omega_0 t).$$