Spectrum Representation cont.

• Using inverse Euler, this signal may be represented as
  $$x(t) = A_0+\sum_{k=1}^{N}\left\{\frac{X_k}{2}e^{j\omega_kt} + \frac{\overline{X}_k}{2}e^{-j\omega_kt}\right\}.$$

• Every signal therefore, can be expressed as a linear combination of complex sinusoids.

• If a signal is the sum of $N$ sinusoids, the spectrum will be composed of $2N + 1$ complex amplitudes and $2N + 1$ complex exponentials of a certain frequency.

  Figure 5: Spectrum of a signal with $N=2$ components.