Conjugate Symmetry (Hermitian)

• A complex sinusoid $e^{j\omega t}$ consists of one frequency $\omega$.

• A real sinusoid $\sin(\omega t)$ consists of two frequencies $\omega$ and $-\omega$.

• Every real signal, therefore, consists of an equal contribution of positive and negative frequency components.

• If $X(\omega)$ denotes the spectrum of the real signal $x(t)$, then $X(\omega)$ is conjugate symmetric (Hermitian), implying
  $$\vert X(-\omega)\vert = \vert X(\omega)\vert$$
  and
  $$\angle X(-\omega) = -\angle X(\omega)$$

• It is sometimes easier to use the ``less complicated'' complex sinusoid when doing signal processing.

• Negative frequencies in a real signal may be ``filtered out'' to produce an analytic signal, a signal which has no negative frequency components.