Complex Conjugate

• The complex conjugate $\overline{z}$ of a complex number $z = x + jy$ is given by
  $$\overline{z} = x - jy.$$

• A real cosine can be represented in the complex plane as the sum of two complex rotating vectors (scaled by 1/2) that are complex conjugates of each other.
  
• The negative frequencies that arise from the complex exponential representation of the signal, will greatly simplify the task of signal analysis and spectrum representation.