• In this appendix, the basic manipulations of complex numbers are reviewed. There ideas are treated:
    • simple algebraic rules: operations on \(z = x+jy\).
    • elimination of trigonometry: Euler's formula for the complex exponential \(z = re^{j\theta}=r\cos\theta+jr\sin\theta\).
    • representation by vectors: a way for visualization.
  • Symbol: \(i\) or \(j\)
    • Physicists and mathematicians use symbol \(i=\sqrt{-1}\).
    • Electrical engineers use symbol \(j=\sqrt{-1}\) instead because \(i\) is left to the current.
  • Algebraic vs. Trigonometric vs. Geometric.

A.1 Introduction

  • The way \(j\) is introduced: \(z^2=-1\) ==> \(z=\pm j\).
  • More general, complex numbers are the roots of quadratic equations.

A.2 Notation for complex numbers

  • There are two types complex number representations:
    • Rectangular form (Cartesian form): \(z = (x, y) = x + jy = \Re \{ z \} + j\Im\{z\}\), where \(\Re\{\}\) and \(\Im\{\}\) represent the real and imaginary parts of the complex number, respectively.
    • Polar form: \(z \leftrightarrow r\angle\theta\) where \(r\) is the amplitude and \(\angle\theta\) is the angle whose principal value belongs to \(-180^{\circ}<\theta < 180^{\circ}\)
    • Conversion:
      • polar --> rectangular: \(z=x + jy\), where \[\begin{equation} \begin{cases} x = r\cos\theta,\\ y = r\sin\theta \end{cases} \end{equation} \label{eq1}\]
      • rectangular --> polar: \(z = re^{j\theta}=|z|e^{j\, \text{arg}|z|}\), where \[\begin{equation} \begin{cases} r = \sqrt{x^2+y^2},\\ \theta = \text{atan}(y, x) \end{cases} \end{equation} \label{eq2}\]

A.3 Euler's formula

  • Euler's formula \[ \begin{equation} e^{j\theta} = \cos{\theta} + j\sin\theta \end{equation}\label{Euler} \]
  • Inverse Euler fomulas \[ \begin{align} \cos\theta &= \frac{e^{j\theta} + e^{-j\theta}}{2}\\ \sin\theta &= \frac{e^{j\theta} - e^{-j\theta}}{2} \end{align}\]

A.4 Algebraic rules for complex numbers

Rectangular form

For \(z_1 = x_1 + jy_1\) and \(z_2 = x_2+jy_2\),

  • addition and subtraction: \(z_1 \pm z_2 = (x_1 \pm x_2) + j(y_1 \pm y_2)\).
  • multiplication: \(z_1 z_2 = (x_1 x_2-y_1 y_2)+j(x_1 y_2+x_2 y_1)\)
  • conjugate: \(z_1^* = x_1 - jy_1\)
  • division: \(\dfrac{z_1}{z_2} = \dfrac{z_1z_2^*}{z_2z_2^*} = \dfrac{z_1z_2^*}{|z_2|^2} = \dfrac{(x_1x_2+y_1y_2) + j(x_2y_1-x_1y_2)}{x_2^2+y_2^2}\)

Polar form

For \(z_1 = r_1e^{j\theta_1}\) and \(z_2 = r_2e^{j\theta_2}\),

  • multiplication: \(z_1z_2 = (r_1r_2)e^{j(\theta_1+\theta_2)}\)
  • conjugate: \(z_1^* = r_1e^{-j\theta_1}\)
  • division: \(\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2}e^{j(\theta_1-\theta_2)}\)
  • addition and subtraction: transfer to rectangular form and do the addition or subtraction, and then, transfer back to polar form.

others

  • \(\Re\{z\} = \dfrac{z+z^*}{2}\)
  • \(\Im\{z\} = \dfrac{z-z^*}{2j}\)
  • \(|z|^2 = zz^*\)

A.5 Geometric views off complex operations

A geometric view provides a convenient visualization for complex number operations.

A.6 Powers and Roots

  • \(z^N = (re^{j\theta})^N = r^Ne^{jN\theta}\)
  • De Moivre's formula: \((\cos\theta + j\sin\theta)^N = \cos N\theta + j\sin N\theta\) (because \((e^{j\theta})^N = e^{jN\theta}\))
  • Roots of unity (\(z^N=1\)): \(z=e^{j2\pi l/N}\) for \(l=0,1,2\dots N-1\)
  • \(z^N=c=|c|e^{j\phi}\): \(z=re^{j\theta}\), where \[\begin{cases} r = |c|^{1/N},\\ \theta = \dfrac{\phi+2\pi l}{N}, \end{cases}\] and \(\theta\) is the angular spacing.