Although complex notation is not complex to understand, one of the themes of this text is to avoid complex notation altogether.

A complex number is defined as an ordered pair of real numbers in $(x,y)$-plane. In that respect, complex numbers can be considered as vectors with initial point on the origin $(0,0)$. Addition of complex numbers is then similar to the addition of vectors in $(x,y)$-plane from this perspective.

However, multiplication is well defined for complex numbers while it is not defined for vectors — the dot product of two vectors is a scalar, not a vector, while the cross product of two vectors in a plane is a vector that is outside of that plane. The product of complex numbers, on the other hand, is a complex number — an extremely useful property.

For our purpose, using the complex numbers but still staying clear of the complex notation means that we focus on a 2-dimensional plane with $x$ or real-axis named as $I$ (which stands for inphase) and $y$ or imaginary-axis named as $Q$ (which stands for quadrature). In the post about frequency, we will learn why the $x$ and $y$ components are called inphase and quadrature, respectively.

## Magnitude and Phase

In polar representation of complex numbers, the magnitude of $V$ in an $IQ$-plane is defined as

|V| = \sqrt{V_I^2 + V_Q^2}

\end{equation*}

Defining the phase $\measuredangle V$ is a little trickier. It is tempting to define it as $\tan^{-1} V_Q/V_I$. However, a problem with this is that

\begin{align*}

\tan^{-1} \frac{+V_Q}{+V_I} ~&=~ \tan^{-1} \frac{-V_Q}{-V_I} \quad \rightarrow \quad \text{in}~ [0,+\pi/2],~ \text{quadrant I} \\

\tan^{-1} \frac{-V_Q}{+V_I} ~&=~ \tan^{-1} \frac{+V_Q}{-V_I} \quad \rightarrow \quad \text{in}~ [0,-\pi/2],~ \text{quadrant IV}

\end{align*}

There is no way to differentiate whether $V$ lies in quadrant I or III. The same holds for $V$ lying in quadrant II or IV. On the other hand, the above Figure tells us that $V$ can lie in any quadrant and its phase should be in the range $[-\pi,\pi)$ because

- Quadrant I: $\measuredangle V$ should be in $[0,\pi/2]$
- Quadrant II: $\measuredangle V$ should be in $[\pi/2,\pi]$
- Quadrant III: $\measuredangle V$ should be in $[-\pi/2,-\pi]$
- Quadrant IV: $\measuredangle V$ should be in $[0,-\pi/2]$

Similarly, from Figure below,

- Case 1: When $V_I<0$ and $V_Q=0$, the phase of $V$ should be $\pi$, not $0$
- Case 2: When $V_I=0$ and $V_Q>0$, the phase of $V$ should be $+\pi/2$
- Case 3: When $V_I=0$ and $V_Q<0$, the phase of $V$ should be $-\pi/2$

Taking into account all four quadrants, $\measuredangle V$ is defined in terms of $\tan^{-1} (V_Q/V_I)$ as

\measuredangle V =

\begin{cases}

\tan^{-1} \frac{V_Q}{V_I} & V_I > 0 \\

\tan^{-1} \frac{V_Q}{V_I} + \pi & V_I < 0 \mbox{ and } V_Q \ge 0\\ \tan^{-1} \frac{V_Q}{V_I} - \pi & V_I < 0 \mbox{ and } V_Q < 0\\ +\pi/2 & V_I = 0 \mbox{ and } V_Q > 0\\

-\pi/2 & V_I = 0 \mbox{ and } V_Q < 0 %\mbox{indeterminate } & V_I = 0 \mbox{ and } V_Q = 0. \end{cases} \end{equation*}

From here onwards, we will call this adjustment as four-quadrant inverse tangent.

## Addition and Multiplication Operations

The addition and multiplication rules for complex numbers are explained below.

Following rules apply to two complex numbers $U$ and $V$ in an $IQ$-plane, which is basically a simpler way to write complex additions and multiplications.

$W = U + V$ implies

\begin{align*}

W_I\: &= U_I + V_I \\

W_Q &= U_Q + V_Q

\end{align*}

Note that one complex addition results in two real additions, one each for $I$ and $Q$, as shown in Figure below.

$W = UV$ implies

\begin{equation}\label{eqIntroductionComplexMultiplication1}

\begin{aligned}

|W| &= |U|.|V| \\

\measuredangle W &= \measuredangle U +\measuredangle V

\end{aligned}

\end{equation}

which in $IQ$ form results in

\begin{equation}\label{eqIntroductionComplexMultiplication2}

\begin{aligned}

W_I\: &= U_I V_I – U_Q V_Q \\

W_Q &= U_Q V_I + U_I V_Q

\end{aligned}

\end{equation}

We will call the above equation the multiplication rule. Note that one complex multiplication results in 4 real multiplications and 2 real additions. This is illustrated in Figure below.

Addition rule above makes perfect sense: both $I$ components are added together as well as both $Q$ components. The multiplication rule seems a little strange though: $I$ is a difference between products of two aligned-axes terms (i.e., $I\cdot I$ – $Q\cdot Q$), while $Q$ is a sum of products of two cross-axes terms (i.e., $Q\cdot I$ + $I\cdot Q$).

To see why multiplication of two complex numbers is perfectly logical, consider that $U_I = |U| \cos \measuredangle U$ and $U_Q = |U| \sin \measuredangle U$, while $V_I = |V| \cos \measuredangle V$ and $V_Q = |V| \sin \measuredangle V$. Therefore,

\begin{align*}

W_I\: &= |U| \cos \measuredangle U \cdot |V| \cos \measuredangle V – |U| \sin \measuredangle U \cdot |V| \sin \measuredangle V \\

W_Q &= |U| \sin \measuredangle U \cdot |V| \cos \measuredangle V + |U| \cos \measuredangle U \cdot |V| \sin \measuredangle V

\end{align*}

which leads to

\begin{align*}

W_I\: &= |U| |V| \left(\cos \measuredangle U \cdot \cos \measuredangle V – \sin \measuredangle U \cdot \sin \measuredangle V \right) \\

W_Q &= |U| |V| \left( \sin \measuredangle U \cdot \cos \measuredangle V + \cos \measuredangle U \cdot \sin \measuredangle V \right)

\end{align*}

Using the identities $\cos A \cos B – \sin A \sin B = \cos (A+B)$ and $\sin A$ $\cos B$ $+$ $\cos A \sin B = \sin (A+B)$,

\begin{align*}

W_I\: &= |U| |V| \cos \left(\measuredangle U + \measuredangle V \right) \\

W_Q &= |U| |V| \sin \left( \measuredangle U + \measuredangle V \right)

\end{align*}

Hence, multiplication of two complex numbers is about multiplying their magnitudes and adding their phases.

## Raising a Complex Number to $M^{\text{th}}$ Power

Another consequence of the definition of complex multiplication described above is that raising a complex number $V$ to a power, say $2$, generates

\begin{equation*}

\begin{aligned}

|W| &= |V|.|V| = |V|^2 \\

\measuredangle W &= \measuredangle V +\measuredangle V = 2 \cdot \measuredangle V

\end{aligned}

\end{equation*}

For a general power $M$,

\begin{equation}

\begin{aligned}\label{eqIntroductionComplexRaisingM}

|W| &= |V|^M \\

\measuredangle W &= M \cdot \measuredangle V

\end{aligned}

\end{equation}

We can conclude that in an $IQ$-plane, raising a complex number to a certain power raises the magnitude to that power while multiplies the phase with that number.

## Phase Rotation

Rotating a complex number in $IQ$-plane by a phase $\theta$ seems very simple in complex notation but a bit complicated in $IQ$ terms. Rotation implies keeping the magnitude constant and adding $\theta$ to the angle of that complex number, as shown in Figure below.

To start, let us multiply a complex number $V$ by a complex number $U$ with magnitude $1$ and angle $\theta$. Using $|U| = 1$ and $\measuredangle U = \theta$,

\begin{equation*}

U_I = \cos \theta \quad and \quad U_Q = \sin \theta

\end{equation*}

Hence,

\begin{equation}

\begin{aligned}\label{eqIntroductionPhaseRotation}

W_I\: &= V_I \cdot \cos \theta – V_Q \cdot \sin \theta \\

W_Q &= V_Q \cdot \cos \theta + V_I \cdot \sin \theta

\end{aligned}

\end{equation}

Using $V_I = |V| \cos \measuredangle V$ and $V_Q = |V| \sin \measuredangle V$,

\begin{equation*}

\begin{aligned}

W_I\: &= |V| \left(\cos \measuredangle V \cdot \cos \theta – \sin \measuredangle V \cdot \sin \theta \right) \\

W_Q &= |V| \left( \sin \measuredangle V \cdot \cos \theta + \cos \measuredangle V \cdot \sin \theta \right)

\end{aligned}

\end{equation*}

Again using the identities $\cos A \cos B – \sin A \sin B = \cos (A+B)$ and $\sin A$ $\cos B$ $+$ $\cos A \sin B = \sin (A+B)$,

\begin{equation}

\begin{aligned}\label{eqIntroductionPhaseRotated}

W_I\: &= |V| \cos \left(\measuredangle V + \theta \right) \\

W_Q &= |V| \sin \left(\measuredangle V + \theta \right)

\end{aligned}

\end{equation}

which keeps the magnitude unchanged and adds the angle $\theta$ to the existing angle.

A fast rule for phase rotation can be devised as follows. For a complex number $V$ and an angle $+\theta$ (i.e., counterclockwise rotation), the phase rotation rule states that

Similarly, for a complex number $V$ and an angle $-\theta$ (i.e., clockwise rotation)

The phase rotation rule above is important because we are not using complex notation in this text. That implies having on our disposal a quick way to recognize an equation if it rotates a complex number by an angle. The above two equations help fulfill that purpose.

## Conjugate

The conjugate $V^*$ of a complex number $V$ is defined as

\begin{align*}

\{V ^*\}_I\: &= V_I \\

\{V ^*\}_Q &= -V_Q

\end{align*}

Since magnitude is the sum of squares of $I$ and $Q$ components, it remains unchanged. On the other hand, phase is $Q$ divided by $I$ which leads to the following definition of the conjugate of a complex number.

\begin{equation}

\begin{aligned}\label{eqIntroductionConjugate2}

|V^*| &= |V| \\

\measuredangle V^* &= – \measuredangle V

\end{aligned}

\end{equation}

A significance of conjugate of a complex number arises from the fact that a complex number multiplied by its complex conjugate cancels the phase and produces its magnitude squared. Using the above relations in multiplication rule,

\begin{equation*}

\begin{aligned}

\{V\cdot V^*\}_I\: &= V_I\cdot V_I – V_Q \cdot \left(-V_Q\right) \\

\{V\cdot V^*\}_Q &= V_Q\cdot V_I + V_I\cdot \left(-V_Q\right)

\end{aligned}

\end{equation*}

which generates the following result.

\begin{equation*}

\begin{aligned}

\{V\cdot V^*\}_I\: &= V_I^2 + V_Q^2 \\

\{V\cdot V^*\}_Q &= 0

\end{aligned}

\end{equation*}

Utilizing the definitions of magnitude and phase of complex numbers,

\begin{equation}

\begin{aligned}\label{eqIntroductionConjugateMultiply}

|V\cdot V^*| &= |V|^2 \\

\measuredangle \left(V\cdot V^*\right) &= 0

\end{aligned}

\end{equation}