Simplify Complex Operations

As compared to real numbers, a new dimension represented by the imaginary number $j$ is added in the complex representation. A complex number $V$ is simply a combination of two real numbers, one on the real axis and one on the imaginary axis, as shown in the figure below. We call the x-component $V_I$ and the y-component $V_Q$.

$$\begin{equation}\begin{aligned} V_I &= A\cos \theta \\ V_Q &= A\sin \theta \end{aligned} \end{equation}$$

where $A$ is the distance from the origin (magnitude) and $\theta$ is the angle from the x-axis.

In complex notation, the equation above can be written as:

$$V=Ae^{j\theta}$$

Magnitude and phase

In the polar representation of complex numbers, the magnitude of $V$ in an $IQ$ plane is defined from Eq (1) as

$$\begin{equation} |V| =A= \sqrt{V_I^2 + V_Q^2} \end{equation}$$

where we have used the property $\cos^2\theta + \sin^2\theta = 1$. Defining the phase $\theta=\measuredangle V$ is a little trickier. It’s tempting to define it as $\tan^{-1} V_Q/V_I$. However, there is a problem here, as illustrated below:

$$\begin{align*} \tan^{-1} \frac{+V_Q}{+V_I} ~&=~ \tan^{-1} \frac{-V_Q}{-V_I} \quad$$ ...