# Introduction

Signal modulation involves changes made to sine waves in order to encode information. The mathematical equation representing a sine wave is as follows:

**Figure 1: Equation of a Sine Wave**

If we think about possible sine wave parameters that we can manipulate‚ the equation above makes it clear we are limited to making changes to the amplitude‚ frequency‚ and phase of a sine wave to encode information. Frequency is simply the rate of change of phase of a sine wave (frequency is the first derivative of phase)‚ so these two components of the sine wave equation can be collectively referred to as the phase angle. Therefore‚ we can represent the instantaneous state of a sine wave with a vector in the complex plane containing amplitude (magnitude) and phase coordinates in a polar coordinate system.

**Figure 2. Polar Representation of a Sine Wave**

In the graphic above‚ the distance from the origin to the black point represents the amplitude (magnitude) of the sine wave‚ and the angle from the horizontal axis represents the phase. Thus, the distance from the origin to the point will remain fixed as long as the amplitude of the sine wave is not changing (modulating). The phase of the point will change according to the current state of the sine wave. For example‚ a sine wave with a frequency of 1 Hz (2∏ radians/second) rotates counter-clockwise around the origin at a rate of one revolution per second. If the amplitude doesn’t change during one revolution, the dot maps out a circle around the origin with radius equal to the amplitude along which the point will travel at a rate of one cycle per second.

Because phase is a relative measurement, imagine that the phase reference used is a sine wave of frequency equal to the sine wave that is being represented by the amplitude and phase points. If the reference sine wave frequency and the plotted sine wave frequency are the same, then the rate of change that the phase of the two signals experience will be the same, and the rotation of the sine wave around the origin will become stationary. In this case, a single amplitude/phase point can be used to represent a sine wave of frequency equal to the reference frequency. Any phase rotation around the origin indicates a frequency difference between the reference sine wave and the sine wave being plotted.

All the concepts discussed above apply to I/Q data, and in fact, I/Q data is merely a translation of amplitude and phase data from a polar coordinate system to a cartesian (X,Y) coordinate system. Using trigonometry the polar coordinate sine wave information can be converted into cartesian I/Q sine wave data. These two representations are equivalent and contain the exact same information, just in different forms. This equivalence is show in Figure 3.

**Figure 3. I and Q Represented in Polar Form**

The figure below shows a LabVIEW example demonstrating the relationship between polar and cartesian coordinates.