Communication Systems/Quadrature Amplitude Modulation

The quadrature amplitude modulation (QAM) system of modulation is the most popular M-ary scheme.

Definition

Consider the case of a system with two carrier waves instead of a single carrier wave as we have considered with modulation schemes so far. One is a sine wave, and the other is a cosine wave of the same frequency. Since these two waves are orthogonal we can use them simultaneously in a single channel without losing the information of either. If both waves have the same frequency f we can write out the equation for a generic symbol, s:

$s(t)=A_k\sin (ft)+B_k\cos (ft)$

In this way, we can create multiple symbols by simply changing the values of A and B. This equation can be broken up into two parts:

• $A_k\sin (ft)$ Which is called the "in-phase" component of the equation.
• $B_k\cos (ft)$ Which is called the "quadrature" component of the equation.

An equation which is written as a sum of a sine plus a cosine is said to be in "quadrature form". If we combine the two components into a single waveform as such:

$s(t)=\sqrt{A_k^2+B_k^2}\cos (ft+{\tan }^{-1}(B_k/A_k))$

This form is called the "Polar Form" of the equation.

Constellation Plots

If we make a graph with the X axis being the values for A, and the Y axis being the values for B, we get what is called a Constellation Plot. These plots are called constellation plots due to the similiarity in shape and layout with astronomical star charts. The A and B values for each symbol are plotted (the "stars") and various measurements between them are used to determine information from the system. On a constellation plot, we can see a number of rules:

1. The further apart the points are on the constellation plot, the less likely they are to be mistaken for each other in the presence of noise.
2. The closer the points are to the origin, the less power it takes to send.
3. The more points there are, the faster the data rate (bit rate) at a fixed symbol rate (more symbols)
4. The fewer points there are, the simpler and cheaper the hardware necessary to distinguish between them (fewer symbols, fewer thresholds in the receiver).

For these reasons there is no single "best" constellation plot, but it is up to the engineer to pick the points that are best for the system. In other words, trade offs need to be made between speed, performance, and cost of hardware. These tradeoffs can be made by placing the constellation points at different locations on the constellation plot.