# QAM Theory and Formulas

## - the basic theory and relevant formulas or equations behind QAM quadrature amplitude modulation give additional insight into its operation.

Quadrature Amplitude Modulation, QAM Tutorial Includes:
Quadrature amplitude modulation, QAM basics     QAM theory     QAM formats     QAM modulators & demodulators

Modulation formats:     Modulation types & techniques     Amplitude modulation     Frequency modulation     Phase modulation

The basic QAM theory aims to express the operation of QAM, quadrature amplitude modulation using some mathematical formulae.

Fortunately it is possible to express some of the basic QAM theory in terms of relatively simple equations that provide some insight into what is actually happening within the QAM signal.

## QAM theory basics

Quadrature amplitude theory states that both amplitude and phase change within a QAM signal.

The basic way in which a QAM signal can be generated is to generate two signals that are 90° out of phase with each other and then sum them. This will generate a signal that is the sum of both waves, which has a certain amplitude resulting from the sum of both signals and a phase which again is dependent upon the sum of the signals.

If the amplitude of one of the signals is adjusted then this affects both the phase and amplitude of the overall signal, the phase tending towards that of the signal with the higher amplitude content.

As there are two RF signals that can be modulated, these are referred to as the I - In-phase and Q - Quadrature signals.

The I and Q signals can be represented by the equations below:

It can be seen that the I and Q components are represented as cosine and sine. This is because the two signals are 90° out of phase with one another.

Using the two equations it is possible to express the signal as:.

$\mathrm{cos}\left(\alpha +\beta \right)=\mathrm{cos}\left(\alpha \right)\mathrm{cos}\left(\beta \right)-\mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right)$

Using the expression A cos(2πft + Ψ) for the carrier signal.

$A\mathrm{cos}\left(2\pi ft+\Psi \right)=I\mathrm{cos}\left(2?ft\right)-Q\mathrm{sin}\left(2\pi ft\right)$

Where f is the carrier frequency.

This expression shows the resulting waveform is a periodic signal for which the phase can be adjusted by changing the amplitude either or both I and Q. This can also result in an amplitude change as well.

Accordingly it is possible to digitally modulate a carrier signal by adjusting the amplitude of the two mixed signals.