[SOLVED] Average energy of square M-QAM constellation

[blade88]

Hello,

In page 58 of the following book,

https://web.stanford.edu/group/cioffi/ee379a/course_reader/chap1.pdf

the average energy of a square MQAM constellation is calculated.

I don't understand why the \[\sqrt{M }\] are introduced in (1.227).

Could someone please explain?

Thank you.

 

[andre_luis]

I´m not sure, but take a look on ML property stated on page 17 and its resulted equation (1.32). I guess could be due to the dismemberment of the two-dimensional sum into 2 separated sums. I think.

 

[blade88]

Thank you for your response.

I think I understand what's happening.

He adds the \[\sqrt{M}\] terms, because he associates MQAM with MPAM modulation, considering also all symbols equally likely to be transmitted, as you suggested.

Specifically, he views the M-QAM system as two separate M-PAM systems with signal constellations of size L=\[\sqrt{M}\], one transmitted over the in phase component and the other transmitted over the quadrature component.

Therefore, the square MQAM constellation equals \[\sqrt{M}\] \[\times\] \[\sqrt{M}\].

In order to calculate the average energy of the MQAM system, you calculate the energy of the separate MPAM systems.

So, for each component we add \[\sqrt{M}\] points, multiplying the sum by \[\sqrt{M}\],
because each dimension has \[\sqrt{M }\] series of points.

Finally, each component is divided by M, so as to find the average energy of the MQAM system.