Help in analytical expression of BER for M-QAM [duplicate]

I am new in learning deriving BER expression. I am familiar for the case of BPSK in AWGN but cannot understand how the BER expression in Eq(12) in this paper, An Enhanced Spectral Efficiency Chaos-Based Symbolic Dynamics Transceiver Design download link

tutorial presents the BER expression for M-QAM in AWGN.

Do you have any reference on how to analytically derive the BER for QAM and Rayleigh in AWGN for this case? How can I derive?

• I know your question has some nuances, but really, you're asking us for a derivation of BET for a QAM modulation, and that's been covered by the question I'm flagging this a duplicate of. I recommend going through the answers, and then asking a new, more precise, question. Oct 24 '16 at 10:31
• @MarcusMüller: the BER expression in the paper that I linked to is for QAM in AWGN. This expression is different from the one that you have linked to. Secondly it is also unclear to me what is the BER for MQAM with Rayleigh channel coefficients and in AWGN noise. I can start to derive for my own if the initial steps are given in a book or somewhere else instead of me just wondering if at all the BER exists for MQAM with Rayleigh r.v and AWGN noise.
– SKM
Oct 24 '16 at 17:14
• The answers Peter and I referred to explain how to generally approach the calculation based on noise PDFs and signal strength PDFs. The rest is just writing pages full of math. Oct 24 '16 at 17:15

You need to read a good book to form a solid foundation first and then read research papers to understand advanced concepts. Eq (12) comes through a standard procedure for BER calculations. See Digital Communications by John Proakis, for example.

• raymaps.com/index.php/… gives the BER of MQAM in rayleigh fading and raymaps.com/index.php/qam-theoretical-ber is the BER of MQAM in AWGN. I want to apply QAM modulation with the channel taps modeled as Rayleigh random variables (dsp.stackexchange.com/questions/35004/… is the link to previous Question in continuation to this one that I had asked)
– SKM
Oct 24 '16 at 6:18
• So, to show the BER performance which analytical expression should I appy since BER for M QAM in AWGN and BER for M QAM in AWGN and RAyleigh are different.
– SKM
Oct 24 '16 at 6:18
• Yes. Both BERs are different, very different. Since you want to apply QAM to a Rayleigh fading channel, you should refer to the article by raymaps.
– QMC
Oct 24 '16 at 10:34
• Thank you once again for your response but this time it is unclear. Firstly, because the BER expression in the paper that I linked to is for QAM in AWGN. This expression is different from the one in the raymaps link. Secondly it is also unclear to me what is the BER for MQAM with Rayleigh channel coefficients and in AWGN noise. I have not found this in any tutorial or book. Should I use the BER for Rayleigh in MQAM? These answers would allow me to get a starting point to begin studying instead of being in the dark.
– SKM
Oct 24 '16 at 17:11
• In Rayleigh channel modeling setup, AWGN is implicitly present. So you should fo by raymaps article and their expression does include both the channel and noise.
– QMC
Oct 24 '16 at 22:29