# Beginner level confusion regarding symbols used in modulation

• For 4-QAM symbol sequence, each symbol can be represented by $k=\log_2(4) = 2$ bits. 4-QAM constellation can be generated using s = round(rand(1,T)*2 -1)

• For 16-QAM, the symbol sequence takes 4 amplitude levels viz. $\{-3,-1,+1,+3\}$. Each symbol is represented by $k=\log_2(16) = 4$ bits.

• For 64-QAM, the symbol sequence takes 8 amplitude levels viz. $\{-7,-5,-3,-1,+1,+3,+5,+7\}$. Each symbol is represented by $k=\log_2(64) = 6$ bits. 64-QAM constellation can be generated using : s = randi(8,1,T)*2 -9) +j*randi(8,1,T)*2 -9)) / sqrt(42)

Question: In general, for M-QAM modulation, what is the relation between $M$ and the type of symbols and the number of symbols. For ex: 64-QAM, why do we need 8 amplitude levels and why these symbols?

An M-QAM modulation contains M different constellation symbols. Hence, in one symbol you can encode $\mu=\log_2(M)$ bits. (i.e. 6 bits for 64-QAM for example). Now, the bits are equally distributed along the real and the imaginary part of the constellation, i.e. you have $\frac{\mu}{2}$ bits for the real and imaginary part. To encode $\frac{\mu}{2}$ bits, you need $2^{\frac{\mu}{2}}$ different amplitude levels (i.e. 2^3=8 for 64QAM).
Another explanation for the relation of amplitude levels and M: Your M constellation points are arranged in a square. Hence, each side of the square contains $\sqrt{M}$ elements. These are your number of different amplitude levels.
• @SrishtiM This is another question and you shoul open a new thread for this one (or look in the existing questions). Essentially, $s_n$ can be anything describing your transmit signal in baseband. So, if you want you can directly transmit QAM symbols, but usually, you would filter them before to limit the bandwidth. – Maximilian Matthé Feb 16 '17 at 7:21