# 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. Feb 16, 2017 at 7:21