Question 8.2: Consider a QAM (Quadrature Amplitude Modulation) constellati......
Consider a QAM (Quadrature Amplitude Modulation) constellation of N = M² signals, M even, regularly deployed on a X Y Cartesian grid. Any point of the constellation has amplitude i−1/2 and j − 1/2 along the X or Y axis, respectively, where −M/2 + 1 < i < M/2 and −M/2 + 1 < j < M/2 are integer numbers. Evaluate the PAPR for such a signal.
First of all, notice that a scale factor common to all constellation points has no influence on the PAPR or the crest factor. The RMS signal value is defined as:
where the factor 1/N² normalizes the power of the symbols, assumed here equiprobable, while the factor 4 accounts for the number of quadrants. We thus have:
\text{RMS}^2=\frac{M^2-1}{6}=\frac{N-1}{6} .
The peak value is:
\text { PEAK }=\sqrt{2} \cdot\left(\frac{M}{2}-\frac{1}{2}\right)=\frac{M-1}{\sqrt{2}} .
The crest factor C_F in term of RMS value is therefore:
C_F=\frac{ \text{PEAK} }{ \text{RMS} }=\sqrt{\frac{6}{M^2-1}} \frac{M-1}{\sqrt{2}}=\sqrt{3} \sqrt{\frac{M-1}{M+1}}=\sqrt{3} \frac{\sqrt{N}-1}{\sqrt{N-1}} .
For example we have: 4QAM, N=4 \rightarrow C_F=1,0\text{ dB} (trivial since the four points are on a circle, hence constant power, non-varying envelope, no PAPR); 16QAM, N = 16