Question

A 4-ary signal constellation is shown below. It is given that ${\Phi }_{1}and{\Phi }_2$ are orthonormal basis functions. Assume the four symbols are equiprobable. This modulated signal is transmitted through an AWGN channel with noise $PSD\frac{N_0}{2}={10}^{-2}W/Hz$.



If the approximate probability of symbol error of the system 2Q(k) when the signal is received by a correlator receiver with exact phase synchronization and optimum detection then the value of k is |Q(x) indicates Q-function and assume $Q^2\left(x\right)<<1|$

Answer 1

For 4-ary PSK $P_e=2Q\left(\sqrt{\frac{d_{min}^2}{2N_0}}\right)$

$d_{min}=\sqrt{(2-2{)}^2+(2(-2){)}^2}$

$d_{min}=\sqrt{4^2}=4$

$P_e=2Q\left(\sqrt{\frac{4\times 4}{4\times \frac{N_0}{2}}}\right)$

$=2Q\left(\sqrt{\frac{8}{N_0}}\right)$

$N_0=2\times {10}^{-2}W/Hz$

$P_e=2Q\left(\sqrt{\frac{800}{2}}\right)=2Q\left(\sqrt{400}\right)=2Q\left(20\right)$

So, k = 20