$[45$ points$]$ In a binary communication system, the receiver structure of the

communication system is given in Figure $2$ where $s(t)=A_c$ksin$(2f_{c}t+m)+$

$A_c\sqrt[\phantom{2}]{1-k^2}cos(2f_{c}t),m=1,2.$

rigure $2$: block alagram or tne system.

a$)$ Find the possible signal levels $r_k$ and noise power observed at the input of the decision

circuit.

b$)$ For equiprobable bits, find the error probability $P_e$ in terms of $Q(*),E_b,N_0,k.$

c$)$ Find the error probability $P_{e|}$ assuming that the extracted carrier is not perfectly

matched to the received signal and has a phase shift of $.$ Then, find the $\frac{E_b}{N_0}$ ratio for

$=\frac{}{4}$ and $k={10}^{-1}.$ Find $P_{e|}$ for $=\frac{}{2}$ and explain your results considering both

the error probability expression and the expression of $r_k.$

d$)$ If $$ follows uniform distribution over the interval $0,,$ find an upper bound for the

average error probability in terms of $I_n(*),E_b,N_0,k,$ where $I_n(*)$ is the $n-$th order

modified Bessel function of the first kind. Plot and compare the $P_{e|}$ and the upper

bound for the average error probability in MATLAB.

$($ Hint: $Q(x)\frac{1}{2}{e}^{-\frac{x^2}{2}},I_n(x)=\frac{1}{}{\displaystyle \underset{0}{\overset{}{}}}cos(n)e^{xcos}d)$