Problem $3[10$ marks$]$ Consider the following measurement model:

$y_k=\sqrt[\phantom{2}]{}{s}_k{r}_k+w_k,k=1,2,$cdots,$N$

where $s_1,$cdots,$s_N$ is a known signal, $0$ is a parameter, and $r_1,$cdots,$r_N,w_1,$cdots,$w_N$ are i$.$i$.$d$.N(0,1)$ random

variables.

$($a$)$ Consider the hypotheses $H_0$:$=0$ versus $H_1$:$={}_1,$ where ${}_1>0$ is known. Derive the structure of

the Neyman$-$Pearson detector. Provide expressions for $P_F$ and $P_D.$ You can leave your answers in terms of

appropriate integrals.

$($b$)$ Now consider the hypotheses $H_0$:$=0$ versus $H_1$:$>0.$ Under what conditions does a UMP test exist?

Find the UMP test under these conditions.

$($c$)$ Derive the GLRT for this problem assuming $s_1=s_2=$cdots$=s_N=s.$Convert this NFA to a DFA. Please post the image big enough so that I can see all steps $/$ the answer. Thanks. Problem $3(10$ marks$)$: Consider the following NFA to a DFA. Clearly mark each DFA state with the corresponding NFA states.