The Next Generation Air Transportation System $($NextGen$)$ will transform America$$s air traffic control system. Unlike the current air traffic control system, which is based on radar, $1$ NextGen relies on Global Positioning System $($GPS$)$ satellite signals to determine each aircraft$$s precise position in the sky$.$ Aircraft then use Automatic Dependent Surveillancebroadcast $($ADS$-$B$)$ technology to periodically broadcast their location information to other aircraft and air traffic control towers in the vicinity. Suppose that an ADS$-$B transmitter broadcasts a binary signal x in $\{1,1\}$ and that an ADS$-$B receiver receives the signal as Y $=$ ax $+$ N$,$ where a $>0$ represents the strength of the received signal after propagating through space and N is additive noise, which is normally distributed with mean $0$ and variance $2.$ For example, if the transmitter broadcasts the signal x $=1$ and a $=0\mathrm{.}1,$ then Y $=$ ax $+$ N $=0\mathrm{.}1+$ N$.$ Notice that the signal attenuation becomes larger as a decreases. At large attenuations $($small values of a$),$ the received signal eventually gets lost in the noise $($i$.$e$.,$ the received signal will eventually become smaller than the noise N$,$ which will make it harder for the receiver to correctly interpret it$).$ The ADS$-$B receiver uses a likelihood ratio test to decode the received message based on the following binary hypothesis testing problem: $$ H$0$: Y $=$ a $+$ N $$ H$1$: Y $=$a $+$ N Note that H$0$ is the hypothesis that the ADS$-$B transmitter broadcasted a $1($i$.$e$.,$ x $=1)$ and H$1$ is the hypothesis that the ADS$-$B transmitter broadcasted a $-1($i$.$e$.,$ x $=1).($a$)$ Determine the likelihood functions fY $($y; H$0)$ and fY $($y; H$1).($b$)$ Construct the log$-$likelihood ratio test for general a$,$ $2,$ and critical value $.($c$)$ Assume that you are designing an ADS$-$B transmitter. Determine the minimum value of a such that the false rejection and false acceptance probabilities at the receiver are less than or equal to $0\mathrm{.}05.$ Assume noise variance $2=1/4$ and critical value $=1.$