NextGen relies on Global Positioning System $($GPS$)$ satellite signals to determine each air$-$

craft$$s precise position in the sky$.$ Aircraft then use Automatic Dependent Surveillance$-$

broadcast $($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 $=1.$

Discussion: In practice, the value of a depends on numerous factors. For instance, a common

propagation model is as follows:

a $=$ d

$$PTX$,$

where d is the distance between the ADS$-$B receiver and the transmitter; d

$$ is the pathloss,

which represents the attenuation of the transmitted signal as it propagates through space;

is the pathloss exponent $($typically$,$ $>2)$; and PTX is the transmission power $($PTX $>0).$

For example, if the transmitter broadcasts the signal x $=1$ with transmission power PTX $=1,$

the receiver is $100$ meters from the transmitter, and the pathloss exponent is $3,$ then Y $=$

d

$$PTXx $+$ N $=1003+$ N $=106+$ N$.$ Notice that the signal attenuation becomes larger

as d increases, such that at large enough distances, the received signal eventually gets lost in

the noise.