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,
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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
\[
Y=a x+N
\]
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 \( \sigma^{2} \). For example, if the transmitter broadcasts the signal \( x=1 \) and \( a=0.1 \), then \( Y=a x+N=0.1+N \). 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 \( f_{Y}\left(y ; H_{0}\right) \) and \( f_{Y}\left(y ; H_{1}\right) \).
(b) Construct the \( \log \)-likelihood ratio test for general \( a, \sigma^{2} \), and critical value \( \xi \).
(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 0.05 . Assume noise variance \( \sigma^{2}=1 / 4 \) and critical value \( \xi=1 \).
Discussion: In practice, the value of \( a \) depends on numerous factors. For instance, a common propagation model is as follows:
\[
a=d^{-\alpha} P_{\mathrm{TX}}
\]
where \( d \) is the distance between the ADS-B receiver and the transmitter; \( d^{-\alpha} \) is the pathloss, which represents the attenuation of the transmitted signal as it propagates through space; \( \alpha \) is the pathloss exponent (typically, \( \alpha>2 \) ); and \( P_{\mathrm{TX}} \) is the transmission power ( \( P_{\mathrm{TX}}>0 \) ). For example, if the transmitter broadcasts the signal \( x=1 \) with transmission power \( P_{\mathrm{TX}}=1 \), the receiver is 100 meters from the transmitter, and the pathloss exponent is 3, then \( Y= \) \( d^{-\alpha} P_{\mathrm{TX} x}+N=100^{-3}+N=10^{-6}+N \). Notice that the signal attenuation becomes larger as \( d \) increases, such that at large enough distances, the received signal eventually gets lost in \( p \) the noise.
