3 Problem 3 First some background. We consider a network of n transmitter /receiver pairs. Transmitter transmits at power level p; (which is positive). The path gain from transmitter j to receiver is G; ; (which are all non-negative, and G, ; are positive). The signal power at receiver i is given by s; = G;;p; . The noise plus interference power at receiver i is given by gi=0+ Gip J#i where o > 0 is the self-noise power of the receivers (assumed to be the same for all receivers). The signal to interference plus noise ratio (SINR) at receiver i is defined as S; = s;/q; . For signal reception to occur, the SINR must exceed some threshold value (which is often in the range 3 10). Various power control algorithms are used to adjust the powers p; to ensure that S; > v (so that each receiver can receive the signal transmitted by its associated transmitter). In this problem, we consider a simple power control update algorithm. The powers are all updated synchronously at a fixed time interval, denoted by = 0,1,2,... Thus the quantities p,q, and S are discrete-time signals, so for example ps(5) denotes the transmit power of transmitter 3 at time epoch = 5. What we'd like is Si(t) = si(t)/ai(t) = ay where o > 1 is SINR safety margin. Note that increasing p; increases S;, but decreases all other S;. A very simple power update algorithm is given by pi(t +1) = pi(t)(av/Si(t))- This scales the power at the next time step to be the power that would achieve S; = ay, if the interference plus noise term were to stay the same. But unfor- tunately, changing the transmit powers also changes the interference powers, so it's not that simple! Finally, we get to the problem. 1. Show that the power control algorithm in (2) can be expressed as a linear dynamical system with constant input, i.e., in the form p(t+1) = Ap(t) +b (2) where p(t) is n-dimensional vector, collection of power {p;(t)}", of in- dividual transmitter. Describe A and b explicitly in the terms of o, 7, . What is the dimension of A and 0?7 1 02 01 2. Use the following data G = |0.1 2 01|, a=12,v=3, 0 =0.01 03 01 3 for power update equation (2). Plot S;(t) (using MATLAB or Python) fori =1,...,3 as a function of and compare it to target value ary. Now repeat the experiment for v = 5. You can assume that the initial power levels are 0.1 (i.e., p;(0) = 0.1). Comment on your observation. You will understand the observation later in the course