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Unformatted text preview: EE263 Autumn 201112 Prof. S. Boyd EE263 homework problems Lecture 2 Linear functions and examples 2.1 A simple power control algorithm for a wireless network. First some background. We consider a network of n transmitter/receiver pairs. Transmitter i transmits at power level p i (which is positive). The path gain from transmitter j to receiver i is G ij (which are all nonnegative, and G ii are positive). The signal power at receiver i is given by s i = G ii p i . The noise plus interference power at receiver i is given by q i = + summationdisplay j negationslash = i G ij p j where > 0 is the selfnoise 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 i = s i /q i . 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 i to ensure that S i (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 t = 0 , 1 , 2 ,... . Thus the quantities p , q , and S are discretetime signals, so for example p 3 (5) denotes the transmit power of transmitter 3 at time epoch t = 5. What wed like is S i ( t ) = s i ( t ) /q i ( t ) = , where > 1 is an SINR safety margin (of, for example, one or two dB). Note that increasing p i ( t ) (power of the i th transmitter) increases S i but decreases all other S j . A very simple power update algorithm is given by p i ( t + 1) = p i ( t )( /S i ( t )) . (1) This scales the power at the next time step to be the power that would achieve S i = , if the interference plus noise term were to stay the same. But unfortunately, changing the transmit powers also changes the interference powers, so its not that simple! Finally, we get to the problem. (a) Show that the power control algorithm ( ?? ) can be expressed as a linear dynamical system with constant input, i.e. , in the form p ( t + 1) = Ap ( t ) + b, where A R n n and b R n are constant. Describe A and b explicitly in terms of ,, and the components of G . (b) matlab simulation. Use matlab to simulate the power control algorithm ( ?? ), starting from various initial (positive) power levels. Use the problem data G = 1 . 2 . 1 . 1 2 . 1 . 3 . 1 3 , = 3 , = 1 . 2 , = 0 . 01 . Plot S i and p as a function of t , and compare it to the target value . Repeat for = 5. Comment briefly on what you observe. Comment: Youll soon understand what you see....
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