EE263 homework problems Lecture 2 – Linear functions and ...

1 EE263 Autumn 2007-08 Prof. S. BoydEE263 homework problemsLecture 2 Linear functions and simple power control algorithm for a wireless some background. We considera network ofntransmitter/receiver pairs. Transmitteritransmits at power levelpi(whichis positive). The path gain from transmitterjto receiveriisGij(which are all nonnegative,andGiiare positive). The signal power at receiveriis given bysi=Giipi. The noise plusinterference power at receiveriis given byqi= +Xj6=iGijpjwhere >0 is the self-noise power of the receivers (assumed to be the same for all receivers).Thesignal to interference plus noise ratio(SINR) at receiveriis defined asSi=si/qi. Forsignal reception to occur, the SINR must exceed some threshold value (which is often inthe range 3 10).

2 Variouspower control algorithmsare used to adjust the powerspitoensure thatSi (so that each receiver can receive the signal transmitted byits associatedtransmitter). In this problem , we consider a simple power control update algorithm. Thepowers are all updated synchronously at a fixed time interval, denoted byt= 0,1,2,.. Thusthe quantitiesp,q, andSare discrete-time signals, so for examplep3(5) denotes the transmitpower of transmitter 3 at time epocht= 5. What we d like isSi(t) =si(t)/qi(t) = where >1 is an SINR safety margin (of, for example, one or two dB). Note that increasingpi(t) (power of theith transmitter) increasesSibut decreases all otherSj. A very simplepower update algorithm is given bypi(t+ 1) =pi(t)( /Si(t)).

3 (1)This scales the power at the next time step to be the power thatwould achieveSi= ,if the interference plus noise term were to stay the same. Butunfortunately, changing thetransmit powers also changes the interference powers, so it s not that simple! Finally, we getto the problem .(a) Show that the power control algorithm (1) can be expressed as a Linear dynamical systemwith constant input, , in the formp(t+ 1) =Ap(t) +b,whereA Rn nandb Rnare constant. DescribeAandbexplicitly in terms of , , and the components ofG.(b)Matlab matlab to simulate the power control algorithm (1), startingfrom various initial (positive) power levels. Use the problem dataG= 3 , = 3, = , = a function oft, and compare it to the target value.

4 Repeat for = 5. Comment briefly on what you :You ll soon understand whatyou equations for a Linear mechanical equations of motion of a lumped me-chanical system undergoing small motions can be expressed asM q+D q+Kq=fwhereq(t) Rkis the vector of deflections,M,D, andKare themass,damping, andstiffnessmatrices, respectively, andf(t) Rkis the vector of externally applied forces. AssumingMis invertible, write Linear system equations for the mechanical system, with statex= [qT qT]T,inputu=f, and outputy= standard time-series time series is just a discrete-time signal, , a functionfromZ+intoR. We think ofu(k) as the value of the signal or quantityuat time (orepoch)k. The study of time series predates the extensive study of state-space Linear systems, and isused in many fields ( , econometrics).

5 Letuandybe two time series (input and output,respectively). The relation (ortime series model )y(k) =a0u(k) +a1u(k 1) + +aru(k r)is called amoving average (MA) model ,since the output at timekis a weighted average ofthe previousrinputs, and the set of variables over which we average slides along with model is given byy(k) =u(k) +b1y(k 1) + +bpy(k p).This model is called anautoregressive (AR) model , since the current output is a Linear com-bination of ( , regression on) the current input and some previous values of the widely used model is theautoregressive moving average (ARMA) model , which com-bines the MA and AR models:y(k) =b1y(k 1) + +bpy(k p) +a0u(k) + +aru(k r).Finally, the problem : Express each of these models as a Linear dynamical system with inputuand outputy.

6 For the MA model , use statex(k) = u(k 1)..u(k r) ,and for the AR model , use statex(k) = y(k 1)..y(k p) .You decide on an appropriate state vector for the ARMA model .(There are many possiblechoices for the state here, even with different dimensions. We recommend you choose a state2for the ARMA model that makes it easy for you to derive the state equations.)Remark:multi-input, multi-output time-series models ( ,u(k) Rm,y(k) Rp) are readily handledby allowing the coefficientsai,bito be Linear functions as matrix thatf:Rn Rmis that there is a matrixA Rm nsuch that for allx Rn,f(x) =Ax. (Explicitlydescribe how you get the coefficientsAijfromf, and then verify thatf(x) =Axfor anyx Rn.) Is the matrixAthat representsfunique?

7 In other words, if A Rm nis anothermatrix such thatf(x) = Axfor allx Rn, then do we have A=A? Either show that thisis so, or give an explicit Linear functions associated with a convolution thatuandyare scalar-valued discrete-time signals ( , sequences) related via convolution:y(k) =Xjhju(k j), k Z,wherehk R. You can assume that the convolution iscausal, ,hj= 0 whenj <0.(a)The input/output (Toeplitz) thatu(k) = 0 fork <0, and defineU= u(0)u(1)..u(N) , Y= y(0)y(1)..y(N) .ThusUandYare vectors that give the firstN+ 1 values of the input and outputsignals, respectively. Find the matrixTsuch thatY=TU. The matrixTdescribes thelinear mapping from (a chunk of) the input to (a chunk of) the called theinput/output or Toeplitz matrix (of sizeN+ 1) associated with the convolution system.

8 (b)The Hankel assume thatu(k) = 0 fork >0 ork < Nand letU= u(0)u( 1)..u( N) , Y= y(0)y(1)..y(N) .HereUgives thepast inputto the system, andYgives (a chunk of) the resulting futureoutput. Find the matrixHsuch thatY= called the Hankel matrix (of sizeN+ 1) associated with the convolution representation of polynomial can represent a polynomial of degreeless thann,p(x) =an 1xn 1+an 2xn 2+ +a1x+a0,as the vector [a0a1 an 1]T Rn. Consider the Linear transformationDthat differentiatespolynomials, ,Dp=dp/dx. Find the matrixDthat representsD( , if the coefficientsofpare given bya, then the coefficients ofdp/dxare given byDa). Consider the (discrete-time) Linear dynamical systemx(t+ 1) =A(t)x(t) +B(t)u(t), y(t) =C(t)x(t) +D(t)u(t).

9 Find a matrixGsuch that y(0)y(1)..y(N) =G x(0)u(0)..u(N) .The matrixGshows how the output att= 0,..,Ndepends on the initial statex(0) andthe sequence of inputsu(0),..,u(N). sparsity patterns.(a) A matrixA Rn nistridiagonalifAij= 0 for|i j|>1. Draw a block diagram ofy=AxforAtridiagonal.(b) Consider a certain Linear mappingy=AxwithA Rm n. Foriodd,yidepends onlyonxjforjeven. Similarly, forieven,yidepends only onxjforjodd. Describe thesparsity structure ofA. Give the structure a reasonable, suggestive and signal flow graphs.(a) FindA R2 2such thaty=Axin the system below:+ (b) FindB R2 2such thatz=Bxin the system below:++++ this two ways: first, by expressing the matrixBin terms ofAfrom the previouspart (explaining why they are related as you claim); and second, by directly evaluatingall possible paths from eachxjto the matrixAfor the mass/force example in the Lecture notes.

10 Forn= 4, find a specific input force sequencexthat moves the mass to final position 1 and finalvelocity force consider the mass/force example in the Lecture notes, andin exer-cise 10, withn= 4, and the requirement that the final position is 1 and final velocity is speaking, you have four variables and two equations, and therefore two extra degreesof freedom. In this problem you use the extra degrees of freedom to achieve other objectives, , minimize some cost functions that are described below.(a) Findfthat meets the specifications and minimizes the sum of squares of the forces, ,f21+f22+f23+f24.(b) Findfthat meets the specifications and minimizes the maximum force applied, ,max{|f1|,|f2|,|f3|,|f4|}.