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.

3 A very simplepower update algorithm is given bypi(t+ 1) =pi(t)( /Si(t)).(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.

4 Use the problem dataG= 3 , = 3, = , = a function oft, and compare it to the target value . 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.

5 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). 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).

6 Finally, the problem: Express each of these models as a Linear dynamical system with inputuand outputy. 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.

7 (Explicitlydescribe how you get the coefficientsAijfromf, and then verify thatf(x) =Axfor anyx Rn.) Is the matrixAthat representsfunique? 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.

8 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.(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.

9 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).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.

10 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. 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.