SOLUTIONS TO HOMEWORK ASSIGNMENT 1

Transcription

1 ECE 559 Fall 007 Handout # 4 Septemer 0, 007 SOLUTIONS TO HOMEWORK ASSIGNMENT. Vanderkulk s Lemma. The complex random variale Z X + jy is zero mean and Gaussian ut not necessarily proper. Show that E expjνz exp ν EZ /, where ν can e assumed to e real though this is not really necessary. This result is known as Vanderkulk s lemma and is similar to the characteristic function result for a real Gaussian random variale. Note that this gives the interesting result that E expjνz when Z is proper complex Gaussian. Solution. By definition p Z z p X,Y x,y p X Y x yp Y y Since X and Y are zero mean and jointly Gaussian, X is conditionally Gaussian given Y, with conditional mean and variance: m y σ X ρ y σ Y, σ σ X ρ where ρ EXY ]/σ X σ Y. Now, we have the following result on the characteristic function of Gaussian random variales. If V is a real-valued Gaussian random variale with mean m and variance σ, then Note that holds even when θ is complex. Now, Applying to the inner expectation, we get which implies that Now apply to Y with θ ν Ee jθv ] e jθm θ σ / Ee jνz ] Ee jνx νy ] E Y EX Y e jνx ] e νy ] E X Y e jνx ] e jνm Y ν σ / Ee jνz ] e ν σ E Y e jν σx ] ρ +j Y σ Y σx ρ σ Y + j and we get Ee jνz ] e ν σ e ν σx ρ σ +j σ Y Y e ν σ X +σ Y ρσ Xσ Y j e ν EZ ]. Optional Properness of a PCG Vector. Prove Result B. in the class notes: Let Y Y I + jy Q e proper complex and Gaussian, i.e. Y I,Y Q are jointly Gaussian. Then p Y y : p Y I Y Q y I,y Q { } π n Σ Y exp y m Y Σ Y y m Y Note: This prolem is optional, and is meant for rave souls who wish to explore the dark world of Matrix Algera.. The solution can e found in the paper y Neeser and Massey. c V. V. Veeravalli, 007

2 3. Cellular Area Reliaility. Read Section.5. of the notes on area reliaility and derive equation.9, i.e., show that F area Qa + ln R + exp a R Q a + ln R ]. Also derive equation.3, i.e., show that F area F edge + e e Q F edge Q Q F edge F area R R 0 Qa + ln ρρdρ R a+ ln R Defining I to e the last integral in the previous equation, we get I a+ ln R a Qt et Qte t dt a+ ln R + π Qte t a dt e a R a+ lnr e t e t dt ln R Qa + ln Rea+ + a+ ln R e Qa + ln RR e a + e Q e t dt π a + ln R a+ lnr where in a we used integration y parts. Sustituting this result ack into the previous equation F area e a I Qa + ln R + e a R R From the definition of F edge we have a + ln R Q F edge. Thus Q a + ln R. Qte t dt and e a R e Q F edge ln R R e Q F edge e lnr F area F edge + e e Q F edge QQ F edge R Note that this is independent of R and a. 4. Signal strength prediction. Consider a moile that is moving on a straight line path not necessarily radial at a constant velocity v. In order to make handoff decisions, the moile periodically takes pilot power measurements from neighoring BS s. Let us assume that these power measurements are averaged to remove multipath fluctuations, so the resulting sampled measurements only have a median component and shadow fading. The k-th sample value of the pilot power in dbm from a particular BS is given y: P r,k dbm] P r d k + Z k A t B log d k + Z k, c V. V. Veeravalli, 007

3 where d k is the distance from the BS at the k-th sampling time, and A t includes the transmitted pilot power. Note that the d k values are not necessarily equally spaced. Let us assume isotropic shadow fading with exponential ACF. Since the velocity vector is constant, the random process {Z k,k,,...} is a stationary first-order auto-regressive AR process with EZ k Z k+m ] σ Za m, where a exp vt s /D c, t s is the sampling time, and D c is the correlation distance. Handoff decisions are often ased on signal strength prediction. Our goal here is to find the MMSE predictor of P r,k+ ased on P r,,p r,,...,p r,k. a Under the assumption that the d k values are known, show that ˆP MMSE r,k+ EP r,k+ P r,,p r,,...,p r,k ] ap r,k + aa t B log and that the corresponding mean-squared error MSE VarP r,k+ P r,,p r,,...,p r,k ] a σ Z. dk+ Hint: You don t need to solve the Yule-Walker equations to find the MMSE solution in this special case. Use the AR- property of the {Z k } to find the solution directly. P r,k+ A t + B log d k+ + Z k+ A t + B log d k+ + az k + σ Z a W k aa t + B log d k + Z k + aa t + B log d k+ ab log d k + σ Z a W k d a k Thus, P r,k+ ap r,k + aa t + B log d k+ d a k + σ Z a W k. By the AR- model construction, W k is independent of P r,k,...,p r,. This implies that P r,k+ given P r,k is independent of P r,k,...,p r,, and Similarly, E P r,k+ P r,k,...,p r, ] E P r,k+ P r,k ] ap r,k + aa t + B log d k+ d a. k ] Var P r,k+ P r,k,...,p r, ] Var P r,k+ P r,k ] E σ Z a W k σz a Discuss how you might address the prediction prolem if the d k values were unknown. The prediction prolem is consideraly harder when d k is unknown. One way to approach the prolem is to use a velocity estimate ased on measurements of doppler frequency to first otain an estimate of the distance ρ etween consecutive samples. We may then use a curve fitting technique to otain an estimate of d k ased on all of the signal strength measurements. Thus, the est estimate of P r,k+ ased on P r,k,...,p r, will e a function of all the measurements, not just the most recent one. c V. V. Veeravalli, 007 3

4 5. Moments of lognormals. Suppose X is a lognormal random variale with mean m X and second moment δ X, and suppose Y 0log X has mean m X and variance σ Y. a Show that βσy m X exp expβm Y and δ X exp βσ Y expβm Y, where β ln0/0. The characteristic function of Y is given y φ Y s Ee sy ] e sm Y + s σ Y Thus, and m X Ee βy ] φ Y β e βm Y + β σ Y δ X EX ] φ Y β e βm Y +β σ Y. Also, show that m Y 0log m X 5log δ X, and that σ Y β 0log δ X 0log m X. Taking the log of the previous equations, we get Solving for m Y and σy, we otain 0log m X ln m X β 0log δ X ln δ X β βσ Y + m Y βσ Y + m Y. m Y 0log m X 5log δ X σ Y β 0log δ X 0log m X. 6. Outage with macrodiversity. Consider a moile at the midpoint etween two ase stations in a cellular network. The received signals in db-w from the ase stations are given y where Z and Z are N0,σ random variales. P r, A t B logd/ + Z, P r, A t B logd/ + Z, We define outage with macrodiversity to e the event that oth P r, and P r, fall elow a pre-specified threshold P thresh. c V. V. Veeravalli, 007 4

5 a If Z and Z are independent, show that the outage proaility is given y where is the fade margin at the edge of the cell. P out ] Q, σ A t B logd/ P thresh P out P{P r, < P thresh } {P r, < P thresh } PP r, < P thresh PP r, < P thresh PZ < P thresh A t B logd/ PZ < P thresh A t B logd/ ] PZ < PZ < Q σ Now suppose Z and Z are correlated in the following way. Z ay + Y and Z ay + Y, where Y, Y and Y are independent N0,σ random variales, and a and are such that a +. Show that P out Q + yσ aσ P out P{aY + Y < } {ay + Y < } ] e y / π dy. P{aY + y < } {ay + y < }p Y ydy PaY < ypay < yp Y ydy πσ Q which is the desired expression. + y aσ e y σ dy Q + wσ aσ e w π dw c Compare the outage proailities of i and ii for the special case of a /, σ 8 and 5. Use a numerical integration routine for P out of ii. P out, P out, 0.36 Thus, correlation reduces diversity gain. c V. V. Veeravalli, 007 5

6 7. Squared-envelope correlation for isotropic fading: a Prove that the squared-envelope covariance function for a Rayleigh fading process {Et} is given y: C α τ R E τ Hint: It may e easier to first show that R α τ R E τ +. You may find the following result to e useful: if X, X, X 3, X 4 are jointly Gaussian zero-mean random variales, then EX X X 3 X 4 ] EX X ]EX 3 X 4 ] + EX X 3 ]EX X 4 ] + EX X 4 ]EX X 3 ]. In particular, when X X and X 3 X 4 we get Thus EX X X 3 X 3 ] EX X ]EX 3 X 3 ] + EX X 3 ]. R α τ E E I t + τ + E Q t + τ E I t + E Q t] EE I t + τe I t] + EE Q t + τe Q t] + EE I t + τe Q t] + EE Q t + τe I t] EEI t + τe I t] + EE I t + τe Q t] EE I t + τ]ee I t] + EE It + τe I t] + EE I t + τ]ee Q t] + EE It + τe Q t] 4 + R E I τ R E I E Q τ + 4 R EI τ + R EI E Q τ + R E τ Prove that the squared-envelop covariance function for a Ricean fading process {Et} is: ]] C α τ RĚτ + κre RĚτe jπfmaxτ cos θ 0, κ + where κ is the Rice factor and θ 0 is the angle of arrival of the specular component. R α τ E α tα t + τ ] E EtE tet + τe t + τ] E β 0 e jφ0t + β 0Ět β 0 e jφ0t + β 0Ě t β 0 e jφ0t+τ + β 0Ět + τ β 0 e jφ0t+τ + β 0Ě t + τ ] a β β 0 Rˇαˇα τ + β 0 β 0Eˇα t] + β 0 β 0Eˇα t + τ] β 0 β 0 ejφ 0t φ 0 t+τ EĚ tět + τ] + β 0 β 0 e jφ 0t φ 0 t+τ EĚtĚ t + τ] β β 0 + RĚτ + β 0 β 0 + β 0 β 0 Re EĚ tět + τ]e jφ 0t+τ φ 0 t ] ] + β0 RĚτ + β0 β 0 Re EĚ tět + τ]e jφ 0t+τ φ 0 t ] + RĚτ + κre RĚτe jπfmaxτ cos θ 0 + κ + κ c V. V. Veeravalli, 007 6

7 Note that there are 6 terms in the expectation. Of these 6 terms, 8 involve only one phase term φ 0. Since the specular component is independent of the diffuse component, and the marginal distriution of the phase is uniform, these eight terms go to zero. Of the six terms containing two phase terms, two vanish ecause they involve phase of the form φ 0 t + φ 0 t + τ. Thus, 6 terms are left in a. C α R α Eα t] ]] RĚτ + κre RĚτe jπfmaxτ cos θ 0 + κ 8. Power Spectrum of Rayleing Fading Process. Consider a Rayleigh fading environment with angular power density pθ. a Show that: S E f Now show that S E f has the closed-form expression: 0 pθ + p θ]δf f max cos θdθ pcos f/f max + p cos f/f max for f < f max S E f f max f 0 otherwise c Specialize this result for isotropic Rayleigh fading, i.e., pθ /π. S E f π fmax π pθe jπfmaxτ cos θ e jπfτ dθdτ pθδf f max cos θdθ 0 π pθ e jπf fmaxτ cos θ dτdθ pθ + p θ]δf f max cos θdθ pcos u/f max + p cos u/f max ] du δu f f max f max u/fmax pcos f/f max+p cos f/f max] if f < f f max, max f 0 otherwise. c V. V. Veeravalli, 007 7