1 if X v I v (X) = 0 if X < v. e sx p(x)dx

Transcription

1 Homework 3 Solution Problem Chernoff Bound: The indicator function I v (X) for a real random variable X i defined a, { if X v I v (X) 0 if X < v For 0 (i) if X < v (ii) if X v E{e X } e v P rx v I v (X) 0 e v I v (X) 0 e X e v I v (X) I v (X) e v I v (X) e v e X e v ( X v) e X e v I v (X) E{e X } e X e v I v (X) e X p(x)dx e X p(x)dx E{e X } e v I v (X)p(X)dX e v I v (X)p(X)dX E{e X } e v p(x)dx ( I v (X) 0 for X < v) E{e X } e v P rx v P rx v e v E{e X } v Notice that the above proof i baed on e X e v I v (X), which i true only for 0 A tighter upper bound i achieved with the mallet poible P rx v min e v E{e X } 3 X i Gauian with zero mean and unit variance P rx v i given by the Gauian Q-function, Q(v) e t dt For v 0 P rx v min e v E{e X } v

2 E{e X } p(x) e X p(x)dx e X E{e X } e E{e X } e e X e X dx e X +X dx e (X X+ ) e dx e (X ) dx ( P rx v min e v E{e X } min e v e min e v+v e v e v Q(v) e v min e ( v) e v (when v) ) e (X ) Problem Antenna pacing and the total antenna length effect the reolvability in angular domain Page (Te and vihwanath), which include figure 76 and 77, have a good dicuion on thi topic MATLAB code %theta, location of main lobe %N, indicate number of receiving antenna %Lt, antenna length %phi, angular range %aphi, patial ignature vector %bv, patial ignature vector with thetapi/ clc; cloe all; clear all;

3 theta 90*pi/80; count ; tx figure; for N,4,6,3; Lt ; bv (/N)*exp(-j**pi*0:(N-)*(Lt/N)*co(theta)) ; phi 0::350*pi/80; bp zero(,length(phi)); for K :length(phi) aphi exp(-j**pi*0:(n-)*(lt/n)*co(phi(k))) ; bp(k) bv *aphi; ubplot(,,count) x polar(phi,(ab(bp)), b - ); et(x, LineWidth,0) title( L t,numtr(lt),, N, numtr(n)) count count+; et(tx, PaperPoition, ); tx figure; theta 90*pi/80; count ; for N 8,6; Lt N/; bv (/N)*exp(-j**pi*0:(N-)*(Lt/N)*co(theta)) ; phi 0::350*pi/80; bp zero(,length(phi)); for K :length(phi) aphi exp(-j**pi*0:(n-)*(lt/n)*co(phi(k))) ; bp(k) bv *aphi; ubplot(,,count) x polar(phi,(ab(bp)), b - ); et(x, LineWidth,0) title( L t,numtr(lt),, N, numtr(n)) count count+; et(tx, PaperPoition, ); 3

4 Figure : Problem 3 Problem 3 Note: The angle hown in the figure are meaured wrt the horizontal axi o in the patial ignature vector we will ee in intead of co In the cla lecture we meaured the angle wrt the vertical axi o we ued co The imulated correlation value in part4 of the problem vary deping on whether you ued co or in The patial ignature of a ignal with a Angle Of Departure (AOD) φ i i given by, a(φ i ) e j d λ in φ i e j d λ in φ i e j(m T ) d λ in φ i The received ignal at the reference antenna i given by, r ref a T (φ i )γ i γ i, catterer amplitude, randomly ditributed a CN (0, N ), tranmitted ymbol vector The patial ignature of a catterer with an Angle Of Arrival (AOA) θ i on the receiving linear array i given by, a(θ i ) e j d λ in θ i e j d λ in θ i e j(m R ) d λ in θ i The repone by the i th catterer with an AOD φ i and (AOA) θ i i given by, h i ( a(θ i )a T (φ i )γ i ) 4

5 The channel H due to all the N catterer i given by H N γ i a(θ i )a T (φ i ) i N γ i A i A i a(θ i )a T (φ i ) i For M T M R and d λ a(φ i ) a(θ i ) e j d λ in φ i e j d λ in θ i e jπ in φ i e jπ in θ i A i a(θ i )a T (φ i ) e jπ in θ i e jπ in φ i e jπ in φ i e jπ in θ i e jπ(in θ i+in φ i ) For N H γ a(θ )a T (φ ) γ e jπ in φ e jπ in θ e jπ(in θ +in φ ) Since H i obtained by the outer product of two vector it rank i 3 r(h) min{m T, M R, N} H N γ i a(θ i )a T (φ i ) i A een in the firt part the rank of channel matrix due to ingle catterer i only A the number of catterer increae the rank of the matrix increae (addition of rank matrice increae the rank of the reulting matrix) However, rank of a matrix fundamentally dep on it ize, the rank of a M R xm T matrix min{m T, M R } If N < min{m T, M R } then r(h) min{m T, M R, N} If N min{m T, M R } then r(h) min{m T, M R }, neceary condition for the matrix to be full rank For the preent cae N 4 N 00 M T M R φ i uniformly ditributed between π 0 and π 0 θ i uniformly ditributed between π and π E{H, H,} (imulated tranmit correlation) () r E{H, H,} 0936 (imulated receive correlation) () 5

6 The angular pread at the tranmitter i much le compared to the angular pread at the receiver o higher correlation i een at the tranmit ide MATLAB code % N, number of catterer % phi, angular pread at the tranmitter % theta, angular pread at the receiver % Mt, number of tranmit antenna % Mr, number of receive antenna % iter, number of iteration % tcff, tranmitter correlation coefficient % rcff, receiver correlation coefficient % gamma, normal ditributed amplitude % aphi, patial ignature at the tranmit ide % atheta, patial ignature at the receive ide % H, imulated channel clc; cloe all; clear all; N 00; Mt ; Mr ; iter 4000; tcff 0; rcff 0; for L :iter phi unifrnd(-pi/0,pi/0,,n); theta unifrnd(-pi,pi,,n); H zero(mt,mr); for K :N gamma (/qrt(*n))*(randn+j*randn); aphi exp(-j**pi*0:(mt-)*(/)*in(phi(k))) ; atheta exp(-j**pi*0:(mr-)*(/)*in(theta(k))) ; H H + gamma*atheta*aphi ; tcff tcff + H(,)*H(,) ; rcff rcff + H(,)*H(,) ; ab(tcff)/iter ab(rcff)/iter 6