Homework 5 Solutions Problem 1 (a Closed form Chernoff upper-bound for the uncoded 4-QAM average symbol error rate over Rayleigh flat fading MISO channel with = 4, assuming transmit-mrc The vector channel for the MISO channel... h = [h 1 h h MT ] The signal at the receiver... y = Es hws + n The weight vector... w = h H h The SNR at the receiver... η = h ρ (ρ = E s /N o ( With ML detection, symbol error probability... P e N e Q 4-QAM with E s = 1... N e =, d min = Symbol error probability... P e Q ( η Chernoff bound for the Q function... Q (x e x Chernoff bound on symbol error probability P e e η ηd min E{P e } E{e h ρ } { } E e s H w 1 F = (1 + s M (Homework 4 T M R E{P e } ( 1 + ρ 4 E{P e } 5 ρ 4 (At high SNR, diversity order 4 (b upper-bound on symbol error rate for a SIMO channel with M R = 4 E{P e } 5 ρ 4 When transmitter has channel knowledge, it can focuss the energy in the only useful direction and it extracts both the array and diversity gains. The performance obtained by MISO (with channel knowledge and SIMO is same in terms of both array and diversity gains. When there is no channel knowledge at the transmitter, the system behaves as a SISO system unless we employ some precoding (space-time codes. With precoding at the transmitter we will be able to extract the diversity gain but not the maximum possible array gain. 1
(c Ergodic capacity when the channel is known to the transmitter { r ( } C MIMO = E log 1 + σi γ opt ρ i For transmit-mrc MISO channel, i=1 r = 1, h = σ 1, γ opt 1 = C MISO = E { log ( 1+ h ρ } Transmit-MRC is an ergodic capacity optimal strategy. As shown above, the optimum strategy for MIMO to achieve capacity with channel knowledge at transmitter is exactly transmit- MRC when the number of receive antennas is 1. Problem P.101, Paulraj book has a table summarizing achievable array and diversity gains under different configurations (a i. M R = 4, = 4, channel known to transmitter and receiver Dominant eigenmode transmission Array Gain E{λ max } = 9.8733 (Analytical expression in [1]. Diversity Gain M R = 16 ii. M R = 1, =, channel unknown to transmitter, known to receiver If we transmit the same symbol on both antennas without space-time coding then we will not be able to achieve array or diversity gains (effectively SISO. With Alamouti coding at transmitter Array Gain 1 Diversity Gain iii. M R = 1, =, channel known to transmitter and receiver Transmit MRC Array Gain Diversity Gain iv. M R = 4, =, channel unknown to transmitter, known to receiver With Alamouti coding at transmitter Array Gain 4 Diversity Gain 8 (b Ergodic capacity when the channel is known to the transmitter { r ( } C MIMO = E log 1 + σi γ opt ρ i i=1
Dominant eigenmode transmission (DET r = 1 γ opt 1 = σ1 = maximum eigenvalue of HH H H F r σ1 H F { ( C DET = E log 1 + σ 1γ opt 1 if γ opt 1 = C DET = E { log ( 1 + σ 1ρ } Dominant eigenmode transmission is optimum iff the waterfilling procedure allocates all the power to the dominant channel. Waterfilling allocates equal power to all modes at high SNRs. So ergodic capacity won t be optimum if we employ DET at high SNRs. There will in general be at least be a few other good channels (from waterfilling at medium SNRs. So even here DET may not be optimum. Very low SNR region might make a DET scheme an optimum one. Rank 1 channel (both SIMO and MISO, with CSI at transmitter is another example where DET is an ergodic optimum scheme. ρ } Problem 3 (a The channel matrix H: ( P e N e Q ηd min [ h11 h H = 1 h 1 h ] BPSK with E s = 1... N e = 1, d min = The SNR at the receiver... η = H F ρ (ρ = E s /N o Chernoff bound on symbol error probability... P e e H F ρ From the above bound it is clear that the Frobenius norm of the channel matrix (at a fixed ρ determines the symbol error probability. The channel with 4 receiving antennas and transmitting antennas H = h 11 h 1 h 1 h h 31 h 3 h 41 h 4 Switching strategy: Calculate the norm of all 4 row vectors of H and Select the rows with maximum norm. 3
(b Example to illustrate the improvement in performance with selection diversity combined with Alamouti coding for a x MIMO system 10 0 10 Alamouti (x, Alamouti with Selection ( out of 4 and MRC (1x8 Alamouti Alamouti with Selection MRC Probability of Symbol Error 10 4 10 6 10 8 10 10 10 1 0 4 6 8 10 1 SNR (db Figure 1: Performance improves with selection Alamouti The Alamouti selection (AS scheme provides a diversity order of 8, the diversity order of regular Alamouti is 4. The change in slope (a direct measure of diversity can be noticed in the curves. Notice the same slope of AS and MRC, indicating that both schemes achieve same diversity order (8. The main difference between SIMO with 8 antennas and AS is the array gain. MATLAB code: % Variables % Mt, number of transmitting antennas % Mr, number of receiving antennas % Mrsel, number of antennas at receiver out of which are selected % Iter, number of trials % Hsel, 4x channel, a x channel is selected from the 4x % Hala, x matrix used for Alamouti implementation % hmrc, 1x8 channel for MRC (SIMO % PeAla, probability of error in Alamouti scheme % Pesel, probability of error with selection and Alamouti scheme % PeMRC, probability of error with MRC in the SIMO system 4
clc; close all; clear all; Mt = ; Mr = ; Mrsel = 4; SNRdB = [0:1:1]; SNR = 10.ˆ(SNRdB/10; PeAla = zeros(1,length(snr; PeSel = zeros(1,length(snr; PeMRC = zeros(1,length(snr; Iter = 10000; for ite = 1:Iter Hsel = (randn(mrsel,mt+j*randn(mrsel,mt/sqrt(; Hala = (randn(mr,mt+j*randn(mr,mt/sqrt(; hmrc = (randn(mrsel*mt,1+j*randn(mrsel*mt,1/sqrt(; for k = 1:length(SNR PeAla(k = PeAla(k +Q(sqrt(SNR(k*norm(Hala, fro ˆ; PeMRC(k = PeMRC(k + Q(sqrt(*SNR(k*norm(hMRCˆ; nh = flipud(sort(abs(hsel(:,1.ˆ+abs(hsel(:,.ˆ; nh = sum(nh(1:,:; PeSel(k = PeSel(k+ Q(sqrt(SNR(k*nH; end end PeAla = PeAla/Iter; PeMRC = PeMRC/Iter; PeSel = PeSel/Iter; semilogy(snrdb,peala, b,snrdb,pesel, r - *,SNRdB,PeMRC, k - o ; legend( Alamouti, Alamouti with Selection, MRC ; xlabel( SNR (db ; ylabel( Probability of Symbol Error ; title( Alamouti (x,alamouti with Selection ( out of 4 and MRC(1x8 ; grid; References [1] P.A Dighe, R.K. Mallik, and S.S. Jamuar, Analysis of transmit-receive diversity in rayleigh fading, IEEE Trans. Commun, vol. 51, no. 4, pp. 694 703, Apr. 003. 5