MIMO Capacity and gain of Optimal power allocation using Water-Filling algorithm EE 575 Information Theory

Transcription

1 ELECTRICAL ENGINEERING DEPARTMENT, KFUPM MIMO Capacity and gain of Optimal power allocation using Water-Filling algorithm EE 575 Information Theory Assignment # 3 Submitted by: Raza Umar Student ID: g /23/2010 In this assignment, capacity of parallel Gaussian channels has been compared for equal power allocation and optimal power allocation based on water-filling algorithm. Mean capacity comparisons, Complementary CDF comparisons and Outage probability comparisons have been analyzed for SISO, SIMO, MISO, MIMO and MIMO using Water-Filling power allocation.

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3 Table of Contents Problem 1: Parallel Gaussian Channels... 4 Part (1a) Total capacity of parallel channels for equal power distribution... 4 Part (1b) Water-filling algorithm... 5 Part (1c) Total capacity of parallel channels for optimal power distribution... 5 Problem 2 + Problem 3: MIMO Capacity + Water filling for 4x4 MIMO Channel... 6 Part (2a+3a) Mean capacity comparisons for MIMO channels... 6 Part (2b+3b) Complementary CDF comparison for flat fading channels... 8 Part (2c+3c) Outage probability vs. SNR for flat adding channels... 9 Appix A Appix B... 13

4 Problem 1: Parallel Gaussian Channels Given received signal: Where, ~ 0, Total Power transmission=p t =20 Part (1a) Total capacity of parallel channels for equal power distribution As we have 4 parallel channels M=4=N t =N r N 1 =1, N 2 =7, N 3 =5, N 4 =3 For equal power distribution: 20 5, 1,2,, 4 4 max ; : 1 2 log log /

5 Part (1b) Water-filling algorithm Matlab implementation of Water filling algorithm is available in Appix A. Part (1c) Total capacity of parallel channels for optimal power distribution For, N 1 =1, N 2 =7, N 3 =5, N 4 =3 Water level obtained from Matlab code = v=9 & P 1 =8, P 2 =2, P 3 =4, P 4 =6 using, With optimal power allocation; 1 2 log / Hence, /

6 Problem 2 + Problem 3: MIMO Capacity + Water filling for 4x4 MIMO Channel Matlab Implementation for MIMO Capacity and Water filling for 4x4 MIMO Channel is available in Appix B. Part (2a+3a) Mean capacity comparisons for MIMO channels

7 Water filing Capacity Gain 0.9 WF gain in capacity WF gain in bps/hz ---> SNR in db --->

8 Part (2b+3b) Complementary CDF comparison for flat fading channels 1 - Outage Probability ---> Complementary CDF comparisons (vs capacity) at SNR=10dB 4x4 MIMO WF 4x4 MIMO 4x1 MISO 1x4 SIMO 1X1 SISO Mean Capacity bps/hz --->

9 Part (2c+3c) Outage probability vs. SNR for flat adding channels 10 0 Outage probability vs SNR for 4 bps/hz 10-1 Outage Probability x4 MIMO 4x1 MISO 1x4 SIMO 1X1 SISO 4x4 MIMO WF SNR in db --->

10 Appix A %% Function: wfill % This routine optimally allocates the power among "m" channel using water % filling algorithm % Input: % 1. Pt: Total Power budget % 2. m: Total available parallel channels % 3. N: Un-correlated Noise variances ( a row vector of lenght m) % Output: % 1. v: water level % 2. P: power levels corresponding to "m" (arranged in the same order as % N) according to water filling algo. function [v P] = wfill(pt,m,n) % Optimum power allocation function res=1; % resolution of step size P=zeros(1,m); % initialize transmitted power over each parallel channel to be 0 [N_sorted,index]=sort(N); % Noise power sorted in ascing order step=(n_sorted(2)-n_sorted(1))/res; N_sorted_temp=N_sorted; for p=1:length(n_sorted_temp)-1 if ((max(n_sorted_temp)-min(n_sorted_temp))>pt) m=m-1; N_sorted_temp=N_sorted_temp(1:-1); if step>pt/2 step=pt/2; if step<0.001 step=.001; j=0; k=0; epsilon=1e-5; epsilon2=1e-4; %threshold for discrepancies in water level over different channels i=1; z=0; q=2; step_old=0; for iter=1:1e6*res if(sum(abs((n_sorted(1).*ones(1,m))-n_sorted(1:m))<1e-6)==m && (sum(p)< Pt)) step=(pt-sum(p))/m; N_sorted(1:m)=N_sorted(1:m)+step; P(1:m)=P(1:m)+step; break;

11 N_sorted(i)=N_sorted(i)+step; P(i)=P(i)+step; if(sum(abs((n_sorted(1).*ones(1,q))-n_sorted(1:q))<1e-6)==q && (sum(p)< Pt)) if q==(length(n)) step=2*step; if(q<m) step=n_sorted(q+1)-n_sorted(q); q=q+1; if (j>0 && step_old>step) step =step_old; j=0; if (sum(p)>pt) k=k+1; N_sorted(1:i)=N_sorted(1:i)-step; P(1:i)=P(1:i)-step; % find how many channels(out of m) are at same level check = check2=find(check); i=max(check2); step=(pt-sum(p))/(i); N_sorted(1:i)=N_sorted(1:i)+step; P(1:i)=P(1:i)+step; abs(n_sorted-n_sorted(1).*ones(1,length(n)))<epsilon; if(k>2) display(['warning: Power alltoment exceeding budget ',num2str(k),'rd time']); sum(p) if i<length(n) if((n_sorted(i)-n_sorted(i+1))>epsilon) %if water has gone above the next level, set the step such that when water %is added to the next level, its level becomes the same as previous level if ((z~=1)&&(step~=n_sorted(i)-n_sorted(i+1))) step_old=step; step=n_sorted(i)-n_sorted(i+1); z=0; if (step< 0.001) j=1; i=i+1; else if i>1 % if water is below next entry of N, %re-initiazlize it to point at 1st entry and start re-filling water i=1; else % if i is pointing at the last entry of N, re intialize it to point at 1st entry

12 i=1; if (abs(sum(p)-pt) < epsilon2) if i>1&&(n_sorted(i) < N_sorted(i-1)) N_sorted(1:i-1)=N_sorted(1:i-1)-step; P(1:i-1)=P(1:i-1)-step; step=(step*(i-1))/i; N_sorted(1:i)=N_sorted(1:i)+step; P(1:i)=P(1:i)+step; j=0; i=1; z=1; if (abs(sum(p)-pt) < epsilon2) flag_vector2=abs(n_sorted(1).*ones(1,length(find(p)))- N_sorted(1:length(find(P))))< epsilon2.*ones(1,length(find(p))); flag_vector=find(flag_vector2-ones(1,length(find(p)))); if isempty(flag_vector) break; if iter>1000 display('warning: wf took more than 1000 iterations'); v= N_sorted(1); % water level N_sorted; N; P=P(index); if(abs(sum(p)-pt) > epsilon2) display('warning: Power above budget');

13 Appix B %% Main function to calculate MIMO channel capacity and associated figures %% of merit like complementary CDF & Outage probability % Objective: (1)This function compares Mean channel capacity for different % MIMO realizations (SISO, SIMO,MISO, MIMO)as a function of SNR % (2) compares complementary CDF for different MIMO realizations % (SISO, SIMO,MISO, MIMO)as a function of capacity in bps/hz % (3) compares Outage probability for different MIMO realizations % (SISO, SIMO,MISO, MIMO)as a function of SNR % Author: Raza Umar as part of EE 575 Information Theory Assignment % Date: May 08, 2010 %% >>>>>>>>>>>>>>>>>>>>>>>>... CLEANING... <<<<<<<<<<<<<<<<<<<<<<<<<<<<<< close all; clear all; clc; %% >>>>>>>>>>>>>>>>>... SIMULTAION PARAMS... <<<<<<<<<<<<<<<<<<<<<<<<<<<< SNR_dB=-10:30; SNR=10.^(SNR_dB./10); ch_realizations=10000; % Monte Carlo sim. of 10,000 channel realizations c_outage=4; epsilon=1e-6; %% >>>>>>>>>>>>>>>>>>>>>>>... SISO... <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< % initialization and param setting c11_i=zeros(length(snr),ch_realizations); c11=zeros(length(snr),1); % matrix initializations to avoid matrix growing inside loop % Mean channel capacity calculations for i1=1:length(snr) SNR_i= SNR(i1); % for one specific value of SNR h11=1/sqrt(2).*complex(randn(1,ch_realizations),randn(1,ch_realizations)); % complex normal r.v. with var=1/2 per dim. h11_mag_sq=abs(h11).^2; c11_i(i1,1:ch_realizations)=log2(1+snr_i.*h11_mag_sq); % instantaneous cap. c11(i1)=mean(c11_i(i1,:)); % mean capacity %complementary CDF at SNR=10dB c11_all=c11_i(21,:); % channel realizations at 10dB SNR range_11=0:0.1:max(c11_all); count11=histc(c11_all,range_11); count11_norm=cumsum(count11)/max(cumsum(count11)); comp_cdf_11=1-count11_norm; %outage probability j1= 1; for i1=find(snr_db==2):find(snr_db==20) c11_all=abs(c11_i(i1,:)); range11=0:0.1:max(c11_all); count11=histc(c11_all,range11);

14 count11_norm=cumsum(count11)/max(cumsum(count11)); if(isempty(find(abs(range11-c_outage)<epsilon))) outage11(j1)=1; else outage11(j1)=count11_norm(find(abs(range11-c_outage)<epsilon)); j1=j1+1; %% >>>>>>>>>>>>>>>>>>>>>... MIMO (4x4)... <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< % initialization and param setting Pt=1; % power budget for MIMO water filling Nr=4; Nt=4; I_Nr=eye(Nr); c44_i=zeros(length(snr),ch_realizations); c44=zeros(length(snr),1); % matrix initializations to avoid matrix growing inside loop % Mean channel capacity calculations for i1=1:length(snr) SNR_i= SNR(i1); % for one specific value of SNR H_vec=1/sqrt(2).*complex(randn(Nr*Nt,ch_realizations),randn(Nr*Nt,ch_realizat ions)); % Channel matrix with elements as complex normal r.v. having var=1/2 per dim. H=reshape(H_vec,Nr,Nt,ch_realizations); for i2=1:ch_realizations H_i=H(1:Nr,1:Nt,i2); H_H_hermt=H_i*H_i'; arg=i_nr+ (SNR_i/Nt).*H_H_hermt; arg2= det(arg); c44_i(i1,i2)=log2(arg2); % instantaneous cap. %% alternate implementation to find inst. cap. (using eigen values) % lamda=eig(h_h_hermt); % arg3=(1+(snr_i/nt)*lamda(1))*(1+(snr_i/nt)*lamda(2))*(1+(snr_i/nt)*lamda(3))* (1+(SNR_i/Nt)*lamda(4)); % c442i(i1,i2)=log2(arg3); % instantaneous cap. %% MIMO water filling lamda=eig(h_h_hermt); N=1./(SNR_i.*lamda); N_wf=N'; [v P_wf]=wfill(Pt,4,N_wf); arg_wf=(1+p_wf(1)/n_wf(1)).*(1+p_wf(2)/n_wf(2)).*(1+p_wf(3)/n_wf(3)).*(1+p_wf (4)/N_wf(4)); c44_i_wf(i1,i2)=log2(arg_wf); % instantaneous cap. using wf %% c44(i1)=mean(c44_i(i1,:)); % mean capacity c44_wf(i1)=mean(c44_i_wf(i1,:)); % mean capacity

15 %% %%complementary CDF at SNR=10dB c44_all=abs(c44_i(21,:)); % channel realizations at 10dB SNR range_44=0:0.1:max(c44_all); count44=histc(c44_all,range_44); count44_norm=cumsum(count44)/max(cumsum(count44)); comp_cdf_44=1-count44_norm; %%MIMO WF c44_all_wf=abs(c44_i_wf(21,:)); % channel realizations at 10dB SNR range_44_wf=0:0.1:max(c44_all_wf); count44_wf=histc(c44_all_wf,range_44_wf); count44_norm_wf=cumsum(count44_wf)/max(cumsum(count44_wf)); comp_cdf_44_wf=1-count44_norm_wf; %% %outage prob j1= 1; for i1=find(snr_db==2):find(snr_db==20) c44_all=abs(c44_i(i1,:)); range44=0:0.1:max(c44_all); count44=histc(c44_all,range44); count44_norm=cumsum(count44)/max(cumsum(count44)); if(isempty(find(abs(range44-c_outage)<epsilon))) outage44(j1)=1; else outage44(j1)=count44_norm(find(abs(range44-c_outage)<epsilon)); j1=j1+1; %%MIMO WF j1= 1; for i1=find(snr_db==2):find(snr_db==20) c44_all_wf=abs(c44_i_wf(i1,:)); range44_wf=0:0.1:max(c44_all_wf); count44_wf=histc(c44_all_wf,range44_wf); count44_norm_wf=cumsum(count44_wf)/max(cumsum(count44_wf)); if(isempty(find(abs(range44_wf-c_outage)<epsilon))) outage44_wf(j1)=1; else outage44_wf(j1)=count44_norm_wf(find(abs(range44_wfc_outage)<epsilon)); j1=j1+1; %% >>>>>>>>>>>>>>>>>>>>>... SIMO (1x4)... <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< % initialization and param setting Nr=4; Nt=1; c14_i=zeros(length(snr),ch_realizations); c14=zeros(length(snr),1); % matrix initializations to avoid matrix growing inside loop % Mean channel capacity calculations for i1=1:length(snr)

16 SNR_i= SNR(i1); % for one specific value of SNR h14=1/sqrt(2).*complex(randn(nr*nt,ch_realizations),randn(nr*nt,ch_realizatio ns)); % complex normal r.v. having var=1/2 per dim. h14_mag_sq=abs(h14).^2; h14_mag_sq_sum=sum(h14_mag_sq); c14_i(i1,1:ch_realizations)=log2(1+snr_i.*h14_mag_sq_sum); % instantaneous cap. c14(i1)=mean(c14_i(i1,:)); % mean capacity %complementary CDF at SNR=10dB c14_all=c14_i(21,:); % channel realizations at 10dB SNR range_14=0:0.1:max(c14_all); count14=histc(c14_all,range_14); count14_norm=cumsum(count14)/max(cumsum(count14)); comp_cdf_14=1-count14_norm; %outage probability j1= 1; for i1=find(snr_db==2):find(snr_db==20) c14_all=abs(c14_i(i1,:)); range14=0:0.1:max(c14_all); count14=histc(c14_all,range14); count14_norm=cumsum(count14)/max(cumsum(count14)); if(isempty(find(abs(range14-c_outage)<epsilon))) outage14(j1)=1; else outage14(j1)=count14_norm(find(abs(range14-c_outage)<epsilon)); j1=j1+1; %% >>>>>>>>>>>>>>>>>>>>>... MISO (4x1)... <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< % initialization and param setting Nr=1; Nt=4; c41_i=zeros(length(snr),ch_realizations); c41=zeros(length(snr),1); % matrix initializations to avoid matrix growing inside loop % Mean channel capacity calculations for i1=1:length(snr) SNR_i= SNR(i1); % for one specific value of SNR h41=1/sqrt(2).*complex(randn(nr*nt,ch_realizations),randn(nr*nt,ch_realizatio ns)); % complex normal r.v. having var=1/2 per dim. h41_mag_sq=abs(h41).^2; h41_mag_sq_sum=sum(h41_mag_sq); c41_i(i1,1:ch_realizations)=log2(1+(snr_i./nt).*h41_mag_sq_sum); % instantaneous cap. c41(i1)=mean(c41_i(i1,:)); % mean capacity

17 %complementary CDF at SNR=10dB c41_all=c41_i(21,:); % channel realizations at 10dB SNR range_41=0:0.1:max(c41_all); count41=histc(c41_all,range_41); count41_norm=cumsum(count41)/max(cumsum(count41)); comp_cdf_41=1-count41_norm; %outage probability j1= 1; for i1=find(snr_db==2):find(snr_db==20) c41_all=abs(c41_i(i1,:)); range41=0:0.1:max(c41_all); count41=histc(c41_all,range41); count41_norm=cumsum(count41)/max(cumsum(count41)); if(isempty(find(abs(range41-c_outage)<epsilon))) outage41(j1)=1; else outage41(j1)=count41_norm(find(abs(range41-c_outage)<epsilon)); j1=j1+1; %% >>>>>>>>>>>>>>>>>>>>... Plotting... <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< % Mean channel capacity figure('name','mean Capacity comparisons (vs SNR) for Flat Fading MIMO channels'); plot(snr_db,abs(c44_wf),'bx-',snr_db,abs(c44),'ro',snr_db,c41,'c+- ',SNR_dB,c14,'gs-',SNR_dB,c11,'b:'); leg('4x4 MIMO WF','4x4 MIMO','4x1 MISO','1x4 SIMO','1X1 SISO','Location','NorthWest'); title('mean Capacity vs SNR'); xlabel('snr (db) --->'); ylabel('mean Capacity bps/hz --->'); axis([ ]); %complementary CDF at SNR=10dB figure('name','complementary CDF at SNR = 10dB'); plot(range_44_wf,comp_cdf_44_wf,'bx- ',range_44,comp_cdf_44,'ro',range_41,comp_cdf_41,'c+- ',range_14,comp_cdf_14,'gs-',range_11,comp_cdf_11,'b:'); leg('4x4 MIMO WF','4x4 MIMO','4x1 MISO','1x4 SIMO','1X1 SISO','Location','NorthEast'); title('complementary CDF comparisons (vs capacity) at SNR=10dB'); xlabel('mean Capacity bps/hz --->'); ylabel('1 - Outage Probability --->'); axis([ ]); %outage probability vs SNR for 4 bps/hz figure('name','outage probability comparisons (vs SNR) for Flat Fading Channels'); %plot(range_44,comp_cdf_44,'bx- ',range_44,comp_cdf_44,'ro',range_41,comp_cdf_41,'c+- ',range_14,comp_cdf_14,'gs-',range_11,comp_cdf_11,'b:'); semilogy(snr_db(find(snr_db==2):find(snr_db==20)),outage44_wf,'bx- ',SNR_dB(find(SNR_dB==2):find(SNR_dB==20)),outage44,'ro',SNR_dB(find(SNR_dB== 2):find(SNR_dB==20)),outage41,'c+-

18 ',SNR_dB(find(SNR_dB==2):find(SNR_dB==20)),outage14,'gs- ',SNR_dB(find(SNR_dB==2):find(SNR_dB==20)),outage11,'b:'); leg('4x4 MIMO WF','4x4 MIMO','4x1 MISO','1x4 SIMO','1X1 SISO','Location','SouthEast'); title('outage probability vs SNR for 4 bps/hz'); xlabel('snr in db --->'); ylabel('outage Probability --->'); axis([2 20 1e-6 1]); % MIMO water-filling capacity gain figure('name','capacity gain of water-filling'); plot(snr_db,abs(c44_wf)'-abs(c44)); grid on; title('wf gain in capacity'); xlabel('snr in db --->'); ylabel('wf gain in bps/hz --->');