advertisement

EE 6332, Spring, 2014 Wireless Communication Zhu Han Department of Electrical and Computer Engineering Class 4 Jan. 27th, 2014 Outline Review (important) – RMS delay vs. coherent bandwidth – Doppler spread vs. coherent time – Slow Fading vs. Fast Fading – Flat Fading vs. Frequency Selective Fading Rayleigh and Ricean Distributions Statistical Models Fading Distributions Describes how the received signal amplitude changes with time. – Remember that the received signal is combination of multiple signals arriving from different directions, phases and amplitudes. – With the received signal we mean the baseband signal, namely the envelope of the received signal (i.e. r(t)). It is a statistical characterization of the multipath fading. Two distributions – Rayleigh Fading – Ricean Fading Rayleigh Distributions Describes the received signal envelope distribution for channels, where all the components are non-LOS: – i.e. there is no line-of–sight (LOS) component. Ricean Distributions Describes the received signal envelope distribution for channels where one of the multipath components is LOS component. – i.e. there is one LOS component. Rayleigh Fading Rayleigh Fading Rayleigh Fading Distribution The Rayleigh distribution is commonly used to describe the statistical time varying nature of the received envelope of a flat fading signal, or the envelope of an individual multipath component. The envelope of the sum of two quadrature Gaussian noise signals obeys a Rayleigh distribution. r r2 ) exp( p ( r ) 2 2 2 0 r 0 0r is the rms value of the received voltage before envelope detection, and 2 is the time-average power of the received signal before envelope detection. Rayleigh Fading Distribution The probability that the envelope of the received signal does not exceed a specified value of R is given by the CDF: 2 R P( R) Pr (r R) p(r )dr 1 e 0 rmean E[r ] rp (r )dr 0 rmedian 2 R 2 2 1.2533 1 1.177 found by solving 2 rmedian p(r )dr 0 rrms 2 rpeak= and p()=0.6065/ r E [ r ] E [ r ] 2 2 2 0 2 r p ( r ) dr 0 . 4292 2 2 2 Rayleigh PDF 0.7 0.6065/ 0.6 mean = 1.2533 median = 1.177 variance = 0.42922 0.5 0.4 0.3 0.2 0.1 0 0 1 2 2 3 3 4 4 5 5 A typical Rayleigh fading envelope at 900MHz. Ricean Distribution When there is a stationary (non-fading) LOS signal present, then the envelope distribution is Ricean. The Ricean distribution degenerates to Rayleigh when the dominant component fades away. Ricean Fading Distribution When there is a dominant stationary signal component present, the smallscale fading envelope distribution is Ricean. The effect of a dominant signal arriving with many weaker multipath signals gives rise to the Ricean distribution. The Ricean distribution degenerates to a Rayleigh distribution when the dominant component fades away. r ( r 2 A2 ) Ar exp[ ] I ( ) 0 r , A0 0 2 2 p ( r ) 2 2 0 r 0 The Ricean distribution is often described in terms of a parameter K which is defined as the ratio between the deterministic signal power and the variance of the multipath. A2 K is known as the Ricean factor K 2 2 As A0, K - dB, Ricean distribution degenerates to Rayleigh distribution. CDF Cumulative distribution for three small-scale fading measurements and their fit to Rayleigh, Ricean, and log-normal distributions. PDF Probability density function of Ricean distributions: K=-∞dB (Rayleigh) and K=6dB. For K>>1, the Ricean pdf is approximately Gaussian about the mean. Rice time series Nakagami Model Nakagami Model m 2 2m r exp( r ) p(r ) ( m ) m m 2 m 1 r: envelope amplitude Ω=<r2>: time-averaged power of received signal m: the inverse of normalized variance of r2 – Get Rayleigh when m=1 Small-scale fading mechanism Assume signals arrive from all angles in the horizontal plane 0<α<360 Signal amplitudes are equal, independent of α Assume further that there is no multipath delay: (flat fading assumption) Doppler shifts fn v cos an Small-scale fading: effect of Doppler in a multipath environment fm, the largest Doppler shift f 1 SbbEz ( f ) k 1 8f m 2 fm 2 Carrier Doppler spectrum Spectrum Empirical investigations show results that deviate from this model Power Model Power goes to infinity at fc+/-fm Baseband Spectrum Doppler Faded Signal Cause baseband spectrum has a maximum frequency of 2fm Simulating Doppler/Small-scale fading Simulating Doppler fading Procedure Level Crossing Rate (LCR) Threshold (R) LCR is defined as the expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified threshold level R in a positive going direction. It is given by: N R 2 f m e 2 where R / rrms (specfied envelope value normalized to rms) N R : crossings per second Average Fade Duration Defined as the average period of time for which the received signal is below a specified level R. For Rayleigh distributed fading signal, it is given by: 1 1 2 P r[r R ] 1 e NR NR 2 e 1 , f m 2 R rrms Fading Model: Gilbert-Elliot Model Fade Period Signal Amplitude Threshold Time t Good Bad (Non-fade) (Fade) Gilbert-Elliot Model 1/AFD Good Bad (Non-fade) (Fade) 1/ANFD The channel is modeled as a Two-State Markov Chain. Each state duration is memory-less and exponentially distributed. The rate going from Good to Bad state is: 1/AFD (AFD: Avg Fade Duration) The rate going from Bad to Good state is: 1/ANFD (ANFD: Avg Non-Fade Duration) Simulating 2-ray multipath a1 and a2 are independent Rayleigh fading 1 and 2 are uniformly distributed over [0,2) Simulating multipath with Doppler-induced Rayleigh fading Review Review Review Review Homework due 2/5 Communication toolbox – TS, sample time, FD Doppler shift, K Rician factor, number of antenna NT=NR=2 – awgn – rayleighchan (TS, FD) – ricianchan(TS, FD, K) – stdchan: select 3 channels – mimochan(NT, NR, TS, FD) Task 1: Plot channel characteristics for above channels Task 2: Plot BER for BPSK for above channels – – – – qammod and qamdemod berawgn berfading biterr Task 1 Example: ts = 0.1e-4; fd = 200; chan = stdchan(ts, fd, 'cost207TUx6'); chan.NormalizePathGains = 1; chan.StoreHistory = 1; y = filter(chan, ones(1,5e4)); plot(chan); Task 2 BER for BPSK modulation in Rayleigh channel AWGN-Theory Rayleigh-Theory Rayleigh-Simulation -1 10 -2 10 Bit Error Rate clear N = 10^6 % number of bits or symbols % Transmitter ip = rand(1,N)>0.5; % generating 0,1 with equal probability s = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 0 Eb_N0_dB = [-3:35]; % multiple Eb/N0 values for ii = 1:length(Eb_N0_dB) n = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % white gaussian noise, 0dB variance h = 1/sqrt(2)*[randn(1,N) + j*randn(1,N)]; % Rayleigh channel % Channel and noise Noise addition y = h.*s + 10^(-Eb_N0_dB(ii)/20)*n; % equalization yHat = y./h; % receiver - hard decision decoding ipHat = real(yHat)>0; % counting the errors nErr(ii) = size(find([ip- ipHat]),2); end simBer = nErr/N; % simulated ber theoryBerAWGN = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))); % theoretical ber EbN0Lin = 10.^(Eb_N0_dB/10); theoryBer = 0.5.*(1-sqrt(EbN0Lin./(EbN0Lin+1))); % plot close all figure semilogy(Eb_N0_dB,theoryBerAWGN,'cd-','LineWidth',2); hold on semilogy(Eb_N0_dB,theoryBer,'bp-','LineWidth',2); semilogy(Eb_N0_dB,simBer,'mx-','LineWidth',2); axis([-3 35 10^-5 0.5]) grid on legend('AWGN-Theory','Rayleigh-Theory', 'Rayleigh-Simulation'); xlabel('Eb/No, dB'); ylabel('Bit Error Rate'); title('BER for BPSK modulation in Rayleigh channel'); -3 10 -4 10 -5 10 0 5 10 15 20 Eb/No, dB 25 30 35