Slide 1

EE 6332, Spring, 2014 Wireless Communication Zhu Han Department of Electrical and Computer Engineering Class 4 Jan. 27 th, 2014 Slide 2 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 Slide 3 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 Slide 4 Rayleigh Distributions Describes the received signal envelope distribution for channels, where all the components are non-LOS: i.e. there is no line-ofsight (LOS) component. Slide 5 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. Slide 6 Rayleigh Fading Slide 7 Rayleigh Fading Slide 8 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. 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. Slide 9 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: r peak = and p( )=0.6065/ Slide 10 Rayleigh PDF mean = 1.2533 median = 1.177 variance = 0.4292 Slide 11 A typical Rayleigh fading envelope at 900MHz. Slide 12 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. Slide 13 Ricean Fading Distribution When there is a dominant stationary signal component present, the small- scale 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. 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. K is known as the Ricean factor As A 0, K - dB, Ricean distribution degenerates to Rayleigh distribution. Slide 14 CDF Cumulative distribution for three small-scale fading measurements and their fit to Rayleigh, Ricean, and log-normal distributions. Slide 15 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. Slide 16 Rice time series Slide 17 Nakagami Model u r: envelope amplitude u = : time-averaged power of received signal u m: the inverse of normalized variance of r 2 Get Rayleigh when m=1 Slide 18 Small-scale fading mechanism Assume signals arrive from all angles in the horizontal plane 0 Task 2 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');