EE 6332, Spring, 2014

Wireless Communication

Zhu Han

Department of Electrical and Computer Engineering

Class 4

Jan. 27th, 2014

                                                           

                                                           

OutlineOutline 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 DistributionsFading 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 DistributionsRayleigh 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 DistributionsRicean 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 FadingRayleigh Fading

                                                           

Rayleigh FadingRayleigh Fading

                                                           

Rayleigh Fading DistributionRayleigh 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.

p rr r

r

r

( )exp( )

2

2

220

0 0

                                                           

Rayleigh Fading DistributionRayleigh Fading Distribution

The probability that the envelope of the received signal does not exceed a specified value of R is given by the CDF:

rpeak= and p()=0.6065/

R R

r edrrpRrPRP0

2 2

2

1)()()(

2

)(2

1177.1

2533.12

)(][

0

0

rms

r

median

mean

r

drrpr

drrrprEr

median

solvingby found

r E r E r r p r dr2 2 2 22

0

2

20 4292

[ ] [ ] ( ) .

                                                           

Rayleigh PDFRayleigh PDF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

mean = 1.2533median = 1.177variance = 0.4292

                                                           

A typical Rayleigh fading envelope at 900MHz.A typical Rayleigh fading envelope at 900MHz.

                                                           

Ricean DistributionRicean 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 DistributionRicean 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 A0, K - dB, Ricean distribution degenerates to Rayleigh

distribution.

p rr r A

IAr

r A

r

( )exp[

( )] ( ) ,

2

2 2

2 0 220 0

0 0

KA

2

22

                                                           

CDF CDF Cumulative distribution for three small-scale fading measurements and their

fit to Rayleigh, Ricean, and log-normal distributions.

                                                           

PDFPDF 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 seriesRice time series

                                                           

Nakagami ModelNakagami Model

Nakagami Model

r: envelope amplitude Ω=<r2>: time-averaged power of received signal m: the inverse of normalized variance of r2

– Get Rayleigh when m=1

m

mm

m

rm

rmrp

)(

)exp(2)(

212

                                                           

Small-scale fading mechanismSmall-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

nn av

f cos

                                                           

Small-scale fading: effect of Doppler in a Small-scale fading: effect of Doppler in a multipath environmentmultipath environment

fm, the largest Doppler shift

2

21

8

1)(

mmbbEz f

fk

ffS

                                                           

Carrier Doppler spectrumCarrier 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 SignalBaseband Spectrum Doppler Faded Signal Cause baseband spectrum has a maximum frequency of 2fm

                                                           

Simulating Doppler/Small-scale fadingSimulating Doppler/Small-scale fading

                                                           

Simulating Doppler fadingSimulating Doppler fading

Procedure

                                                           

Level Crossing Rate (LCR)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 directionpositive going direction. It is given by:

second per crossings

rms) to normalized value envelope (specfied

where

:

/

22

R

rms

mR

N

rR

efN

                                                           

Average Fade DurationAverage Fade Duration

Defined as the average period of time for which the received signal isbelow a specified level R.

For Rayleigh distributed fading signal, it is given by:

rmsm

RR

r

R

f

e

eN

RrN

,2

1

11

]Pr[1

2

2

                                                           

Fading Model: Gilbert-Elliot ModelFading Model: Gilbert-Elliot Model

Fade Period

Time t

SignalAmplitude

Threshold

Good(Non-fade)

Bad(Fade)

                                                           

Gilbert-Elliot ModelGilbert-Elliot Model

Good(Non-fade)

Bad(Fade)

1/ANFD

1/AFD

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 multipathSimulating 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 fadingSimulating multipath with Doppler-induced Rayleigh fading

                                                           

Review Review

                                                           

Review Review

                                                           

Review Review

                                                           

Review Review

                                                           

Homework due 2/5Homework 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 1Task 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 2Task 2clear

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');0 5 10 15 20 25 30 35

10-5

10-4

10-3

10-2

10-1

Eb/No, dB

Bit

Err

or R

ate

BER for BPSK modulation in Rayleigh channel

AWGN-Theory

Rayleigh-TheoryRayleigh-Simulation