Stochastic small fading model

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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


0r 
 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 = 0.42922
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  ,
A0

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 A0, 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  
8f 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
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