EE 6332, Spring, 2014 Wireless Communication
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Transcript of EE 6332, Spring, 2014 Wireless Communication
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EE 6332, Spring, 2014
Wireless Communication
Zhu Han
Department of Electrical and Computer Engineering
Class 4
Jan. 27th, 2014
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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
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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
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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.
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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.
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Rayleigh FadingRayleigh Fading
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Rayleigh FadingRayleigh Fading
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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
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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
[ ] [ ] ( ) .
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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
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A typical Rayleigh fading envelope at 900MHz.A typical Rayleigh fading envelope at 900MHz.
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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.
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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
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CDF CDF Cumulative distribution for three small-scale fading measurements and their
fit to Rayleigh, Ricean, and log-normal distributions.
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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.
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Rice time seriesRice time series
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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
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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
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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
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Carrier Doppler spectrumCarrier Doppler spectrum Spectrum Empirical investigations show results that deviate
from this model Power Model Power goes to infinity at fc+/-fm
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Baseband Spectrum Doppler Faded SignalBaseband Spectrum Doppler Faded Signal Cause baseband spectrum has a maximum frequency of 2fm
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Simulating Doppler/Small-scale fadingSimulating Doppler/Small-scale fading
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Simulating Doppler fadingSimulating Doppler fading
Procedure
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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
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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
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Fading Model: Gilbert-Elliot ModelFading Model: Gilbert-Elliot Model
Fade Period
Time t
SignalAmplitude
Threshold
Good(Non-fade)
Bad(Fade)
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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)
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Simulating 2-ray multipathSimulating 2-ray multipath
a1 and a2 are independent Rayleigh fading
1 and 2 are uniformly distributed over [0,2)
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Simulating multipath with Doppler-induced Rayleigh fadingSimulating multipath with Doppler-induced Rayleigh fading
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Review Review
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Review Review
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Review Review
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Review Review
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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
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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);
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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