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Mobile Radio PropagationChannel Models
Dr. Chih-Peng Li ()
2
Table of ContentsIntroductionPropagation Path Loss Model
Friis Free Space ModelHata Model
Large Scale Propagation Model Long Term FadingLognormal Distribution
Small scale Propagation Model Short Term FadingMultipath
Delay Spread vs. Coherent BandwidthFlat Fading vs. Frequency Selective FadingNarrowband vs. Wideband
3
Table of ContentsDopplers Effect
Dopplers Shift vs. Coherent TimeFast Fading vs. Slow Fading.
Rayleigh DistributionLevel Crossing Rate
Ricean DistributionComputer Simulation of Multipath InterferenceChannel Models in WCDMAFading Counteraction Diversity Schemes
Introduction
5
Main Components of Radio Propagation
Propagation Path Loss. ( ~ 1/r2 in free space)Large Scale: Propagation models that predict the strength for an arbitrary separation distance.Small Scale: Propagation models that characterize the rapid fluctuation of the received signal strength over very short travel distance (~ ) or short time duration (~s).
6
Large-Scale Propagation ModelsPropagation models that predict the mean signal strength for an arbitrary transmitter-receiver (T-R) separation distance are useful in estimating the radio coverage area of a transmitter.
They characterize signal strength over large T-R (transmitter receiver) separation distances (several hundreds or thousands of meters).
7
Small-Scale Propagation ModelsSmall-scale fading is used to describe the rapid fluctuation of the amplitude of a radio signal over a short period of time or travel distance.Fading is caused by interference between two or more versions of the transmitted signal which arrive at the receiver at slightly different time.These waves, called multi-path waves, combine at the receiver antenna to give a resultant signal which can vary widely in amplitude and phase.
8
Small-Scale Propagation Models
The received signal power may vary by as much as three or four orders of magnitude (30 or 40 dB) when the receiver is moved by only a fraction of a wavelength.Typically, the local average received power is computed by averaging signal measurements over a measurement track of 5 to 40 .
Hint : with f = 1 GHz ~ 2 GHz,
cmcmfc 15~30==
9
Small-Scale vs. Large-Scale Fading
Propagation Path Loss Model
11
Free Space Propagation Model
The free space propagation model is used to predict received signal strength when the transmitter and receiver have a clear, unobstructed line-of-sight (LOS)path between them.
e.g. satellite, microwave ling-of-sight radio link.
As with most large-scale radio wave propagation models, the free space model predicts that received power decays as a function of the T-R separation distance raised to some power.
12
Friis Free Space Equation
meters.in h wavelengt the:n.propagatio torelatednot factor loss system the:
meters.in distance separation R-T the:gain. antennareceiver the:
gain. antennaer transmitt the:.separation R-T
theoffunction a ish power whic received :)(power. ed transmitt:
)4()( 22
2
LdGG
dPP
LdGGPdP
r
t
r
t
rttr =
13
The gain of an antenna is related to its effective aperture, Ae, by:
The effective aperture Ae is related to the physical size of the antenna.The miscellaneous losses L (L1) are usually due to transmission line attenuation , filter losses, and antenna losses in the communication system.L=1 indicates no loss in the system hardware.
Friis Free Space Equation
2
4 eAG =
14
Log-distance Path Loss Model
( )0
ndPL dd
( ) ( )00
dB 10 log dPL PL d nd
= +
15
Hata Model
An empirical formulation of the graphical path loss data provided by OkumuraValid from 150M to 1500Mhz.hte : 30m ~ 200m (base station antenna height)hre : 1m~10m (mobile antenna height)d : T-R separation distance (in km)a (hre) : correction factor for effective mobile antenna height.
dhhahfdburbanL
terete
c
log)log55.69.44()(log82.13log16.2655.69))((50
++=
16
Hata Model
For small to medium sized city:
For large city:
For suburban area:
For open rural areas:
dBfhfha crecre )8.0log56.1()7.0log1.1()( =
2
2
( ) 8.29(log1.54 ) 1.1 for 300
( ) 3.2(log11.75 ) 4.97 for 300re re c
re re c
a h h dB f MHz
a h h dB f MHz
=
=
4.5)]28/[log(2)()( 25050 = cfurbanLdbL
98.40log33.18)(log78.4)()( 25050 = cc ffurbanLdbL
17
Hata Model in PCS BandExtension of Hata model to 2 GHz.
Mtere
tec
CdhhahfdburbanL
+++=
10
50
log)log55.69.44()(log82.13log9.333.46))((
centersan metropolitfor 3dB areassuburban andcity sized mediumfor 0dB
==MC
kmkmdmmh
mmhMHzMHzf
re
te
20~1:10~1:
200~30:2000~1500:
Large Scale Propagation ModelLong-Term Fading
19
Shadow Fading Lognormal DistributionWhen reaching the mobile station, the radio wave will have traveled through different obstructions such as buildings, tunnels, hills, trees, etc. This is called the shadowing effect.The received signal R, when measured in decibels, has a normal density function. Thus R is described by lognormal distribution.
The PDF of R is:
X: a zero-mean Gaussian distributed random variable (dB) with standard deviation 4 - 10 dB.
( ) ( ) ( ) XddndPLXdPLdPL +
+=+=
00 log10
( )( ) ( )
( )
2 2ln / 21 02
0 0
r me rp r r
r
=
20
Correlation of Path Loss
Shadow Fading:
Correlation of Path Loss: (from Viterbi, Principles of Spread Spectrum Communications)
1010nattenuatio
2 2
2
1,
( ) ( ) ( ) 0,
( ) ( ) ( ) .
mobile base station
mobile base station
mobile base station
a b
a b
E E E
Var Var Var
= +
+ =
= = =
= = =
Small Scale Propagation ModelShort-Term Fading
22
Small Scale Fading -- 1Problem 1: multi-path induces inter-symbol interference(ISI) and delay spread.
23
Impulse Response Model of a MultipathChannel
A mobile radio channel may be modeled as a linear filter with a time varying impulse response, where the time variation is due to receiver motion in space.The filtering nature of the channel is caused by the summation of amplitudes and delays of the multiple arriving waves at any instant of time.
24
Channel Impulse ResponseDue to the different multipath waves which have propagation delays which vary over different spatial locations of the receiver, the impulse response of the linear time invariant channel shouldbe a function of the position of the receiver.
( , ) ( ) ( , ) ( ) ( , )
where ( , ) is the channel impulse resonse. ( ) is the transmitted signal. ( , ) is the received signal at position .
y d t x t h d t x h d t d
h d tx ty d t d
= =
25
Multipath Radio Channel
26
Multipath Propagation Effect
27
Measured Multipath Power Delay Profiles
From a 900 MHz cellular system in San Fancisco.
28
Measured Multipath Power Delay Profiles
Inside a grocery store at 4 GHz.
29
Delay SpreadDelay spread and coherence bandwidth are used to describe the time dispersive nature of the channel.Received Signal:
Delay Spread corresponds to standard deviation of Ti .
Excess delay is the relative delay of the i-th multipathcomponent as compared to the first arriving component.
)()(1
=
=n
iii Ttath
30
Time Dispersion Parameters
Mean Excess Delay: the first moment of the power delay profile.
RMS (room mean square) Delay Spread: the square root of the second central moment of the power delay profile.
==
kk
kkk
kk
kkk
P
P
a
a
)(
)(
2
2
22 )( =
==
kk
kkk
kk
kkk
P
P
a
a
)(
)( 2
2
22
2
31
Time Dispersion Parameters
These delays are measured relative to the first detectable signal arriving at the receiver at 0=0.The equations in the previous page do not rely on the absolute power level of P(), but only the relative amplitudes of the multipath components within P().Typical values of rms delay spread are on the order of microseconds in outdoor mobile radio channels and on the order of nanoseconcds in indoor radio channels.
32
Typical Measured Values of RMS Delay Spread
33
Maximum Excess DelayMaximum Excess Delay (dB) = the time delay during which multipath energy falls to dB below maximum.Maximum Excess Delay (dB) can also be defined as x-0, where 0 is the first arriving signal; x is the maximum delay at which a multipath component is within X dB of the strongest arriving multipath signal.The value of x is sometimes called the excess delay spread of a power delay profile.
34
Example of An Indoor Power Delay Profile
35
Coherence BandwidthTime domain focus on excess delay.Frequency domain focus on coherence bandwidth Bc.Bc is defined related to rms delay spread 1/(Bc)Bc (Coherence Bandwidth)
A statistical measure of the range of frequencies over which the channel can be considered flat (i.e. a channel which passes all spectral components with approximately equal gain and linear phase.).The range of frequencies over which two frequency components have a strong potential for amplitude correlation.Two sinusoids with frequency separation greater than Bc are affected quite differently by the channel.
36
Coherence BandwidthVersion 1: the bandwidth over which the frequency correlation is above 0.9
Version 2: the bandwidth over which the frequency correlation is above 0.5
150c
B
=
15c
B
=
37
Types of Small-Scale Fading
Based on multi-path time delay spread
Flat Fading (narrowband system)BW of signal < BW of channelDelay spread < Symbol period
Frequency Selective Fading (wideband system)BW of signal > BW of channelDelay spread > Symbol period
38
Wideband v.s. Narrowband
f f
t1
t2 Signal Bandwidth
wideband narrow band
Signal Bandwidth
39
Flat Fading
Signal undergoes flat fading if CS BB >STTs : reciprocal BW (e.g. symbol period)Bs : BW of the TX modulation : rms delay spread Bc : Coherence BW
40
Flat Fading
The mobile radio channel has a constant gain and linear phase response over a bandwidth which is greater than the bandwidth of the transmitted signal.The multipath structure of the channel is such that the spectral characteristics of the transmitted signal are preserved at the receiver.The strength of the received signal changes with time, due to fluctuations in the gain of the channel caused by multipath.
41
Flat Fading
Typical flat fading channels cause deep fades, and thus may require 20 or 30 dB more transmitter power to achieve low bit error rates during times of deep fades as compared to systems operating over non-fading channels.Also known as amplitude varying channel.Also referred to as narrowband channels since the bandwidth of the applied signal is narrow as compared to the channel flat fading bandwidth.The most common amplitude distribution of flat fading channel is the Rayleigh distribution.
42
Frequency Selective Fading
Signal undergoes frequency selective fading if CS BB > and
43
Frequency Selective FadingThe channel possesses a constant-gain and linear phase response over a bandwidth that is smaller than the bandwidth of transmitted signal.The received signal includes multiple versions of the transmitted waveform which are attenuated and delayed in time.The channel induces inter-symbol interference.Certain frequency components in the received signal spectrum have greater gains than others.Also known as wideband channels.M-ray Rayleigh fading model is usually used for analyzing frequency selective small-scale fading.
44
Small Scale Fading -- 2Problem 2: moving receiver induces fading effects (Doppler shift) for each ray path.
45
Doppler Effect
Doppler is a frequency shift, cause by movement of the mobile antenna relative to the base station-f = V/ (at 250 km/h and 900 MHz, f = 208 Hz)
V
f
f + ff - f
46
Doppler Shift / Spread
coscos tvdl ==
cos22 :Change Phase tvl ==
cos21 :ShiftDoppler v
tfd =
=
47
Doppler Spread and Coherence Time
Delay spread and coherence bandwidth do not offer information about the time varying nature of the channel caused by either relative motion between the mobile and base station, of by movement of objects in the channel.Doppler spread and coherence time are parameters which describe the time varying nature of the channel in a small-scale.Doppler spread BD is a measure of the spectral broadening caused by the time rate of change of the mobile radio channel and is defined as the range of frequencies over which the received Doppler spectrum is essentially non-zero.When a pure sinusoidal tone of frequency fc is transmitted, the received signal spectrum, called the Doppler spectrum, will be in the range fc-fd to fc+fd , where fd is Doppler shift.
48
Coherence Time
Coherence time Tc is the time domain dual of Doppler spread and is used to characterize the time varying nature of the frequency dispersiveness of the channel in the time domain.Coherence time is a statistical measure of the time duration over which the channel impulse response is essentially invariant.Coherence time is the time duration over which two received signals have a strong potential for amplitude correlation.
49
Coherence TimeVersion 1:m : the maximum Doppler shift, m = /
Version 2: the time over which the time correlation > 0.5
Version 3: Geometric mean of version 1 and version 2.
mc fT 1
mc fT 169
mmc ff
T 423.016
92 ==
50
Doppler Power Spectrum
51
Types of Small-Scale FadingBased on Doppler Spread
Fast FadingHigh Doppler spread.Coherence time < Symbol Period.Channel variations faster than base-band signal variations
Slow FadingLow Doppler spread.Coherence time > Symbol period.Channel variations slower than base-band signal variations
52
Fast FadingTS>TC and BS
53
Slow Fading
TSBDChannel impulse response changes at a rate much slower than the transmitted baseband signal s(t).The channel may be assumed to be static over one or several reciprocal bandwidth intervals.The Doppler spread of the channel is much less than the bandwidth of the baseband signal.The velocity of the mobile and the basebandsignaling determines whether a signal undergoes fast fading or slow fading.
54
Type of Small-Scale Fading
55
Type of Small-Scale Fading
Rayleigh Distribution
57
Rayleigh Distribution
Rayleigh distributions are 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.
Envelope of the sum of two quadrature Gaussian noise signals obeys a Rayleigh distribution.
58
Typical Rayleigh Fading Envelope
59
Rayleigh DistributionConsider a carrier signal s at a frequency 0 and with an amplitude a:
The received signal sr is the sum of n waves:
)exp( 0tjas =
[ ] [ ]
sin cos :where
sin and cos :have We
sincos)exp( :Define
)exp()exp( where
)(exp)(exp
22211
11
1
01
0
ryrxyxr
ayax
jyxajajr
jajr
tjrtjas
n
iii
n
iii
n
iii
n
iii
n
iii
n
iiir
==+=
++=
=
++=
==
==
=
=
60
Rayleigh DistributionBecause (1) n is usually very large, (2) the individual amplitudes ai are random, and (3) the phases i have a uniform distribution, it can be assumed that (from the central limit theorem) x and yare both Gaussian variables with means equal to zero and variance
Because x and y are independent random variables, the joint distribution p(x,y) is
The distribution p(r,) can be written as a function of p(x,y) :
222 = yx
+== 2
22
2 2exp
21)()(),(
yxypxpyxp
),(),( yxpJrp =
61
Rayleigh Distribution
The probability that the envelope of the received signal does not exceed a specified value R is given by the corresponding cumulative distribution function (CDF)
2
2
22
2 2
0
/ / cos sin; ( , ) exp
/ / sin cos 2 2Thus, the Rayleigh distribution has a pdf:
exp 0 ( ) ( , ) 2
0 ot
x r x r r rJ r p ry r y r
r r rp r p r d
= = =
= =
herwise
=== 2
2
0 2exp1)()Pr()(
RdrrpRrRP
R
62
Rayleigh DistributionMean:
Variance:
Median value of r is found by solving:
Mean squared value:
Most likely value = max { p(r) } =
2533.12
)(][0
====
drrrprErmean
22
2
0
2222
4292.02
2
2)(][][
=
=
==
drrprrErEr
=medianr
drrp0
)(21
177.1=medianr
==0
222 2)(][ drrprrE
63
Rayleigh Probability Density Function
64
Level Crossing and Fading Statistics
The level crossing rate (LCR) is defined as the expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified level in a positive-going direction.
Useful for designing error control codes and diversity.Relate the time rate of change of the received signal to the signal level and velocity of the mobile.
65
Level Crossing Rate (LCR)
The number of level crossings per second is given by:2
0
( , ) 2
is the tim e derivative of ( ) (i.e. the slope)
( , ) is the joint density function of and at is the m axim um D oppler frequency
/ is the value of the sp
R m
m
rm s
N r p R r d r f e
r r t
p R r r r r Rf
R R
= =
=
=
ecified level , norm alized to the local rm s am plitude of the fading envelope.
R
Clarke, R. H., A Statistical Theory of Mobile-Radio Reception,Bell Systems Technical Journal, Vol. 47, pp. 957-1000, 1968.
66
Average Fade DurationAverage fade duration is defined as the average period of time for which the received signal is below a specified level R. For a Rayleigh fading signal, this is given by:
where Pr[rR] is the probability that the received signal r is less than R and is given by
where i is the duration of the fade and T is the observation interval of the fading signal.
[ ]1 PrR
r RN
=
[ ] 1Pr ii
r RT
=
67
Average Fade DurationThe probability that the received signal r is less than the threshold R is found fro the Rayleigh distribution as:
where p(r) is the pdf of a Rayleigh distribution.The average fade duration as a function of and fm can be expressed as
)exp(1)(]Pr[ 20
== R
drrpRr2
22
2 R=
2
12m
ef
=
68
Average Fade DurationThe average duration of a signal fade helps determine the most likely number of signaling bits that may be lost during a fade.
Average fade duration primarily depends upon the speed of the mobile, and decreases as the maximum Doppler frequency fm becomes large.
Ricean Distribution
70
Ricean Fading Distribution
When there is a dominant stationary (non-fading) signal component present, such as a line-of-sight propagation path, the small-scale fading envelope distribution is Ricean.
sincos
)(
)](exp[)exp(])[( )exp()](exp['
22200
esdirect wav
0
wavesscattered
0
ryrAx
yAxr
tjrtjjyAxtjAtjrsr
==+
++=
+++++=
71
Ricean Fading DistributionBy following similar steps described in Rayleigh distribution, we obtain:
( )
=
=
=