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Chapter 5Small-scale Multipath Propagation
Multi-path Propagation10
-100
time or wavelength
dB
Wireless Communication
Chapter 5 –Small-scale multipath propagation 1 Dr. Sheng-Chou Lin
Small-scale Fading and Multipath
Rapid fluctuation of the amplitude of a radio signal over a short period oftime or travel distance
Fading is caused by multipath waves•Transmitted signal which arrive at the receiver at slightly different times
Effects: factors influencing small-scale fading•Rapid changes in signal strength over a small travel distance or time interval
–Random frequency modulation varying Doppler shift–Speed of the mobile or speed of surrounding objects.
•Time dispersion Multipath delay : depends on bandwidth of the signal. Fading
•No single line-of-sight (LOS): mobile antennas are below the height of surroundstructures
•With LOS, multipath still occurs•Multipath random distributed amplitude, phases and angles.•A mobile is stationary, the signal may fade due to movement of surrounding objects.•A receiver moving at high speed can pass through several fades in a small of time.•Doppler shift
Page 2
Wireless Communication
Chapter 5 –Small-scale multipath propagation 2 Dr. Sheng-Chou Lin
Multipath Fading
Slow Fading•over large distances, due to gross changes in path•also called shadowing, log-normal fading
Fast Fading•over distances on the order of a wavelength•also called Rayleigh fading
Assumptions for above types:•many waves of roughly equal amplitude arrive•Rayleigh distributed amplitude•uniformly distributed phase•spatial angle of arrival•azimuth is uniformly distributed•elevation: PDF has mean of 0o, biased towards small angles, does not
extend to infinity, and has no discontinuities Rician Fading
•there is a LOS or dominant path, producing fewer deep fades occurs insmall cells
Wireless Communication
Chapter 5 –Small-scale multipath propagation 3 Dr. Sheng-Chou Lin
Multipath Fading: Duration andFrequency
n = average number of level crossings at 10 dB below average power
t = average fade duration at 10 dB below RMS power
n = 0.75 crossings/second and t = 0.132 secondsV
V
V = vehicle speed, = carrier wavelength
e.g. at 850 mHz and 15 miles per hour, n = 16 crossings/second, and t = 6 msec
for a Rayleigh signal 95% of the amplitude is above -10 dB
10
-100
AT LEAST 18 dB C/I
time or wavelength
dBTHRESHOLD FORESTIMATING n and t
Page 3
Wireless Communication
Chapter 5 –Small-scale multipath propagation 4 Dr. Sheng-Chou Lin
Multi-path Propagation Effects
Signal levels vary as user moves Slow variations come from blockage
and shadowing by large objectssuch as hills and buildings
Rapid Fading comes as signalsreceived from many paths drift intoand out of phase•phase cancellation occurs, causing
rapid fades that are occasionally deep
•the fades are roughly /2 apart:7 inches apart at 800 MHz.3 inches apart at 1900 MHz
•called Rayleigh fading, after thestatistical model that describes it
A
t
10-15 dB
Rayleigh Fading
Multi-path Propagation
Wireless Communication
Chapter 5 –Small-scale multipath propagation 5 Dr. Sheng-Chou Lin
Selective Diversity
Use a diversity scheme to take advantage of uncorrelated fadingUse the dominant instantaneous amplitudeThis eliminates most of the deep nulls
A
t
. . ...... ... ... . ..... .. . . .. ..
.. .... ...
path 1
path 2
. maximum amplitude
Page 4
Wireless Communication
Chapter 5 –Small-scale multipath propagation 6 Dr. Sheng-Chou Lin
Space DiversityA Method for Combating Rayleigh Fading
Fortunately, Rayleigh fades arevery short and last a smallpercentage of the time
Two antennas separated by severalwavelengths will not generallyexperience fades at the same time
“Space Diversity”can be obtainedby using two receiving antennasand switching instant-by-instant towhichever is best
Required separation D for gooddecorrelation is 10-20(BS)•12-24 ft. @ 800 MHz.•5-10 ft. @ 1900 MHz.
Required separation D is (MS)
Signal received byAntenna 1
Signal received byAntenna 2
Combined Signal
D
Wireless Communication
Chapter 5 –Small-scale multipath propagation 7 Dr. Sheng-Chou Lin
Space DiversityApplication Limitations
Space Diversity can be appliedonly on the receiving end of a link.
Transmitting on two antennaswould:•fail to produce diversity, since the
two signals combine to produce onlyone value of signal level at a givenpoint -- no diversity results.
•produce objectionable nulls in theradiation at some angles
Therefore, space diversity isapplied only on the “uplink”, i.e.,reverse path•there isn’t room for two sufficiently
separated antennas on a mobile orhandheld
Signal receivedby Antenna 1
Signal receivedby Antenna 2
CombinedSignal
D
Page 5
Wireless Communication
Chapter 5 –Small-scale multipath propagation 8 Dr. Sheng-Chou Lin
Doppler Shift
Doppler spreading increases the signal bandwidth•fd : moving toward, –moving away•fd = cos() (v/ )
Example: fc 1850 MHz, 60mile/hour (mph)•= c / fc = 3 × 108 / 1850 × 106 = 0.162 m•v = 60 mph = 26.82 m/s•The mobile is moving toward the transmitter, fd = 26.82 / 0.162 = 1850.0 Hz•The mobile is moving away the transmitter, fd = - 1850.0 Hz•fd = 0, as = 90 cos() = 0
D
v
Wireless Communication
Chapter 5 –Small-scale multipath propagation 9 Dr. Sheng-Chou Lin
Impulse Response Model of amultipath Channel
The small-scale variations of a mobile radio signal ; assumptions•The velocity may be assumed constant over a short time (or distance) interval.•The multipath channel is a band-limited band-pass channel•The high frequency variations caused by carrier is removed. (baseband).
ai ( t, ) : real amp. and excess delay ofIth multipath component at time t.
i (t, ) = 2fc I ( t ) + i (t, ) : thephase shift due to free spacepropagations of the Ith multipathcomponent + additional phase shifts
i (t, ) : lumps together all themechanisms for phase shifts of asingle multipath component withinthe ith excess delay bin.
hb (t, ) = ai ( t, ) exp [ j2fc i ( t ) + i (t, )] (-i ( t )]I = 0
N-1
i (t, )A time-varying discrete-time impulse response for a multipath radio channel
Page 6
Wireless Communication
Chapter 5 –Small-scale multipath propagation 10 Dr. Sheng-Chou Lin
Impulse Reponses Model (time invariant)
If the channel impulse response is assumed to be time invariantor is at WSS over a small-scale time or distance interval, then
Power delay profile: the spatial of | hb (t, ) |2 over a local area.
P ( t ; ) = k | hb (t, ) |2 base band
• average over a local area toprovide a single time -invariantmultipath power make severallocal area measurement indifferent location P ()
hb () = ai exp [ ji ] (- i )i = 0
N-1
x + i y
Measurement : hb ( ) can bepredicted by a probing pulse p( t )
p( t ) ( t - )
, K relates the transmitted power
x2(t) = 1/2c(t)2
Wireless Communication
Chapter 5 –Small-scale multipath propagation 11 Dr. Sheng-Chou Lin
Bandwidth and Received power
Wideband signal : a very narrow pulse, p( t ), does not fluctuate when areceiver is moved about a local area The received power varies very little
Narrowband signal : the CW signal strength will vary at a rate governed bythe fluctuations of ai and i large signal fluctuations (fading) occur•ai varies little over local area•I varies greatly due to changes
in propagation distance•When the path amplitudes are
uncorrelated, multipath phasesare I.I.d over [ 0 , 2]
•multipath is not resolved• fading due to the phase shifts of
the many unresolved multipathcomponents
– Ex: Tb = 10ns wideband signal andCW signal, fc = 4GHz
Wideband
Narrowband
Page 7
Wireless Communication
Chapter 5 –Small-scale multipath propagation 12 Dr. Sheng-Chou Lin
Bandpass and Baseband channelimpulse response
Mobile radio channel as a function of time and space.•Channel impulse response = h(d, t), x(t) = transmitted signal• the received signal y(d,t) = x(t) h(d,t), d: position of the receiver•d = vt, v : assumed constant over a short time interval.•h(d, t) h( t, ) , t : time due to motion, : multipath delay for a fixed t.
Bandpass channel Complex baseband impulse response
Bandpass channel
fc-fc
Complex Baseband
factor of 1/2 are due to the properties of the complex envelope
1
2
y(t)
f
f
f
fy(t)~
y(t) = 1/2 x(t ) h(t ),~ ~ ~ x(t ) = c(t ), h(t ) = hb(t ), y(t ) = r(t )~ ~ ~
Wireless Communication
Chapter 5 –Small-scale multipath propagation 13 Dr. Sheng-Chou Lin
Complex Envelop of Bandpass System
fc-fc
1
2
x(t)
f
fc(t)
fc-fc
1h(t)
f
2hb(t)
f
r(t)= ½ (c(t) hb(t)
f
f
2
1 y(t)
tjhthth,ethReth IRbtfj
bc 2
c.cedetch
c.cedtch
dtxhthtxty
tfjfjb
tfjb
cc
c
22
2
41
41
c.cethth tfjb
c 2
21 Immediate complex
term
c.cetctx tfj c 2
21
0
tcth21
tretcthRety btfj
bc
2
21
Cos(2fct )+ j sin(2fct )
Page 8
Wireless Communication
Chapter 5 –Small-scale multipath propagation 14 Dr. Sheng-Chou Lin
Channel Baseband Complex Envelope
Baseand impulse of a multipath channel
hb ( ) = ai exp [ ji ] (- i )i = 0
N-1
x + i y
Time invariant baseand impulse
d(t)
t
•Power delay profile : the spatialaverage of hb (t, ) 2 over alocal area ~ d ~ t.
•To provide a single time-invariantmultipath power delay profile P ()
•Maximum bandwidth that thismodel can accurately represent isequal to 1 / 2
hb ( t, ) = ai ( t, ) exp [ j2fc i ( t ) + i (t, ) ] (- i ( t ) )I = 0
N-1
i (t, )
Initial phase
hbth Re[)(
Wireless Communication
Chapter 5 –Small-scale multipath propagation 15 Dr. Sheng-Chou Lin
Wideband Signal in Mutipath Channel A pulsed, transmitted RF signal
x( t ) = Re { p( t ) exp [ j2 fc( t ) ],
• a repetitive baseband pulsetrain with very narrow pulse width Tbb andperiod TREP Wideband signal
• TREP >> max , max : maximum excess delay• Low-pass channel output r(t) hb (t)
To determine the received power atsome time t0 Complex Base Band•The measured power if the multipath
components are resolved
r (t ) = ai exp [ ji ] p(t - i )i = 0
N-1Measure at t0 ~ d0
t 1 2 3 i
a1
a2 a3ai
Resolved
Tbb
TREP
Sum of powers received in each multipath bins receivedpower does not fluctuate with d ~ ai ,
PWB = |r (t0 ) |2 = ak2(t0 ) Ea, [PWB ]= ak
2
k= 0
N-1 N-1
k= 0
f
d0
d1
a1
a2
Instantaneous power delay profile
d2average
bbmax Ttp 2
2
Page 9
Wireless Communication
Chapter 5 –Small-scale multipath propagation 16 Dr. Sheng-Chou Lin
Narrow Signal in Mutipath Channel A CW, transmitted RF signal
x( t ) = Re { p( t ) exp [ j2 fc( t ) ],
• The complex envelope is given by c(t) = 2• Instantaneous complex envelope of the received
signal
Instantaneous power (Complex base band )
r (t ) = ai exp [ ji ( t, ) ]i = 0
N-1
Tbb
f
d0
d1
a1
a2
• r i, j = Ea [ai aj ]: path amp. Correlation coefficient
• Ea,[Pcw ]= Ea, [PWB ] as r i, j and/or cos(i - j ) =0
This can occur i are i.i.d over [0,2] or path amplitudesare uncorrelated
Instantaneous envelope
N-1
Pcw = |r (t0 ) |2 = |ai exp [ ji ( t, ) ] |2
Ea,[Pcw ] = ai2 + 2 r i, j cos(i - j )
i = 0N-1
i = 0 i = 0
N-1
j i
N
a1 + a2
fading
Measured at t0 ~ d0
d
Wireless Communication
Chapter 5 –Small-scale multipath propagation 17 Dr. Sheng-Chou Lin
An Example (SMRCIM)
This technique of quantizing the delay bins determines thetime delay resolution of the channel model•Maximum bandwidth that the SMRCIM model (Simulation of Mobile
Radio Channel Impulse-response Models) can accurately representis equal to 1 / 2 (useful frequency span of the model)
Example: A discrete channel impulse response model, Ifnumber of multipath bins is 64,•urban radio cahannel with excess delays up to 100 s.
–= N / N = 100/ 64 =1.5625 s–1 / 2 = 1/ (2(1.5625 s)) = 0.32 MHz
•microcellular channels with excess delays < 4 s.–= N / N = 4/ 64 =62.5 ns–1 / 2 = 1/ (2(62.5s)) = 8 MHz
•indoor channels with excess delays < 500ns–= N / N = 500 10-9 / 64 =7.8125 ns–1 / 2 = 1/ (2(7.8125 ns)) = 64 MHz
DELAY SPREAD FUNCTION
N =
Page 10
Wireless Communication
Chapter 5 –Small-scale multipath propagation 18 Dr. Sheng-Chou Lin
An Example (Narrow band v.s.Wideband)
A mobile traveling at a velocity of 10 m/s, two multipathcomponents, fc = 1000MHz, The first path with 0 = 0 and power = -70dBm, Second path with 1 = 1s and power = -73dBm. Mobilemoves directly towards the first path and away from the second.•0 = 0, 1 = 0, = c / f = 0.3m•P0 = -70dBm = 100pW, P1 = -73dBm = 50pW complex•at t =0, the narrow instantaneous power = r(t)2
=100pW exp(0)+ 50pW exp(0)2 = 291 pW•at t = 0.1s, 0 = 2d / = 2vt / = 210 0.1 / 0.3 = 20.94 rad = 2.09 rad= 120
– 1 = -120 , since mobile moves away from the second component.–r(t)2 = 100pW exp(j120)+ 50pW exp(-j120)2 = 79.3 pW
•at t = 0.3s, 0 = 360= 0, 1 = -360= 0= r(t)2 = 291 pW•at t = 0.4s, r(t)2 = 79.3 pW, at t = 0.5s, r(t)2 = 79.3 pW.•The average narrowband received power = (2)(291)+(4)(79.3)/6 = 149 pW•The average wideband received power = P0 + P1 = 100+50 = 150 pw
–PW,B PN,B, The wideband signal power remains constant over the same interval
first
second
t = 0
t = 0.1
Wireless Communication
Chapter 5 –Small-scale multipath propagation 19 Dr. Sheng-Chou Lin
Delay Profile
Measured multipath power delay profiles•900 MHz cellular in San Francisco•Inside a grocery store at 4GHz
Page 11
Wireless Communication
Chapter 5 –Small-scale multipath propagation 20 Dr. Sheng-Chou Lin
Time Dispersion Parameters
Power delay profile•Mean excess delay•RMS delay spread•Excess delay spread
k
ak2 k
k
ak2
=k
P( k ) k
k
P( k )
RMS deplay spread
Mean excess delay
= 2 - ( )2 Where 2
k
ak2 k
2
k
ak2
=k
P( k ) k2
k
P( k )
• In outdoor mobile: RMS ~ s • In indoor mobile: RMS ~ ns
Wireless Communication
Chapter 5 –Small-scale multipath propagation 21 Dr. Sheng-Chou Lin
An Example
Maximum excess delay ( xdB ):•time delay during which multipath energy falls to X dB below the maximum.•I.e. x - 0 , where0 is the first arriving signal, x is the maximum delay at
which a multipath component X dB of the strongest arriving component(which does not necessarily arrive at 0 )
Threshold level: , 2 , depend on the choice of noise threshold•noise threshold , , 2 ,
Example:
= 4.38 sec
2 = 21.07 sec
= 1.37 sec
Pr()0dB
-10dB
-20dB
-30dB
0 1 2 5
( s)
Bc =1
5
= 146 kHz
Page 12
Wireless Communication
Chapter 5 –Small-scale multipath propagation 22 Dr. Sheng-Chou Lin
Typical measured RMS delay spread
•Outdoor mobile channel : RMS is on the order of s• Indoor radio channel : RMS is on the order of ns
Wireless Communication
Chapter 5 –Small-scale multipath propagation 23 Dr. Sheng-Chou Lin
Coherence Bandwidth
Relation derived from RMS delay spread•BW Bc , the channel can be considered as “flat”•Flat channel: a channel which passes all spectral components with
equal gain and linear phase•Two frequency components have a strong potential for amplitude
correlation over the range of frequencies.
Relation between Frequency correlation function and Bc
•correlation function > 0.9 Bc
•correlation function > 0.5 Bc
Ex:= 1.37 sec, Bc 1/ 5= 146 kHz•AMP BW = 30k no equalizer required.•GSM 200 k equalizer required
1
50
1
5
CR > 0.5
CR > 0.9
Page 13
Wireless Communication
Chapter 5 –Small-scale multipath propagation 24 Dr. Sheng-Chou Lin
Signal BW v.s. Coherent Bandwidth
t f
t f
Narrowband Channel
Wideband Channel
1
Bc
BW > Bc
Flat channelBW Bc
Freq. Selective channel
BW
T
1/T
Wireless Communication
Chapter 5 –Small-scale multipath propagation 25 Dr. Sheng-Chou Lin
Flat and Frequency-selective Fading
Flat fading Frequency-selective fading
2-ray multipath channel (point-point)
i
Page 14
Wireless Communication
Chapter 5 –Small-scale multipath propagation 26 Dr. Sheng-Chou Lin
Channel Delay Spread, APhenomenological Model
•The delay spread of a channel d is the RMS value of the channelimpulse response (delay spread function)
• In a mobile environment, the delay spread function is constantlychanging (i.e., |h(f)| is a nonlinear time-varying filter)
•The channel transfer function |h(f)| has a lowpass characteristic withmultiple delays (time dispersion)
•The delay spread represents the time it takes most of the energy fromthe transmitter to propagate (at c = 3e+8 m/s) to the receiver
•can be considered the group delay of the channel model |h(f)|•For in-building propagation, = 0.1 s; for urban propagation = 3s
RECEIVERTRANSMITTER
| H(f) |
CHANNELTRANSFERFUNCTION
DELAY SPREAD FUNCTION
Wireless Communication
Chapter 5 –Small-scale multipath propagation 27 Dr. Sheng-Chou Lin
Coherence Bandwidth, FrequencyDiversity Gain and Delay Spread
The channel coherence bandwidth BC can be computed from thedelay spread d of a channel:
If a signal has a bandwidth b greater than BC , then the signal hasfrequency components that fade independently. the signal has afrequency diversity gain, G
Signals with bandwidths greater than BC are more resistant tochannel fading effects
EXAMPLES:
•Compute the coherence bandwidth of a channel with = 3s (Bc = 53 khz)
•Show there is no frequency diversity gain for amps.(AMPS = 30 khz < 53 khz)
•Compute the frequency diversity gain for CDMA. ( g = 1 + 1.25 x 3 = 4.75)
BC =1
2
G = 1+B , B : Bandwidth of signal
Page 15
Wireless Communication
Chapter 5 –Small-scale multipath propagation 28 Dr. Sheng-Chou Lin
INTERSYMBOL INTERFERENCE (ISI)AND DELAY SPREAD
To avoid isi in the standstill (nonfading) case, the maximum data rateRB is related to the delay spread d of the channel
To avoid ISI in mobile environments (fading case), the maximum datarate RB is given by:
EXAMPLE:
•You are interested in buying a wireless modem from a vendor for indoor datatransmission at rates less than 300 kbits/sec.
• the vendor insists that you buy modems equipped with equalizers whichdoubles the price.
• is this necessary? no. assume a fading case with = 0.5s, then RB = 318kbits/sec
RB =1
RB =1
2
Wireless Communication
Chapter 5 –Small-scale multipath propagation 29 Dr. Sheng-Chou Lin
Doppler Spread
To describe time varying nature of the channel in asmall-scale region.•Doppler spread BD : a measure of the spectral boarding
caused by the time rate of change of the channel.•Doppler spectrum : components in the range fc-fd to fc-fd.•Effect of Doppler spread are negligible, as BWsignal BD
Coherence time TC is the time domaindual of Doppler•To characterize time varying nature•Tc 1/ fm under Rayleigh fading•Ts Tc channel will change during the
transmission of the baseband message distortion
•Time correlation function > 0.5, Tc 9 / 16fm , fm: the max. Doppler shift
tTc
channel
Page 16
Wireless Communication
Chapter 5 –Small-scale multipath propagation 30 Dr. Sheng-Chou Lin
A thumb rule
A popular rule of thumb for modern digital communications is
•Tc 1/ fm suggests a time duration during Rayleigh fading•Tc 9 / 16fm is often too restrictive•Definition of coherence time : two signals arriving with a time separation > Tc
are affected differently by the channel.
Example: A vehicle, speed = 60 mile/per hour, fc = 900 MHz•Tc = 9 / 16fm = 2.22ms• If a digital transmission is used, max. symbol rate Rc = 1/ Tc = 454bps.
–Distortion could result from multipath time delay spread
•Using the practical rule, Tc = 0.423/fm = 6.77ms , max. symbol rate Rc = 1/ Tc =150bps
TC = 9
16 fm2
=0.423
fm
Wireless Communication
Chapter 5 –Small-scale multipath propagation 31 Dr. Sheng-Chou Lin
An Example
Small-scale propagation measurements•Determine the proper spatial sampling interval•consecutive samples are highly correlated in time•fc = 1900 MHz and v = 50m/s.•For correlation, the sampling time is Tc /2. Use the smallest Tc for
conservative design.–Tc 9 / 16fm = 9/16 v = 565 s Tc /2 = 282.5 s
•How many samples is required over 10m travel distance.–Spatial sampling interval: x = vTc /2 = 50 565 s /2 = 1.41 cm–Required samples = Nx = 10 / x =708 samples
•How long would it take to make these measurements–d / v = 10m/50 = 0.2 seccond
•The Doppler spread BD = fm = vfc / c = 316.66 Hz
Page 17
Wireless Communication
Chapter 5 –Small-scale multipath propagation 32 Dr. Sheng-Chou Lin
Types of small-scale fading
Depending on the relationbetween the signal andchannel parameters,different transmittedsignals will undergodifferent types of fading
•Signal parameters:Bandwidth, symbol period
•Channel parameters: RMSdelay spread, Dopplerspread
Small-Scale fading(based on multipath time delay spread)
Flat Fading
1. BW of signal < BW of channel2. Delay spread < Symbol period
Frequency selective Fading
1. BW of signal > BW of channel2. Delay spread > Symbol period
Small-Scale fading(based on Doppler spread)
Fast Fading
1. Low Doppler spread2. Coherence time > Symbol period3. Channel variations slower than
baseband signal variations
Slow Fading
1. High Doppler spread2. Coherence time < Symbol period3. Channel variations faster than
baseband signal variations
Wireless Communication
Chapter 5 –Small-scale multipath propagation 33 Dr. Sheng-Chou Lin
Types of fading
Type of fading experiencedby a signal as a function of
•Signal parameters:–Symbol period (Ts )–Baseband signal
bandwidth ( Bs )•Channel parameters:
–RMS delay spread ()Coherent BW ( Bc )–Doppler spread ( BD )Coherent Time ( Tc )
Flat SlowFading
Flat FastFading
Ts
Ts
Tc
Transmitted Symbol Period
Sym
bo
lPer
iod
of
Tra
nsm
itti
ng
Sym
bo
l
Frequency SelectiveSlow Fading
Frequency SelectiveFast Fading
Bs
Flat SlowFading
Flat FastFading
Transmitted baseband signal bandwidth
Tra
nsm
itte
db
aseb
and
sig
nal
ban
dwid
th
Frequency SelectiveSlow Fading
Frequency SelectiveFast Fading
Bs
BD
Bc
Page 18
Wireless Communication
Chapter 5 –Small-scale multipath propagation 34 Dr. Sheng-Chou Lin
Rayleigh Distribution
To describe statistical time varyingnature of the received envelope•A flat fading signal•An individual multipath component•The envelope of the two quadrature
Gaussian noise
t
x + i y
Zero-mean Gaussian dist. with 2
r =x2 + y2
: rms before envelope2 : time-average power before
envelope
P(r) =r
2exp ( r 2
22)
0
, 0 r
, r 0
x, y
Rayleighfading beams
Wireless Communication
Chapter 5 –Small-scale multipath propagation 35 Dr. Sheng-Chou Lin
Rayleigh Distribution Parameters
Cumulative distribution function (CDF)
P( R ) = Pr( r R) = 1- exp ( R2
22)
Mean value of Rayleigh distribution
rmean = E [ r ] =
0
r p( r ) dr = / 2 = 1.2533
1.2533
Variance of Rayleigh distribution
E [ r 2 ] = E [ x 2 ] + E [ y 2 ] = 2 2
r2 = E [ r 2 ] –(E [ r ] )2 = 2 2 - (/2) = 0.4292 2
rms of the envelope = square root of the mean square = E [ r 2 ] = 2
Page 19
Wireless Communication
Chapter 5 –Small-scale multipath propagation 36 Dr. Sheng-Chou Lin
Ricean Fading Distribution
There is a dominant stationary (nonfading) signal component•line-of-sight ( LOS )•small-scale fading envelope distribution is Ricean•Ricean Rayleigh as LOS fades away
LOSRandom multipath
P(r) =
r
2exp ( ( r 2 +A2 )
22)
0
, 0 r
, r 0
I0 (Ar2 ) , A 0
A : peak amplitude of LOSI0 ( ) : Modified Bessel function of the first kind
of zero-order
K = A2 / ( 2 2 ) : describe Ricean distribution
K (dB) = 10 log [ A2 / ( 2 2 ) ] dB
• A 0, K dB, Ricean Rayleigh
Rayleigh
Wireless Communication
Chapter 5 –Small-scale multipath propagation 37 Dr. Sheng-Chou Lin
Clarke’s Model for Flat Fading
Assumptions•A fixed transmitter with a vertically polarized antenna•The field on the mobile antenna comprises of N azimuthal plane waves with
–arbitrary carrier phases–arbitrary azimuthal angles of arrival–each wave having equal average amplitude
in absence of a direct LOSexperience similar attenuation over small-scale distances
Vertically polarized plane waves at BS
Ez = E0 Cn cos ( 2 fct + n )n =1
N
• Doppler shift is very small• The phase angles uniformly distributed on [0, 2]
Ez = Tc( t ) cos ( 2 fct ) + Ts( t ) sin( 2 fct )
Tc( t ) and Ts( t ): GaussianRandom processes
r ( t ) = Tc2( t ) + Ts
2( t )
Rayleigh distribution = p (r)Tc2 = Tc
2 = Ez 2 = E02 / 2 = 2
x
z
Azimuthal plane
vertically polarizedy
Page 20
Wireless Communication
Chapter 5 –Small-scale multipath propagation 38 Dr. Sheng-Chou Lin
Spectral-Shape with Doppler-spread
Spectral analysis for Clark’s model•Total received power
G() = Antenna Azimuthal gain patternA : average received power w.r.t an
isotropic antennap() = incoming power of the angle
•instantaneous freq. Of the received signal (CW, freq.= fc) componentarriving at an angle
fm : maximum Doppler shift, an even function f () = f (-)
•The received power with frequency
x
y
z
Azimuthal plane
vertically polarized
y
Pr = A G() p() d2
0
f () = f = cos() + fc = fm cos() + fc df = d-sinfm
S( f ) df = A [ G() p() + G(-) p(-)] d
v
Wireless Communication
Chapter 5 –Small-scale multipath propagation 39 Dr. Sheng-Chou Lin
Doppler power spread
Doppler power spectrum(unmodulated CW carrier)
•For the case, vertical /4 Antenna G()=1.5 and p() = 1/2over [0, 2]
A Baseband power spectral density
K [ ] : complete elliptical integral of the first kind•not intuitive
= cos -1 [ ( f-fc ) / fm ]
sin = 1-[( f-fc ) / fm ] 2
SEz( f ) =1.5
fm1-[( f-fc ) / fm ] 2
0º 180º
SbbEz( f ) =1
8fm1- ( f / 2fm ] 2K
Page 21
Wireless Communication
Chapter 5 –Small-scale multipath propagation 40 Dr. Sheng-Chou Lin
Two-Rayleigh Fading Model To consider multipath time delay spread as well as fading
hb ( t ) = 1 exp [ j1 ] ( t ) + 2 exp [ j2 ] ( t - )
Rayleigh fading beam1st Rayleigh fading neam
2nd Rayleigh fading neam
•1 , 2 : independent and Rayleigh distributed•1 , 2 : independent and uniformly distributed
over distributed•: time delay between two rays•To create a wide range of frequency selective
fading effects by varying
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Chapter 5 –Small-scale multipath propagation 41 Dr. Sheng-Chou Lin
Simulation of Flat Fading Model
Quadrature Amplitude modulation•RF Doppler filter•Baseband Doppler filter
Quadrature Amplitude modulation Two indep. Gaussian low-pass noise
for in-phase and quadrature fading Spectral filter
Accurate time domain of Dopplerfading IFFT at the last stage
Construct negative components of the noise source
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Chapter 5 –Small-scale multipath propagation 42 Dr. Sheng-Chou Lin
Simulation of frequency-selective Fading
To produce both flat andfrequency-selective fadingeffects•Several Rayleigh fading
simulators•Variable gains•Time delays
Rayleigh Ricean
•Add a single freq. Componentdominant in amplitude withinDoppler fading spectrum
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Chapter 5 –Small-scale multipath propagation 43 Dr. Sheng-Chou Lin
Small-scale multipath measurements
Measurements•Direct RF pulse system•Spread spectrum sliding
correlator–Narrow BW wideband
•Frequency Domain–FFT, IFFT
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Chapter 5 –Small-scale multipath propagation 44 Dr. Sheng-Chou Lin
Digital Modulation under flat fading
Performance in slow, flat fading channels in AWGN
•Binary modulation•Rayleigh fading•To average the error
probability in AWGN overthe possible ranges of signalstrength due to fading
• = [ Eb/ No] 2 is theaverage value of signal-to-noise ratio, has a Rayleighdistribution
•Mean SNR is significantlylarger than that requiredwhen operating over anonfading channel (~20-50dB)
Fading v.s. nonfading
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Chapter 5 –Small-scale multipath propagation 45 Dr. Sheng-Chou Lin
: joint density function of and at r = R
Level Crossing Rate (LCR)
Level Crossing Rate (LCR): the rate at which the Rayleighfading envelope, normalized to RMS level (2 ), crosses aspecified level.•NR : the number of level crossings per second ( r = R )
-10
dB
100
R
NR
1 sec.
fm : the maximum Doppler frequency
= R / Rrms : the value of the specifiedR normalized to RMS amplitude offading envelope
Example: For a Rayleigh fading signal, = 1, maximum Doppler freq.fm= 20Hz, fc = 900 MHz.•NR = 2(20) 1e -1 = 18.44 crossings per second• f d,max = v / v = 20 (1/3) = 6.66 ms = 24 km/hr
0
2
2 efrdr,RprN mR
r: time derivative of r(t)
r,Rp r r
• Few crossings at both high and low levels• Maximum at = 2
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Chapter 5 –Small-scale multipath propagation 46 Dr. Sheng-Chou Lin
Average fade duration
Average fade duration: average period of time for which thereceived signal a specified R. For a Rayleigh fading signal
100
R
NR
1 sec.Pr [ r R ] = fading time in 1 sec.
• Pr [ r R ] = 1- exp( 2 ) : probabilitythat the received signal less than R =the fading time in one second.
• Helps determine the most likelynumber of signaling bits that may belost during a fade
= Pr [ r R ] =1NR
e 2- 1
fm2
Example: Threshold level = 0.707, Doppler freq. = fm = 20Hz, Binarydigital modulation with bit duration of 50 bps, bit error occurs for 0.1•The average fade duration = (e 0.7072
-1) / (0.707)20 2= 18.3 ms•bit period = 1/50 = 20ms the signal undergoes than fast Rayleigh fading• for =0.1, = 0.002 s = 20ms one bit will be lost during a fade, NR = 4.96 the
total number of bits in error is 5/sec. BER = 5/50 = 0.1
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Chapter 5 –Small-scale multipath propagation 47 Dr. Sheng-Chou Lin
Lesson 5 Complete