Chapter 5 Small-scale Multipath · PDF filePage 4 Wireless Communication Chapter 5 –Small-scale multipath propagation 6 Dr. Sheng-Chou Lin Space Diversity A Method for Combating

Page 1

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



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



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



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



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

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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



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



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



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

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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



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



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 )~ ~ ~



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



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[)(



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

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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



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



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



Delay Profile

Measured multipath power delay profiles•900 MHz cellular in San Francisco•Inside a grocery store at 4GHz

Page 11



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



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



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



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



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



Flat and Frequency-selective Fading

Flat fading Frequency-selective fading

2-ray multipath channel (point-point)

i

Page 14



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



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



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



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



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



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



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



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



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



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



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



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



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



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



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



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

Page 22



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



Small-scale multipath measurements

Measurements•Direct RF pulse system•Spread spectrum sliding

correlator–Narrow BW wideband

•Frequency Domain–FFT, IFFT

Page 23



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



: 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

Page 24



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



Lesson 5 Complete

Documents

Chapter 5 Small-scale Multipath · PDF filePage 4 Wireless Communication Chapter 5 –Small-scale multipath propagation 6 Dr. Sheng-Chou Lin Space Diversity A Method for Combating