WC-Chapter 2-Statistical Multipath Channel Models

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Chapter 2

Statistical Multipath Channel ModelsHa Hoang Kha, Ph.DHo Chi Minh City University of TechnologyEmail: [email protected]

Outline

1) Small scale multipath propagationMultipath componentp pDoppler shift

2) Impulse response model of a multipath channel3) Types of small-scale fading

Fat/frequency selective fadingSlow/fast fading

4) Statistical models for multipath fading channels4) Statistical models for multipath fading channels

2 H. H. Kha, Ph.DStatistical Multipath Channel Models

Review

What we discussed last lecture:The large-scale fading because of path lossTh i i l h l f lThe empirical path loss formulas

Now, we will discuss about the small-scale fading and the statistical models represent it

Statistical Multipath Channel Models 3 H. H. Kha, Ph.D

1. Small scale multipath propagation

The small-scale fading is usually called “fading”It is caused by multipath signal, so it is also called “multipath fading”multipath fadingMultipath signal causes constructive and destructiveaddition of the received signal


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Small scale multipath propagation

If a single pulse is transmitted in the multipath channel, it will yield a train of pulses with delay time

Delay spread ( ): the time delay between the arrival of the first received signal component and the last

i d i l i d i h i l

LOSReflected components

mT

received signal component associated with a single transmitted pulse



If the delay spread is small compared to 1/B (B is the signal bandwidth), then there is little time spreading in the received signalthe received signalIf the delay spread is relatively large, there is little time spreading of the received signal, i.e. signal distortionMultipath channel is also time-varying that means either the transmitter or the receiver is movingIt also causes the location of the reflectors will changeIt also causes the location of the reflectors will change over timeWe will limit the model to be narrowband fading, i.e. the bandwidth B is small compared to (1/delay spread)



Physical factors influencing fading:Multipath propagationS d f h bilSpeed of the mobileSpeed of surrounding objectsThe transmision bandwidth of the signal


Doppler Shift

Consider a mobile moving at a constant velocity v, along a path segment having length d between

i X d Y hil i i i l fpoints X and Y, while it receives signal from a remote source S


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Doppler Shift

The difference in path lengths traveled by the wave from source S to mobile at points X and Y

is the time required for the mobile travel from X to Yθ is assumed to be the same at points X and Y

The phase change in received signal due to the difference in path lengths is

cos cosl d v tθ θΔ = = ΔtΔ

2 2 cosl v tπ πφ θλ λΔ Δ

Δ = =


Doppler Shift

The apparent change in frequency, or Doppler shift, is given by

1 φΔ

The Doppler shift relates to the mobile velocity and the spatial angle between the direction of the mobile and the direction of arrival of the wave

1 cos2d

vftφ θ

π λΔ

= =Δ

the direction of arrival of the waveIf the receiver is moving towards the transmitter, the Doppler freq is positive, otherwise it is negative


Example

Consider a transmitter which radiates a carrier of 1850 MHz. For a vehicle moving 26.82 mps, compute the received carrier frequency if the mobile is moving:received carrier frequency if the mobile is moving:

Directly towards the transmitterDirectly away from the transmitterIn a direction which is perpendicular to the direction of arrival of the transmitted signal


Carrier freq = 1850 MHz Wavelength =Vehicle speed = 26.82 m/sV hi l i d h i i i

Solution

8

6

3 10 0.1621850 10c

c mf

λ ×= = =

×

Vehicle moving towards the transmitter means positive Doppler frequency

Vehicle moving directly away the transmitter means negative Doppler frequency

26.821850 cos0 1850.000160.162c df f f MHz= + = + =

26 82

Vehicle is moving perpendicular means

26.821850 cos0 1849.9998340.162c df f f MHz= − = − =

90θ = °26.821850 cos90 18500.162c df f f MHz= + = + =


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Doppler Spread and Coherence Time

Doppler spread is given by

2 1:d d dB f f= −

Where and

E.g. If the mobile is moving at 60 kmph and fc = 900 MHz, the the Doppler spread is

1c

df vfc

= − 2c

df vfc

= +

6

8

2

900 10 16.672 1003 10

c c cd

f v f v f vBc c c

Hz

⎛ ⎞ ⎛ ⎞= − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

× ×= =

×


Doppler Spread and Coherence Time

Coherence time is actually a statistical measure of the time duration over which the channel impulse response is essentially invariant.is essentially invariant. The coherence time is related with Doppler spread (Doppler shift)

0.423 0.423c

D

Tf v

λ= =

where fD=the maxium Dopper frequency

Df


2. Impulse response model of a multipath channel

We have already known that the transmitted signal is

Th h i d i l i l i h h l i( ) ( ){ } ( ){ } ( ) ( ){ } ( )2 cos 2 sin 2cj f t

c cs t u t e u t f t u t f tπ π π= ℜ =ℜ −ℑ

Then, the received signal in multipath channel is

n = 0 corresponds to the LOS pathN(t) is the number of resolvable multipath components( ) p p

is corresponding delayis Doppler phase shiftis amplitude


The n-th resolvable multipath component may correspond to the multipath associated with a singlereflector or multiple reflectors clustered togetherreflector or multiple reflectors clustered together


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Time-Varying Channel Impulse Response

If single reflector exists, the amplitude is based on the path loss and shadowing, its phase change associated with delay

and Doppler phase shift of ( ) ( )2 c nj f tn t e π ττ −→ ( )2

N ND Df t dtφ π= ∫

If reflector cluster exists, two multipath components with delay and are resolvable if If the criteria is not satisfied, then it is nonresolvable since

Th l bl t bi d i t i l

( )n N Nt∫

1τ 2τ 11 2 uBτ τ −−

( ) ( )1 2u t u tτ τ− ≈ −The nonresolvable components are combined into a singlemultipath component with delay and an amplitude and phase corresponding to the sum of different components

1 2τ τ τ≈ ≈



The amplitude of the summed signal will undergo fastvariations due to the constructive and destructivecombining of the nonresolvable multipath componentscombining of the nonresolvable multipath componentsWideband channels have resolvable multipath components the parameters change slowlyNarrowband channels tend to have nonresolvablemultipath components the parameters change quicklyquickly



We can simplify by letting

The received signal is then

( )r t

( ) ( )2nn c n Dt f tφ π τ φ= −

The received signal is then

The received signal is obtained by convolving the baseband input signal with equivalent lowpass time-

i h l i l f h h l dvarying channel impulse response of the channel, and then upconverting the carrier frequency



The represents the equivalent lowpass response of the channel at time t to an impulse at time

( ),c tτt τ−

Parameters of Mobile Multipath ChannelsTime dispersion parameters and coherence bandwidthDoppler spread and coherence time


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Time-invariant channels


Time Dispersion Parameters

The time dispersive properties of wideband multipath channels are most commonly quantified by their meanexcess delay and rms delay spreadexcess delay and rms delay spreadThe mean excess delay:

The rms delay spread is the square root of the second l f h d l fil

( )

( )

2

2

k k k kk k

k kk k

P

P

α τ τ ττ

α τ= =∑ ∑∑ ∑

central moment of the power delay profile

( )22τσ τ τ= −

( )

( )

2 2 2

22

k k k kk k

k kk k

P

P

α τ τ ττ

α τ= =∑ ∑∑ ∑



The delays are measured relative to the first detectable signal arriving at the receiver atThe maximum excess delay (X dB) of the power delay

0τ =

The maximum excess delay (X dB) of the power delay profile is defined to be the time delay during which multipath energy falls to X dB below the maximum. The maximum excess delay sometimes called excess delay spread, which can be expressed asWhere is the maximum delay at which a multipath

0Xτ τ−

τWhere is the maximum delay at which a multipath component is within X dB of the strongest arriving multipath signal and is the first arriving signal

Xτ

0τ




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Coherence Bandwidth

Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered “flat”considered flatFlat fading is a channel which passes all spectral components with approximately equal gain and linear phaseThe coherence bandwidth can be expressed as

(above 90% correlation)1B ≈ (above 90% correlation)

(above 50% correlation)1

5cBτσ

≈

50cBτσ

≈


Example

Compute the mean excess delay, rms delay spread, and the maximum excess delay for the following power delay profiledelay profileEstimate the 50% coherence bandwidth of the channel


Solution

Using the definition of maximum excess delay (10 dB), it can be seen thatThe mean excess delay:

10 4dB sτ μ=

( )( ) ( )( ) ( )( ) ( )( )1 5 0.1 1 0.1 2 0.01 04 38

+ + +The mean excess delay:The second moment

The rms delay spread:The coherence bandwidth:

( )( ) ( )( ) ( )( ) ( )( )( )

4.380.01 0.1 0.1 1

sτ μ= =+ + +

( )( ) ( )( ) ( )( ) ( )( )( )

2 2 2 22 21 5 0.1 1 0.1 2 0.01 0

21.070.01 0.1 0.1 1

sτ μ+ + +

= =+ + +

( )221.07 4.38 1.37 sτσ μ= − =

1 1 146B kHz= = =

Note that GSM requires 200 kHz bandwidth which exceeds Bc, thus, an equalizer would be needed for this channel

( )146

5 5 1.37cB kHzsτσ μ

= = =


Measured values of RMS Delay Spread


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3. Type of Small-scale Fading


Flat Fading

If the mobile radio channel has a constant gain and linear phase response over a bandwidth which is greaterthan the bandwidth of the transmitted signal, then thethan the bandwidth of the transmitted signal, then the received signal will undergo flat fading


Flat Fading

Flat fading channels are also known as amplitude varying channelsIt is also sometimes referred to as narrowband channelsIt is also sometimes referred to as narrowband channelsThe most common amplitude distributions are: Rayleigh, Rician, and NakagamiSummarize: a signal undergoes flat fading if

B Bs cB B

sT τσ


Frequency Selective Fading

If the channel has a constant-gain and linear phase response over a bandwidth that is smaller than the bandwidth of transmitted signal, then the channelbandwidth of transmitted signal, then the channel creates frequency selective fading on the received signal


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The received signal includes multiple versions of the transmited waveform which are attenuated and delayedin time, and hence the received signal is distortedin time, and hence the received signal is distortedFrequency selective fading is due to time dispersion of the transmitted symbols within the channelThus, the channel induces intersymbol interference (ISI)The modeling for this kind of channel is more difficultThe modeling for this kind of channel is more difficult since each multipath signal must be modeled and channel must be considered to be a linear filter



It is sometimes called wideband channels since the bandwidth of the signal is wider than the bandwidth of the channel impulse responsethe channel impulse responseSummarize: a signal undergoes frequency selective fading if

s cB B>

T σ<sT τσ<


Fast Fading

In a fast fading channel, the channel impulse response changes rapidly within the symbol durationIn other words the coherence time of the channel isIn other words, the coherence time of the channel is smaller than the symbol period of the transmitted signalThis causes frequency dispersion (time selective fading) due to Doppler spread, which lead to signal distortionSignal distortion due to fast fading increases with increasing Doppler spread relative to the bandwidth ofincreasing Doppler spread relative to the bandwidth of the transmitted signalSummarize: a signal undergoes fast fading if

s cT T>

s DB B<


Slow Fading

In a slow fading channel, the channel impulse response changes at a rate much slower than the transmitted signalsignalThe channel may be assumed to be static over one or several reciprocal bandwidth intervalThe Doppler spread of the channel is much less than the bandwidth of the baseband signalSummarize: a signal undergoes slow fading ifSummarize: a signal undergoes slow fading if

s cT T s dB B


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Summary


Remarks

When a channel is specified as a fast or slow fading channel, it does not specify whether the channel is flat fading or frequency selectiveFast fading only deals with the rate of change of theFast fading only deals with the rate of change of the channel due to motionIn flat fading channel, we can approximate the impulse response to be simply delta functionA flat fading, fast fading channel is a channel in which the amplitude of the delta function varies faster that the rate of the transmitted baseband signalA frequency selective, fast fading channel, the amplitudes, phases, and time delays of any one of the multipath components vary faster than the rate of change of the transmitted signal


4. Narrow fading models

Assume delay spread maxm,n|τn(t)-τm(t)|<<1/BThen u(t)≈u(t-τ).Received signal given by

No signal distortion (spreading in time)⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ℜ= ∑

=

)(

0

)(2 )()()(tN

n

tjn

tfj nc etetutr φπ α

Multipath affects complex scale factor in brackets.Extreme narrow band case: u(t)=1


I-phase and Q-phase: Gaussian

Extreme Narrowband

In phase and quadrature components

and : independent among different n’s.and : independent among different n s.With N large, and CLT: jointly Gaussian random processes


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Rayleigh Fading

The Rayleigh distribution is commonly used to describe the statistical time varying nature of the received envelope of a flat fading signalenvelope of a flat fading signalRayleigh distributed signal:


Rayleigh Fading

The Rayleigh distribution has pdf

Signal Envelope A variance of in-phase and quadature components is σ2

An average received signal power

The probability that the envelope of the received signal

2 2( ) | ( ) | ( ) ( )I Qz t r t r t r t= = +

p y p gdoes not exceed a specified value R is


Rayleigh Fading

The mean value of Rayleigh distribution is

The variance of the Rayleigh distribution (represent the ac power)

The median value isfound by solving

The median is often used in practice


0

1 ( )2

medianr

p r dr= ∫

Rayleigh Fading

The corresponding Rayleigh pdf is


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Example

Consider a channel with Rayleigh fading and average received power Pr=20 dB. Find the probability that the received power is below 10 dB.received power is below 10 dB.

Ans: 0.095


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 ato the local rms signal level, crosses a specified level in a positive-going directionThe number of level crossing per second is given by

Where ( ) 2

0

, 2R DN rp R r dr f e ρπ ρ∞

−= =∫

is time derivative of r(t) (the slope)is the joint density function of r and at r = R ( ),p R r

rr

rmsR Rρ =


Example

For a Rayleigh fading signal, compute the positive-going level crossing rate for when the maximum Doppler frequency is 20 Hz

1ρ =Doppler frequency is 20 HzWhat is the maximum velocity of the mobile for this Doppler frequency if the carrier frequency is 900 MHz?

Solution:Use the equation for LCR

( )( ) 1

Use equation of Doppler frequency


( )( ) 12 20 1 18.44RN eπ −= =

( )20 1 3 6.66 /Dv f m sλ= = =

Level Crossing and Fading Statistics

The average fade duration is defined as the average period of time for which the received signal is below a specified level R. 1p

For a Rayleigh fading signal, it is given by

[ ]1 PrR

r RN

τ = ≤

[ ]

( ) ( )2

1Pr

1 exp

ii

R

r RT

p r dr

τ

ρ

≤ =

= = − −

∑

∫

So, the average fade duration can be expressed as

( ) ( )0∫

2

12D

ef

ρ

τρ π

−=


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Example

Find the average fade duration for threshold levelswhen the Doppler frequency is 200 Hz

0.01ρ =

SolutionAverage fade duration is

( )

20.01 1 19.90.01 200 2

e sτ μπ

−= =( )


Rician Fading Distribution

When there is a dominant stationary (nonfading) signal component present, such as line-of-sight propagation path, the small-scale fading envelope distribution ispath, the small scale fading envelope distribution is RicianThe Rician distribution is given by

( )( )2 2

2202 2 for 0

0 for 0

r sr rsp r e I r

r

σ

σ σ

+− ⎛ ⎞= ≥⎜ ⎟

⎝ ⎠= <

I0 is the modified Bessel function of 0th order.

0 for 0r= <


Bessel function



The Rician distribution is described in terms of a parameter K

2s

is the average power in the NLOSis the average power of LOS component

is the average received signal

22sKσ

=

As we have Rayleigh fadingAs we have no fading, channel has no multipath, only LOS component

0K =K = ∞


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The Rician pdf is


Remarks

Small-scale fading is variation of signal strength over distances of the order of the carrier wavelengthIt is due to constructive and destructive interference ofIt is due to constructive and destructive interference of multipathKey parameters:

Doppler spread coherence timeDelay spread coherence bandwidth

Statistical small-scale fading: Rayleigh fading and Ricianfading flat fading


I-phase and Q-phase: Gaussian

Extreme Narrowband

In phase and quadracture components

and : independent among different n’s.and : independent among different n s.With N large, and CLT: jointly Gaussian random processes


Key assumption: for each path (from one reflector)

I-phase and Q-phase: mean

Random phase is uniform (no LOS)

With all the above independent, we have the mean:


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I-phase and Q-phase: autocorrelation

Since changes rapidly relative to all other phases theSince changes rapidly relative to all other phases, the second term goes to zero.


I-phase and Q-phase: Cross-correlation

For the passband signal:p ssb d s g


Uniform Scattering-Clarke/Jakes Model


Uniform Scattering: Autocorrelation


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De-correlation over distance

Zero correlation


Power Spectral Density

Doppler spread


References

[1] A. Goldsmith, Wireless Communications, Cambridge University Press, 2005.

[2] T S Rappaport Wireless Communications Prentice Hall 2002[2] T. S. Rappaport, Wireless Communications, Prentice Hall, 2002.[3] A. Goldsmith, Wireless Communications, Lecture Notes,

http://www.stanford.edu/class/ee359/archived_material/2010/lectures.html


Homework

Problems: 3, 6, 7, 8, 12 in Chapter 3 of [Goldsmith 2005]Problems: 5.1, 5.2, 5.7, 5.8, 5.11 in Chapter 5 of [Rappaport

2002]2002]


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WC-Chapter 2-Statistical Multipath Channel Models