Mobile_Lec5

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Lecture 5Multipath Fading Channels

Mobile Communication Systems

Dr Charan Litchfield [email protected]

22nd October 2008

Resources:

Mobile Communication Systems Oct09

2

� Reading:Chapter 3 of Goldsmith “Wireless Communications”.

� Supplemental Reading:Parsons: “The Mobile Radio Propagation Channel”

� References:“On the correlation and scattering functions of the WSSUS channel for mobilecommunications”. Sadowsky, J.S.; Katedziski, V., IEEE Trans. Vehic. Technol., Feb. 1998.

“The WSSUS channel model: comments and a generalilisation”. Sadowsky, J.S.; Katedziski, V., IEEE Trans. Vehic. Technol. Feb. 1998.

“Fading channels: information-theoretic and communications aspects”, Biglieri, E.; Proakis, J.; Shamai, S. IEEE Transactions on Information Theory, Oct. 1998.

“Dynamic characteristics of a narrowband land mobile communication channels”, H.A. Barger, IEEE Trans. Vehic. Technol., Feb. 1998.

Wiley; 2nd

edition,

August 2000

University of

Cambridge,

2005

Multipath Fading

Game Plan:


3

4.1 Shadowing.

4.2 Fast Fading Channels.

4.3 Mathematical Models.

4.4 Probability Models.

Multipath Fading 4

4.1 Shadowing


Lognormal DistributionMultipath Fading

� Shadowing (Slow Fading) is a statistical variable accounting for absorption in a medium or multiple reflections and diffractions. Shadowing can be incorporated with a large scale path loss model.

� Usually occurs due to transmission and reflection through multiple structures causing large absorption.

� It is treated in a statistical manner due to unpredictability and nature of environment. Modelled with Lognormal PDF.

5

4.1 ShadowingMobile Communication Systems Oct09

Hexagonal cell shape:- fictitious.Uniform path loss:- circular cells.Non-uniform path loss- amoeba cells (realistic).Non-uniform path loss+shadowing- amoeba cells with holes in coverage (realistic).

Multipath Fading

4.1 ShadowingMobile Communication Systems Oct09

6Multipath Fading

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� Assumption: shadowing is dominated by the attenuation from blocking objects.

� Attenuation of for depth d:s(d) = e−αd,

(α: attenuation constant). � Many objects:

s(dt) = e−α∑ di = e−αdt , dt = ∑ di is the sum of the random object depths

� Cental Limit Theorem (CLT): αdt = log s(dt) ~ N(µ, σ). � log s(dt) is therefore log-normal

7

4.1 Shadowing


Multipath Fading

� General Proof: In measurements, the shadowing process as a dB variable is a Gaussian RV, Xi. This means that shadowing is a process that is Lognormal. Represent Lognormal variable as Zi.

{ }( )

{ }( ).

2σ

ZEZLogexp

σ2πZ

1ZX

(X)P(Z)Pthen

2σXEX

expσ2π

1(X)PSince

X.ZLogi.e.,eZisVariabletheoftionTransforma

2Z

2e

Z

XL

2X

2

X

X

eX

−−=

∂

∂=

−−=

==

Gaussian PDF.

Lognormal PDF.

8

4.1 Shadowing


= ∏∑

iie

ii ZlogX

Shadowing:

Multipath Fading

Fit model to data:Fit model to data:

1. Path loss (K, γγγγ), d0 known:

• “Best fit” line through dB data.

• K obtained from measurements at d0.

• Exponent is MMSE estimate based on data.

• Captures mean due to shadowing.

2. Shadowing variance:

• Variance of data relative to path loss model (straight line) with MMSE estimate for γγγγ.

9


Multipath Fading

4.1 Shadowing

Why is ψdB a normally distributed? Hint: See slide 8.10


Multipath Fading

4.1 Shadowing

� Outage probability:� Probability that Pr(d) (RX power at a distance d) < Pmin.

� If outage occurs, it can be assumed user disconnected or service cancelled.

� For combined Path Loss and Shadowing Model, can write:

( ) ( )

γ−+−σ−= −

ψd

dLog10KLog10PpQ1d,pp 0

1010Tmin

1

minout

( ) ( ) dB0

1010

T

r

d

dLog10KLog10dB

P

PΨ+

γ−=

),0(N~ 2

dB ψσΨRandom Variable

Mean

11


Multipath Fading

4.1 Shadowing

Gaussian Q function

∫∞

∂

−π

=>=S

2

y2

yexp

2

1)ZX(p)S(Q

Z

p(X)

X

)S(Q1)ZX(p −=<

Useful function for Gaussian Distribution

{ }

∫

∑∞

∞−

=∂

=∀

1X)X(p

1X)X(p Discrete

Continuous

Xm

12


Multipath Fading

4.1 Shadowing

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Gaussian Q functionProof:

( )∫∞

∂

σ

−−

σπ=>

Z

2

2

m X2

XXexp

2

1)ZX(p

σ

−= mXX

y

),X(N~X 2

m σ

( )SQ)ZX(p

y2

yexp

2

1)ZX(p

mXZ

2

=>⇒

∂

−π

=>⇒ ∫∞

σ

−

σ

−= mXZ

S

Gaussian RV

Change the Variable to

where

noting

)X(p

y

X)X(p)y(p

⋅σ=

∂

∂=

13


Multipath Fading

4.1 Shadowing

� In cellular communication, can define two outages – that due to thermal noise, and that due to interference (from other cells).

� Interference non trivial problem (since co-channel interference from large number of additive components depend on statistical path loss and shadowing properties of each component).

� Define noise: N = kTBw. N0 = kT = -174dBm/Hz for T = 290K.

� A Noise outage at edge of cell can be defined as PN(R)=P(Pr(dBm)(d)<N(dBm))

where is the shadow margin.

( )

σ=∂

σ

µ−−

σπ= ∫

∞−

shad

N

2

2

P MQX

2

)R(xexp

2

1)dBm(

)dBm(r

)dBm(Pshad N)R(M)dBm(r

−µ=

14


Multipath Fading

4.1 Shadowing

4.2 Fast Fading Channels


15

Network Analyzer

Multipath Fading


4.2.1 Basic Wave Concepts

16

Sum of sinusoidal components results in another sinusoidal component with certain amplitude and phase. This is basic mechanism of fading. Assumption: Components same frequency.

1φ1A

2A2φ


Multipath Fading



17

( ) ω∆tωφ = ( )∆t

ωωφ

=∂

∂

Important relation – first derivative (w.r.t. ω) of phase yields time delay – it is also called group delay.

∆t

Phase response

GTωφ

=∂

∂

Simple Example: Adding two tones with Time shift.

∆tΣ

( ) t-jω∆tjω ee1o/p −+=

Group Delay


Multipath Fading



18

Phase delay dependent on frequency. In case of wave packet (i.e. Such as a pulse) consisting of infinite record of sinusoidal components with different frequencies, have phase spectrum – i.e. A single time delay does not yield equal phase delay on all sinusoidal tones in wave packet. Brings us towards concepts like filtering.

∆t


Multipath Fading

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4.2.2 The Wireless Channel

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Distortion:-Fading,-ISI due to propagation delays. Channel behaves like a filter.-Channel usually non linear phase response = Phase distortion, ISI.

Model


Multipath Fading 20

4.2.3 Two Ray Model



Typically a propagation channel – I.e. Fading large scale. For large d, path difference is:

λd

hh4πφ∆

d

h2hd∆ rtrt =⇒≈

cd

h2hT

t∆

φ∆ rtd == Time delay

Multipath Fading


21

Wireless Channel

•EM Models:-Environment specific.-Too detailed to be traceable.

•Behavioural Models:-Estimate Physically Meaningful Parameters.-Apply to Theoretical Model.-Very Detailed but traceable.

•Measurement:-Transmit well structured signal.-Take “average” measurements.-Leads to Linear System Approach.

4.2.4 Black Box Approach to Channel

Black BoxX(t) Y(t)Deterministic signal

Observed output signal


Multipath Fading

� Path loss is a natural phenomenon occurring due to spreading of electromagnetic waves radiated by the transmitter (where the gain of the antenna is non-singular).

� The effect of path loss is such that the SNR at the receiver decreases monotonically with distance from the transmitter.

Waves spread from the radiator in space where the power flux density decreases per unit distance.


4.2.5 Introduction to Fast Fading


22Multipath Fading

Small-scale multipath fading� Wireless communication typically happens at very high carrier frequency. (eg.

fc = 900 MHz or 1.9 GHz for cellular)

� Multipath fading due to constructive and destructive interference of the transmitted waves. In the case of large scale and slow fading, we usually deal with a small number of reflected waves. Means the fading is slowly varying when traversing in environment.

� If, however, due to scattering or many reflecting objects where the number of waves constituting a wave bundle is very large (and scattering parameters are random), then get fast fading in nonstationary environment.

� Channel varies when mobile moves a distance of the order of the carrier wavelength. This is about 0.3 m for 900 Mhz cellular. For vehicular speeds, this translates to channel variation of the order of 100 Hz.

� Primary driver behind wireless communication system design.

23




Multipath Fading

Typical Urban Radio Channel:

Solid Line = Fast Fading.

Dashed Line = Variation in

Statistical Mean (Shadowing

or Slow Fading)

Fast Fading Rayleigh Channel

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Multipath Fading

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� Path loss, shadowing => average signal power loss

� Fading around this average.

� Subtract out average => fading modeled as a zero-mean random process

� Narrowband Fading channel: Each symbol is long in time

� The channel h(t) is assumed to be uncorrelated across symbols => single “tap” in time domain.

� Fading w/ many scatterers: Central Limit Theorem

� In-phase (cosine) and quadrature (sine) components of the snapshot r(0), denoted as rI (0) and rQ(0) are independent Gaussian random variables.

� Envelope Amplitude:

� Received Power:Single Tap Fading

Base Station (BS)Mobile Station (MS)

multi-path propagation

Path Delay

Pow

er

path-2

path-2

path-3

path-3

path-1

path-1

Channel Impulse Response: Channel amplitude |h| correlated at delays ττττ. Each “tap” value @ kTs Rayleigh distributed(actually the sum of several sub-paths)

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Multipath Fading

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Multipath Fading

RMS Delay Spread:

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Multipath Fading

Multipath: Time-Dispersion => Frequency Selectivity� The impulse response of the channel is correlated in the time-domain (sum of

“echoes”)� Manifests as a power-delay profile, dispersion in channel autocorrelation

function A(∆τ)� Equivalent to “selectivity” or “deep fades” in the frequency domain� Delay spread: τ ~ 50ns (indoor) – 1µs (outdoor/cellular).� Coherence Bandwidth: Bc = 500kHz (outdoor/cellular) – 20MHz (indoor)� Implications: High data rate: symbol smears onto the adjacent ones (ISI).

Multipath effects

~ O(1µs)

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Multipath Fading 30


Doppler Shift


tVX ∆=

Distance on x axis

( )cosθtV∆d∆ =

Phase difference

( )[ ]

cosθc

Vf2π

t∆

φ∆λ

cosθtV∆2πdK∆φ∆

=

==

cosθcV

ffd

=

Doppler Frequency FrequencyMultipath Fading


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� Doppler spread:

� Note: opposite sign for doppler shift for the two waves� Effect is roughly like the product of two sinusoids


Doppler Shift


31Multipath Fading


Doppler Spread: Effect

� Fast oscillations of the order of GHz� Slow envelope oscillations order of 50 Hz => peak-to-zero every 5 ms � A.k.a. Channel coherence time (Tc) = c/4fv

5ms

32



Multipath Fading




33

Doppler Spread: Effect

Multipath Fading


Doppler: Non-Stationary Impulse Response.

Set of multipaths changes ~ O(5 ms)

34




Multipath Fading

35

� The doppler power spectrum shows dispersion/flatness ~ doppler spread (100-200 Hz for vehicular speeds) � Equivalent to “selectivity” or “deep fades” in the time domain correlation

envelope. � Each envelope point in time-domain is drawn from Rayleigh distribution. But

because of Doppler, it is not IID, but correlated for a time period ~ Tc(correlation time).

� Doppler Spread: Ds ~ 100 Hz (vehicular speeds @ 1GHz) � Coherence Time: Tc = 2.5-5ms.� Implications: A deep fade on a tone can persist for 2.5-5 ms! Closed-loop

estimation is valid only for 2.5-5 ms.




Doppler: Dispersion (Frequency) => Time-Selectivity

Multipath Fading

Fading Summary: Time-Varying Channel Impulse Response

� #1: At each tap, channel gain |h| is a Rayleigh distributed r.v.. The random process is not IID.

� #2: Response spreads out in the time-domain (τ), leading to inter-symbol interference and deep fades in the frequency domain: “frequency-selectivity” caused by multi-path fading

� #3: Response completely vanish (deep fade) for certain values of t: “Time-selectivity” caused by doppler effects (frequency-domain dispersion/spreading) 36




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Dispersion-Selectivity Duality

37




Dispersion-Selectivity Duality

38




Power Delay Profile => Inter-Symbol interference

� Higher bandwidth => higher symbol rate, and smaller time per-symbol� Lower symbol rate, more time, energy per-symbol� If the delay spread is longer than the symbol-duration, symbols will “smear” onto

adjacent symbols and cause symbol errors

Symbol

TimeSymbol Time

path-2

path-3

path-1

Path Delay

Pow

er

Delay spread

~ 1 µµµµs

Symbol Error!

If symbol rate

~ Mbps

No Symbol Error! (~kbps)

(energy is collected

over the full symbol period

for detection)39




Effect of Bandwidth (No. taps) on MultiPath Fading

Effective channel depends on both physical environment and bandwidth.




Multipath Fading

41

4.3 Mathematical Models


Multipath Fading 42


4.3.1 Summary


Multipath Fading

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4.3.1 Summary


43Multipath Fading

� Wireless channels can be modeled as linear time-varying systems:

where ai(t) and τi(t) are the gain and delay of path i.

� The time-varying impulse response is:

� Consider first the special case when the channel is time-invariant:

44


4.3.2 Physical Models


Multipath Fading

� Communication takes place at

� Processing takes place at baseband

45




Multipath Fading

� The frequency response of the system is shifted from the passband to the baseband.

� Each path is associated with a delay and a complex gain.




46Multipath Fading

Sampled baseband-equivalent channel model:

where hl is the lth complex channel tap,

and the sum is over all paths that fall in the delay bin

System resolves the multipaths up to delays of 1/W .

Delay bin. Some paths cannot be resolved and the receiver would thus merge paths on similar delay components.

This is discrete convolution.

47




Multipath Fading

� Fading occurs when there is destructive interference of the multipaths that contribute to a tap.

Delay spread

Coherence bandwidth

single tap, flat fading

multiple taps, frequency selective

Narrowband Model

Wideband Model

Usually calculated with

statistics

48




Multipath Fading

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� Discrete symbol x[m] is the mth sample of the transmitted signal; there are W samples per second.

� Continuous time signal x(t), 1 s ≡W discrete symbols� Each discrete symbol is a complex number;

� It represents one (complex) dimension or degree of freedom.

� Bandlimited x(t) has W degrees of freedom per second.� Signal space of complex continuous time signals of duration T which have most of their energy within the frequency band [−W/2,W/2] has dimension approximately WT.

� Continuous time signal with bandwidth W can be represented by W complex dimensions per second.

� Degrees of freedom of the channel to be the dimension of the received signal space of y[m]




49Multipath Fading

Ideal Baseband Channel:

Multipath Fading Channel:

50




Multipath Fading

� Wideband Fading:

� Received signal experiences:� Large-scale path losses

� Shadowing

� Small-scale rapid fading

� Phase distortions

� Doppler shift

� Time dispersions




51Multipath Fading

Doppler shift of the ith path

Doppler spread

Coherence time

Doppler spread is proportional to: The carrier frequency fc and the

angular spread of arriving paths.

where θi is the angle the direction of motion makes with the ith path52




Time Variations:

Multipath Fading

� Channel Model as an FIR Filter.

( ) [ ]∑=n

-nZnhZH

ωjeZ =

( ) ( )∑=n

-nTs

sZnThZH




53

Sample Continuous Time signal at integer intervals of Ts

Multipath Fading




54Multipath Fading

3/23/2010

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55




Multipath Fading

� Coherence time Tc depends on carrier frequency and vehicular speed, of the order of milliseconds or more.

� Delay spread Td depends on distance to scatterers, of the order of nanoseconds (indoor) to microseconds (outdoor).

� Channel can be considered as time-invariant over a long time scale.

56


4.3.3 Typical Channels Underspread


Multipath Fading

57

4.4 Probability Models


� Used for analysis purposes in a wireless link.

� Design and performance analysis based on statistical ensemble of channels rather than specific physical channel.

� Flat Fading Models: Many small Scattered Paths – Time delays are not resolvable and the receiver sees many paths merging on same (similar) time reference.

� Selective Fading Models: Many small Scattered Paths where the bandwidth of signal means components arriving at different times are potentially resolvable. Causes Baseband Signal distortion, ISI etc.

� Can Combine Path loss and Shadowing with fast fading.

Multipath Fading

Has no Line of Sight component.Start with Complex circular symmetric Gaussian Random Variable N=X+jY. Let X = RcosΦ and Y=RsinΦ be I.I.D with E{X}=0 and E{Y}=0. The envelope, R, given by the chi-squared random variable with two degrees of freedom.

22YXR +=

( )

−−= 22

22yx

2σ

1exp

2π

1y)(x,p YX,

σ

y),(x,pJΦ)(R,p YX,ΦR, ⋅=RcosΦRsinΦ

sinΦcosΦ

Φ

Y

Φ

XR

Y

R

X

Φ)(R,

Y)(X,J

−=

∂

∂

∂

∂∂

∂

∂

∂

=∂

∂=

Change the Variables.

Joint PDF of x and y.

Jacobian of the transform.


4.4.1 Rayleigh Flat Fading


58Multipath Fading

2π

1RΦ)(R,p)(p

0

ΦR,Φ =∂= ∫∞

Φ

−=⇒2

2

2 2σ

Rexp

2π

RΦ)(R,p

ΦR,σ

ΦΦ)(R,p(R)p

2π

0

ΦR,R ∫ ∂=

−=2

2

22σ

Rexp

σ

R(R)pR⇒

Rayleigh Fading Distribution

The Phase of a Rayleigh RV is Uniformly distributed.

59




Multipath Fading

−=

2

2

22σ

Rexp

σ

R(R)pR

Rayleigh Fading Distribution:

•Describes the statistics in a fading channel with no line of sight component.

•Also referred to as Fast Fading due to scattering and Doppler spectrum.

2π

1RΦ)(R,p)(p

0

ΦR,Φ =∂= ∫∞

Φ




60

3/23/2010

11

Level Crossing Rates and Fade Duration

•The LCR is defined as the

expected rate at which the

fading envelope (normalized to

RMS signal) crosses a specified

threshold in a positive

direction.

•The average Fade Duration is

the average period of time the

received signal is below a

threshold.LCR:Number of level crossing per sec: r)rp(R,rN

0

R&&&∫

∞

∂⋅=

−⋅

⋅⋅π=

2

RMSRMS

mR

Rexp

R

Rf2NR

Joint density function

of r and at r = R.

Time derivative of

envelope r(t).

r&

fm = Doppler frequency, R = target value /

threshold, RRMS = Average Signal Power.


4.4.1 Rayleigh Flat Fading4.4 Probability Models

61

Average Fade Duration:

Fade duration in seconds:

−=∂= ∫

RMSR

RR

exp-1rp(r)0

⋅⋅π

−

−

=τ

RMS

m

2

RMS

R

Rf

1R

Rexp

2

[ ]RrPr ≤=τRN

1

[ ] ∑τ=≤i

irT

1RrP

62

Level Crossing Rates and Fade Duration



( ) ( )∫γ

γ γ∂γ=γ<γ=0

0

0SOUT PPP

Defined as:

γ

γ−−=γ∂

γ

γ−

γ= ∫

γ

s

0

0

s

s

s

s

OUT exp1exp1

P0

{ }OUTe

0s

P1Log −−

γ=γ

In Rayleigh fading:

Instantaneous SNR Minimum SNR for acceptable performance – in

BPSK with Pe = 10-3, Min SNR = 7dB.

Average SNR

63

Outage Probability



Multipath Fading

� Will concentrate on Clarks and Jakes Model. Also a Lognormal fading model will be studied.

� These models essentially drive how we simulate fast and slow fading channels on computer (using Matlab for example). Essence is to capture the baseband models:

∑=

π⋅=N

1i

)cos(θj2-i

ieAh(t)

∑=

⋅=

N

0ii

light

ci )tcos(θ

v

vfj2-expAtUh(t) π)(

Where { }{ } )(2πh(t)E

τ)(thh(t)E)R(

d0

2

*

τf

τ

ℑ⋅=

−⋅=

≤

−=

otherwise0,

ff,

f

f1

1

πf

1

(f)P

d2

d

d

hhwith

64


4.4.2 Simulation Models


Multipath Fading

� Scattering based model with N sources.

� At any point, the received field is

Can be written in terms of I and Q

components

with

ωn = 2πfn, with fn the Doppler frequency, θn = 0 unless 3D model.

Cn a random scalar.




65

� Model works on the assumption that I(t) and Q(t) result in IID Gaussian Random variables. If the number of sources, N, are large, central limit theorem comes into play and hence both I(t) and Q(t) are Gaussian variables.

� Results in a Rayleigh fading distribution for the envelope.

66




Multipath Fading

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