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Muhammad Ali Jinnah University, Islamabad Campus, Pakistan

© Dr. Noor M Khan

EE, MAJU Multi-Path Fading

Channel

1 EE5713 Advanced Digital Communications

Week 6-7; Spring - 2013

Fading Channel

Base

Station

Mobile

Station

Lecturer: Assoc. Prof. Dr. Noor M Khan

Department of Electronic Engineering,

Muhammad Ali Jinnah University,

Islamabad Campus, Islamabad, PAKISTAN

Ph: +92 (51) 111-878787, Ext. 129 (Office), 186 (ARWiC Lab)

Fax: +92 (51) 2822743

email: [email protected], [email protected]

mailto:[email protected]

mailto:[email protected]


© Dr. Noor M Khan


Channel



Mobile Channel Parameters

• Time delay spread |

• Coherence Bandwidth | -> ISI

• Doppler Spread |

• Coherence Time | -> Unstable channel

• Flat fading

• Frequency selective fading

• Fast fading

• Slow fading


© Dr. Noor M Khan


Channel



Multi-path Propagation

• Multi-path smears or spreads out the signal

– delay spread

• Causes inter-symbol interference

– limits the maximum symbol rate

Transmitted

Symbol Received

Symbol

Base

Statio

n

Mobile

Statio

n

t


© Dr. Noor M Khan


Channel



Delay Spread

Transmitted

Symbol Received

Symbol

Base

Station

Mobile

Station

t

Time

Space


© Dr. Noor M Khan


Channel



Intersymbol Interference

Transmitted

Symbol of

Interest

Received

Symbol of

Interest

t

Received

Symbols Transmitted

Symbols

t


© Dr. Noor M Khan


Channel



Average Delay Spread

• Average delay spread

k

k

k

kk

k

k

k

kk

P

P

a

a

)(τ

)τ(ττ

2

2

τ

τ

Multi-path

Profile

(Discrete)

t

)(τkP

0τ0 1τ 2τ kτ

)(τ1P

)(τ0P

)(τ2P

)(τkP


© Dr. Noor M Khan


Channel



RMS Delay Spread

(Discrete)

• RMS delay spread

22

τ ττσ

k

k

k

kk

k

k

k

kk

P

P

a

a

)(τ

)τ(ττ

τ

2

2

22

2


© Dr. Noor M Khan


Channel



Average Delay Spread

(Continuous Delay Profile)

• Average delay spread

• Representative delay functions

0

0

)(

)(

τdttP

dttPt

elsewherezeroandttP

etP

d

t

202

)(uniform

1)(exp

τ


© Dr. Noor M Khan


Channel



Measurements

Type of

EnvironmentDelay Spread d

(s)

Open area <0.2

Suburban area 0.5

Urban area 3


© Dr. Noor M Khan


Channel



Coherence Bandwidth

• Coherence bandwidth Bc is a range of frequencies

over which the channel can be considered flat

– passes all spectral components with approximately equal

gain and liner phase

• Bandwidth where the correlation function RT() for

signal envelopes is high

• Therefore two sinusoidal signals with frequencies

that are farther apart than the coherence bandwidth

will fade independently.


© Dr. Noor M Khan


Channel



Coherence Bandwidth

• If RT() > 0.9

• If RT() > 0.5

• An exact relationship between coherence

bandwidth & delay spread does not exist

50

1CB

5

1CB


© Dr. Noor M Khan


Channel



Doppler Shift

• fc broadening from fc to (fc + fm )

BS1 v

c

vff

cm

v


© Dr. Noor M Khan


Channel



Doppler Spread & Coherence

Time

• Describes the time varying nature of the channel

in a local area

• Doppler Spread BD, is a measure of the spectral

broadening caused by the time rate of change

• fc broadening from (fc - fm) to (fc + fm )

• If the base-band signal bandwidth is much greater

than BD, the effects of Doppler spread are

negligible at the receiver


© Dr. Noor M Khan


Channel



Coherence Time

• Coherence Time is the time domain dual of Doppler spread

• Doppler spread and coherence time are inversely proportional

• TC = 1/fm

• Statistical measure of the time duration over which the channel impulse response is invariant


© Dr. Noor M Khan


Channel



Coherence Time

• If the coherence time is defined as the time over

which the correlation function is above 0.5, then

• Rule of thumb for modern digital communication

defines TC as the geometric mean of the above two

expressions for TC

m

Cf

T16

9

216

9

m

Cf

T


© Dr. Noor M Khan


Channel



Types of Small-Scale Fading


© Dr. Noor M Khan


Channel



Flat Fading 1

• If the mobile radio channel has a constant gain and

linear phase over a bandwidth greater than the

bandwidth of the transmitted signal - the received

signal will undergo flat fading

• Please, observe that the fading is flat (or frequency

selective) depending on the signal bandwidth

relative to the channel coherence bandwidth.


© Dr. Noor M Khan


Channel



Flat Fading 2

• BS << BC & TS >>


© Dr. Noor M Khan


Channel



Frequency Selective Fading 1

• If the mobile radio channel as a constant gain and

linear phase over a coherence bandwidth, smaller

than the bandwidth of the transmitted signal - the

received signal will undergo frequency selective

fading

• Again, the signal bandwidth is wider then the

channel coherence bandwidth, causing one or

more areas of attenuation of the signal within the

signal bandwidth


© Dr. Noor M Khan


Channel



Frequency Selective Fading 2

• BS > BC & TS <


© Dr. Noor M Khan


Channel



Fast Fading

• The channel impulse response changes rapidly within the symbol duration - coherence time < symbol period

• TS > Tc and BS < BD

• Channel specifies as a fast or slow fading channel does not specify whether the channel is flat fading or frequency selective fading


© Dr. Noor M Khan


Channel



Slow Fading

• The channel impulse response changes at a rate much slower than the transmitted base-band signal.

• Doppler spread is much less than the bandwidth of the base-band signal

• TS << Tc and BS >> BD

• Velocity of the MS and the base-band signaling determines whether a signal undergoes fast or slow fading


© Dr. Noor M Khan


Channel



Fading in Short

Large Doppler Spread

Time-Selective Fading

Large Delay Spread

Frequency-Selective Fading

Large Angle Spread

Space-Selective Fading


© Dr. Noor M Khan


Channel



Fading in Short

Delay Spread Coherence Bandwidth

Frequency separation at which two

frequency components of Tx signal undergo

independent attenuations

Doppler Spread Coherence Time

Time separation at which two time

components of Tx signal undergo

independent attenuations


© Dr. Noor M Khan


Channel



Fading (Continued)

Time

Bc

Tc

Bandwidth

Flat in Time and

Selective in Frequency

Flat in Time and

Frequency

Flat in Frequency and

Selective in Time

Selective in both

Time and Frequency


© Dr. Noor M Khan


Channel



Fading (Continued)

Fast and Slow Fading

If the channel response changes within a symbol

interval, then the channel is regarded FAST FADING

Otherwise

the channel is regarded as SLOW FADING


© Dr. Noor M Khan


Channel



Fast Fading

When?

The channel impulse response changes rapidly within

the symbol period of the transmitted signal.

What?

The Doppler Spread causes frequency dispersion

which leads to signal distortion.


© Dr. Noor M Khan


Channel



Doppler Spread

The Doppler effect (in addition to the fading effect)

renders the received pulse to be time-varying

The State Transitions are determined from the

dynamics of the fading channel (Fading Correlation

Function or The Doppler Spectrum)


© Dr. Noor M Khan


Channel



Doppler Spread (Continued)

f: carrier frequency

c: speed of light

v: mobile speed

θ: Angle of motion with

incoming multipath

Scattering Point

Incoming

multipath

BS

MS

Line of

Sight

y

x

v

θ


© Dr. Noor M Khan


Channel




cosd

f vf

c

f: carrier frequency

c: speed of light

v: mobile speed

θ: Angle of motion with

incoming multipath


© Dr. Noor M Khan


Channel




For the land mobile fading spectrum,

The Auto-Correlation Function Doppler Fading Spectrum


© Dr. Noor M Khan


Channel




h is the channel impulse response

h has a complex normal distribution with zero mean

|h| is Raleigh distributed

Phase φ is uniformly distributed between 0 and 2π

is Chi-square distributed


© Dr. Noor M Khan


Channel



Rayleigh Fading 1

• The received envelope (amplitude) of a flat

fading signal is described as a Rayleigh

distribution

– Square root sum r, of two quadrature Gaussian

noise signals xI and yQ has a Rayleigh

distribution (Papoulis65)

22

QI yxr )0(;2

exp)(2

2

2

r

rrrp


© Dr. Noor M Khan


Channel



Rayleigh Fading 3

• - rms value of the received voltage signal before

envelope detection

• 2 - time average power before envelope detection

• The probability that the received signal envelope

does not exceed R is given by:

)0(2

exp)(2

2

2

r

rrrp

2

2

02

exp1)()Pr()(

RdrrpRrRP

R


© Dr. Noor M Khan


Channel



Rayleigh Fading 4

• The median value of r is found by solving

• Mean and median differ by only 0.55dB

77.1

)(2

1

0

median

r

r

drrpmedian


© Dr. Noor M Khan


Channel



Ricean Fading 1

• When there is a dominant stationary signal

component

• At the output of an envelope detector -

adding a DC component ot the random

multi-path

)0,0(;)(20

2

)(2

22

rAforAr

Ier

rp

Ar

e


© Dr. Noor M Khan


Channel



Ricean Fading 2

• A - peak amplitude of the dominant signal

• I0 () - modified Bessel function of the first

kind and zero order

• Described in terms of a Ricean factor, K

)(2

log10)(2

2

dBA

dBK

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