The Wireless Channel (2)

Wireless Communication

www.ee.ui.ac.id/wasp

Review What we discussed last lecture:

The large-scale fading because of path loss The empirical path loss formulas

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

Introduction The small-scale fading is usually called

“fading” It is caused by multipath signal, so it is also

called “multipath fading” Multipath signal causes constructive and

destructive addition of the received signal

Introduction 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 received signal component associated with a single transmitted pulse

LOS

Reflected components

dT

Introduction If the delay spread is small compared to the

(1/BW), then there is little time spreading in the received signal

If the delay spread is relatively large, there is little time spreading of the received signal, i.e. signal distortion

Multipath channel is also time-varying that means either the transmitter or the receiver is moving

It also causes the location of the reflectors will change over time

We will limit the model to be narrowband fading, i.e. the BW is small compared to (1/delay spread)

Introduction Physical factors influencing fading:

Multipath propagation Speed of the mobile Speed of surrounding objects The transmision bandwidth of the signal

Review of Doppler Shift The received signal may experience Doppler shift

If the receiver is moving towards the transmitter, the Doppler freq is positive, otherwise it is negative

v eff

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: Directly towards the transmitter Directly away from the transmitter In a direction which is perpendicular to the

direction of arrival of the transmitted signal

Carrier freq = 1850 MHz Wavelength = Vehicle speed = 26.82 m/s Vehicle moving towards the transmitter

means positive Doppler frequency

Vehicle moving towards the transmitter means negative Doppler frequency

Vehicle is moving perpendicular means

Solution

8

6

3 100.162

1850 10c

cm

f

26.821850 cos 0 1850.00016

0.162c df f f MHz

26.821850 cos 0 1849.999834

0.162c df f f MHz

90 26.82

1850 cos90 18500.162c df f f MHz

Doppler Spread Doppler spread is given by

Where, and

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

2 1:sD D D

1

fvD

c 2

fvD

c

6

8

2

900 10 16.672 100

3 10

s

fv fv fvD

c c c

Hz

Time-Varying Channel Impulse Response We have already known that the transmitted

signal is

Then, the received signal in multipath channel is

n = 0 corresponds to the LOS pathN(t) is the number of resolvable multipath components is corresponding delay is Doppler phase shift

is amplitude

2 cos 2 sin 2cj f tc cs t u t e u t f t u t f t

Time-Varying Channel Impulse Response The n-th resolvable multipath component may

correspond to the multipath associated with a single reflector or multiple reflectors clustered together

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

If reflector cluster exists, two multipath components with delay and are resolvable if

If the criteria is not satisfied, then it is nonresolvable since

The nonresolvable components are combined into a single multipath component with delay and an amplitude and phase corresponding to the sum of different components

2 c nj f tn t e

2N ND D

t

f t dt

1 21

1 2 uB

1 2u t u t

1 2

Time-Varying Channel Impulse Response The amplitude of the summed signal will

undergo fast variations due to the constructive and destructive combining of the nonresolvable multipath components

Wideband channels have resolvable multipath components the parameters change slowly

Narrowband channels tend to have nonresolvable multipath components the parameters change quickly

Time-Varying Channel Impulse Response We can simplify by letting

The received signal is then

The received signal is obtained by convolving the baseband input signal with equivalent lowpass time-varying channel impulse response of the channel, and then upconverting the carrier frequency

r t

2nn c n Dt f t

Time-Varying Channel Impulse Response The represents the equivalent lowpass

response of the channel at time t to an impulse at time

,c t

t

Parameters of Mobile Multipath Channels Time dispersion parameters Coherence bandwidth Doppler spread and coherence time

Time Dispersion Parameters The time dispersive properties of wideband

multipath channels are most commonly quantified by their mean excess delay and rms delay spread

The mean excess delay:

The rms delay spread is the square root of the second central moment of the power delay profile

2

2

k k k kk k

k kk k

a P

a P

22

2 2 2

22

k k k kk k

k kk k

a P

a P

Time Dispersion Parameters The delays are measured relative to the first

detectable signal arriving at the receiver at 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 component is within X dB of the strongest arriving multipath signal and is the first arriving signal

0

0X

X

0

Time Dispersion Parameters

Coherence Bandwidth Coherence bandwidth is a statistical measure

of the range of frequencies over which the channel can be considered “flat”

Flat fading is a channel which passes all spectral components with approximately equal gain and linear phase

The coherence bandwidth can be expressed as

(above 90% correlation)

(above 50% correlation)

1

5cB

1

50cB

Example Compute the mean excess delay, rms delay

spread, and the maximum excess delay for the following power delay profile

Estimate the 50% coherence bandwidth of the channel

Solution Using the definition of maximum excess delay

(10 dB), it can be seen that The mean excess delay:

The second moment

The rms delay spread:

The coherence bandwidth:

10 4dB s

1 5 0.1 1 0.1 2 0.01 04.38

0.01 0.1 0.1 1s

2 2 2 2

2 21 5 0.1 1 0.1 2 0.01 021.07

0.01 0.1 0.1 1s

221.07 4.38 1.37 s

1 1

1465 5 1.37cB kHz

s

Doppler Spread and Coherence Time Doppler spread has been discussed before The coherence time is related with Doppler

spread (Doppler shift)

0.423cT v

Types of Small-Scale Fading

Flat Fading If the mobile radio channel has a constant

gain and linear phase response over a bandwidth which is greater than 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 channels It is also sometimes referred to as narrowband

channels The most common amplitude distributions

are: Rayleigh, Rician, and Nakagami Summarize: a signal undergoes flat fading if

s cB B

sT

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 channel creates frequency selective fading on the received signal

Frequency Selective Fading The received signal includes multiple versions

of the transmited waveform which are attenuated and delayed in time, and hence the received signal is distorted

Frequency selective fading is due to time dispersion of the transmitted symbols within the channel

Thus, the channel induces intersymbol interference (ISI)

The 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

The common model: 2-ray Rayleigh fading

Frequency Selective Fading It is sometimes called wideband channels

since the bandwidth of the signal is wider than the bandwidth of the channel impulse response

Summarize: a signal undergoes frequency selective fading if s cB B

sT

Fast Fading In a fast fading channel, the channel impulse

response changes rapidly within the symbol duration

In other words, the coherence time of the channel is smaller than the symbol period of the transmitted signal

This causes frequency dispersion (time selective fading) due to Doppler spread, which lead to signal distortion

Signal distortion due to fast fading increases with increasing Doppler spread relative to the bandwidth of the transmitted signal

Summarize: 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 signal

The channel may be assumed to be static over one or several reciprocal bandwidth interval

The Doppler spread of the channel is much less than the bandwidth of the baseband signal

Summarize: a signal undergoes slow fading if

s cT T

s DB B

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 selective

Fast fading only deals with the rate of change of the channel due to motion

In flat fading channel, we can approximate the impulse response to be simply delta function

A 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 signal

A 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

Rayleigh Fading The Rayleigh distribution is commonly used to

describe the statistical time varying nature of the received envelope of a flat fading signal

Rayleigh distributed signal:

Rayleigh Fading The Rayleigh distribution has pdf

The probability that the envelope of the received signal does not exceed a specified value R is

the rms value of the received voltage signal before envelope detection

2 the time-average power of the received signal before envelope detection

Rayleigh Fading The mean value of Rayleigh distribution is

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

The median value is

The median is often used in practice

Rayleigh Fading The corresponding Rayleigh pdf is

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 a positive-going direction

The number of level crossing per second is given by

Where is time derivative of r(t) (the slope)

is the joint density function of r and at r = R

2

0

, 2R DN rp R r dr f e

,p R rr

r

rmsR R

Example For a Rayleigh fading signal, compute the

positive-going level crossing rate for when the maximum Doppler frequency is 20 Hz

What is the maximum velocity of the mobile for this Doppler frequency if the carrier frequency is 900 MHz?

1

Solution Use the equation for LCR

Use equation of Doppler frequency

12 20 1 18.44RN e

20 1 3 6.66 /Dv f m s

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.

For a Rayleigh fading signal, it is given by

So, the average fade duration can be expressed as

1Pr

R

r RN

2

0

1Pr

1 exp

ii

R

r RT

p r dr

2

1

2D

e

f

Example Find the average fade duration for threshold

levelswhen the Doppler frequency is 200 Hz

Solution Average fade duration is

0.01

20.01 119.9

0.01 200 2

es

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 is Rician

Random multipath componnets arriving at different angles are superimposed on a stationary dominant signal

The Rician distribution is given by

2 2

2202 2

for 0, 0

0 for 0

r Ar Ar

p r e I A r

r

Rician Fading Distribution The Rician distribution is described in terms of

a parameter K

As we have Rayleigh fading As we have no fading, channel has no

multipath, only LOS component

2

22

AK

2

210log

2

AK dB

0K K

Rician Fading Distribution The Rician pdf is

Conclusions Small-scale fading is variation of signal

strength over distances of the order of the carrier wavelength

It is due to constructive and destructive interference of multipath

Key parameters:Doppler spread coherence timeDelay spread coherence

bandwidth Statistical small-scale fading: Rayleigh fading

and Rician fading flat fading