wireless communicationsreview: power delay profile ! a plot of relative received power as a function...

Electrical & Computer Engineering

Wireless Communications

Small Scale Fading

Hamid Bahrami

Small-Scale Fading

l  Describe the rapid fluctuations of the amplitudes, phases, or multipath delays of a radio signal over a short period of time or travel distance l  The large-scale path loss effects might be ignored l  Fading is caused by multipath waves

l  Rapid changes in signal strength l  Random frequency shift l  Time dispersion (echoes)

Factors Influencing Small-Scale Fading

l  Multipath propagation l  Multiple signal paths l  Intersymbol interference

l  Speed of the mobile l  Doppler shift l  Random frequency modulation

l  Speed of surrounding objects l  Surrounding objects move at a greater rate than the MS

l  The transmission bandwidth of the signal l  Comparing to the bandwidth of the multipath channel

Doppler Shift

l  Named after Chris Doppler, an Australian scientist l  The apparent change in frequency and wavelength of a

due to the moving receiver or the waving transmitter

A mobile moving at a constant velocity v Along a path between X and Y with the distance d

Signals received from a remote source S

The source is very far away, same

The difference in path lengths θθ coscos tvdl Δ==Δ

The phase change θλπ

λπφ cos22 tvl Δ=Δ=Δ

Doppler Shift

l  Doppler shift, the apparent change in frequency

l  Moving toward the direction of arrival of the wave à the Doppler shift is positive (increasing frequency)

l  Moving away from the direction of arrival of the wave à the Doppler shift is negative (decreasing frequency)

θλ

φπ

cos21 v

tfd =

ΔΔ=

Enhance our understanding…

l  Consider a transmitter which radiates a sinusoidal carrier frequency of 1850 MHz. For a vehicle moving 60 mph, compute the received carrier frequency if the mobile is moving (a) directly toward the transmitter, (b) directly away from the transmitter, (C) in a direction which is perpendicular to the direction of arrival of the transmitted signal

Impulse Response Model

l  The impulse response model can be used for the small-scale variations of a mobile radio signal ?!

l  So far, we know l  The impulse response is a wideband channel

characterization and contains all information l  The mobile radio channel may be modeled as a linear

filter with a time varying impulse response l  We need the proof…


l  For a fixed position d, the channel between the Tx and the Rx can be modeled as a linear time invariant system h(t)

l  Due the multipath fading, the channel transfer function should be a function of the position of the Rx, h(d, t)

( )∫∞

∞−−=⊗= τττ dtdhxtdhtxtdy ),(),()(),(

The received signal The transmitted signal

For a causal system, ( )∫ ∞−−=

tdtdhxtdy τττ ),(),(

d=vt

( )∫ ∞−−=

tdtvthxtvty τττ ),(),(


l  The mobile channel can be modeled as a linear time varying channel, changing with time and distance

( ) ),()(),()(),()( tdhtxtvthtxdtvthxtyt

⊗=⊗=−= ∫ ∞−τττ

Since v is constant over a short time or time interval,

( ) ),()(),()( ττττ thtxdthxty ⊗== ∫∞

∞−

x(t): the transmitted bandpass waveform y(t): the received waveform h(t, τ): the impulse response of the time varying multipath radio channel τ: the channel multipath delay for a fixed value of t


l  Discrete model l  Equal time delay segments àexcess delay bins l  τ0: the first arriving signal l  N: the total number of possible multipath components

t1

t2

h tb ,τ( )

τ t2( )

τ t1( )

τ to( )τ 1 τ 4τ 3τ 2 τ N −1τ N − 2

t

delaybin i iΔτ τ τ= −+1


l  Discrete model (cont.) l  The model may be used to analyze transmitted RF

signals having bandwidths less than 2/ l  Excess delay τi: the relative delay of the ith multipath

component as compared to the first arriving component l  Maximum excess delay of the channel: N

τΔ

τΔ


l  Discrete model (cont.) l  If the channel impulse response is assumed to be time

invariant or at least wide sense stationary

l  Power delay profile of the channel over a local area

[ ] ( )iN

iiib jah ττδθτ −=∑

−

=

1

0

exp)(

( ) 2),( ττ thkP b=

Parameters of Mobile Multipath Channels

l  Review: Power delay profile l  A plot of relative received power as a function of excess delay

with respect to a fixed time delay reference l  Averaging instantaneous power delay profile measurements over

a local-area

∑−

−

=1

00

22

0 )()(N

kk tatr

Time Dispersion Parameter

l  The mean excess delay

( )( )∑

∑∑∑

==

kk

kkk

kk

kkk

P

P

a

a

τ

ττττ 2

2

The rms delay spread

( )( )( )∑

∑∑∑

==

−=

kk

kkk

kk

kkk

P

P

a

a

τ

ττττ

ττστ2

2

22

2

22

The maximum excess delay

dB X within iscomponent multipath aat which delay max. the: arrivalfirst the:0

0max

X

X

ττ

τττ −=


l  The rms delay spread and mean excess delay are defined from a single power delay profile, which is averaged over a local area

l  Many measurements are made at many local areas in order to determine a statistical range of multipath channel parameters

l  In practice, such parameters depend on the choice of noise threshold used to process P() analysis

l  …Related to bandwidth

Example: RMS Delay Spread

Environment Freq. MHz RMS Delay Spread Notes

Urban 910 1300 ns avg. New York City

Urban 892 10~25 µs San Francisco

Suburban 910 200~310 ns Typical case

Suburban 910 1960~2110 ns Extreme case

Indoor 1500 10~50 ns Office building

Indoor 850 270 ns max. Office building

Indoor 1900 70~94 ns avg. Office building

Practice: page 201, ex. 5.4 l  Compute the RMS delay spread for the following power delay

profile: l  If BPSK modulation is used, what is the maximum bit rate that can

be sent through the channel without needing an equalizer?

0 dB 0 dB

0 1 us

1.0≤sTτσ

Coherence Bandwidth

l  A defined relation derived from the rms delay spread l  A statistical measure of the range of frequencies over

which the channel can be considered “flat” l  The range of frequencies over which two frequency

components have a strong potential for amplitude correlation

τσ501≈cB

τσ51≈cB

As the bandwidth over which the frequency correlation function is above 0.9

As the bandwidth over which the frequency correlation function is above 0.5

Doppler Spread and Coherence Time

l  Describe the time varying nature of the channel in a small-scale region

l  Doppler spread BD l  A measure of the spectral broadening caused by the time rate of

change of mobile radio channel l  As the range of frequencies over [fc-fd, fc+fd] which the received

Doppler spectrum is essentially non-zero l  Slow fading channel: if the baseband signal bandwidth is much

greater than BD, the effect of Doppler spread are negligible at the receiver


l  Coherence time TC l  The time domain dual of Doppler spread l  Characterize the time varying nature of the frequency

dispersiveness of the channel

mmc

mc

mc

ffT

fT

fT

423.0169

169

1

2 ==

≈

≈

π

π

The maximum Doppler shift fm=v/

The time over which the time correlation function is above 0.5

Practical


l  Coherence time is actually a statistical measure of the time duration over which the channel impulse response is essentially invariant

l  Coherence time is the time duration over which two received signals have a strong potential for amplitude correlation

l  Two signals arriving with a time separation greater than TC are affected differently by the channel

l  If the reciprocal bandwidth of the baseband signal is greater than the coherence time of the channel, then the channel will change during the transmission of the message, thus causing distortion at the receiver.

Practice: page 202, ex. 5.5

l  Calculate the mean excess delay, rms delay spread, and the max. excess delay (10dB) for the multipath of the channel. Calculate the coherence bandwidth.

-20 dB

-10 dB

0 dB

Pr()

0 1 2 5 (us)

Types of Small-Scale Fading

l  Depending on the relation between the signal parameters: bandwidth, symbol period

l  Depending on the relation of the channel parameters: delay spread, Doppler spread

l  Types l  Time dispersion fading l  Frequency selective fading l  Frequency dispersion fading l  Time selective fading

Multipath Time Delay Spread

l  Flat fading l  A constant gain and linear phase response over a bandwidth l  A bandwidth is greater than the BW of the transmitted signal l  Most common type l  The spectral characteristics of the transmitted signal are preserved at

the receiver, even though the strength of the received signal changes with time

l  Narrowband channels or amplitude varying channels

ττ σσ 10,>><<

s

cs

TBB

Multipath Time Delay Spread

l  Frequency selective fading l  A constant gain and linear phase response over a bandwidth l  A bandwidth is smaller than the BW of the transmitted signal l  More difficult to model l  The received signal includes multiple versions of the transmitter

waveform which are faded and delayed l  Wideband Channel or Intersymbol Interference (ISI)

ττ σσ 10<>

s

cs

TBB

Flat Fading

Frequency Selective Fading

Due to Doppler Shift

l  Fast fading l  The channel impulse response changes rapidly within the

symbol duration l  Resulting in frequency dispersion à signal distortion l  Only dealing with the rate of the change of the channel due to

the motion. It does not imply flat/frequency selective fading l  In practice, fast fading only occurs for very low data rates

Ds

cs

BBTT

<>

Due to Doppler Shift

l  Slow fading l  The channel impulse response changes slower than the

transmitted baseband signal l  The channel may be assumed to be static over one or several

reciprocal bandwidth intervals à the Doppler spread of the channel is much less than the bandwidth of the baseband

Ds

cs

BBTT

>><<

Types of Small-scale Fading

Based on mulitpath time delay spread

Flat fading BW of signal < BW of channel Delay spread < Symbol period

Frequency selective fading BW of signal > BW of channel Delay spread > Symbol period

Based on Doppler spread

Fast fading High Doppler spread

Coherent time < Symbol period Channel variation faster than

baseband signal variations

Slow fading Low Doppler spread

Coherent time > Symbol period Channel variation slower than


Types of Small-scale Fading Sym

bol Period of Transm

itting symbol

Transmitted B

aseband Signal B

andwidth

Transmitted Symbol Period

Transmitted Baseband Signal Bandwdith

Flat Slow Fading Flat Fast Fading

Frequency Selective Slow Fading

Frequency Selective Fast Fading

TS

BS

BS

TS TC

BC

Bd

Frequency Selective Fast Fading

Frequency Selective Slow Fading

Flat Slow Fading Flat Fast Fading

Practice: page 253, 5.30

l  For each of the three scenarios below, decide if the received signal is best described as undergoing fast fading, frequency selective fading or flat fading. l  A binary modulation has a data rate of 500 kbps, fc=1 GHz and a

typical urban radio channel is used to provide communications to cars moving on a highway.

l  A binary modulation has a data rate of 5 kbps, fc=1GHz and a typical urban radio channel is used to provide communications to cars moving on a highway.

l  A binary modulation has data rate of 10bps, fc=1GHz and a typical urban radio channel is used to provide communications to cars moving on a highway.

Rayleigh Distribution

l  To describe the statistical time varying nature of l  The received envelope of a flat fading signal l  The envelope of an individual multipath component l  The envelope of the sum of two quadrature Gaussian

noise signals l  Probability Density Function (PDF)

⎪⎩

⎪⎨⎧

>

∞≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−=

r

rrrrP

00

02

exp)( 2

2

2 σσ

: the rms value of the received voltage signal before envelope detection 2: the time average power of the received signal before envelope detection, variance

σ2

Rayleigh Distribution l  Cumulative Density Function (CDF): the probability of the

received signal does not exceed a specified value R

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−==≤= ∫ 2

2

0 2exp1)(Pr)(

σRdrrPRrRP

R

l  The mean value and variance of the Rayleigh distribution

2

0

2222

0

4292.02

)(][][

2533.12

)(][

σπσσ

σπσ

=−=−=

====

∫

∫∞

∞

drrPrrErE

drrrPrEr

r

mean

l  The median value of r

σ177.1)(21

0=⇒= ∫ median

rrdrrPmedian


PDF CDF

l  Without LOS - there are many objects in the environment that scatter the radio signal before it arrives at the receiver

l  The urban environment l  Build-up city center

l  Small-scale fading effect


Ricean (Rice) Distribution

l  The small-scale fading envelope distribution l  When there is a dominant stationary (nonfading) signal

component present (LOS) l  As the dominant signal becomes weaker à Rayleigh

⎪⎩

⎪⎨⎧

<

≥∞≤≤⎟⎠⎞⎜

⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=ro

ArArIArrrP

0

0,02

exp)( 202

22

2 σσσ

A: the peak amplitude of the dominant signal I(.): Bessel function of the first kind and zero-order

Ricean Distribution

l  Ricean factor K – the ratio between the deterministic signal power and the variance of the multipath l  Aà0, Kà- dB, the dominant path decreases à

Rayleigh l  K>>1: less severe fading l  K<<1: severe fading

2

2

2

2

2log10)(

2

σ

σAdBK

AK

=

=

Ricean Distribution

Nakagami Distribution l  An empirical model that fits measured data in many mobile

environment better than Rayleigh or Ricean model

( )( )

22 2 1mm mmf x exp xx mα⎛ ⎞ ⎛ ⎞−= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠Γ Ω Ω

[ ] [ ]( )[ ]

222 1

2E xE x ; mVar x

Ω = = ≥

12112

, one sided Gaussian, Rayleigh pdf

m, approximates Ricean, no fading

>

⎧⎪⎪= ⎨⎪⎪→∞⎩

Nakagami Distribution l  Power distribution

( )( )2

1m mx mmf exp xxmα

− ⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎝ ⎠Γ Ω⎝ ⎠Ω

( ) [ ]

( )

( )

( )

2

2

2

0

10

1

1

x

mx m

F Px x

f dyy

mm y exp y dym

mG m, xm

α

α

α

−

= ≤

= ∫

⎛ ⎞⎛ ⎞= −⎜ ⎟∫⎜ ⎟ ⎝ ⎠Γ Ω⎝ ⎠Ω⎛ ⎞= ⎜ ⎟Γ ⎝ ⎠Ω ( ) ( ) ( ) 1

0x a yG y e dya,x a,xa − −= Γ −Γ = ∫


l  Show that the magnitude (envelope) of the sum of two independent identically distributed complex (quadrature) Gaussian sources is Rayleigh distributed. Assume that the Gaussian sources are zero mean and have unit variance.

Practical Models

l  Clark model l  Saleh and Valenzuela indoor statistical model l  Two-ray Rayleigh fading model l  SIRCIM and SMRCIM indoor and outdoor

statistical models l  And more…

Level crossing rate by Rice

l  Level crossing rate (LCR): 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

2

2),(0

ρρπ −∞== ∫ efrdrRprN mR

The time derivative of r(t)

The joint density function of r and at r=R

The value of specified level R, normalized to the local rms amplitude

R/Rrms

The Average Duration

l  The Average Duration: the average period of time for which the received signal is below a specified level R.

[ ]

[ ] [ ] ( )

signal fading theof intervaln observatio the:T fade theofduration the:

exp1Pr1Pr

Pr1

2

i

ii

R

RrT

Rr

RrN

τ

ρτ

τ

−−=≤=≤

≤=

∑

πρτ

ρ

21

2

mfe −=


l  For a Rayleigh fading signal, compute the positive-going level crossing rate for ρ=1, 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?


l  Find the average fade duration for threshold levels ρ=0.01, 0.1 and 1, when the Doppler frequency is 200 Hz.

l  Find the average fade duration for a threshold levels of ρ=0.707 when the Doppler frequency is 20 Hz. For a binary digital modulation with bit duration of 50 bps, is the Rayleigh fading slow or fast? What is the average number of bit errors per second for the given data rate. Assume that a bit error occurs whenever any portion of a bit encounters a fade for which ρ<0.1.

The average duration of a signal fade helps determine the most likely number of signal bits that may be lost during a fade.

Summary: Small Scale Fading

l  Parameters of mobile multipath channels l  Time dispersion parameters and coherence bandwidth l  Doppler spread and coherence time

( )( )∑

∑∑∑

==

kk

kkk

kk

kkk

P

P

a

a

τ

ττττ 2

2

( )( )( )∑

∑∑∑

==

−=

kk

kkk

kk

kkk

P

P

a

a

τ

ττττ

ττστ2

2

22

2

22

τσ501≈cB

τσ51≈cB

mmc

mc

mc

ffT

fT

fT

423.0169

1691

2 ==

≈≈

π

π

θλ

φπ

cos21 v

tfd =

ΔΔ=


l  Types of small-scale fading l  Flat fading vs. frequency selective fading l  Fast fading vs. slow fading

Flat fading BW of signal < BW of channel Delay spread < Symbol period

Frequency selective fading BW of signal > BW of channel Delay spread > Symbol period

Fast fading High Doppler spread

Coherent time < Symbol period Channel variation faster than


Slow fading Low Doppler spread

Coherent time > Symbol period Channel variation slower than



l  Rayleigh fading distribution

⎪⎩

⎪⎨⎧

>

∞≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−=

r

rrrrP

00

02

exp)( 2

2

2 σσ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−==≤= ∫ 2

2

0 2exp1)(Pr)(

σRdrrPRrRP

R

l  Level crossing rate and average duration

Practice l  Page 250, example 5.12: The fading characteristics of a CW

carrier in an urban area are to be measured. The following assumptions are made (1) The mobile receiver uses a simple vertical monopole. (2) Large-scale fading due to path loss is ignored. (3) The mobile has no line-of-sight path to the base station. (4) The pdf of the received signal follows a Rayleigh distribution. l  Derive the ratio of the desired signal level to the rms signal level that

maximizes the level crossing rate. Express your answer in dB. l  Assuming the maximum velocity of the mobile is 50 km/hr, and the

carrier frequency is 900 MHz, determine the maximum number of times the signal envelope will fade below the level found in (a) during a 1 minutes test.

l  How long on average will each fade in (b) last?


l  Describe all the physical circumstances that relate to a stationary transmitter and a moving receiver such that the Doppler shift at the receiver is equal to (a) 0 Hz, (b) fdmax , (c) -fdmax and (d) fdmax/2

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