temporal characteristics of fading

Ghassan Dahman / Fredrik Tufvesson

Department of Electrical and Information Technology

Lund University, Sweden

Channel Modelling – ETIN10

Lecture no:

Fredrik Tufvesson - ETIN10 1

4

Temporal Characteristics of

Fading


Small-scale fading

Doppler shifts

rv

Frequency of received signal:

𝑓 = 𝑓0 + 𝜈

where the Doppler shift is

𝜈 = − 𝑓0

𝑣𝑟

𝑐cos (𝜃)

The maximum Doppler shift

𝜈𝑚𝑎𝑥 = 𝑓0

𝑣𝑟

𝑐

Receiving antenna moves with

speed 𝑣𝑟 at an angle θ relative

to the propagation direction

of the incoming wave, which

has frequency 𝑓0.

k

The relationship 𝜈𝑚𝑎𝑥 = 𝑓0 𝑣𝑟 𝑐 is based on several assumptions –

e.g., static IOs, no double reflections on moving objects, etc.


Small-scale fading

Doppler shifts

• f0=5.2 109 Hz, vr=5 km/h, (1.4 m/s) 24 Hz

• f0=900 106 Hz, vr=110 km/h, (30.6 m/s) 92 Hz

How large is the maximum Doppler frequency in the

following cases:

• at pedestrian speeds for 5.2 GHz WLAN,

• and at highway speeds using GSM 900?

𝜈𝑚𝑎𝑥 = 𝑓0

𝑣𝑟

𝑐


• f0=900 106 Hz, v=40 km/h, (11.1 m/s) 33.33 Hz

Each 30 ms the envelop of the received signal will go to zero.

max 0

vf

c

Consider a GSM 900, and an Rx moving at 40 km/h.

The beating effect

0 0.05 0.1 0.15 0.2 0.25-2

0

2

time (s)

E(t

)

1

𝑣𝑚𝑎𝑥

Superposition of two carriers with different frequencies (beating)


Small-scale fading

Example: the time-variant two-path model

Wave 2

Wave 1

v

The two components have different Doppler shifts!

The Doppler shifts will cause a random frequency modulation

𝐸1 = 𝐸 1 (𝑗2𝜋𝑡𝑓𝑐 1 −𝑣

𝑐cos 𝜃1 )

𝑘1

𝑘2 v

𝑘1

𝜃1 𝐸2 = 𝐸 2 (𝑗2𝜋𝑡𝑓𝑐 1 −

𝑣

𝑐cos 𝜃2 )

𝑑01, 𝑑

02: distances at t=0

𝑘1, 𝑘2: wavenumbers

𝑘1 = 𝑘2 = 𝑘0 = 2𝜋 𝜆

𝐸𝑇(𝑡) = 𝐸 𝑒𝑥𝑝 𝑗2𝜋𝑓𝑐𝑡


Equivalent cases:

• Two superimposed incident waves + moving Rx

• Superimposing two signals with different Doppler shifts at the Rx antenna

Besides the distribution, the temporal behavior is needed to characterize

small-scale fading.

Small-scale fading

Example: the time-variant two-path model


Radio Channels

(some properties)

Path loss

Large-scale

fading

Small-scale fading

Attenuation

due to

distance

Variation

around the

mean due to

IOs

Frequency dispersion of the

signal due to the time-varying

nature of the channel

Time dispersion of the signal due

to the multipaths

Fast

fading Slow

fading

Flat

fading

Frequency-

selective

fading

The final effect is a function of the characteristics of

BOTH: the channel, and the transmitted signal


0 maxf 0 maxf 0f

Spectrum of received signal

when a f0 Hz signal is transmitted.

RX

RX movement

Incoming waves from several directions

(relative to movement or RX)

All waves of equal strength in

this example, for simplicity.

1

1

2

2

3

3

4

4

The received signal may occupy the 𝑓0 − 𝑣𝑚𝑎𝑥 to 𝑓0 + 𝑣𝑚𝑎𝑥 frequency band

Doppler spectra


Jakes spectrum

Isotropic uncorrelated

scattering

RX

Uniform incoming

power distribution

(isotropic)

Uncorrelated

amplitudes

and phases

RX movement

0 maxf 0 maxf 0f

0DS f

Doppler spectrum

at center frequency f0.

Jakes spectrum


Condensed parameters

Coherence time Doppler spectrum vs. the time-autocorrelation function

time-autocorrelation function Doppler power spectrum

0 maxf 0 maxf 0f

𝑆𝑣


• Time correlation – how static is the channel?

• For a uniform scattering environment the time correlation of the Re-

and Im-components is described by a “zeroth-order Bessel function of

the first kind”

• The time correlation for the amplitude is

*

0 max2t E a t a t t J t

2

0 max2t J t

The channel time correlation

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

d/

J 02(2

d/

)


The channel time correlation (example)

Example: Assume that an MS is located in a fading dip. On average, what

distance should the MS move so that it is no longer influenced by this

fading dip?

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d/

J 02(2

d/

)0.38𝜆

0.18𝜆

𝜌 = 0 (𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑑𝑒𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛)

𝜌 = 0.5


*r

*r

Time

Received amplitude [dB]

rmsr|dBM

What about the length and the frequency

of fading dips ?

Important for system design

block length

code design

choice of modulation

Temporal Dependence of small-scale fading


Frequency of the fading dips Length of fading dips

These curves are for Rayleigh fading & isotropic uncorrelated scattering (Jakes’ doppler spectrum).

Temporal Dependence of small-scale fading (cont.)

Example: assume a multipath environment where the received signal has a Rayleigh

distribution and the Doppler spectrum has Jakes shape. Compute the LCR and the

ADF for a maximum Doppler frequency=50Hz, and r/rmin=0.1.

a) 0.25 = LCR/50 => LCR=12.5

b) 0.04 = ADF×50 => ADF=0.8 msec


• Caused by the motion of the Tx, Rx, or IOs.

• Characterizes the time-varying nature of the channel in a small-

scale region (How fast does the channel change?) The

Coherence Time 𝑇𝑐(defined as the time during which the channel is

time invariant) is used as a measure.

• Characterizes the frequency-dispersion of the channel ( How does

the Doppler shifts spread the frequency components of the

transmitted signal?) The Doppler spread 𝐷𝑠 (defined as the

Maximum Doppler Shift) is used as a measure.

• As a general approximation :

Tc =1

Ds

Temporal Nature of the Channel



Coherence time

Given the time correlation of a channel, we can define the

Coherence Time TC.

t t

t

0t

0

2

t

CT

TC shows us over how long time

we can assume that the channel is

fairly constant.

If the Symbol Time 𝑇𝑠 is much

shorter than TC , it will

experience the same channel

(i.e., it will get attenuated by the

channel, however, it will not get

distorted due to fading).

The coherence time



Coherence time

If the bandwidth of the base band

signal W is much greater than

𝐷𝑆, the effect of the Doppler spread

on the received signal is negligible (no

distortion).

Bandwidth vs. Doppler spread

Doppler power spectrum

0 maxf 0 maxf 0f

𝑆𝑣



Coherence time Slow fading vs. Fast fading

𝑇𝑐 ≫ 𝑇𝑠 OR 𝐷𝑠 ≪ 𝑊

Slow Fading (i.e., the channel fading rate is less than the symbol

rate which results only in SNR degradation)

𝑇𝑐 < 𝑇𝑠 OR 𝐷𝑠 > 𝑊 (or comparable)

Fast Fading (i.e., the channel fading rate is greater than the symbol

rate which results in distortion degradation)

Mitigation Techniques:

• Robust modulation (schemes that don’t require phase tracking, and

reduces the detector’s integration time)

• Increasing the symbol rate

• Coding and Interleaving


Radio Channels

(some properties)

Path loss

Large-scale

fading

Small-scale fading

Attenuation

due to

distance

Variation

around the

mean due to

IOs The final effect is a function of the characteristics of

BOTH: the channel, and the transmitted signal

or comparable values

Time dispersion of the signal due

to the multipaths

Flat

fading Frequency-

selective

fading


Measurement example

• Measurement in the lab

• Center frequency 3.2 GHz

• Measurement bandwidth 200 MHz, 201 frequency points

• 60 measurement positions, spaced 1 cm apart

• Measured with a vector network analyzer


Coherence time, measured

0 0.1 0.2 0.3 0.4 0.5 0.6-90

-85

-80

-75

-70

-65

-60

-55

-50

-45

-40

-35

Position (m)

Fre

quency r

esp (

dB

)

Assume 1 m/s, max 0

vf

c =10.7 Hz Compare 1/(2**vmax)=0.014 s

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.5

0

0.5

1

Tim

e c

orr

Position (m)

measured

theoretical

Pr(

dB

)


Probability density function

0 0.002 0.004 0.006 0.008 0.01 0.0120

20

40

60

80

100

120

140

160

180

amplitude

pdf

measured

theoretical

Vehicle-to-vehicle communication


This channel has highly dynamic scatterers, both Tx and Rx are moving

Vehicle-to-Vehicle channel characteristics

24 / 12

Time-delay characteristics:

0 100 200 300 400Propagation distance [m]

Po

wer

t = 0 s


Po

wer

t = 0.2 s


Po

wer

t = 0.4 s


Po

wer

t = 0.6 s


Po

wer

t = 0.8 s


Po

wer

t = 1 s


Po

wer

t = 1.3 s


Po

wer

t = 1.5 s


Po

wer

t = 1.7 s


Po

wer

t = 1.9 s


Po

wer

t = 2.1 s


Po

wer

t = 2.3 s


Po

wer

t = 2.5 s


Po

wer

t = 2.8 s


Po

wer

t = 3 s


Po

wer

t = 3.2 s


Po

wer

t = 3.4 s


Po

wer

t = 3.6 s


Po

wer

t = 3.8 s


Po

wer

t = 4.1 s


Po

wer

t = 4.3 s


Po

wer

t = 4.5 s


Po

wer

t = 4.7 s


Po

wer

t = 4.9 s


Po

wer

t = 5.1 s


Po

wer

t = 5.3 s


Po

wer

t = 5.6 sRX

TX


Po

wer

t = 5.8 s


Po

wer

t = 6 s


Po

wer

t = 6.2 sLOS


Po

wer

t = 6.4 s


Po

wer

t = 6.6 s


Po

wer

t = 6.9 s


Po

wer

t = 7.1 s


Po

wer

t = 7.3 s


Po

wer

t = 7.5 s


Po

wer

t = 7.7 s


Po

wer

t = 7.9 s


Po

wer

t = 8.1 sDiscrete comp.

Diffuse

comp. Other vehicles

Houses, road

signs etc.

• Rapidly varying channel

• Discrete components carry significant energy and change delay bin with time

• Diffuse components following LOS

Fredrik Tufvesson - ETIN10


Time variant impulse response

Let’s take a closer

look at the Doppler

shifts here


Scattering function, t=8.5-8.65 s

27 / 12

Doppler-delay characteristics

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Doppler frequency [Hz]

t = 7.4 s

Delay [ns]

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dB

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Doppler-delay characteristics:

• Discrete components: small Doppler spread, but can change delay bin rapidly

• Diffuse components: large delay and Doppler spread

• Time-variant Doppler spectrum → WSSUS conditions are violated

Fredrik Tufvesson - ETIN10

Impulse response, fading of scatterer


300 350 400 450 500-105

-100

-95

-90

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-75

Propagation distance [m]

Rec

eiv

ed p

ow

er [

dB

]Impulse response,

two cars, same direction

90 km/h

Fading of a single scatterer

temporal characteristics of fading

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