hertzian links - unige.it · 2020. 4. 26. · references [1] theodore s. rappaport, wireless...

Physical layer models and

techniques for software radio

HERTZIAN LINKS

Carlo Regazzoni

Sistemi di Radiocomunicazione

DITEN - Department of Electrical, Electronic, Telecommunications Engineering and

Naval Architecture

1

References

[1] Theodore S. Rappaport, Wireless Communications, 2ed,

Prentice Hall, 2002

[2] J.G. Proakis, Communication Systems (Fifth edition),

McGraw-Hill, New York, 2008.

[3] G. Maral, M. Bousquet, “Satellite Communications

Systems” (Terza Edizione), Wiley, 1998.

[4] A. Bernardini, “Sistemi di telecomunicazione - Lezioni”,

Edizioni Ingegneria 2000, 1989.

2

Contents

Circuital model of a transmission line for the hertzian

link.

Long range channel model: Free space Friis model,

Link Budget.

Multipath channel.

Optimal receiver for frequency selective channels.

3

Generality

An hertzian link can be considered as a radio TLC

system (e.g. for a PTP TLC system) connecting two

radios.

The transmitted signal is transduced by an antenna

into a proportional electromagnetic field (e.m.). This

component of the hertzian link is defined as

TRANSMITTER

Such field propagates, by changing some of its

characteristics, through a channel, the HERTZIAN

link, until it reaches a RECEIVER

The received signal is inversely transduced by the

receiver antenna that senses the e.m. field at

receiver premises.

4

Generality

The transmission device and/or the receiving device

can be defined as RADIO STATIONs or as RADIO

TERMINALS (transmission or receiving

station/terminal)

Each radio device is associated with an antenna that

represents the sensor/actuator allowing the radio

interacting among them through the shared channel.

Antenna are basically transductors that convert

information in appropriate ways to be processed in

the respective context (e.g. e.m wave in the channel,

electric signal in the radio circuits)

5

Circuital model of hertzian link (1/6)

Antenna can optimally transduce information at its

own operating points.

In many antennas it is required that the transmission

signal has low (narrow) fractional bandwidth

(defined as ratio between: total spectrum occupation

of the signal transmitted and central frequency)

Hertzian links allow to transmit only modulated

signals i.e. e.m waves that are well suited to

propagate through a medium.

6

Circuital model of hertzian link (2/6)

A circuital model can be used as a first analogic

model to describe an Hertzian link.

Some hypothesis are necessary

In particular, only narrowband signals are

considered, e.g. sinusoidal signals with carrier

frequency and amplitude and phase slowly time

varying.

A pure sinusoid is the limit example of a

narrowband signal, has it can be described by a pure

pulse in frequency domain.

7

Circuital model for hertzian link

(3/6) The carrier frequency of the sinusoid, can be

defined through the corresponding wave length

Where c is propagation velocity of the e.m. wave into the space

(c = 3 108 [m/s] for free space).

The link satisfies two gates network model (see the

figure of next slide), with the following notations:

Input impedance:

Output impedance:

Intrinsic transfer function: ( )

pp fc

fZAT fZi

fZAR fZu

fHP fHQ

8


(4/6)

Two gates networkEquivalent circuit of the Two gates

network (passive), it’s powered by a

generator and it’s connected to a load

(this is a model for the hertzian link)

)()()()()( 2121111 fIfZfIfZfV

)()()()()( 2221212 fIfZfIfZfV

The sinusoid can be modeled as a Voltage electric

signal transmitted at the input of the circuit and its

modifications along the hertzian link can be described

in different circuits parts using a transmission line

model.

9


(5/6)

The distance between the antennas is so large that

the presence of receiving antennas don’t alter the

e.m. field generated by transmitting antenna so the

term can be considered almost zero:

can be considered only as transmitting antenna

characteristic quantity and possibility as external

environment structure where it’s inserted; it’s called INPUT

IMPEDENCE of transmitting antenna.

has similar proprieties, it’s called OUTPUT

IMPEDENCE of receiving antenna.

)(12 fZ

)( fZAT

)( fZAR

10


(6/6)

HP(f)

ZAT(f)

ZAR(f)

Transmitter Riceiver

ZG(f) VG(f)

so,

This channel term does not depend from adaptation of input or

output antennas; it depends only on coupling between antennas

and medium propagation characteristics.

)()()(11 fZfZfZ ATi

)(

)()(

)()(

)()( 21

fZ

fZfZ

fZfZ

fZfH

AT

GAT

GAT

p

represents a

voltage controlled

amplifier

It is an ideal component

because its input

impedance is infinite

)( fH p

11

Circuital model for hertzian link and

free space model

• The circuital model of a wireless

telecommunication systems can be related to

the classical simplest model that describes

transformations of a signal from the transmitter

to the receiver in a wireless channel

• The free space model is such a model

• The free space model equation was described

by Friis in 1946

• H. T. Friis, Proceedings of the IEEE, vol. 34,

p.254, 1946.

[1] H. T. Friis, Proceedings of the

IEEE, vol. 34, p.254, 1946.12

Free space model

Hypothesis: There aren’t other objects in the space.

The space is isotropic and no-dissipative.

The links between the antennas and the receiver are adaptedto maximum power transfer (the output impedances of thetransmitter and input impedances of the receiver must be equalto and respectively into the bandwidth of thetransmitted signal.

Under these conditions the hertzian behaves like a perfect(ideal) channel, that modifies the transmitted signal byintroducing only frequency dependent:

Delay due to finite speed of the e.m. field.

Loss, i.e. amplitude reduction due to distance betweentransmitter and receiver.

)(* fZAT

)(* fZAR

13

Free space model and antennas

Two basic cathegories of antennas can be described

parametrically in free space model:

Isotropic antenna: an ideal antenna that radiates power

uniformly in all direction.

Directional antenna: an antenna that does not radiate

power in uniformly in the space, but it radiates power

differently as a function of the transmission main direction.

The antennas used in real application are directional.

14

Antenna Gain

Gain of an antenna is a parameter used to describe directionalpower emission/reception characteristics of an antenna,

Directive gain function GT(q,f) is the ratio between radiation power (or received power) for solid angle unit of a

directional antenna (in given direction, (q,f))and

radiation power (or received power) for solid angle unit of aisotropic antenna (uniform for all directions).

Antenna directive gain (Gmax) is the maximum value of thegain function i.e. 𝐺𝑚𝑎𝑥= max

𝜃,𝜑𝐺𝑇 𝜃, 𝜑

Directivity D corresponds to the electromagnetic axis of theantenna, i.e. 𝐷 = 𝑎𝑟𝑔max

𝜃,𝜑𝐺𝑇 𝜃, 𝜑

The directive gain value is GT=1 for an isotropic antenna.

15

Antenna Gain

The directive gain function (called radiation

pattern) of an antenna can be expressed in

Polar coordinates (a)

Cartesian coordinates (b).

q3dB is an angle which corresponds to a loss of 3dB than Gmax)

16

Directive antennas: examples

Antenna a bandiera Antenna circolare Antenna a spillo

f

q

ff 0;20:,R

f 0:,R

222:, ff R

Flag Antenna (Antenna a

bandiera):

Circular Antenna (Antenna

circolare):

Pin Antenna (Antenna a

spillo):

17

Antenna Efficiency

Antenna efficiency 𝜂 can be defined as the ratiobetween power of signal before and after the

transmitting antenna, i.e.,

𝑃𝑇 = 𝜂𝑃𝑖𝑛 h can be seen as the product of different efficiency

factors, accounting a different physical effect:

lightness, power loss, antenna resistivity, etc.

khhhh ............21

This product provides a value in the range: 0.55 and 0.75.

18

Antenna sensitivity The power of the e.m. field 𝑊 𝜃,𝜑 at the receiver

antenna is transduced into the power of the signal at

the receiver 𝑃𝑅 i.e.

𝑃𝑅 = 𝐴 𝜃, 𝜑 𝑊 𝜃,𝜑 𝐴 𝜃, 𝜑 is called antenna aperture of effective area

Effective area describes quantitatively the area the antenna

would need to occupy in order to intercept the observed

captured power

For a good reception the antenna size should be inversely

proportional to the square of signal frequency

qhq ,, GAA

Physical Aperture (depends from antenna

geometry)Receiving antenna

efficiency

19

Transmitters vs Receivers A antenna can work both as a transmitter and as a

receiver

The receiving pattern (sensitivity as a function of

direction) of an antenna when used for receiving is

identical to the far-field radiation pattern of the

antenna when used for transmitting. (Reciprocity

theorem)

It can be shown that the ratio between directive gain

and effective area ratio is a constant independent

from directions, i.e.

𝐺 𝜃, 𝜑 = 𝐴 𝜃, 𝜑4𝜋

𝜆2where

𝜆2 is the free-space wavelength

20

Directive antennas

For directional antennas, direction of maximum

emission corresponds to the direction of the

maximum received power.

Maximum value of SR, can be related with maximum

gain in a dB relation, i.e.:

So while the increase of signal frequency could make

antenna more compact, more attenuation will be

present for such a more compact systems

pdBRRdBeffR fGSA 10)(10

,)( log205.21,log10max fq

fq

21

Free space model Generic geometrical model

Let us consider a geometrical model of the wireless link as

follows. Let us define

a polar coordinate reference system (r,q,f) centered at thetransmitter antenna

a polar coordinate reference system 𝑟′, 𝜃′, 𝜑′ centered atthe receiver antenna

O

L

r

(q,f) (q’,f’)

22

Free space model

Isotropic antenna.

GT(q,f ) = GT =1.

O is a point in the space where is a transmission antenna, PTis output power from the transmitter. Let transmitter

antenna efficiency 𝜼𝑻 = 𝟏 , i.e. all signal power P0 isradiated.

Using no-dissipative free-space assumption with isotropic

antennas, the radiated power for solid angle unit can be

written as:

4TPW

(W/steradianti)

23

Free space model

Directional antenna delivers the radiated power in differentangular directions in accordance with gain function, GT(q,f ).

The radiated power for solid angle unit 𝑊 𝜃,𝜑 for directionalantennas becomes:

𝑊 𝜃,𝜑 is defined as: Equivalent (or Effective) Isotropically Radiated Power

(EIRP).

fqfq

4

,),( TT

GPW (W/steradians)

24

Free space model

Due to no dissipative hypothesis, the total power

through a spherical surface, with at its center the

transmission antenna and radius r, will be equal to PT

As a consequence the radiation pattern area is

constrained by the following equation to be constant

fqfqqf

4, ddGT

TPddW fqfqqf

,

25

Free space model

Let us place a receiver R at a distance L from a

receiver using an antenna with a effective area

𝐴 𝜃, 𝜑

A/L2 can be defined as the underlying solid angle.

This surface receives a power equal to:

Φ is the power flow density, and is measured in

W/m2.

A

L

AGPW TTR 24

,

fq

(W)

26

Free space model

Let the receiving antenna be at a distance L in adirection (ϑ,ϕ) wrt transmitting antenna, looking at thetransmitting antenna in the direction (ϑ’,ϕ’).

Only a fraction of the EIRP is transduced by receivingantenna to the receiver.

The fraction is usually expressed as a surface parameter,i.e. the effective area 𝑺𝑹 𝜽,𝝋 of the receiving antenna.

O

L

r

(q,f) (q’,f’)

27

Free space model

The available power at the output of the receiving

antenna is expressed by:

SR(ϑ’,ϕ’) is the receiver effective area function.

By recalling relationship between effective area and

gain function one can write

2

''''

4

,,),(

L

SGWSW RTdTRdR

fqfqfq

ff

4,,

2

p

RR GS Gain function ofthe receiving

antenna

28

Free space model equation

Power Loss of the hertzian link under the free space

hypothesis depends from

Alignment between transmitting and receiving antennas

Distance between transmitting and receiving antennas

Frequency (wavelength) of the modulated signal

and can be expressed a :

ff

,,

142

RTp

dGG

LA

dBRdBTMHzpkmdBd GGfLA 1010 log20log204.32

29


If antenna gains are zero (in dB), i.e. in case of

isotropic antennas are used at transmitter and

receiver, this relation is called

BASIC FREE SPACE LOSS. i.e.

𝐴𝑑 =𝑃𝑇

𝑃𝑅=

4𝜋𝐿

𝜆

2or 𝐴𝑑𝑑𝑏 = 20 log10 𝐿 + 20 log10 𝑓𝑝𝑀ℎ𝑧

The loss of the hertzian link grows with

square of the length of the link, and not in an exponential

way as for coaxial cables or optical fibers.

square of the carrier frequency

30


A simplified model can be written by fixing a

reference distance within the far-field validity range of

the model, i.e. 𝑑0 = 100𝑚

As a consequence a constant 𝐾𝑑𝑏 can be computedfor omnidirectional isotropic antennas as

𝐾𝑑𝑏 = 20 log10 𝑑0 + 20 log10 𝑓𝑝𝑀ℎ𝑧

The free space model can be rewritten in this case as

a simplified model:

𝐴𝑑𝑑𝑏 = 𝐾𝑑𝑏 − 10𝛾 log10𝐿

𝑑0

where 𝛾 = 2

31


The simplified free space model can be adapted to

represent other models computed starting from

different environmental assumptions. The same

expression holds in those cases


𝑑0

However, the parameters 𝐾𝑑𝑏 and 𝛾 can vary forother deterministic models

32

Other models

The free space model assumptions related to

absence of objects in the space between transmitter

and receivers is valid in the limit case only in space

where atoms/molecula are not present.

Otherwise the transmission channel cannot be

isotropic and interaction with objects and e.m

waveforms make the space dissipative.

So other models can complement the free space

model to better allow predicting characteristics of a

received signal

Such models can be deterministic or probabilistic

33

Other models Deterministic models (as the free space one) assume signal

and environment parameters like distance, frequency are

perfectly known at the moment of computing path loss.

Statistical models are more realistic, as they describe a variety

of possible motivations that can cause not precisely forecastable

variations and oscillations in the received signal

For example, variations of material by which objects in the

medium are composed can result in different interactions with

e.m wave.

As the composition of space is unknown and time variant only

a average statistical representation of its effect on propagated

signal can be used.

34

Deterministic models:

Surface wave Additional loss

Earth absorption.

The earthly surface can be considered as a conductor bodywith conductivity and dielectric constant depending on thecomposition of the ground below the link.

If the antennas are very close to the ground it can beexpected that all or part of the e.m. field will propagate at theinterface between air and ground media.

Such a propagation along the earth surface (SURFACE WAVE)will cause a additional loss increasing with signal frequencyand distance wrt the one described by the free space model;

Such a loss can be modeled as a multiplicative factor

as= as(L),

that takes into account the power dissipated by the surfacewave .

35

Additional loss

Up to 3Mhz waves travel efficiently using also surface

wave propagation

For frequency less than 10 MHz:

Where σ≈5∙10-5 is the ground conductivity.

When either the distance and/or carrier frequency

grow, surface wave becomes unusable to build

hertzian link (see the figure).

In that case skywave is the information carrier

a ps 1 22

p f L f Lp MHz km p MHz km

5810

106

2 2.

36

Additional loss of the surface wave as a

function of distance and frequency

37

Statistical models

However, to better predict variability of situations,

only statistical models can be used.

Such models can be progressively more complex

depending on the type of varying conditions they

model that could affect the transmitted signal

Rain attenuation and Shadowing are more simple

statistical models concentrating on less complex

possible causes of signal modifications.

Multipath models are more complex, but more

complete, statistical models that better allow to

predict signal variability at the receiver.

38

Direct link – Line-On-Sight

The antennas are allocated to a sufficient height

above the ground, in this way the ground doesn’t

significantly affect e.m. field propagation, an the link

is along a DIRECT WAY (LOS link = Line-On-Sight).

To this end two antennas must be placed in

visibility; considering the earthly radius is about

6380 km:

(km)

Where h1 and h2 are the heights of the antennas

above the ground, expressed by km.

216.3 hhLvis

39

LOS Link

Atmospheric absorption and rain.

The hypothesis of not-dissipative space is valid upto about 10 GHz: above, the loss increases by amultiplicative term (ADDITIONAL LOSS) that canoriginate by exchange of energy between thewave and from oxygen, water vapor and rain.

The additional loss depending on oxygen absorptionis relevant at carrier frequencies above 30 GHz,with a maximum around 60 GHz (20 dB/km).

The water vapor absorption has a maximum valueat about 22 GHz (12dB/km).

40

Statistical model for

Rain Absorption

For modeling the additional loss due to the rain, let us

consider a infinitesimal link of length ∆x.

If the wavelength is comparable with the diameter of

rain drops, the power absorption is:

r0 is rainfall intensity (mm/hour)

K and α are two parameters depending on different

factors (as temperature, wind velocity, carrier

frequency, shape of rain drops, etc.).

xKra p 0 0log10log/ rKa KmdBp

41

Dependence of the K and α parameters

by carrier frequency value

Carrier

frequency (GHz)

K

10 1.27 0.010

15 1.14 0.036

20 1.08 0.072

25 1.05 0.120

30 1.01 0.177

35 0.97 0.248

42

Additional loss due to rainfall

43


We take a reference X on the line joining the centers

of the TX and RX antennas. The origin is at the

transmitting antenna while the receiving antenna is

placed at L:

rL is average rainfall in the distance;

r0 is punctual rainfall.

L

Lpp tKLrdxtxrKtaa0

0 ,

r tL

r x t dxL

L

10

0

1

,Where:

(1)

44


rL (t) is a distribution function of additional loss as a function oflink length; it can be considered as a realization of a randomprocess RL(t), whose characteristics vary from zone to zone.

For one zone, we can define it as:

Inverse function provides the attenuation threshold as afunction of the selected probability level and the distance.

apdB represents the additional loss level that can be exceededwith a determinate probability D in function of link length L.

D a L A a LApdB pdB pdB pdB; Pr |

a a D LpdB pdB ;

45


From (1), we obtain the probability that rainfull

exceeds a given loss value on the link)

Knowing DRL and L, the inverse function rl* = rl(D;L)

can be expressed as. The additional loss that will be

exceeded with probability D can so be computed as:

LKLarDLrKLaRLaALaD pdBRLpdBLpdBpdBpdBApdB LL ;|Pr|Pr;1**1

a D L KLr D LpdB L, ;

46


Available knowledge is generally the local distribution

of the rainfall intensity DR0(r0) (e.g. based on

pluviometer measurements)

The inverse local function can then be computed as

r0=r0(D); this is equivalent to a level of local rainfall

that can be exceeded with probability D

To compute the link DRL(rL;L) that depends on L an

interpolation is needed.

This can be done in the simplest way, by

hypothesizing that rain will fall with constant intensity

above all the linkextension, i.e.:

r D r D LL0 ;

47


However, rainfall intensity is not generallyuniform, so we should adjust the model to keepinto account such variability

To this end an equivalent link length Le=L∙f(D,L)can be used

f(D,L)


For example r0=r0(10-4) means that r0 is the rainfall

intensity value in mm/hour that is exceeded with

probability equal to 10-4.

r0(D) decreases with D.

If D=1, then r0=0, because it is certain that the rainfall level r0=0 (that corresponds at a no rain situation is always

exceeded (r0


Accordingly the rainfall intensity distribution function can be

expressed as in the following function

graphically:

or analitycally

The discontinuity in 0 and the related dirac function in the

corresponding pdf of amplitude (1-Q) represents the

probability that it is not raining

f(r0) is so the distribution function of a r0 rainfall, conditioned

to the hypothesis that it is raining. The global probability that

it is raining is so equal to Q.

0

0

0 1

Pr

00

0

0

0000

rperrfQ

rperQ

rper

rRrDR

50


In many European Country conditional rainfall

intensity distribution is of log-normal type:

Probability density:

Distribution function:

22lnexp 022000 rrrprf LN

D x erfc xLN 05 2. ln

locality Q Nation

Fucino 0.048 1.223 .157 Italy

Graz 0.0317 1.466 .047 Austria

Kjeller 0.0293 1.438 .033 Norway

51

Additional Loss Distribution in relation

with the frequency changes (1/2)

Rainfall intensity distribution

function in the location

Fucino

Additional Loss Distribution corresponds to 15 and 30 GHz (curves b e c)

52


Shadowing

Variability in received power can happen whether

blockage occur when objects occlude the path

between the transmitting and the receiving antenna

Such situation cannot be a-priori predicted and can

give rise to variations of the received power at a

given distance

This situation can be modeled by considering the

received power as a random process depending on

the random nature of the channel, in a similar way as

done for rain

The corresponding model is called shadowing

model

53


Shadowing As for rain, attenuation is considered as a random

variable, i.e..𝜉 = 𝐴𝑑 =𝑃𝑇

𝑃𝑅becomes a random variable

In most common shadowing models this variable is

assumed to be log-normal, i.e. for 𝜉 > 0

𝑝 𝜉 =𝑛

𝜎𝜉𝑑𝐵𝑒𝑥𝑝

−log10 𝜉−𝜇𝜉𝑑𝐵

2

𝜎2𝑑𝐵

Shadowing can be combined with free space models

using the following modified expression where 𝜉𝑑𝐵 is a Gaussian zero mean random variable


𝑑0− 𝜉𝑑𝐵

54

Multipath Statistical model

Physical aspects

The rain and shadowing models are statistical models

relatively simple that act mainly by modeling additional

losses as a random variable

However, the presence of objects in space that interfere

with e.m.waves is related with more complex phenomena

like Reflection and scattering.

Due to such phenomena propagation does not follow

rectilinear trajectories and can follow multiple paths In

general

If there is region with different refraction index, the e.m. radius

turns to zone has higher index (REFLECTION or REFRACTION);

If the e.m. wave passes through irregular structures, it can suffer

absorption and re-irradiation phenomena in all directions

(SCATTERING).55

Atmosphere and multipath The atmosphere composition determines different zones with different refraction

index and scattering properties can be defined

TROPOSPHERE is the lower layer of the atmosphere, It is extended up to an altitude of about 20 km.

Refraction index can be variable, in particular it decreases with altitude.

Scattering phenomena are possible for frequencies between 100 MHz and 10 GHz.

IONOSPHERE is placed to an altitude from 80 km to 1000km;

the high concentration of electrons and free ions produce absorption and reflection phenomena for frequencies between 5 and 30 MHz.

Scattering phenomena are possible for frequencies from 35 to 50MHz, due to the presence of meteorites. Otherwise such frequencies tend to penetrate ionosphere.

Ionosphere can be furtherly divided into the following layers:

Layer D (50-90km), absorbing (daylight hours only);

Layer E (110km), (night time only) reflecting and quite stable;

Layer F1 (220km), (daylight hours) reflecting and quite stable;

Layer F2 (300km), (daylight hours) reflecting and quite stable;

Layer F (night time merging F1 and F2) sporadic, reflecting.

56

Atmosphere and multipath Propagation of skywaves in atmosphere vary

depending on transmitted signal carrier frequencies

value.The following situations happen:

For carrier frequencies between some tens of KHz to a few

MHz, propagation is by ground wave, with limited delivery

capacity.

Between some MHz up to some tens of MHz, radio waves

can extend above the earth curvature as propagation can

happen in different ways, i.e.

• by direct LOS beam (the delivery capacity is equal

to optics),

• by subsequent ionospheric reflections between the

E and F layers and on the earthly surface,

• by IONOSPHERIC scattering.

57

Atmosphere and multipath

Example of non LOS

propagation for ionospheric

reflections at frequencies

between some MHz and

some tens of MHz.

58

Atmosphere and multipath

For values larger than some tens of MHz,

propagation is

by direct beam so between two point in visibility,

by TROPOSPHERIC SCATTERING.

by tropospheric reflection phenomena due to variations of

the refractive index.

So when a modulated signal generated by the

transmitting antenna is above few Mhz, there can

exist infinite path between the transmitting and the

receiving antennas

59

Multiple paths Consequent propagation abnormality, i.e. an

observed signal at the receiver including variations

not forecasted by free space and other previous

models, requires a new more complex model to be

defined where the rays connecting the antennas:

are multiple (MULTIPLE PATHS, called MULTIPLE

CHANNEL) due to reflections or scattering

(TROPOSPHERIC SCATTERING or IONOSPHERIC).

are described in a statistical way as they are unknown,

time variant and not stable during the connection time

60

Multiple paths

During the basic telecommunication courses, is beenconsidered the radio AWGN channel as Additive,

Gaussian,

with limited bandwidth

This approximation can be accepted only in few particularcases.

Actually the propagations rules of the signals pass through thewireless channels are quite far from a characterization of thechannel as a AWGN channel.

The radio channel is characterized by multipath propagation,that produces fading phenomena of the received signal.

The radio channel distortion wrt the free loss and shadowingmodels, due to reflection and scattering phenomena is calledmultipath fading

61

Time-varying impulse response

The AWGN channel is characterized by a time-invariant impulseresponse, in the case of the Nyquist condition is respected: thesignaling rate is less than the double of the channel bandwidth.

Instead a multipath channel is characterized by time-varyingimpulse response.

When a very narrow impulse is transmitted (ideally a Dirac delta),the channel response will typically be a impulse train, spaced oneto the other by time interval called time spread (t).

Impulse amplitudes will be also distorted wrt expected free spaceloss.

When transmission experiments are repeated for different timesboth the time spread and the loss will assume differentcharacteristics, as shown in the next slide.

Such variability requires a probabilistic statistical model to bedescribed.

62

Example:

impulse response in a multipath

non stationary channel

63

Multipath channel response

A generic band pass signal 𝑠 𝑡 can be representedin terms of its complex envelope 𝑠𝑙 𝑡 (equivalentlowpass) as follows:

If multipath propagation is considered, each path is

assigned a different propagation delay and loss

factor. Band pass received signal r 𝑡 can beexpressed as follows:

tfjl cetsts 2)(Re)( (i)

n

nn ttsttr )()()( t (ii)

64


Replacing (i) in (ii), the following equation is given:

Equivalent low pass complex envelope of the

received signal, 𝒓𝒍 𝒕 , can be written as :

rl(t) can be modeled as the response of an equivalent

low pass channel at the equivalent low pass signal

sl(t).

tfj

n

nl

tfj

ncnc ettsettr

t t 2)(2 )()(Re)( (iii)

n

nl

tfj

nl ttsettrnc )()()(

)(2 t t(iv)

65


The equivalent low pass channel model is also a

complex valued function that describes how the in

phase and quadrature components of the signal are

modified by the channel.

Such a complex function 𝑐 𝜏, 𝑡 represents the time-varying impulse response of a channel characterized

by the presence of a finite number of multiple paths

and it can be written as:

n

n

tfj

n tettcnc )()();(

)(2 ttt t (v)

66


By convolving the impulse response in (v) with the

low pass equivalent 𝒔𝒍 𝒕 one can obtain againequation (iv) that is the low pass equivalent of the

received signal, 𝒓𝒍 𝒕 , so confirming that the chosenimpulse response provides a meaningful linear

operator in the transformed lowpass domain:

𝑟𝑙 𝑡 = 𝑐 𝜏, 𝑡 𝑠𝑙 𝑡 − 𝜏 𝑑𝑡

(vii)

67


For some channel types, e.g., for tropospheric channel, thenumber of paths can be modeled as infinite, so a continuous(and not discrete as for finite paths) representation of thechannel response is more appropriate, i.e.

𝑐 𝜏, 𝑡 = ∞−+∞

𝛼 𝜉, 𝑡 𝑒−𝑗𝜑 𝜉,𝑡 𝛿 𝜏 − 𝜉 𝑑𝜉 = 𝛼 𝜏, 𝑡 𝑒−𝑗𝜑 𝜏,𝑡

Where α(t; t) and 𝜑 𝜏, 𝑡 are the loss and the delay of the tmultipath signal component at to time t.

Equation (vi) represents the equivalent low pass impulseresponse for continuous type multipath channel.

(v) represents the equivalent low pass impulse response for amultipath channel characterized by a discrete number of paths

(vi)

68

Narrowband Statistical multipath

channel representation (1/4)

Let us consider the case of multipath channel withdiscrete paths. Let us suppose that signal transmittedis a sinusoid modulated at carrier fc (frequency):

In this case, the signal received is reduced tofollowing equation:

Rewriting the signal received in vector form, it’spossible to represent it as sum of phasors,characterized by time-varying amplitude and phase.

ttsl 1)( (ix)

)()()()()(2

n n n

tj

n

tfj

lnnc etettr

qt 2ˆ)( (t)ft ncn tq (x)

69



Im

Re

1

2

q2

q1

rl(t)

Vector representation of the multipath channel characterized by two

paths.

70


channel representation (3/4) If the propagation medium is stable, n(t) value will suffer

of small oscillations over time.

However, qn(t) can vary by a factor of 2 if tn(t) varies by afactor of 1/fc, that generally is very small number.

Therefore, qn(t) is a very sensible parameter with respectto small oscillations of the time-varying multipathchannel.

As the propagation delay tn(t), associated with each pathvaries in a deterministically unpredictable way, thanwe consider this type of propagation as a randomprocess.

Consequently the received signal rl(t) too is modeled asa random process.

71



If the number of the paths is very high (as we canreasonably assume for real radio channels), it’s possibleto use the central limit theorem

Accordingly we can suppose rl(t) as complex valuesGaussian random process.

For this the multipath channel impulse response c(t ; t)can be modeled as a complex values Gaussian randomprocess.

Multipath propagation of real radio channels, representedin equation (x) can be translated to physical layer in afading phenomena of the received signal.

This phenomena is known as multipath fading.

72

Narrowband Multipath fading:

vector graphic representation

Im

Re

1

2

q2q1

rl(t)

instant t1

Im

Re

q1q2

2

instant t+ t

rl(t+ t)

FADING (received signal

suffers a very high loss

than previous instant)

73

Narrowband Multipath fading effects

on communication quality

From the previous slide, one can observe that multipathfading causes many temporal fluctuations of thereceived signal power.

At subsequent time instants the signal sometime can be: Amplified (vector constructive effect of the multipath)

Attenuated (vector destructive effect of the multipath),

all this caused by the time-varying phase shift value qn(t).

This effect can be observed, at macroscopic level (andanalogical level), for example when the signal is receivedfrom a common car radio.

In digital system transmission, one will observesignificant increase of BER at of “subsidence” of signalfor multipath fading, for this it can be considered as a self-interference.

74

Rayleigh fading When channel multipath impulse response c(t ; t) is a complex

Gaussian random process with average equal to 0, the

envelope |c(t ; t)| at every instant t is distributed as Rayleigh, forthis we can call it as Rayleigh fading.

Ω is the average received power based on free space loss andshadowing. Phase is distributed uniformly.

altrimenti 0)(

0 2

)(

2

rf

rer

rf

R

r

R

)(ˆ2RE

Example of Rayleigh

distribution with variance

equal to unit

75

Rice fading When channel multipath impulse response c(t ; t) is given as

complex Gaussian random process with average different from

to 0, the envelope |c(t ; t)| at every instant t is distributed as

Rice, for this we can call it as Rayleigh fading, or multipath

Ricean channel.

altrimenti 0)(

0 I )(2'0

22

2

22

rf

rrs

er

rf

R

sr

R

2

2

2

1ˆ XXr

11)( mXE

22)( mXE

2

21 )var()var( XX

2

2

2

1ˆ mms

I0(.) = Bessel function

zero order Example of Rice distribution with

m1 = m2 = 1 and variance = 1

76

Nakagami fading

Signal received envelope from multipath fading

channel can be modeled statistically with another

distribution: m distribution of Nakagami.

altrimenti 0)(

0 )(

2)(

2

12

rf

rerm

mrf

R

mr

m

m

R

)(ˆ2RE

21

ˆ22

2

m

REm

m is called fading

figure

m =2

m =1 (Rayleigh)

m =0.5

Example of Nakagami distribution

with = 1

77

Use of different fading models (1/3)

The model of the Rayileigh fading is generally used in theradio-channel where It doesn’t exist the line of sight (LOS) signal component

signal is received only through its delayed and phase shifted replicas.

This situation mostly occurs in the case of radio channel,where the antennas are placed at a height lower than reflectionmeans and dispersion of the signal (tree, building).

The model of the Rice fading is useful when: There exists a LOS signal component in the received signal.

The received signal also contains delayed and phase shifted replicasgenerated by secondary reflections of direct component. Suchreplicas are pure self-interference signals.

𝐾 =𝑠

2𝜎2is the ratio between the LOS and the not LOS component

78

Use of different fading model (2/3)

Example of airplane-control tower link, by a direct linkand only one multipath component generated by areflection, it is received with t0 than LOS component.

This multipath channel type can be effectively representedby Rice model.

))(()();( 0 tttc tttt Channel impulse

response

Multipath component

(modelling by Rayleigh)

Transfer Function )(2 0)();(tfj

ettfCt

Direct path (specular

fixing component)

79

Use of different fading model (3/3)

The model that used the m distribution of Nakagami,

that as particular case includes the Rayleigh model, is

effectively used in urban radio mobile channel.

This model is parameterized in dual way: than and m,for this is guaranteed more flexibility and accuracy of

channel statistical representation.

In the literature is shown as the Nakagami model gives

more efficient statistical representation for multipath

fading, in particular for urban radio mobile environment,

where the multipath fading is very important phenomena.

80

Wideband Fading Models A digital wideband signal can be here considered as a signal whose

bandwidth W is due to the presence of time pulses whose duration T issignificantly spread in time by the channel

In such cases, some mathematical simplifications used for narrowbandmodels do not hold.

However, the time varying impulse response of the channel, when thenumber of multipath components is high can still be represented as aGaussian complex random process due to Central Limit Theorem

Rayleigh fading models can still hold but are not sufficient to describeeffects.

The phase can still be assumed uniformly distributed

So a channel model can be designed by analyzing properties of aGaussian , 𝑐 𝜏, 𝑡 with zero mean components.

So LOS effects cannot be modeled in his way (zero mean assumptionwould not be compatible.

Autocorrelation of 𝑐 𝜏, 𝑡 can be analyzed

81

Autocorrelation function for multipath

channel and power spectrum (1/2)

Autocorrelation function of the multipath channel is

defined (assuming random process 𝑐 𝜏, 𝑡 as widesense stationary WSS) as follows:

In many radio channels, loss and phase shift

associated with a path characterized by time delay

(t1) are uncorrelated with respect to loss and phase

shift associated with a path with (t2). This fading is

called Uncorrelated Scattering – US.

In WSSUS channels it is possible to write as follows:

ttctcEtc ;;2

1ˆ);,( 21

*

21 ttttf(xi)

21121* ;;;2

1tttftt tttctcE c (xii)

82

Autocorrelation function for multipath

channel and power spectrum (2/2) If t = 0 and 𝜏 = 𝜏1 = 𝜏2, then the autocorrelation

depends only on path 𝜏, i.e.:

This can be viewed as the average received power in

function of time t.

fc(t) is the multipath intensity profile or delay

power spectrum. A typical profile is shown in the

figure at below:

tftf cc 0;

The range of t for which

fc(t) isn’t null is called

multipath spread of the

channel and it’s

indicated as Tm..

83

Frequency representation for

multipath channel (1/4)

Now will be discussed multipath channel in frequencydomain. Using the Fourier transform applied to di c(t;t),will be obtained channel time-varying transfer functionC(f;t), where the frequency is variable. In this way:

If c(t;t) is modelled as complex Gaussian random process,the C(f;t) has the same statistical modelling. Assuming thechannel as WSS and for uncorrelated scattering (thechannel is defined by the abbreviation WSSUS) it’spossible to demonstrate that (cfr Proakis, Ch. 14, pp.763):

tt t detctfC fj

2) ;();( (xiii)

ttfttfCtfCEtff cCC ; ;;;2

1);,( 21

*

21 tf

(xiv)84



In (xiv) is been placed f = f2 - f1 and we can note thatC(f,t) is the Fourier transform of the function fc(t;t),it gives output average power of the channel as function ofthe delay t and of the difference t between two observingtime instant (in t = 0 is multipath intensity profile).

The function C(f,t), is called channel correlationfunction spaced in time-frequency, and it can measuretransmitting a couple of sinusoids spaced between themof f and doing an operation of cross-correlation betweenthese two signal separately received with a delay t.

Assuming to take t=0 in (xiv). In this way we obtain therelation follows:

ttf t def fjcC

2)()( (xv)

85



The physical means of relation in (xv) is graphically

represented in figure follows:

C(f) is a cross-correlation function in frequency

variable, it gives a measure of frequency response

coherence of multipath channel.

86



The function C(f), is the result of Fourier transform ofthe multipath intensity profile function fc(t), for this it willassume significant values at the interval of the space f,this frequency interval is reverse-proportional to multipathspread Tm. This interval is called coherence bandwidthof the channel is defined as follows:

In practice the coherence bandwidth is frequency intervalwithin which the multipath channel effects, at differentfrequencies of the spectrum, are correlate betweenthem a so can be considered as similar.

Two sinusoids transmitted and frequency spacedbetween them of an interval higher than (f)c suffer adifferent “processing” by the channel.

m

cT

f1

87

Frequency selectivity of multipath

channel

If coherence bandwidth (f )c is narrower than

bandwidth of the signal transmitted, the channel is

called frequency selective (frequency selective

fading). In this case the signal transmitted is

subjected to different and multiple distortions

cased by multipath fading.

If coherence bandwidth is larger than bandwidth of

the signal transmitted, the channel is called not

frequency selective (frequency non-selective fading

or flat fading). In this case the signal transmitted is

subjected to uniform distortion in its bandwidth

88

Temporal representation of


Time-varying effect of multipath channel can be observed

as spread spectrum and Doppler shift spectrum, they

work on the sinusoidal tone transmitted (spectrum row).

To create a relation between Doppler effect and time-

varying of the multipath channel, is necessary to define

the following function, that is Fourier transform, on the

time variable t, of the channel time-frequency spaced

cross-correlation function C(f;t):

We take f = 0 and the relation (xvi) becomes:

tdetffS tjCC

2);(ˆ);( (xvi)

tdetSS tjCcC

2)()();0( (xvii)

89



The Sc() is a power spectrum function, an it given

signal intensity in function of Doppler frequency .

For this Sc() is called Doppler power spectrum. In

the figure at below is highlighted the graphical form

and the physically meaning of the relation in (xvii).

90



From the relation (xvi) is shown that, if the channel is time-invariant, C(t)=1 and the function Sc() becomes aDirac delta. In this case there aren’t time variations inthe channel and there aren’t spread spectrum in thetransmission of one sinusoidal tone.

The range of for which Sc() has non zero values iscalled channel Doppler-spread Bd .

Sc() is put in relation with, C(t) from Fourier transform,the inverse of Bd is a measure of channel coherencetime, that is time interval of observation during whichchannel effects on signal transmitted are correlatedbetween them and can be considered as similar.

d

cB

t1

Multipath channel is characterize by slowly

time variations and it has high coherence

time which corresponds to low Doppler

spread (slow fading channel).

91

Scattering function of multipath

channel (1/2)

In the first it’s been established a relation based onFourier, transform between the functions C(f;t) andfc(t;t), with the variables (t,f) which are interested, thena same relation between C(f;t) and SC(f; ), withvariables (t,) which are interested.

There are two Fourier relations to defined too, betweenfc(t;t) and SC(f;) to close le loop.

For this target we define a new function: Fouriertransform of fc(t;t) on variable t:

We can observe that exists the following relation:

tdetS tjc

tft 2);(ˆ);( (xviii)

fdefSS fjC

tt 2);();( (xix)

92

Scattering function of multipath

channel (2/2)

Moreover S(t;) can be put on relation with C(f;t)

by two variables Fourier transform:

The function S(t;) is called channel scattering

function. It gives a measure of channel output

average power, in function of delay time t and of the

Doppler frequency .Example of scattering

function for

tropospheric channel

ftddeetfS fjtjC

tt 22);();( (xx)

93

Close loop of Fourier relations

among the function considered

tc ;tfTime-varying delay power

spectrum

tfC ;

FT

Spaced-frequency correlation

function

FT

;fSC Spaced frequency Doppler

power spectrum

t ;SScattering function

IFT

IFT

94

Choice of multipath channel model depending

on modulated signal properties (1/4)

Channel model to be used depends on the

characteristics of the signal transmitted.

In general, the equivalent low pass 𝑟𝑙 𝑡 of thereceived signal can be written both using the

equivalent low pass sl(t) of the transmitted signal

and its spectrum Sl(f), as follows:

Using convolution in the time domain :

Using multiplication and IFT from the frequency domain :

ttt dtstctr ll )(;)(

dfefStfCtr ftjll2)(;)(

(xxi)

(xxii)

95

Choice of multipath channel model depending on

modulated signal properties (2/4)

Let us transmit a short duration amplitude or phase modulated

pulse with a rate equal to 1/T, where T is signaling rate.

Modulated signal bandwidth 𝑊 ≈1

𝑇

From (xxii), channel distorts signal spectrum Sl(f).

If W is larger than channel coherence bandwidth, signal

spectrum Sl(f) will suffer different loss and different phase

shift through its bandwidth.

So if 𝑊 > ∆𝑓 𝑐 the channel is frequency selective andfrequency fading can be observed.

Frequency

selective

channel

96



Distortion is also time variant due to time fluctuations

of channel frequency response C(f; t).

Time varying amplitude distortion of the received

signal occurs as a combined result of multipath and

time variance.

This distortion is called temporal fading.

Temporal fading of

signal, measured

experimentally at New

York.

97



Frequency fading and temporal fading are different

distortion effects.

Frequency selectivity is directly connected with

multipath spread and to channel coherence

bandwidth.

Temporal fading is connected to both multipath

effects and temporal variations of the global channel

frequency response, i.e. also with channel

coherence time and Doppler spread.

Frequency selectivity mc Tf ,

Temporal fading dc Bt ,

98

Frequency Not-Selective channel

(1/2) Channel effects on the transmitted signal sl(t) depend on

the bandwidth and on signal rate (which depend in turnon the chosen modulation).

For example, if a signal rate T is chosen as T >>Tm,multipath will not introduce a significant intersymbolicinterference due to replicas of signal transmitted.

If T >>Tm, then:

i.e. bandwidth W of transmitted signal is much lower thanchannel coherence bandwidth.

This is the definition of frequency not-selective channel.

c

m

fT

W 1

99

Frequency Not-Selective channel

(2/2)

The transmitted signal spectrum Sl(f) in such case experiences the same loss andthe same phase shift for all frequencies in W. As the model is expressed in lowpassfrequency domain, frequencies of the lowpass equivalent transmitted signal willinclude also f = 0. So:

As a consequence (xxii) can be rewritten here as follows:

So for frequency not selective channels it can be said that multipath components ofreceived signal replicas cannot be solved, because in this case the received signalcannot be written as a weighted sum of signal replicas (such replicas are relativelytoo close each other as concentrated in a small time range, as Tm is low).

As a consequence, multipath structure (i.e. each signal replica) cannot be estimatedby observing the received signal and the receiver cannot gain any benefit fromknowledge on the loss and phase shift of individual paths

tCtfC ;0;

)(;0)(;0)( 2 tstCdfefStCtr lftj

ll

100

Slow fading frequency not-selective

channels (1/3) Transfer function of frequency not-selective channel

can be expressed as follows:

The speed of temporal variation of fading in frequencynot-selective channels is related to time-spacedcorrelation function C(t) or by Doppler powerspectrum Sc() namely on its parameter values (t)c e Bd.

)(;0 tjettC f

Envelope: random

process with Rayleigh

distribution

Phase: random

process with uniform

distribution between

(-,)

101


channels (2/3)

A slowly fading channel occurs when

𝑇 ≪ ∆𝑡 𝑐 =1

𝐵𝑑

i.e when signal rate is lower than time coherence. In such case channelloss and phase shift are stationary for all duration of the transmittedsymbol.

As 𝑊 ≈1

𝑇and W


channels (3/3)

In the table, coherence time, Doppler spread andspread factor are provided for some radio underspreadchannels.

For such channels, a modulated signal sl(t) can be chosenso that frequency not-selective and slowly fadingeffects appear on the received signal.

Such a chammel is quite similar to AWGN channel and inany case characterized by constant and measurablefading, so this channel can be easily equalized.

103

Frequency Selective channel

There are many practical interesting cases in which not-selective andslowly fading transmission is not applicable but for narrow band, low ratesignal modulation

In multimedia transmission (wide band transmission), high bit rate isnecessary;

Moreover, (spread spectrum) techniques there exist that increase signalbandwidth beyond the minimum (used often in narrowband modulations)there exist motivated by several reasons.

As a consequence, in such cases, W>>(f)c.

Different frequencies of the same signal can be managed independently(statistically) by the channel, generating so called frequency selectivity

In this case however, T

Diversity concept (1/3)

Availability of multiple replicas of a signal that can be solved at thereceived is an example of the diversity concept

Being the channel statistically characterized, there can existsignal frequencies affected by heavy fading at the the receiver,in correspondence of fade depths.

However each signal spectrum component separated more thanthe coherence bandwidth can be affected by a diverse(constructive or destructive) fading

So in a wideband frequency selective channel, the receiver can relyon a higher probability that at least part of the signal informationcontent is received with a sufficiently high Signal-to-noise ratio,based on the consideration that the sub-channels affecting thewideband signal separated by more than te coherence bandwidthare characterized by interdependent fading.

The probability that all the replicas received are affected by fadingis extremely reduced

105


Diversity is a more general concept and can be alsoinduced at the source during modulation. Forexample:

Frequency diversity: is obtained by the sameinformation transmits on different carriers L, spacedof one or more coherence bandwidth (f)c of thechannel.

Time diversity: is obtained by transmission of thesignal into L subsequent temporal slots, every onespaced of one or more coherence time (t)c of thechannel.

106


Spatial diversity: the first two methods aren’t very

effective, because they generate waste of bandwidth.

Multiple antennas at transmiytter and at receiver

(MIMO) are nowadays used to allow to generate (and

receive) multiple replicas of the same signals in

points separated by an appropriate distance.

Generally it is possible to space antennas in an array

one from the other at least by 10 wavelength, to

allow them to receive information that can be

considered to come from uncorrelated paths.

107

Other techniques based on diversity

concept Examples of diversity generated by modulations can be: Frequency-hopping (FH): the signal transmitted hops from one carrier

frequency to another, according to a predefined temporal sequence,within a given larger bandwidth. At the receiver it will be necessary tosynchronize on the hop sequence to demodulate the signal. This concept isused in Spread Spectrum techniques based on Frequency Hopping(FH/SS). The target could be to minimize the number of the hops that aredistorted by the selective sequence channel.

Signal transmission on multiple carriers sufficiently spaced betweenthem including orthogonality properties to minimize self-interference :this concept is used in OFDM e DTM (where different symbols of thesame signal can be transmitted on different carriers) and MC-CDMA(where the same symbol is transmitted on different carriers). Theorthogonal spacing between different carriers (equal to k/T, k = 0,1,..,N),allows high efficient to information recovery, robustness wrt multipathfading and easy implementation (“full digital” with architecture based onFFT realized by DSP technology”).

108

Wideband channel model (1/6)

The wideband techniques are characterized bytransmission of a signal sl(t) with a bandwidth muchlarger than coherence bandwidth of the channel, i.e.W>> (f)c.

A receiver observing a wideband channel can be able toresolve different multipath signal components, and tobetter recover signal information if a sufficiently highnumber of paths is characterized by independentconstructive fading.

The number of resolvable paths L in a multipathchannel is given, by definition, Tm/T = TmW. Because Tm isinversely proportional to coherence channel bandwidth, :

c

f

WL

109

Wideband channel model(2/6)

Let us assume the channel is a slowly fading

channel, so that T


The signal sl(t) is limited bandwidth, so that the

sampling theorem allows following representation:

The Fourier transform of the signal sl(t) is :

The received signal can ben represented as follows

(see next slide):

WntWsincW

nsts

n

ll

)(

otherwise 0

2 1

)(2

WfeW

ns

WfSWfnj

n

ll

(xxiii)

(xxiv)

111


Where c(t;t) is the impulse response of the time-

varying channel. (xxv) has the form of a convolution

sum, and can be written as follows:

Where the time-varying path coefficients are:

tW

ntc

W

ns

WdfetfC

W

ns

Wtr

n

l

Wntj

n

ll ;1

;1

)( /2 (xxv)

n

ll tW

nc

W

nts

Wtr ;

1)( (xxvi)

t

W

nc

Wtcn ;

1ˆ)( (xxvii)

112

Wideband channel model (5/6) The final input output relation can be written where

resolvable channel components can be identified:

From (xxviii) the frequency selective multipath

channel can be represented as a temporal delay line

with a set of L taps, each delaying the signal of a time

Dt=1/W. Weight coefficients {cn(t)} attenuate and

delay independently signal replicas on each path

W

ntstctr l

n

nl )()((xxviii)

113

Wideband channel model(6/6)

So if the equivalent low pass has bandwidth equal to W, where W >> (f)c one canobtain a profile resolution of the multipath for the received signal with resolutionequal to 1/W.

The delay line does not have to be composed by a infinite number of taps, but can bebe truncated at L = [TmW]+1 tap. Paths beyond thtat time can be considered not tohave sufficient power to be solved.

The received signal (except channel noise) can be written as:

Time-varying channel coefficients {cn(t)} are stationary random process withcomplex values,.

Under uncorrelated scattering hypothesis, {cn(t)} coefficients are mutuallyuncorrelated.

For example in case of Rayleigh fading, the norm of {cn(t)} could be distributed asRayleigh, while the phase can be considered uniformly distributed between (-,).

W

ntstctr l

L

n

nl

1

)()( (xxix)

114

Narrow band signal format for transmission

on selective frequency channels

A typical wideband signal used to realize the situationW>>(f)c is Direct Sequences Spread Spectrum (DS/SS).

This signal is obtained by multiplying the information bit flowwith binary pseudo-random signal, characterized by signal ratemuch high than original information signal.

To transmit the information on very large bandwidth by DS/SStechniques, in way to oppose the degradations introduce bymultipath fading, is a typical approach common used by manystandard for wireless digital communications:

IEEE 802.11 for data transmission on WLAN local network;

IS-95 USA standard for mobile phone;

UMTS future European standard for radio-mobile communication;

Satellite System low orbit GLOBALSTAR for mobile phone.

115

Optimum receiver for selective

frequency channels (1/4)

Let us consider the optimal receiver for digitally modulated signals on afrequency selective channel. Such a channel can be modeled as a delayline with time-varying, statistically independent weights on each branch, asshown before.

The received signal is composed by L replicas of the original transmittedsignal. The larger the transmission bandwidth W, the higher theprobability to have at least a few not-distorted replica characterized by notself destructive interference, at the receiver. Such replicas could be usedto extract the information transmitted.

Let us consider a binary signal (that could be modulated by a BPSKmodulation). The related waveforms for the two base-signals sl1(t) and sl2(t)representing the binary digits, can be selected as antipodal.

Their duration is originally such that T>>Tm, It could be possible to use thechannel as not selective.

However, here we will consider what happens if one uses a spreadspectrum DS/SS modulation based on a narrowband BPSK modulation,to increase the bandwidth of the modulated signal W in such a way that themodulated channel becomes frequency selective.

116



When W>> (f)c, the received signal may be written

as follows:

2,1 ,0

)()()()/()()(

1

iTt

tztvtzWktstctr

L

k

ilikl(xxx)

sl1(t)

t0 T

sl2(t)

t0 T

Antipodal Signals

(BPSK/SS)

AWGN

Spreading

sequence

PN

117



Let channel coefficients [cn(t)] be all known, or let they can be estimated atthe receiver (in the case of slowly fading channel low complexityequalization methods can be used).

The matched filter is the optimal receiver, where the two pseudo randomsignals v1(t) e v2(t) are the waveforms to be matched ay the receiver.

The adaptive filter consists of parallel bank of integrators of the product ofthe received signal by the corresponding signal waveform, followed by asampler and by a decision module. Such module selects the filterproviding as output the largest correlation output.

However, as we can have multiple replicas solved at the receiver thematched filter can be applied on each of such replicas

The practical implementation of this receiver is obtained by a delay line,through which the received signal rl(t) goes.

After each tap, the delayed copy of the signal is correlated with ck(t)[slm(t)]*that represent a distorted copy of each expected waveform to be matched,where k = 1,2…L and m = 1,2.

The structure of this receiver, called RAKE receiver as it extracts energy ineach of the replicas is shown in the next slide.

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This delay line receiver tries to rake all the energy

carried to the receiver by the channel through

different paths, i.e.,replicas of the transmitted signal.

For this reason is defined as RAKE RECEIVER by

Price and Green who proposed it in 1958.

RAKE Receiver

(Canonical

Schema)

119

Notes on RAKE receiver

performances (1/2)

The capacity of RAKE to extract the replicas from signal, tocompensate fading effects, depends in first from bandwidth W.

A RAKE receiver reduces to a matched filter on the receivedsignal when the transmission bandwidth W is comparable withchannel coherence bandwidth.

Canonical RAKE receiver performances are conditional to reliabilityof channel coefficient estimates. Reliable estimates of coefficientscan be obtained by low complexity algorithm if fading is slowenough, e.g., if (t)c>100T.

If channel coefficients cannot be estimated accurately, or fading istoo fast (e.g., in urban radio mobile channel), it’s possible to usealternative RAKE structures.

In this case soft integrator outputs envelope of signal waveformsmatched integrators output is estimated before a binary decision istaken through a square law module (as shown in the next slide).

120

Notes on RAKE receiver

performances (2/2)

RAKE receiver

(alternative schema)

121

hertzian links - unige.it · 2020. 4. 26. · references [1] theodore s. rappaport, wireless...

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