2. Channel Characterization and...

Channel Characterization 1

2. Channel Characterization

and Measurement

Contents:

• Representation of impinging plane waves• Small scale/fast/short term fading• Shadowing, long-term fading• Large-scale/long-term fluctuations• Path loss• The mobile radio channel as a linear system• Time-variant linear systems

ure 2:


2. Channel Characterization

and Measurement

Contents (cont’d):

• Random time-variant linear systems• WSSUS channels• Direction dispersion• Space-variant linear systems• WSSUS space-variant channels• Wideband channel measurement methods• Dispersion at the BS in macrocellular environments


Geometrical configuration at the receive antenna location:

r1

r2

r3

ar ΩN( )

f Ω( )

r

Ωi

O

kN

φi

θi

MS

Ω1

k1

ki

ΩN

ar Ωi( )

E1

Ei

EN

y r( )

S

ar Ω1( )

Definitions/remarks:

• : origin of an arbitrary coor-

dinate system

• : sphere centred at with

unit radius

• : direction of the th wave

• : azimuth and coeleva-

tion

• : propaga-

tion vector

•We assume that plane waves

are incident in a neighborhood

of the MS

O

S O

Ωi i

φi θi,

ki2πλ

------ ar Ωi( )–=

N

Representation of impinging plane waves


Electric field of an incident plane wave [no modulation]:

Ei r( ) Ei ej– ki r⟨ ⟩

=

Complex electric field of at O

Ei ej2πλ

------ ar Ωi( ) r⟨ ⟩=

E h i, jϕh i,( )exp⋅

E v i, jϕv i,( )exp⋅e

j2πλ

------ ar Ωi( ) r⟨ ⟩=

Eh i,

Ev i,

ej2πλ

------ ar Ωi( ) r⟨ ⟩=



Electric field of an incident plane wave (cont’d):

About the location-dependency of the phase:

O

ki

ar Ωi( )

r

ar Ωi( ) r⟨ | ⟩

λ

ki2πλ

------ar Ωi( )–=

Effective displacement with respect

to the direction of propagation



Signal contributed by an incident plane wave [single polarization]:Contribution from the -th incident plane wave to the antenna output:

The (non-negative) proportionality factor depends on the antenna characteristics, among others onthe antenna gain.

Interpretation of :

Comment: Considering both polarization, we have instead.

i

yi r( ) bf Ωi( )Ei ej2πλ

------ ar Ωi( ) r⟨ ⟩=

⎧ ⎪ ⎨ ⎪ ⎩

hi

b

hi

Ei ki f Ωi( )hi f Ωi( )Ei∝yi r( ) hi e

j2πλ

------ ar Ωi( ) r⟨ ⟩= with

hi b f h Ωi( )Ei h, f v Ωi( )Ei v,+[ ]=



Signal contributed by an incident plane wave (cont’d):

Complex representation:

yi r( ) Ri ejϕi r( )

=

Real part

Imag

inar

y

yi r( )

ϕi r( )

Ri

ℜe xi r( ){ }

ℑm

y ir()

{}

par

t

ϕi r( ) ϕi f Ωi( )( )arg2πλ

------ ar Ωi( ) r⟨ ⟩+ +≡

Ri hi≡ b f Ωi( ) E i=⎩⎪⎨⎪⎧



Distance dependency for a rectilinear displacement:

yi d( ) yi r( )r dar Ω( )=

=

hi ej2πλ

------ αi( )dcos

=

r dar Ω( )=

O

ar Ωi( )

r dar Ω( )=

d αi( )cos

λ

ar Ω( )

αi

k i



Time dependency resulting from a MS movement with constantvelocity:

r r t( ) vt= =

hi ej2πv

λ--- αi( )tcos

=

hi ej2πνit

=

O

ar Ωi( )vt αi( )cos

λ

ar Ω( )

αi

k id d t( ) vt= =

yi t( ) yi d( )d vt=

=

v var Ω( )=



Time dependency resulting from a MS movement with constantvelocity:

Doppler frequency of the th wave:

Maximum Doppler frequency:

i

νi1

λ--- ar Ωi( ) v⟨ ⟩≡ v

λ--- αi( )cos=

νDvλ---=



Small-scale displacements:

1 12 2

3

3

4

45

5

BA

Some tens ofwavelengths

Small scale/fast/short-term fading


Total received signal:

where can be any of the variables , , and .

y z( ) yi z( )i 1=

N

∑=y1

y2

yNy

yN 1–

Re

Im

z r d t

f Ω1( )E1

Eiki

k1

f Ωi( )y z( ) yi z( )

i 1=

N

∑=

ENkN

f ΩN( )



Fluctuations of the received signal:

distance [m]d

Location A

y1

y2y3

y

y1

y2 y3

y

As a function of the location

Location B

yd()

[dB

]

λ 2⁄≈20 30 [dB]–≈

Re

Im

Re

Im



Rayleigh distribution:

•Theoretical justification:

- is large.

- The amplitudes 's are small and of the same order of magnitude.

- The phases 's are uniformly distributed over .

=> Central Limit Theorem (CLT):

are approximately independent Gaussianrandom variables with zero-mean and same variance .

y yii 1=

N

∑= y1

y2

yNy

yN 1–

Re

Im

N

yi

yiarg 0[ 2π ),

ℜe y{ } ℑm y{ },σ2



Rayleigh distribution (cont’d):

• Probability density:

• Applicability:

- Non-Line-Of-Sight (NLOS) situation with many scatterers- Diffuse scattering

Comment: is a complex circular-symmetric Gaussian random variable with zero-mean and

variance .

p y z( ) z

σ2------

z2

2σ2---------–

⎩ ⎭⎨ ⎬⎧ ⎫

exp≈ pRayleigh z( )= z 0≥( )

p y( )arg z( ) 1

2π------≈ z 0[ 2π ),∈( )

y

E y2[ ] 2σ2

=



Rice distribution

• Theoretical justification:

- is the electric field of a strong wave, e.g. a line-of-sight (LOS) wave or a strong

reflection.

- The sum is approximately zero-mean circular-symmetric Gaussian (CLT)

with variance .

- -factor:

y

y0

y1

y2

yN

yN 1–

y y0 yii 1=

N

∑+=

Re

Im

Δy

y0

Δy yii 1=

N∑≡

E Δy2[ ] 2σ2

=

K K y02

2σ2⁄≡



Rice distribution (cont’d)


• Applicability: LOS situation or situation with strong reflectors

p y z( ) z

σ2------

z2

y02

+

2σ2----------------------–

⎩ ⎭⎨ ⎬⎧ ⎫

exp I 0

z y0

2σ-----------⎝ ⎠

⎛ ⎞≈ pRice z( )= z 0≥( )

distance [m]d

λ 2⁄≈y0 dB

≈

is the Bessel function

0th order of the first kind:

I 0 z( )

I 0 z( ) 1

2π------ e

z w( )cosdw

π–

π

∫≡

yd() d

BSmall scale/fast/short-term fading


Rayleigh and Rice distributions

−25 −20 −15 −10 −5 0 5 10

10−4

10−3

10−2

10−1

100

K=0 (R

ayle

igh)

K=2

K=5

K=1

0

K=

30

z [dB]

P[|y

|≤ z]



Large-scale displacements:

1

1

3

45

5

A

Variation of theangle of incidence

Path 2 isobstructed

Transition dif-fraction-LOS

2Variation of the angle of

incidence and of thepropagation delay

Path to beobstructed

3

C

Some hundredsof wavelengths

Large-scale/long-term fluctuations


Received signal (no modulation):

Location A

y1

y2y3

yy1

y3

y

As a function of the location

Location Cy

d()

[dB

]

Mean value

distance [m]d

Re

Im

Re

Im



Lognormal distribution (cont’d):


is a lognormal random variable <=> is Gaussian

where-

-

Comment: Usually is given in dB, i.e.

y yln

p y z( ) pLognormal z( )≈ 1

2πςz----------------

zln μ–( )2

2ς2------------------------–

⎩ ⎭⎨ ⎬⎧ ⎫

exp= z 0≥( )

μ E yln[ ]≡

ς Var yln[ ]≡

yy dB 20 ylog 20 elog( ) yln= =



Lognormal distribution:

• Theoretical justification (one path only):

is sufficiently large so that the random variable is approxi-

mately Gaussian (CLT).

Tx

Rx

A1

A2

AM

E0

Er

Er Aii 1=

M∏( )E0=

Er

E0

------dB

Ai dBi 1=

M

∑=

…

M Ai dBi 1=

M

∑



Effects contributing to the large-scale/long-term fluctuations:• Shadowing/long-term fading

• Transitions where waves arise and disappear

• Variation of the number of impinging waves

• Variations of the propagation delays

• Variation of the incidence directions

Impact on communication systems:

+ Hand-over

+ Tracking, acquisition

+ Power control

+ Dynamic channel allocation



Measured space-variant delay SF in a transition NLOS-LOS [outdoor]:

Sourc

e: D

euts

che

Tel

ekom

Ag D

arm

stad

t,

AT

DM

A r

eport

: C

han

nel

Model

s Is

sue

2

NLO

S

->

LOS



Measured space-variant delay SF in a transition LOS->NLOS [indoor]:

0

5

10

15

20

25

0

50

100

150

200

0

0.5

1

Sourc

e: E

TH

Z, C

TL

Am

pli

tude

[lin

. re

lati

ve

to m

axim

um

]

Delay [ns]

Dista

nce

[m]



Dynamic evolution of the propagation constellation:

DistanceDelay

Transitions

Path loss &Shadowing

Variation ofthe delays

Range wherethe wave is

active

Pow

er



Distance-dependency of path loss:Resultant field:

A widely used path loss equation (See Section III: Prediction models)

where is the decay exponent.Typically, .

Free-space propagation: .

distance [m]d

yd()

[dB

] y⟨ ⟩ d( ): spacial averaging over

several wavelengths⟨ ⟩

Er2⟨ ⟩ d( )

E02

------------------------- K dn–⋅= L d( )

Er2⟨ ⟩ d( )

E02

-------------------------⎝ ⎠⎜ ⎟⎛ ⎞

dB

– 10n dlog 10 Klog–=≡⇔

nn 1.5 (corridor)…4 (densely built-up areas, multifloor propagation)=

n 2=

Path loss


Geometrical configuration at the antenna location:

The waves are now

modulated with the

signal .x t( )

r1

r2

r3

ar ΩN( )

f Ω( )

r

Ωi

O

kN

φi

θi

MS

Ω1

k1

ki

ΩN

ar Ωi( )

E1

Ei

EN

y r t,( )

S

ar Ω1( )

The mobile radio channel as a linear system


Electric field of the -th incident wave:

•at location :

•at an arbitrary location :

i

O

Ei 0 t,( ) Ei x t τi–( )⋅=

Complex electric field atof the unmodulated wave

O Modulatingsignal

Propagationdelay at O

r

Ei r t,( ) Ei ej2πλ

------ ar Ωi( ) r⟨ ⟩x t τi r( )–( )⋅= xxxxxxxxxx

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Electric field of the

unmodulated wave at rPropagation delay

at r



Electric field of the -th incident wave (cont’d):

About the location-dependency of the propagation delay:

i

O

ki

ar Ωi( )

r

ar Ωi( ) r⟨ | ⟩

λ

Effective displacement with respect

to the direction of propagation

τi r( ) τi

ar Ωi( ) r⟨ ⟩

c----------------------------–= : velocity of lightc



Received signal:

(✹)

We can rewrite the above sum as an integral according to

y r t,( ) b f h Ωi( )Ei h, f v Ωi( )Ei v,+[ ] ej2πλ

------ ar Ωi( ) r⟨ ⟩x t τi r( )–( )⋅

i 1=

N

∑=⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

hi

y r t,( ) g r τ;( ) x t τ–( ) τd⋅∫= (✥)[Convolution]



Space-variant delay spread function:The function

is called the space-variant delay spread function (SF) of the channel. It

coincides with the location-dependent impulse response of the channel.

gi r( )xxxxxxxxxxx

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

g r τ;( ) hi ej2πλ

------ ar Ωi( ) r⟨ ⟩δ τ τi r( )–( )⋅

i 1=

N

∑≡

h1h2

hihN

τ1 r( ) τ2 r( ) τi r( ) τN r( ) τ

g r τ;( ) Contribution of wave

to the delay SFi

… …



Linear time-invariant systems:

For fixed , (✹) describes the input-output relationship of a linear time-

invariant system with space-variant delay SF .

r

L g r τ;( )

O

MSBS

x t( ) y r t,( )

L g r τ;( )⁄

r

1

2i

N



Linear system:

Superposition principle⇒

Lx1 t( ) y1 t( )

Lx2 t( ) y2 t( )

La1x1 t( ) a2x2 t( )+ a1 y1 t( ) a2 y2 t( )+



Time-invariant system:

x t( )

t

x t t0–( )

tt0

y t( )

t

y t t0–( )

tt0

⇒

Lx t( ) y t( )

Lx t t0–( ) y t t0–( )



Interpreting (✹) as the input-output relation of a transversal filter:

x t( )

y r t,( ) gi r( ) x t τi r( )–( )⋅i 1=

N

∑=

∑

g1 r( )

τ1 r( )

g2 r( )

Δτ2 r( )

gi r( )

Δτi r( )

gN r( )

ΔτN r( )Δτ2 r( ) τ2 r( ) τ1 r( )–=

Δτi r( ) τi r( ) τi 1– r( )–=

ΔτN r( ) τN r( ) τN 1– r( )–=

x t τ1 r( )–( ) x t τ2 r( )–( ) x t τi r( )–( ) x t τN r( )–( )

……

… …



Discrete and continuous components of the delay SF:

Usually, embodies a continuous component as well:g r τ;( )

τ

gd r τ;( ) gc r τ;( )

τ

τ

g r τ;( )

+

Discrete component Continuous component



Interpreting (✥) as the input-output relation of a transversal filter:

General case:

τ

g r τ,( )

x t( )

y r t,( ) g r τ;( ) x t τ–( ) τd∫=

∫

τ

x t τ–( )

τd

g r τ;( ) τd

g r τ;( ) x t τ–( ) τd

Continuoustapped delay line



Estimated space-variant delay SF [indoor environment]:

0500

10001500

20002500

30003500

400450

500550

600650

700750

800

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Framenumber

Delay [ns]

Frame number n

Distance [cm]d 2n=Delay [ns]τ

Am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL



System functions of time-invariant linear systems:

Impulse response

Transfer function

x t( ) y t( )h τ( )

X f( ) Y f( )H f( )

X f( ) H f( )⋅ Y f( )=

h τ( ) x t τ–( ) τd⋅∫ y t( )=Time domain

Frequency domain

w t( ) W f( ) j2πft( ) fdexp∫=

W f( ) w t( ) j2πft–( ) tdexp∫=

f

t

f

t

f

t

f

t



Impulse response and transfer function of the channel:

Delay [ns]τ Frequency [MHz]f

Rel

ativ

e am

pli

tude

[lin

]

Rel

ativ

e am

pli

tude

[lin

]

Impulse response h τ( ) Transfer function H f( )

Sourc

e: E

TH

Z, C

TL



Delay dispersion:

Delay [ns]τ

Rel

ativ

e am

pli

tude

[lin

]

Impulse response h τ( )

Delay dispersion:Differently delayed (and

weighted) replicas of the

transmitted signal are

received.

Sourc

e: E

TH

Z, C

TL



Frequency selectivity:

Frequency [MHz]f

Rel

ativ

e am

pli

tude

[lin

]

Transfer function H f( )

Frequency selectivity:Distinct spectral components

of the transmitted signal are

differently affected by the

channel transfer function.

Sourc

e: E

TH

Z, C

TL



Duality delay dispersion <-> frequency selectivity:

Because and form a Fourier pair, i.e.

,

delay dispersion and frequency selectivity are dual expressions in the

-domain and -domain respectively of the same effect.

h τ( ) H f( )

h τ( ) H f( )ft

τ f



Small-scale representation:

If is sufficiently confined in a domain around the reference point we

can use the approximation in (✹):

with the space-variant delay SF

r O

τi r( ) τi

ar Ωi( ) r⟨ ⟩

c----------------------------– τi≈=

y r t,( ) hi ej2πλ

------ ar Ωi( ) r⟨ ⟩x t τi–( )⋅

i 1=

N

∑=

g r τ;( ) x t τ–( ) τd⋅∫=

g r τ;( ) hi ej2πλ

------ ar Ωi( ) r⟨ ⟩δ τ τi–( )⋅

i 1=

N

∑≡



Time fluctuations induced by the movement of the MS:

When the mobile station (MS) moves, the impulse response and the transfer

function of the channel fluctuate with time according to their spatial

dependency along the MS trajectory .

Movement with constant velocity:

r t( )

r r t( ) vt= =O

vt αi( )cos λ

αi

k i

r t( ) vt=

Time-variant linear systems


Time fluctuations induced by the movement of the MS (cont’d):

Received signal:

where

• is the Doppler frequency of the th wave.

•

y t( ) y r t,( ) r r t( ) vt= =≡ hi e

j2πλ

------ ar Ωi( ) r⟨ ⟩x t τi r( )–( )⋅

i 1=

N

∑r r t( ) vt= =

=

y t( ) hi ej2πνit x t τi t( )–( )⋅

i 1=

N

∑= (❈)

νivλ--- αi( )cos≡ i

τi t( ) τi

νi

f----t–=



Time-variant delay SF of the channel:

We can rewrite the above sum as an integral according to

where

is called the time-variant delay SF of the channel. By abuse of language it is

also called channel impulse response even though it is not a response to an

impulse. Indeed, the integral in (■) is not a convolution.

y t( ) g t τ;( ) x t τ–( ) τd⋅∫= (■)

g t τ;( ) g r t;( ) r r t( ) vt= =≡

hi ej2πνit δ t τi t( )–( )⋅

i 1=

N

∑=



Time-variant delay SF of the channel (cont’d):Example of an estimated time-variant delay SF:

Delay [ns]τ

Time [s]t

Tim

e [s]

t

Delay [ns]τ

Rel

ativ

e am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL



Short-term representation:

If is sufficiently confined in an interval around the time reference ,

we can use the approximation in (❈):

where

t t 0=

τi t( ) τi

νi

f----t– τi≈=

y t( ) hi ej2πνit x t τi–( )⋅

i 1=

N

∑=

g t τ;( ) x t τ–( ) τd⋅∫=

g t τ;( ) hi ej2πνit δ t τi–( )⋅

i 1=

N

∑≡



Other source leading to time fluctuations of the channel:•Mobile scatterers

-> Cars in outdoor environments

-> Persons in indoor environments

•Time-variant electrical properties of certain scatterers

-> fluorescent tubes:

0

0.02

0.04

0.06

0.08

450

500

550

600

000

0.2

0.4

0.6

0.8

1.0

Time Delay

Delay [ns]Time [s]

Am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL



Input-output relation ship of a time-variant linear system:

The time-variant delay SF entirely determines the time-variant

linear system .

y t( ) g t τ;( ) x t τ–( ) τd⋅∫=

g t τ;( )L

O

MSBS

x t( ) y t( )

L g t τ;( )⁄

r t( )



System functions of a time-variant linear system:

y t( ) H t f,( )X f( ) j2πft( ) fdexp∫= Y f( ) k ν f; ν–( )X f ν–( ) νd∫=

g t τ;( )

H t f,( )

h ν τ,( )

k ν f;( )

y t( ) g t τ;( ) x t τ–( ) τd⋅∫= y t( ) h ν τ,( )x t τ–( ) j2πνt( ) τd νdexp∫∫=

Doppler-delay spread functionTime-variant delay spread function

Time-variant transfer function Output Doppler spread function

τ

f

νt

νt

τ

f



System functions of a time-variant linear system (cont’d):

Comments:

•Any one of the four system functions entirely characterizes the time-

variant linear system .

•In this sense, the four system functions are fully equivalent.

•Effectively used system functions: , , and .

L

g t τ;( ) H t f,( ) h ν τ,( )



Estimated time-variant delay spread function:

Delay [ns]τ

Time [s]t

Tim

e [s]

t

Delay [ns]τ

Rel

ativ

e am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL



Estimated time-variant transfer function:

Time [s]t

Tim

e [s]

t

Frequency [MHz]f

Frequency [MHz]f

Sourc

e: E

TH

Z, C

TL

Rel

ativ

e am

pli

tude

[lin

]



Dispersion in Doppler frequency and in delay:

Received signal:

MSBS

x t( ) y t( ) hi ej2πνit x t τi–( )⋅

i 1=

N

∑=

h1 ej2πν1t

x t τ1–( )⋅

hN ej2πνN t

x t τN–( )⋅

hi ej2πνit x t τi–( )⋅

y t( ) hi ej2πνit x t τi–( )⋅

i 1=

N

∑=

h ν τ,( )x t τ–( ) j2πνt( ) τd νdexp∫∫=



Dispersion in Doppler frequency and in delay (cont’d):

where

h ν τ,( ) hiδ ν νi–( )δ τ τi–( )i 1=

N

∑=

τ1

h ν τ,( )

τ

ννD+

νD– ν1

νi

τi

νN

h1

h2

hihN

describes how

the channel spreads the

transmitted signal

jointly in Doppler fre-

quency and delay.

h ν τ,( )

τN



Dispersion in Doppler and in delay frequency (cont’d):

Delay [ns]τ

Doppler frequency [Hz]ν

Rel

ativ

e am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL

Estimated squared Doppler-delay spread function :h ν τ,( ) 2



Dispersion in Doppler frequency and in delay (cont’d):

Sourc

e: E

TH

Z, C

TL

Estimated normalized squared Doppler-delay spread function:

Delay [ns]τ

Doppler frequency [Hz]νR

elat

ive

ampli

tude

[lin

]

Normalized squared Doppler-delayspread function:

where

hn ν τ,( ) 2 1

q τ( )---------- h ν τ,( ) 2≡

q τ( ) h ν τ,( ) 2dν∫≡



Sources of randomness:

•The scattering environment is random:The features of the reflecting, diffracting and scattering objects [location, dimension, electromag-

netic properties] are random.

•Usually, for a given scattering environment, many waves are incident

which cannot be resolved:The resolution of the measurement/communications system is limited.

•The trajectory of the mobile station (MS) is at least partly random:

Consequence:

The system functions of the mobile radio channel are random [or stochas-

tic] processes.

Random time-variant linear systems


First- and second-moment characterization of random time-variantsystems:

Reasonable assumption:(Especially under the above assumption on the wave’s phase)

Expected value Correlation functions

E g t τ;( )[ ] E g∗ t1 τ1;( ) g t2 τ2;( )[ ]

E H t f,( )[ ] E H∗ t1 f 1,( ) H t2 f 2,( )[ ]

E h ν τ,( )[ ] E h∗ ν1 τ,1

( ) h ν2 τ,2

( )[ ]

E k ν f;( )[ ] E k∗ ν1 f;1

( ) k ν2 f;2

( )[ ]

E g t τ;( )[ ] E H t f,( )[ ] E h ν τ,( )[ ] E k ν f;( )[ ] 0= = = =

Random time-variant linear systems


Meaning of the acronym WSSUS:

WSSUS Wide-Sense-Stationary and Uncorrelated-Scattering

Applicability of the WSSUS assumption:

The WSSUS assumption is realistic to describe the short-term variations of

the radio channel.

Due to the rapid fluctuations of the phase of the electric field of the imping-

ing waves ( over one wavelength), the components contributed by two

distinct waves in the system functions can be reasonably assumed to be

uncorrelated.

≡

2π

WSSUS channels


Correlation functions [CF] of WSSUS systems:

E k∗ ν1 f 1;( ) k ν2 f 2;( )[ ] S ν1 f;2

f 1–( )δ ν2 ν1–( )=

E h∗ ν1 τ1,( ) s ν2 τ2,( )[ ] P ν1 τ1,( )δ ν2 ν1–( )δ τ2 τ1–( )=

E H∗ t1 f 1,( ) H t2 f 2,( )[ ] R t2 t1– f 2 f 1–,( )=

E g∗ t1 τ1;( ) g t2 τ2;( )[ ] Q t2 t1– τ1;( )δ τ2 τ1–( )=

WSSUS channels


Wideband time correlation function :Q Δt τ;( )

E g∗ t τ1;( ) g t Δt+ τ2;( )[ ] Q Δt τ1;( )δ τ2 τ1–( )=

For any the process is wide-sense-

stationary with CF .

the WSS property holds

τ1 g t τ1;( )

Q Δt τ1;( )

⇒

g t τ1;( )

τ

t

g t τ;( )

τ2

g t τ2;( )

t2g t2 τ;( )

τ1

g t1 τ;( )t1

If , the processes and

are uncorrelated.

The US property holds

τ1 τ2≠ g t τ1;( ) g t τ2;( )

⇒

WSSUS channels


Wideband time correlation function (cont’d):Q Δt τ;( )

Normalized wideband time correla-tion function:

where

Qn Δt τ;( ) 1

P τ( )-----------Q Δt τ;( )≡

P τ( ) Q 0 τ;( )≡ E g t τ;( ) 2[ ]=

Sourc

e: E

TH

Z, C

TL

Estimated normalized wideband time correlation

function:

Am

pli

tude

[lin

]

Time lag [s]Δt

Delay [ns]τ

WSSUS channels


Time-frequency correlation function :R Δt Δf,( )

T t f 2,( )

f 1 f

t

T t f,( )

f 2t1

T t1 f,( ) The 2D-parameter process

is WSS with CF .

H t f,( )R Δt Δf,( )

T t f 1,( )

t2

T t2 f,( )

E H∗ t f,( ) H t Δt+ f Δf+,( )[ ] R Δt Δf,( )=

WSSUS channels


Doppler-delay scattering function (or power spectrum) :

D

P ν τ,( )

E h∗ ν1 τ,1

( ) h ν2 τ2,( )[ ] P ν1 τ1,( )δ ν2 ν1–( )δ τ2 τ1–( )=

ν1ν

τ

h ν τ,( )

ν2

τ1

For any and ,τ ν

E h ν τ,( ) 2[ ] P ν τ,( )=

τ2

h ν1 τ1,( )

h ν2 τ,2

( )If and the random

variables and are

uncorrelated.

τ1 τ2≠ ν1 ν2≠

h ν1 τ1,( ) h ν2 τ2,( )

WSSUS channels


Doppler-delay scattering function (cont’d):P ν τ,( )

Delay [ns]τ

Doppler frequency [Hz]ν

Rel

ativ

e am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL

Estimated Doppler-delay scattering function

:P ν τ,( ) h ν τ,( )2

≡

WSSUS channels


Doppler-delay scattering function (cont’d):P ν τ,( )

Sourc

e: E

TH

Z, C

TL

Estimated normalized Doppler-delay scattering

function:

Delay [ns]τ

Doppler frequency [Hz]νR

elat

ive

ampli

tude

[lin

]

Normalized Doppler-delay scatteringfunction:

where

Pn ν τ,( ) 1

P τ( )-----------P ν τ,( )≡

P τ( ) Q 0 τ;( )≡ E g t τ;( ) 2[ ]=

WSSUS channels


Doppler-delay scattering function (cont’d):

Comments:

•Total received power:

• is proportional to the distribution with respect to and of themean received power.In this sense, describes the dispersive behaviour in Doppler frequency and delay of thechannel.

•We define

P ν τ,( )

PR P ν τ,( ) νd τd∫[ ]PT=

P ν τ,( ) ν τ

P ν τ,( )

P P ν τ,( ) νd τd∫≡ PR PT⁄=

WSSUS channels


Fourier relationship between the channel characterizing functions:

S ν Δ; f( )R Δt Δf,( )

E H∗ t f,( ) H t Δt+ f Δf+,( )[ ] R Δt Δf,( )= E k∗ ν1 f;( ) k ν2 f; Δf+( )[ ] =

Q Δt τ;( ) P ν τ,( )

E g∗ t τ1;( ) g t Δt+ τ2;( )[ ] = E h∗ ν1 τ,1

( ) h ν2 τ2,( )[ ] =

Delay-Doppler-delay scattering functionWideband time correlation function

Time-frequency correlation function Doppler cross-power spectral density

Q Δt τ1;( )δ τ2 τ1–( )= P ν1 τ1,( )δ ν2 ν1–( )δ τ2 τ1–( )=

S ν1 Δ; f( )δ ν2 ν1–( )=

τ

Δf

νΔt

νΔt

τ

Δf

WSSUS channels


Delay dispersion -- frequency selectivity:

Delay scattering function:

describes the distribution w.r.t. of the

mean received power.

P τ( ) P ν τ,( ) νd∫≡

Q 0 τ;( ) E h t τ;( ) 2[ ]==

P τ( ) τ

P τ( )

τ

Delay dispersion

Frequency CF:

For any fixed , the processes is

WSS with CF

f

t

H t f,( ) [dB]

t1

H t1 f,( )[dB]

t2H t2 f,( )

[dB]

Frequency selectivity

t1 H t1 f,( )

R 0 Δf,( ) R Δf( ) Channel frequency CF≡

Delay dispersion Frequency selectivity⇔

P τ( ) R Δf( )τ ΔfDelay domain Frequency domain

WSSUS channels


Delay dispersion -- frequency selectivity (cont’d):Estimated delay scattering function (typical urban (TU)):

Exponential decaying delayscattering function:

P τ( )τστ------–⎝ ⎠

⎛ ⎞exp ; τ 0≥

0 ; τ 0<⎩⎪⎨⎪⎧

∝

-5 0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0

Relative Delay /� Ts

Po

we

r[d

B]

Aarhus,low Antenna position

Aarhus,High Antenna position

Stockholm

Exponential decaying function

T S 0.923 μs=( ) Source: AAU, CPK

WSSUS channels


Delay dispersion -- frequency selectivity (cont’d):

Delay [ns]τ

Rel

ativ

e am

pli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL

Estimated delay scattering and frequency correlation functions (indoor):

Am

pli

tude

[lin

]

Frequency lag [MHz]Δf

WSSUS channels


Delay spread -- coherence bandwidth:

Normalized delay scattering function:

Mean excess delay, delay spread:

is a measure of the

extent of .

Delay domain

Pn τ( ) 1

P---P τ( )≡ Pn τ( ) τd∫⇒ 1=( )

μτ τPn τ( ) τd∫≡ στ τ μτ–( )2Pn τ( ) τd∫≡,

Pn τ( )

τμτ

2στστ

Pn τ( )

Normalized frequency CF:

Coherence bandwidth at level :

is a measure of the width of

the main lobe of .

Frequency domain

Rn Δf( ) 1

P---R Δf( )≡ Rn 0( )⇒ 1=( )

c 0 1 ),[∈

Δf( )c min Δf 0> : Rn Δf( ) c={ }arg≡

1c

Δf( )c Δf

Rn Δf( )

Δf( )cRn Δf( )

Uncertainty relation:

στ Δf( )c⋅ 1

2π------ arccos c( )≥

WSSUS channels


Delay spread -- coherence bandwidth (cont’d):

Typical values for the delay spread:

Cell types

Cell dimension /

Base station height(with respect to the average heights of the

surrounding buildings)

Delay spread

Pico-cell 10-100 m / low, indoor 1 - 100 ns

Micro-cell 100-1000 m / about the same 10 - 1000 ns

Macro-cell 1-35km / high 0.1 - 10 sμ

WSSUS channels


Delay spread -- coherence bandwidth (cont’d):Empirical distribution of the estimated delay spreads (TU):

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aarhus,low antenna position

Aarhus,high antenna position

Stockholm

RMS Delays Spread (DS) [ s]�

Cu

mu

lative

dis

trib

utio

n

Sourc

e: A

AU

, C

PK

WSSUS channels



Meaning of the coherence bandwidth:

H t1 f 1,( )H t1 f 2,( )

H t1 f 3,( )f 1

f 2

f 3

H t1 f,( )

f

Δf( )0.9

Δf( )0.5

H t1 f 1,( ) H t1 f 2,( )≈

E H∗ t1 f 1,( )H t1 f 3,( )[ ] 0≈

WSSUS channels



Scatter plot of estimated versus estimated (indoor):Δf( )0.5 στ

The lower bound

is achieved if, and only ifis of the form

στ Δf( )0.5⋅ 1

6---=

Q τ( )

P τ( )

τμτ στ–

P2---

P2---

μτ σ+ τ

μτ

WSSUS channels


Doppler dispersion -- time selectivity:

Doppler scattering function:

describes the spread w.r.t. of the mean

received power.

P ν( ) P ν τ,( ) τd∫≡

P ν( ) ν

P ν( )

ν

Doppler dispersion

Time CF:

For any fixed , the process is WSS

with CF

f 1f

t

H t f,( ) [dB]

f 2

H t f 1,( )[dB]

H t f 2,( )[dB]

Time selectivity

f 1 H t f 1,( )

R Δt 0,( ) R Δt( ) Channel time CF≡

Doppler dispersion Time selectivity⇔

P ν( ) R Δt( )ν ΔtDoppler domain Time domain

WSSUS channels


Doppler dispersion -- time selectivity (cont’d):R

elat

ive

ampli

tude

[lin

]

Sourc

e: E

TH

Z, C

TL

Estimated Doppler scattering function:

Am

pli

tude

[lin

]

Estimated time correlation function:

Doppler frequency [Hz]ν Time lag [s]Δt

WSSUS channels


Doppler spread -- coherence time:

Normalized Doppler scattering function:

Mean excess Doppler, Doppler spread:

is a measure of the

extent of .

Doppler domain

Pn ν( ) 1

P---P ν( )≡ Pn ν( ) νd∫⇒ 1=( )

μν νPn ν( ) νd∫≡ σν ν μν–( )2Pn ν( ) νd∫≡,

Pn ν( )

νμν

2σνσν

Pn ν( )

Normalized frequency CF:

Coherence time at level :

is a measure of the width of

the main lobe of .

Time domain

Rn Δt( ) 1

P---R Δt( )≡ Rn 0( )⇒ 1=( )

c 0 1 ),[∈

Δt( )c min Δt 0> : Rn Δt( ) c={ }arg≡

1c

Δt( )c Δt

Rn Δt( )

Δt( )cRn Δt( )


σν Δt( )c⋅ 1

2π------ arccos c( )≥

WSSUS channels


Motivation:We seek a system function which incorporates dispersion in direction.

Mag

nit

ude

Time delayIncidence

direction

1

2

3

1

23

Time delayM

agnit

ude

12

3

Propagation environment Direction-delay SF

Delay SF

Direction dispersion


Motivation (cont’d):The BS and the MS of third generation mobile radio communicationsystems utilize more or less advanced spatial diversity techniques exploiting

- direction dispersion

- spatial decorrelation

in addition to Doppler and delay dispersion.

“Smart antennas” is a collective name embracing these techniques. Mostadvanced spatial diversity techniques are implemented at the BS due to thequasi absence of power consumption constraint for the signal processors.

Design of such techniques requires SCMs incorporating dispersion in direc-tion, in delay, and Doppler frequency at the BS and at the MS site.

To simplify the presentation, we shall first focus the presentation on direc-tion and delay dispersion. The full characterization of channel dispersionwill be addressed in Lecture 5.



Delay-direction spread function:

We start from the space-variant delay SF:

Invoking the identity,

we can express the space-variant delay SF as

g r τ;( ) f Ωi( )bEi ej2πλ

------ ar Ωi( ) r⟨ ⟩δ τ τi–( )⋅

i 1=

N

∑=

⎧ ⎪ ⎨ ⎪ ⎩

hi

u z0( ) u z( )δ z z0–( )dz∫=

g r τ;( )

g r τ;( ) bf Ω( ) ej2πλ

------ ar Ω( ) r⟨ ⟩h Ω τ,( )dΩ∫=



Delay-direction spread function (cont’d):

where

(@)

is the sought direction-delay SF of the channel.

h Ω τ,( ) hiδ Ω Ωi–( )δ τ τi–( )i 1=

N

∑≡



Direction (azimuth)-delay spread function:

τ1

τ2

φ1

φ2

h1

h2

h φ τ,( )

: Azimuthφ



System functions of a space-variant linear system:

g r τ;( )

y r t,( ) H r f,( )X f( ) j2πft( ) fdexp∫= …

H r f,( )

bf Ω( )h Ω τ,( )

k Ω f;( )

y r t,( ) g r τ;( ) x t τ–( ) τd⋅∫=

Ωx

y r t,( ) bf Ω( )h Ω τ,( )ej2πλ

------ ar Ω( ) r⟨ ⟩x t τ–( ) Ω τdd∫∫=

Direction-delay spread functionSpace-variant delay spreadfunction

Space-frequency transfer function Output direction spread function

τ

f

τ

f

Space-variant linear systems


Correlation functions of WSSUS systems:

Comment: The direction-delay scattering function describes how the averagereceived power is scattered jointly in delay and direction in the same

fashion as the Doppler delay scattering function characterizes thedispersion of the average incident power jointly in Doppler frequency anddelay.

E k∗ Ω1 f;1

( ) k Ω2 f;2

( )[ ] S Ω1 Δ; f( )δ Ω2 Ω1–( )=

E h∗ Ω1 τ,1

( ) h Ω2 τ,2

( )[ ] P Ω1 τ1,( )δ Ω2 Ω1–( )δ τ2 τ1–( )=

E H∗ r1 f 1,( ) H r2 f 2,( )[ ] R r2 r1– f 2 f 1–,( )=

E g∗ r1 τ1;( ) g r2 τ2;( )[ ] Q r2 r1– τ1;( )δ τ2 τ1–( )=

P Ω τ,( )

P ν τ,( )

WSSUS space-variant channels


Direction-delay scattering function:

Example of an estimated azimuth-delay scattering function at the MS site:

Delay [ s]μDelay [ s]μ

Rel

. am

pli

tude

[dB

]

Azimuth

Delay

Sourc

e: A

AU

, C

PK



Fourier relationship between the channel characterizing functions:

S Ω Δ; f( )R Δr Δf,( )

E H∗ r f,( ) H r Δr+ f Δf+,( )[ ] R Δr Δf,( )= E k∗ Ω1 f;( ) k Ω2 f; Δf+( )[ ] =

Q Δr τ;( ) b2

f Ω( ) 2P Ω τ,( )

E g∗ r τ1;( ) g r Δr+ τ2;( )[ ] = E h∗ Ω1 τ,1

( ) h Ω2 τ,2

( )[ ] =

Direction-delay scattering functionWideband space correlation function

Space-frequency correlation function Direction cross-power spectral density

Q Δr τ1;( )δ τ2 τ1–( )= P Ω τ,( )δ Ω2 Ω1–( )δ τ2 τ1–( )=

S Ω1 Δ; f( )δ Ω2 Ω1–( )=

τ

Δf

ΩΔx

ΩΔx

τ

Δf



Delay dispersion -- frequency selectivity:

Delay scattering function:

describes the distribution w.r.t. of the

mean received power.

P τ( ) b2

f Ω( ) 2P Ω τ,( ) Ωd∫≡

Q 0 τ,( ) E g r τ;( )2

[ ]==

P τ( ) τ

P τ( )

τ

Delay dispersion

Frequency CF:

For any fixed , the processes isWSS with CF

f

r

H r f,( ) [dB]

r1

H r1 f,( ) [dB]

r2H r2 f,( ) [dB]

Frequency selectivity

r1 H r1 f,( )

R 0 Δf,( ) R Δf( ) Channel frequency CF≡

Delay dispersion Frequency selectivity⇔

P τ( ) R Δf( )τ ΔfDelay domain Frequency domain



Delay scattering function:Estimated delay scattering function at the BS site in typical urban macrocel-

lular environments:

-5 0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0


Po

we

r[d

B]



Stockholm


T S 0.923 μs=( ) Source: AAU, CPK



Direction dispersion -- space selectivity:

Direction scattering function:

describes the (weighted) spread w.r.t.

of the mean received power.

P Ω( ) b2

f Ω( ) 2P Ω τ,( )∫ τd≡

P Ω( ) Ω

P φ( )

φ


ππ–

Space CF:

For any fixed , the process is WSS

with CF

f 1f

r

H r f,( ) [dB]

f 2

H r f 1,( ) [dB]

H r f 2,( ) [dB]

Space selectivity

f 1 H r f 1,( )

R Δr 0,( ) R Δr( ) Channel space CF≡

Direction dispersion Space selectivity⇔

P Ω( ) R Δr( )Ω ΔrDirection domain Spatial domain



Direction scattering functions:Estimated azimuth scattering function at the BS site in typical urban

macrocellular environments:

-30 -20 -10 0 10 20 30-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Aarhus,high antennaposition

Stockholm

Azimuth [ ]0

Po

we

r[d

B]

Source: AAU, CPK

Laplacian scattering function:

180°– φ 180°<≤

Pσφφ( ) 1

q 2σφ----------------- 2 φ σφ⁄–( )exp∝

q1

2σφ-------------- 2 φ σφ⁄–( )exp φd

180–

180∫≡



Delay and direction scattering functions (cont’d):Constellation of propagation paths at the BS and MS sites in macro-cellular

environments -> Different delay scattering function at the BS and MS sites.

Propagation path #N

BS MS

v

Propagation path #i

Propagation path #2

Propagation path #1



Azimuth spread -- coherence distance:

Normalized azimuth scattering function:

Mean azimuth, azimuth spread:

Azimuth domain

Pn φ( ) 1

P---P φ( )≡ Pn φ( ) φd∫⇒ 1=( )

μφ ejφ

Pn φ( ) φd∫⎩ ⎭⎨ ⎬⎧ ⎫

arg≡

σφ ejφ μφ–

2Pn φ( ) φd∫≡

Pn φ( )

μφ

2σφφππ–

Normalized space CF:

Coherence distance at level :

Rn Δr( ) 1

P---R Δr( )≡ Rn 0( )⇒ 1=( )

c 0 1 ),[∈

Δd( )c min Δr : Rn Δr( ) c=⎩ ⎭⎨ ⎬⎧ ⎫

arg≡


σφ Δd( )c⋅ 1

2π------ arccos c( )≥



Azimuth spread (cont’d):Empirical probability distribution of estimated azimuth spreads in typicalurban (TU) macrocellular environments:

1.0

0.9

0 5 10 15 20 25 30

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Cum

ula

tive

dis

trib

ution

Aarhus, lowantenna position

Aarhus, highantenna position

Azimuth Spread [ ]

Stockholm

Sourc

e: A

AU

, C

PK



Coherence distance:

Rn Δr( )

Δr2

Δr1

Δdar Ω( )

1

c

Δd( )Ω c,

Δr Rn Δr( ); c=⎩ ⎭⎨ ⎬⎧ ⎫

Rn Δdar Ω( )( )

Δd( )c minΩ Δd( )Ω c,{ }≡



Coherence distance (cont’d):

Δd:

Δr2

Δr1

Δdar Ω( )

Δd( )Ω c,

Δd( )c

Δdar Ωm( )

is the minimum spacing

between the elements of an

antenna array such that the

correlation of the channel

transfer function at the

element outputs is at most .

Δd( )c

c

Δr Rn Δr( ); c=⎩ ⎭⎨ ⎬⎧ ⎫

Δr Rn Δr( ); c>⎩ ⎭⎨ ⎬⎧ ⎫



Pulse sounding technique:

Principle:

t0

x t( )

y t( )

t

t

RxTx

g t τ;( )

y t( )x t( )

t0

T c

g t0 t;( )

t

g t0 t; t0–( )

p t t0–( )

y t( ) g t0 t;( ) * p t t0–( )≈

g t0 τ;( ) p t t0– τ–( ) τd⋅∫= g t0 t; t0–( )≡

acts as a smoothing functionp t( )

Wideband channel measurement methods


Pulse sounding technique (cont’d):

Transmitter:

Frequency

standard

Pulse

generator

MixerAmplifier

Tx filterR. F.

generator



Pulse sounding technique (cont’d):

Frequency

standard

Amplifier

LP filter

R. F.

generator

LP filter

Data

storage

90°

Rx filter

Mixer

Mixer

Receiver:

ℜe g t0 t; t0–( ){ }

ℑm g t0 t; t0–( ){ }

Quadrature component

In-phase component



Pulse compression sounding technique:

x t( ) y t( )

tt

RxTx

p T t–( )

t

Matched filter (MF)g t τ,( ) t0

t0

x t( )y t( )

R p t( ) p τ( ) p τ t+( ) τd∫≡

t

Principle:

T c

T

t0 t0 T+≡

p t t0–( )g t0 t; t0–( )

y t( ) g t0 t;( ) * p t t0–( )[ ] * p T t–( )≈ g t0 t;( ) * p t t0–( ) * p T t–( )[ ]=

g t0 t;( ) * R p t t0–( )= g t0 τ;( ) R p t t0– τ–( ) τd⋅∫ g t0 t; t0–( )= =



Pulse compression sounding technique (cont’d):

Transmitter:

Frequency

standard

Sounding

sequence

generator

MixerAmplifier

Tx filterR. F.

generator

Two modes:

- Period = T:

- Period >T:




Correlator

or MF

Correlator

or MF

Sounding

sequence

generator

Frequency

standard

Amplifier

LP filter

R. F.

generator

Data

storage

90°

Rx filter

LP filter

Mixer

Receiver: Mixer

ℜe g t0 t; t0–( ){ }

ℑm g t0 t; t0–( ){ }




Sounding sequences:

•Linear FM (frequency chirp):

t

x t( )

t

f t( )

f min

f max

B

T T



Sounding sequences (cont’d):

•Phase-coded waveform:

- Barker codes

T c

t

x t( )+1

T

Rx t( )

tT

NT cAperiodic ACF:

•Good properties of the aperiodic autocorrelation function (ACF):

for

•Only short sequences are known: max. length = 13 chips

Rx t( ) T c≤ t T c≥

T c

: chip length of the sequenceN T T c⁄=




•Phase-coded waveform (cont’d):

-Pseudo-noise (PN) sequences

Periodic ACF

tTT c

•PN sequences are easily generated by maximal-length linear feedback shift registers

•The periodic ACF of PN sequences exhibits good properties:

NT c

T c

Rx t( )

tT

NT c

Aperiodic ACF:




•Sequences with flat amplitude spectrum and small CR factor

f

X f( )

B

Crest (CR) factor:

CR u[ ]maxt T∈ x t( )

1

T--- x t( ) 2

dt0

T∫------------------------------------ 1≥≡

•Optimal sounding signal for least-square channel estimation

flat amplitude spectrum

•Amplifier non-linearity

minimize the CR factor

⇒

⇒



Frequency stepping sounding technique:

Principle:

xi t( ) 2π f Δf i+( )t( )cos=t

RxTx

H f( )

y t( )

H Δf 2( )H Δf 1( )t

y2 t( )y1 t( )

x2 t( )x1 t( )x t( )

y t( )

yi t( ) H Δf i( ) 2π f Δf i+( )t H Δf i( )( )arg+[ ]cos⋅=

ℜe H Δf i( ) j2π f Δf i+( )t[ ]exp{ }=



Frequency stepping sounding technique (cont’d):

Transmitter:

Frequency

standard

Amplifier

Tx filterVariable

frequencygenerator



Frequency stepping sounding technique (cont’d):

Frequency

standard

Amplifier

LP filter

LP filter

Data

storage

90°

Rx filter

Variablefrequencygenerator

Mixer

Receiver: Mixer

ℜe H Δf i( ){ }

ℑm H Δf i( ){ }



Comparison of the sounding techniques:

Sounding technique + -Delay

resolution

Dynamic

range

Pulse

sounding

Time variant channel High peak to

mean power ratio

Low dynamic range

8-100 ns ~ 20 dB

Pulse

compression

Time variant channel

High dynamic range

Complex signal processing

(matched or inverse filtering)

5-100 ns 60-100 dB

Frequency

stepping

Simple technique

(network analyzer)

Time invariant channel ns ?0.5≤



Estimation of the direction-delay scattering function:•Wideband pencil-beam antenna combined with one of the previously

described method:

ar Ω( )

Tx

Rx

gΩ r τ;( )r

P Ω τ,( ) gΩ r τ;( )2

⟨ ⟩ x∝

gΩ r τ;( ) bf Ω' Ω–( )h Ω' τ',( )ej2πλ

------ ar Ω'( ) r⟨ ⟩x τ τ'–( ) Ω' τ'dd∫∫ h Ω τ,( )e

j2πλ

------ ar Ω( ) r⟨ ⟩∝=

f Ω' Ω–( )



Example of an estimated direction-delay scattering function:

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6−20

−10

0

10

20

Delay1Delay2

PD

PA

mpli

tude

[dB

]

Delay [s]μ

Delay [s]

μ

Sourc

e: A

AU

, C

PK



Estimation of the direction-direction scattering function (cont’d):

•Smart antennas combined with a pulse compression technique:

Tx

Rx

P Ω τ,( )…

High resolu-

tion signal

processing

algorithm

(MUSIC,

ESPRIT,

ML, EM,

etc.)

xxx⎧ ⎨ ⎩

Antenna array



Estimation of the direction-delay scattering function with an EM-basedalgorithm:

2

3

4

tr

Propagation environment:

1

2

3

TxRx Direction of the

linear array

Sourc

e: E

TH

Z, C

TL



Estimation of the direction-delay scattering function with an EM-basedalgorithm (cont’d):

600400

2000

200400

6000

200

400

600

−40

−30

−20

−10

0

Am

plit

udein

dB

AngleinDeg

120 90

150 60

18030

ETZ

0

ETF

Direct

Estimated direction-delay scattering function:A

mpli

tude

rel.

to m

ax. [d

B]

Azi

mut

hal a

ngle

[deg

]

1

2

3

Delay [ns] Sourc

e: E

TH

Z, C

TL



Delay scattering function (typical urban (TU)):

-5 0 5 10 15 20 25 30-30

-25

-20

-15

-10

-5

0


Po

we

r[d

B]



Stockholm


T S 0.923 μs=( ) Sourc

e: A

AU

, C

PK

Dispersion at the BS in macrocell. environments


Empirical distribution of the estimated delay spreads (TU):

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aarhus,low antenna position

Aarhus,high antenna position

Stockholm

RMS Delays Spread (DS) [ s]�

Cum

ula

tive

dis

trib

ution

Sourc

e: A

AU

, C

PK



Azimuth scattering function (TU):

-30 -20 -10 0 10 20 30-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Aarhus,high antennaposition

Stockholm

Azimuth [ ]0

Pow

er

[dB

]

Sourc

e: A

AU

, C

PK



Empirical distribution of the estimated azimuth spreads (TU):

1.0

0.9

0 5 10 15 20 25 30

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Cum

ula

tive

dis

trib

ution

Aarhus, lowantenna position

Aarhus, highantenna position

Azimuth Spread [ ]

Stockholm

Sourc

e: A

AU

, C

PK



Delay and azimuth scattering functions (Bad urban (BU)):

0 10 20 30 40 50 60 70 80 90-25

-20

-15

-10

-5

0


Pow

er

[dB

]

Cluster #2

Cluster #1


Measured

-60 -40 -20 0 20 40 60-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Cluster #1

Cluster #2

Laplacian model

Azimuth [ ]0

Po

we

r[d

B]

T S 0.923 μs=( )Source: AAU, CPK



Map of one of the investigated BU environment:

BS

Dire

ctio

n#2

Direction #1

River

Stockholm city, Sweeden North

Direction #1 -> BU

Direction #2 -> TU

Sourc

e: A

AU

, C

PK


2. Channel Characterization and...

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