COMPACT LECTURE NOTES on ADVANCED COMMUNICATION THEORY. 5 2005 - LN - TRANS... · IMPERIAL COLLEGE LONDON, DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING. COMPACT LECTURE NOTES

IMPERIAL COLLEGE LONDON,DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING.

COMPACT LECTURE NOTES on ADVANCED COMMUNICATION THEORY.Prof. Athanassios Manikas, version Autumn 2005

Space-Time Communications

MSc, MEng.

Outline:ì Definitions, notation, spaces and projection operator.ì Modelling of array received signal-vector and the concept of the

array manifoldì Multidimensional correlatorsì The Detection Problemì The Estimation Problem: Directions-of-Arrival, Signal Powers,

Cross-correlation etcì The Reception Problem: Array Pattern & Beamforming, Popular

Beamformers.ì Performance Evaluation - SNIR9?>

ì Outage Probabilityì MIMO Systemsì Array-CDMA: Signal Modelling and Channel Effectsì STAR Receiver Architectures

Advanced Communication Theory Compact Lecture Notes

Space-Time Communications 2 A. Manikas

1. Introduction

ì is defined as a general technique thatThe 'Diversity' Concept: Diversityutilizes two or more copies of a signal with varying degrees ofnoise/interference effects to achieve, by selection or a combination scheme,higher degree of message-recovery performance than that achievable by anyone of the individual copies separately.

ì Classification of Diversity Techniques: Time Diversity ˆ Frequency Diversityˆ Space Diversityˆ Polarization Diversityˆ others (e.g. code diversity or combination of the above)ˆExamples:1) In , the same information (message) is transmitted onFrequency Diversityfour (say) parallel channels using four different carrier frequencies; then at thereceiver the four signals are combined before a decision is made.



Subcarrier #

Symbolstream

Subcarrier "

Subcarrier R=-

BasebandMT-CDMA

signal

PN-signal

+Serial toparallel

converter Subcarrier #

DC signal(PN bit 1)

DC signal(PN bit#)

DC signal(PN bit R=-)

Subcarrier "

Subcarrier R=-

BasebandMC-CDMAsignal+Symbol

stream

Subcarrier #

Subcarrier "

Subcarrier R=-

BasebandMC-DS-CDMAsignal+

PN-signal

Symbolstream

2) Multi-tone (MT) CDMA 3) Multi-Carrier (MC) CDMA

4) Multi-Carrier (MC) DS-CDMA



Tx Rx

info(i) info(i)

info(i-1) info(i-1)

Time Diversity:

Tx Rx

f1

f2

info(i)

info(i)

Frequency Diversity:

Tx Rx

info(i)

info(i)

Multi-Path Diversity:



Space Diversity (or Antenna Diversity):

Tx Rx

info(i)

info(i)

a) Tx Diversity:

Tx Rx

info(i)

info(i)

b) Multiple-Antenna (Rx) Diversity:



c) Array Diversity:

TxRx

info(i)

N x info(i)

NArray System

(Array at the Rx)

similarly: Array at the Tx, or, Array-to-Array_______________________Array systems/techniques can be seen as themost sophisticated and advanced space diversitysystems/techniques.(This type of systems/techniques will be considered in this course.)



2. Space-Only Examples (Array at the Rx)

G œ F "R ‚log#a bSNIRoutF Ä _ Ê G Ä R ‚"Þ%% T

R=

!



$ R À-Dim. Gain Pattern for =1 (two different viewing angles)















3. Classification of Space-Time Communication System ArchitecturesSESE

MEME

MESE

SEMETx

Space-TimeTx

Rx

Space-TimeRx

SESE: from Single-Element (SE) Tx to Single-Element (SE) RxSEME: from Single-Element (SE) Tx to Multiple-Element (ME) RxMESE: from Multiple-Element (ME) Tx to Single-Element (SE) RxMEME: from Multiple-Element (SE) Tx to Multiple-Element (ME) Rx (known also as MIMO i.e. Multiple-Input Multiple-Output System)



A COMMENT:

ìA receiver is said to be a ' ' if it operates onspace-time receivermore than one antennas, processing the received signals both in'space' and 'time'.(A similar statement can be made for a space-time transmitter)

Advantage:suppression of co-channel interf. and noise= system capacity=

quality=ÅÅ

ÅÊ œ



ì Capacity of a Spatio-Temporal Link (SEME)



4. CLASSIFICATION OF MULTIPLE ACCESS (MA) CHANNELS

TxTx

Tx

Rx

VISOChannel

TxTx

TxTx

Tx

VIVOChannel

TxTxSpace-Time

Rx

1) 2)

MIVOChannel

Space-TimeRx

Space-TimeTx

Space-TimeTx

Space-TimeTx

3)



Notes: 1) VISO: Vector Input - Scalar Output

2) VIVO: Vector Input - Vector Output

3) MIVO: Matrix Input - Vector Output

4) VISO is the standard ('time' only) MA Channel

5) VIVO and MIVO are space-time MA Channel

6)N.B.: The following channels are not Multiple-Access Channels SISO (Scalar Input - Scalar Output) Channel and SIVO (Scalar Input - Vector Output) Channel

Tx

Rx

SISOChannel Tx

SIVOChannel

Space-TimeRx



5. Wireless Channels (Ignoring Space)

5.1. Important Wireless Channel Parameters:G œ Channel Capacity (inf. bits/sec)F œ œ Tx-Signal Bandwidth (Hz) "

X-=

F œcoh œ Coherent Bandwidth of the Channel (Hz) "Xspread

ˆ

pp

typical examples of coherent bandwidth:

= 3MHz wireless channels100MHz wireless channelsFcoh œ indoor

outdoor

F œDop Doppler Spread of the Channel (Hz) œ"

Xcoh



Initially wireless systems were used to cover very large distances(e.g. in Marconi's wireless transmission across the Atlantic andPacific Oceans)

Due to their flexibilty and comfort, today wireless systems are usedto cover very small distances (short range wireless links)

Wireless Channels are much more difficult and hostile than wiredchannels.

5.2. Multipaths





5.3.Propagation Loss



In a wireless system the received signal is the summation of anumber of paths.

impulse response: (baseband)

2Ð>Ñ œ Ð4 4# J Ñ Ð> Ñ

.

!ðóóóóóóóóóóñóóóóóóóóóóòŠ ‹ ç5œ"

P".

+

5 - 5 5

#5

5

5

-

exp : 1 7 $ 7

"

1-



5.4. Fading



Sample of a "fading" signal envelope: amplitude in dB versustime or location of the antenna. Wave interference of multiplereflected waves, each with a different amplitude and phase,causes fluctuations of the received signal amplitude.Changing the antenna location or the carrier frequency alsochanges the signal amplitude.

This is known as fadinghttp://www.wireless.per.nl/reference/chaptr03/rayjava/rayjava.htm



5.5. DELAY SPREAD, :Xspread

is the time it takes for light to travel a distance equal to the longestpath minus the shortest path

i.e. Xspread ´max mina4

Ö × Ö ×a4

-

. .4 4

ˆ : typical examples of delay spread fraction of µs many µsŸ ŸXspread



5.6. Classification of Wireless Channelsˆ X X Ñ X Xby comparing (or with and/or cs spread coh-



ˆ or, by comparing or ) with and/or F Ð F F Fss coh Dop



ì Comments: Multipath Fading in a Conventional System

ˆ In a conventional mobile cellular system (TDM/FDM)the destructive interference is known asmultipath or Rayleigh fading.

This occurs when the mobile .more frequently is moving

This fading is to the system performance.detrimental



Thus, in a conventional system is compared to Xspread X-=

if then paths can be separated(i.e. )œ X Xspread -=

F Fcoh else signals are distorted FLAT FADINGÊ

ˆ resolvable pathsNumber of in a conventional system: P œ "ª «X

Xspread

-=

ì Comments: Multipath Fading in a Spread Spectrum System Multipath fading exists in Spread Spectrum (or CDMA) Systemsas well (but it is )significantly lowerN.B.: P œ "ª «X

Xspread

-



5.7. Channel Selectivity and Channel Coherenceì Channel A channel has if as aSelectivity: selectivity it varies

function of either time, frequency, or space

ì Channel opposite of Channel Coherence Selectivity: ( )ˆ A channel has if as a function ofcoherence it does not vary

either time, frequency, or space over a specified 'window' ofinterest.

ˆ This is the concept in describing wirelessmost importantchannels

ˆ coherence:ÚÛÜ

temporalfrequencyspatial

coherence - coherence time coherence - coherence bandwidth

( coherence - coherence

XF

coh

cohdistance )Hcoh



5.8. Examples:







ì Example of effect of transmitting a rect pulse over aTime Selective Fading Channel



ì Example of effect of transmitting a rect pulse over aFrequency Selective Fading Channel



5.9. WIRELESS CHANNEL DESCRIPTION/ANALYSIS - (TIME ONLY):



6. Conventional Multipath Channel Modelling (SISO Channel)

Tx

Rx

SISOChannel

i/p o/p

SISO (Scalar Input - Scalar Output) Channel is not a Multiple-Access Channel



i.e etc & etc are random" " " 7 7 7" # $ # $ß ß ß ß1

impulse response: , (baseband)

2Ð> Ñ œ Ð> Ñ7 " $ 7!5œ"

P

5 5

where number of paths that can be resolvedP œ

i.e. , is a random variable which is function of time and 2Ð> Ñ >7 7



`











7. Comments on Wireless Channels:



8. Wireless Channels - with Space InformationWe need 'vector' and 'matrix' notation

8.1. Notationa denotes a column vector (or A) = denotes a matrix

9: elevation angle): azimuth angle

? œ Þ ß Þ ßc dcos cos sin cos sin) 9 ) 9 9 X (9) a (3 1) real unit-vector pointing towards the direction ( , )œ ‚ ) 9

N.B.: wavenumber-vector? ? 5 ?X #œ " œ ´ and 1-

_[ ] represents the linear space/subspace spanned by the columns of .



Origin

N-dim complex observation space

ì

Denotes a -dimensional complexR(or real) observation space

Note that any vector in this spacehas R /6/7/8>=

aì _[ ] a denotes a one-dimensional

subspace/space spanned by the vector a

L[ ]Aì denotes an -dimensional subspace/spaceQ (with spanned by the columnsQ #Ñ of the matrix

ìNote that any vector [ ] can be written as a linearB − _ combination of the columns of the matrix



8.2. The concept of "Projection Operator"ì R ‚QÑ Q Ÿ RConsider an ( matrix with

(i.e. the matrix has columns)Q

ìLet the columns of be linearly independent(i.e. a column of cannot be written as a linear combination of the remainingQ " columns)

Then the columns of span a subspace [ ] of dimensionality _ Q(i.e. dim [ ] ) lying in a -dimensional space observation space e f_ œ Q R H

L[ ]AOrigin




ì −Any vector can be projected on to by using theB H _[ ]concept of the projection operator

Notation: œ projection operator on to subspace spanned by the columns of œ ˆ ‰L L" (10)

L[ ]AOrigin


x

PAx

i.e.

dim <a b_[ ] œ Q R



ì

R ‚ R

Properties of : ( ) matrix

ÚÛÜ . œ

œ L(11)

ì_ _[ ] [ ] ¼ denotes the subspace to complement

L[ ]AOrigin

N-dim complex observation spaceL[ ]A

dima b_[ ] œ Q dima b_[ ] ¼ œ R Q

ì ¼ ¼represents the projection operator of and is defined as_[ ]

ˆ ¼

Rœ (12)



ì −Any vector can be projected on to [ ] and [ ] asB H _ _ ¼

follows

L[ ]AOrigin

N-dim complex observation spaceL[ ]Ax

PAx

PAx

ì



8.3. Space-Selective FadingìA wireless receiver is located (and moves) in our 3D real

space.

ì In addition to delay-spread (causing frequency-selectivefading) and doppler-spread (causing time-selective fading)there is also angle-spread

ìAngle Spread causes Space-Selective fading

l < l ¸ Z < < l ŸH( ) for | -! !H#-92



ìSpatial Coherence

A wireless channel has spatial coherence if the magnitude ofthe carrier remains constant over a spatial displacement ofthe receiver.

H-92 represents the largest distance that a wireless receivercan move with the channel appearing to be static.

If the displacement of the receiver is greater than thenH-92

the channel experiences small-scale fading(and if this displacement is of the order of many wavelengthsthen channel experiences large-scale fading)





8.4. Some CommentsLÐ0ß >ß Ñ<

F ? ? ?LÐ 0 ß >ß Ñ<

W Ð ß 0 ß ÑL 7 5

frequency time spacedependencycoherenceSpectral domain delay, doppler, wavenumber, Spectral width delay spread,

0 >F X H

0X

<

5-92 -92 -92

7

=:</+. H9:Doppler spread, wavenumber spread, F 5=:</+.

W Ð Ñ œ ‚ :Ð ß Ñ ´LÐ# Ñ Ð Ñ

Ð Ñ5

1 $$ #

# #

l l5 1-

1-

) 9 wavenumber spectrum as a function of the angle spectrumNB: 5 5 ?œ Þ Ð ß Ñl l ) 9



Types (and Examples) of Angle Spectrum

Angle SpectrumSpecular

Angle SpectrumDiffuse

of Specular & Diffused Angle SpectrumCombination



8.5. The Concept of the Local Area

This is the largest volume of free-space about a specific point< œ Ò< ß < ß < ÓB C D

X in which the wireless channel can be modelledas the summation of homogeneous plane waves

!æ3œ"

PX3Š ‹ ç" #

.

+

3

3 - 3

#3

3 -

-

œ Z

Ð4 4# J Ñ

.˜

exp : 1 7

1-

1-4 ? Ñ Ð4# J >Ñexp 1 -

propagating from the Tx to the receiver Rx through our 3D realspace. This planewave arrives at the receiver's antenna and producesa constant-amplitude voltage

7Ð> Ñ 4 4# J Ð> Ñ œ7 9 1 7Š ‹ a b a b"+

-3 exp exp

7Ð> Ñ 4 4# J 4# J >3 33- -"

+

- -Š ‹ a b a bˆ ‰exp exp exp9 1 1

7Ð> Ñ 4 4 4# J >33 -

1-

" #+

-Š ‹ Š ‹a b a bexp exp exp9 3 1-





If, however, the receiver is displaced at a specific point and the<5direction of the planewave propagation is described by the vector?( )) 93 3ß

where ( ) ? ) 9 ) 9 ) 9 9ß œ Þ ß Þ ßc dcos cos sin cos sin X(13) a (3 1) real unit-vector pointing towardsœ ‚ the propagation direction ( , )) 9

then

7Ð> Ñ 4 4 4# J 4# J >

œ

- - - -33 -

1-

" #+

- -7 735 35ðóóóóóóóóóóóñóóóóóóóóóóóòŠ ‹ Š ‹a b a b a bexp exp exp exp9 3 1 1

"

-

˜

(14)



Origin

(0,0,0)

k- pointth

rk

ui

c=velocity of propagation

Travell

ing plav

e wave

at time t 1

Travelli

ng plave w

ave a

t time t 2

Travellin

g plav

e wave

at the

-th

kpoin

t

Travellin

g plav

e wave

at the o

rigin

lik

LocalArea

ì Estimation of 735 À



7 356-œ œ35

¼ ¼ É É ? 53 5X X

?3 ? 53 5X

? 53< < < < <

- - -œ œ œ

œ œÉ a b< ? ? ? ? <5

X X X3 3 53 3

"

-

œ œ œÉ Éa b< ? ? < < ? ? <5

X X3 53 5

X3

#

3X

5

- - -

i.e. 735 œ? <3X

5

- (15)

(where denotes the Cartesian coordinates of the receiver i<5$‚"− V

metersÑ



ì based on the above, Equ.14 may be rewritten as:7Ð> Ñ 4 4 4# J 4# J >

œ

- - - -33 -

1-

" #+

- -? < ? <3 3X X

5 5Þ Þ- -ðóóóóóóóóóóóñóóóóóóóóóóóòŠ ‹ Š ‹ Š ‹a b a bexp exp exp exp9 3 1 1

"

-

˜

ðóóóñóóóòðóóóóóóóóóóóñóóóóóóóóóóóòŠ ‹ Š ‹ Š ‹a b7Ð> Ñ 4 4

œ

- - - -3

3

3 -1

-

-

" #+

? <3X

5

-

Þ-

3

#3X

5

¸ 7 Ð> Ñ

4 Þ

(narrowband assumption)

exp exp exp exp9 3

"

-

˜

1- ? < a b4# J >1 -

(16)



8.6. Array SystemsIn the "local area" of the previous section consider that we have anarray system.

An array system is a collection of > sensors (transducingR "elements, receivers, antennas, etc) distributed in the 3-dimensionalcartesian space, with a common reference point.

ì œ Ò ß ß ÞÞÞß Ó œ Ð$ ‚ RÑ Let rœ < < <" # R ‘< ß < ß 25 a5 œ "ß #ß ÞÞÞß R

The region over which the sensors are distributed is called theaperture of the array. In particular array aperture œ

a34? max l l< <3 4 (17)



Then, the planewave of Equation-16 arrives at each antenna of thearray and produces a constant-amplitude voltage-vector as follows:

Equ-16

Ê

7Ð> Ñ 4 4# J >

7Ð> Ñ 4 4# J >

ÞÞÞ

7Ð> Ñ 4

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

Š ‹ a bŠ ‹ a bŠ ‹

3 1-

3 1-

3 1

-# X

3 " -

-# X

3 # -

-#

" 1

" 1

"

exp exp

exp exp

exp

-

-

? <

? <

-

3 1-

-

-

? <

? <

X3 5 -

-# X

3 R -

exp

exp exp

a bŠ ‹ a b

4# J >

ÞÞÞ

7Ð> Ñ 4 4# J >

1

" 1



After down-conversion the above vector becomes Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

Š ‹Š ‹Š ‹Š ‹

7Ð> Ñ 4

7Ð> Ñ 4

ÞÞÞ

7Ð> Ñ 4

ÞÞÞ

7Ð> Ñ 4

3 1-

3 1-

3 1-

3 1-

-# X

3 "

-# X

3 #

-# X

3 5

-#

"

"

"

"

exp

exp

exp

exp

-

-

-

-

? <

? <

? <

?X3 R

-

# X3 "

# X3 #

# X3 5

#<

? <

? <

? <=

7Ð> Ñ

4

4

ÞÞÞ

4

ÞÞÞ

4

3

1-

1-

1-

"

ðóóóóóóóóóóñóóóóóóóóóóò


Š ‹Š ‹Š ‹Š ‹

exp

exp

exp

exp

-

-

-

1--? <

W

X3 R

3 3œ Ð ß Ñ˜ ) 9



An Important CommentIf the assumption used in Equation7Ð> Ñ 7Ð> Ñ3 3

- -? <3X

5Þ- ¸

(16) is not valid then the received baseband array signal-vectorcan be modelled as follows:

"7 W3 3 3Ð> Ñ7 5 (18)

with - - -73 3Ð> Ñ œ Ò7 7 Ð> Ñß7 Ð> Ñß ÞÞÞß7 Ð> ÑÓ3 3 3 3 3 3RX3 3 3

- - -- - -7 7 71 2

where 73 œ Ò ß ß ÞÞÞß Ó7 7 73" 3# 3RX



8.7. Summary: Compact Modelling of Array Manifold Vectors(see also Chapter-1 of my book):

( ) ( ) W < < < 5) 9 ) 9ß ßœ 4 Ò ß ß ÞÞÞß Óexpˆ ‰" # RX (19)

or ( ) ( )W < < < 5) 9 ) 9ß ßœ 4 Ò ß ß Óexpa bB C D ( 1) complex vectorœ R ‚

where wavenumber vector5 ? ?Ð Ñ œ Þ Ð Ñ œ Þ Ð Ñ œ) 9 ) 9 ) 9ß ß ß# J-

#1 1-

-

-

( ) ? ) 9 ) 9 ) 9 9ß œ Þ ß Þ ßc dcos cos sin cos sin X (20)



9. Scalar-Input Vector-Output (SIVO) Channel

Tx

SIVOChannel

Space-TimeRx

ìLet us assume that the transmitted signal arrives at the referencepoint of an array receiver via paths (multipaths).P

ìConsider that the path arrives at the array from direction4>2

( ) ) 94 4ß with channel propagation parameters and " 74 4

representing the complex path gain and path-delay, respectively.



ìNote that and represent the azimuth and elevation angles) 94 4

respectively associated with -th path.4

ìLet us assume that the paths are arranged such thatP 7 7 7" # PŸ Ÿ ÞÞÞ Ÿ

ìFurthermore, the path coefficients model the effects of path"4

losses and shadowing, in addition to random phase shifts due toreflection; they also encompass the effects of the phase offsetbetween the modulating carrier at the transmitter and thedemodulating carrier at the receiver, as well as differences in thetransmitter powers.



ì The vector W W4œ Ð ß Ñ −) 9 V4 4

R , is the array manifold vector ofthe th path (to be defined later).4



ì The impulse response (vector) of the SIVO channel is

,hÐ> Ñ œ Ð> Ñ7 " $ 7!4œ"

P

4 4WÐ ß Ñ) 94 4

i.e. SIVO: hÐ> Ñ,7 œ >’ " $diagˆ ‰ a b (21) where ’ œ œÒ ß ß ÞÞÞß Ó Ò ß ß ÞÞÞß ÓW W W" # P " # P

X; ;" " " "

$a b> œ Ò Ð> Ñß Ð> Ñß ÞÞÞß Ð> ÑÓ$ 7 $ 7 $ 7" # PX

ìBased on the above SIVO model, the received complex signal-basebandvector at the antenna array can be represented as:BÐ>Ñ BÐ>Ñ œ 7Ð>Ñ‡h nÐ> Ñ >, ( )7

( )œ 7Ð> Ñ > !4œ"

P

4 4W4 " 7 n (22)

SIVO: Ê BÐ>Ñ œ >’ "diagˆ ‰ a b7 > n( ) (23) where 7 > œa b Ò7Ð> Ñß 7Ð> Ñß ÞÞÞß 7Ð> ÑÓ7 7 7" # P

X



10. Vector-Input Vector-Output (VIVO) Channel

Space-TimeRx

1st Tx

2nd Tx

M-th Tx

Array of elementsN

Vector Input - Vector OutputMA Channel

(VIVO)



ìLet users and element array system Consider that the number of pathsQ R Þfor the -th user is denoted by Then3 P Þ3



where



ìBased on the above VIVO model, the received complex basebandsignal-vector at the antenna array can be represented as:BÐ>Ñ

ìVIVO: BÐ>Ñ œ Ð> Ñ >!3œ"

Q

7 Ð>Ñ‡3 h n3 , ( )7

œ ! ðóóóóóóóñóóóóóóóò3œ"

Q

7 Ð>Ñ‡3 34!4œ"

P

34 34

3

W " $ 7

7

Ð> Ñ >

>h

n

3( , )

( ) (24)

œ !3œ"

Q !4œ"

P

34 3 34

3

W34 " 77 Ð> Ñ >n( )



VIVO: Ê BÐ>Ñ œ >" ˆ ‰ a b3œ"

Q

3 3 3’ "diag 7 > n( ) (25)

where ’3 œ Ò ß ß ÞÞÞß ÓW W W3" 3# 3P3

"3 œ Ò ß ß ÞÞÞß Ó" " "3" 3# 3PX

3

7 > œ3a b Ò7Ð> Ñß 7Ð> Ñß ÞÞÞß 7Ð> ÑÓ7 7 73" 3# 3PX

3

ìEquation-25 can be rewritten in a more compact form as follows:

BÐ>Ñ œ >’ "diagˆ ‰ a b7 > n( ) (26) where ’ ’ ’ ’œ Ò ß ß ÞÞÞß Ó1 2 Q

" œ Ò ß ß ÞÞÞß Ó" " "X X X X" # Q

7 > œa b Ò Ð>Ñß Ð>Ñß ÞÞÞß Ð>ÑÓ7 7 7X X X X" # Q



11. Summary Modelling of Array Signal-Vector BÐ>Ñ

ìSIVO: BÐ>Ñ œ >’ "diagˆ ‰ a b7 > n( ) (27)where ’ œ Ò ß ß ÞÞÞß ÓW W W" # P

" œ Ò ß ß ÞÞÞß Ó" " "" # PX

ìVIVO: BÐ>Ñ œ >’ "diagˆ ‰ a b7 > n( ) (28)where ’ ’ ’ ’œ Ò ß ß ÞÞÞß Ó1 2 Q



with ’3 œ Ò ß ß ÞÞÞß ÓW W W3" 3# 3P3

"3 œ Ò ß ß ÞÞÞß Ó" " "3" 3# 3PX

3

7 > œ3a b Ò7Ð> Ñß 7Ð> Ñß ÞÞÞß 7Ð> ÑÓ7 7 73" 3# 3PX

3

ì In the previous expression n denotes a complex white Gaussian baseband noise vectora b>with covariance matrix 5 ˆn

#R



12. ARRAY PROCESSINGì R Consider an array of -sensors with locations

rœ œ Ò ß ß ÞÞÞß Ó œ Ð$ ‚ RÑ< < <" # R ‘< ß < ß 25

a5 œ "ß #ß ÞÞÞß Rì We have seen ( that if the array operates in thesee also Chapter-1 of my book)

presence on narrowband point co-channel signals, thenQ

the observed array signal-vector BÐ>Ñ can be modelled as (29)B 7Ð>Ñ œ Þ Ð>Ñ Ð>Ñ? ’ n

where

unknown matrix( ( (

denotes the unknown message signal-vector

is

ÚÝÝÝÝÛÝÝÝÝÜ

c d’ ?œ œ R ‚Qß Ñß ß Ñß ÞÞÞß ß Ñ

Ð>Ñ ÐQ ‚ "Ñ

Ð>Ñ

W W W

7

) 9 ) 9 ) 9" Q1 22 Q

n an ( complex noise-vector representing the AWGN (power )R ‚ "Ñ 5n2

with the columns of known as 'array manifold vectors'’





ì Modelling of Array Manifold Vectors :(see Chapter-1 of my book) ( ) j ( ) W < < < 5) 9 ) 9ß ßœ Ò ß ß ÞÞÞß Óexpˆ ‰" # R

X (30) ( 1) complex vectorœ R ‚

where wavenumber vector5 ? ?Ð Ñ œ Þ Ð Ñ œ Þ Ð Ñ œ) 9 ) 9 ) 9ß ß ß# J-

#1 1-

-

-

( ) ? ) 9 ) 9 ) 9 9ß œ Þ ß Þ ßc dcos cos sin cos sin X (31) a (3 1) real unit-vector pointing towards the direction ( , )œ ‚ ) 9

ì In many cases the signals are assumed to be on the (x,y) plane i.e. .Ð 9 œ 0°)In this case the manifold vector is simplified to ( ) j ( )W < < < 5) )œ Ò ß ß ÞÞÞß Óexpˆ ‰" # R

X ,0° j œ Ð Ñexp cos sina b1 < <B C) ) (32)(with sensor locations measured in units /2-- Ñ

ì A polular class of arrays is that and, in this of linear arrays: < <C Dœ œ !Rcase Equation 30 is simplified toß

( ) j W <) )œ exp cosa b1 B (33)



ì Summary:An array maps one or more real directional parameters or ( to an : :ß ;Ñ R ‚ "a bcomplex vector , known as array manifold vector, or arrayW WÐ: Ð:ß ;Ñ) orresponse vector, or source position vector.

That is : Ð : ;Ñ Ðœ œ− − − −e V e V" R " RØ Ø

r r) or ( , )W Wp p,q

Note: This should be an ' ' mappingone-to-one

ì :ObjectiveThe general is concerned with array processing problem obtaininginformation about a signal environment by observing the receivedarray signal-vector .BÐ>Ñ



In particular, by observing , an array system aims to solve theB 7 8Ð>Ñ œ Ð>Ñ Ð>Ñ’following general problems:three

1. Detection problem: ? Q œ(i.e. to detect the presence of emitting sources)Q co-channel

2. Estimation problem:to estimate various signal and channel parameters e.g. DOAs ? œ a3 ? T œ 7 Ð>Ñ œ a37

#33

Xe f ? , with 3 X

34œ 7 Ð>ÑÞ7 Ð>Ñ œ a3ß 4 3 Á 4˜ ™3

‡4

n ?T œ œ Ð>Ñ œn n5 X# #e f polarization parameters, fading coefficients, signal spread

3. Reception problem:to receive one signal (desired signal) and suppress theremaining as unwanted cochannel interferenceQ "





13. General Problem FormulationConsider an observed complex signal-vector modelled as followsÐR ‚ "Ñ Ð>ÑB

( ) (34)B 7Ð>Ñ œ Þ Ð>Ñ Ð>Ñ? ’ p n

where

( ) unknown matrix( ( (

denotes the unknown signal-vector

is an ( comp

ÚÝÝÝÝÛÝÝÝÝÜ

c d’ œ œ œ R ‚Q: Ñß : Ñß ÞÞÞß : Ñ

ÐQ ‚ "Ñ

R ‚ "Ñ

? ’ p W W W" # Q

7Ð>Ñ

Ð>Ñn lex noise-vector representing the AWGN (power )5n2

with the set (vector) of generic (unknown) parameters p œ : ß : ß ÞÞÞß :" # Q

known R œ (this is a system parameter) unknown Q œ (this is a signal parameter - number of signals) with Q R (later this condition will be removed)

Estimate etc.Qß: ß : ß ÞÞÞß : ß Ð>Ñ" # Q statistics of ,7 5n#ß







ì Note: The array processing problem is a largethe most general amongstnumber of problems in Comms and Signal Processing - involvingsuperposition of signals - that fit the above general formulation.Representative examples of some specific problems are given below:

i) The harmonic retrieval problem:Consider that is a signal composed of sinusoids with angularBÐ>Ñ Qfrequencies - plus noise.= = =" Qß ß ÞÞÞß

#

i.e. BÐ>Ñ œ Ð > Ñ !3œ"

Q

3E Ð>Ñ3 3cos = : n

where denote the amplitude and phase of the -th sinusoid.E3 3, : 3If the signal is applied at the input of a tapped-delay line with equallyBÐ>Ñ Rspaced taps delay-units apart then at the -th tap (see figureX 5= (this is the system)below) we have

BÐ> 5X Ñ œ Ð Ð> Ñ Ñ = 33œ"

Q!E Ð>Ñ3 3 5cos = 5X= : n

or, using complex notation (for convenience)



BÐ> 5X Ñ œ Ð Ð> Ñ Ñ = 33œ"

Q!E Ð>Ñ3 3 5exp = 5X= : n

Ts Ts Ts Ts

x t( ) x t-( )Ts x t-( 2 )Ts x t- N-( ( 1) )TsThe signal-vector at the output of the whole tapped-delay-lineÐR ‚ "Ñ Ð>ÑBcan be modelled as )B 7Ð> œ Þ Ð>Ñ Ð>Ñ’ n

where

( ( ( unknown matrix

ÚÝÝÝÝÝÝÛÝÝÝÝÝÝÜ

c d’ ’? ( )œ

Ð>Ñ

= œ œ R ‚QÑß Ñß ÞÞÞß Ñ

œ ÒE Ð4 > ÑßE Ð4 > Ñß ÞÞÞ ß E Ð4 > ÑÓ

W W W= = =

= = =

" # Q

" " # # Q QX7 exp exp exp: : :" # Q


is an ( complex noise-vector representing the AWGN (power )

ÐQ ‚ "Ñ

R ‚ "ÑnÐ>Ñ 5n2



In this case the manifold vector, associated with the -th sinusoid, is defined as3

( j j jW = = = =3 3 3 3Ñ œ Ò"ß Ð X Ñß Ð #X Ñß ÞÞÞÞß Ð ÐR "ÑX ÑÓexp exp exp= = =X

..., ]p œ œ Ò ß ß= = = =" # QX

Thus, the Harmonic-retrieval problem is defined as follows:Given the observed signal-vector

) BÐ> œ ÒBÐ>Ñß BÐ> X Ñß BÐ> #X Ñß ÞÞÞß BÐ> ÐR "ÑX ÑÓ= = =X

at the output of the TDL, estimate the number of sinusoids ( ) , theirQfrequencies and their amplitudes = = =" # Q " # Qß ß ÞÞÞß E ß E ß ÞÞÞÞß E



ii) The echo retrieval problem:Consider that a known pulse is transmitted by a sonar or radar system and- >( )is received back via a number ( of delayed echos - plus noise. In this caseQ Ñthe received signal can be modelled asBÐ>Ñ

BÐ>Ñ œ -Ð> Ñ !3œ"

Q

3E Ð>Ñ3 7 n

where , denote the and of the -th echo.73 E3 delay amplitude 3

Ts Ts Ts Ts

x t( ) x t-( )Ts x t-( 2 )Ts x t- N-( ( 1) )Ts

If the signal is applied at the input of a tapped-delay line with equallyRspaced taps delay-units apart, then the signal-vector at theX Ð Ñ= R ‚ " Ð>ÑBoutput of the whole tapped-delay-line can be modelled as follows:



)B 7Ð> œ Þ Ð>Ñ Ð>Ñ’ n

where



ÚÝÝÝÝÝÛÝÝÝÝÝÜ

c d’ ’? ( )œ

Ð>Ñ

7 œ œ R ‚QÑß Ñß ÞÞÞß Ñ

œ ÒE ßE ß ÞÞÞ ß E Ó ÐQ ‚ "Ñ

W W W7 7 7" # Q

" # QX7

nÐ>Ñ is an ( complex noise-vector representing the AWGN (power )R ‚ "Ñ 5n2

In this case the manifold vector associated with the -th echo is3

W(7 7 7 7 73 3 3 3 3Ñ œ Ò-Ð> Ñß -Ð> X Ñß -Ð> #X Ñß ÞÞÞÞß -Ð> ÐR "ÑX ÑÓ= = =X

Thus the echo-retrieval problem is defined as follows:given the observed signal-vector ) BÐ> œ ÒBÐ>Ñß BÐ> X Ñß BÐ> #X Ñß ÞÞÞß BÐ> ÐR "ÑX ÑÓ= = =

X

at the output of the TDL estimate the number of received echos ( ) , theirQdelay and their amplitudes 7 7 7" # Q " # Qß ß ÞÞÞß E ß E ß ÞÞÞÞß E



ii) Direction Finding Problem:

ìConsider co-channel signals plus noise) incident at a linear array of Q Ð Rsensors with unknown directions .) ) )1ß ß ÞÞÞß# Q

The sensor locations are = rœ œ Ò ß ß ÞÞÞß Ó Ð$ ‚ RÑ< < <" # R c d< ß ßB

X0N N0

The array received signal-vector can be modelled asÐR ‚ "Ñ Ð>ÑB

)BÐ> œ Þ7Ð>Ñ 8Ð>Ñ’

where

( ) unknown matrix( ( (

denotes the unknown sign


c d’ ’? ) ) )œ œ œ R ‚QÑß Ñß ÞÞÞß Ñ

Ð>Ñ œ Ò7 Ð>Ñß7 Ð>Ñß ÞÞÞ ß 7 Ð>ÑÓÐQ ‚ "Ñ

) W W W

7

" # Q

" # QX

al-vector is an ( complex noise-vector representing the AWGN (power )8Ð>Ñ R ‚ "Ñ 52

and in this case the manifold vector is associated with the -th signal is3



( exp(-j W )3Ñ œ <BR‚"cosÐ ÑÑ −) V

..., ]p œ œ Ò ß ß) ) ) )" # QX

Thus, the DF array problem is defined as follows:Given the observed signal-vector

) BÐ> œ ÒB Ð>Ñß B Ð>Ñß B Ð>Ñß ÞÞÞß B Ð>ÑÓ" # $ RX

at the output of the array estimate the number of incident signals ( ) , theirQassociated azimuth angles and their powers .) ) )" # Q " # Qß ß ÞÞÞß T ß T ß ÞÞÞÞß T



14. Second Order Statistics of the observed signal vector BÐ>ÑCASE-1: If the signal is observed over BÐ>Ñ œ Ð>Ñ Ð>Ñ’7 8 infiniteobservation interval then its 2nd order statitics can be calculated and are givenby the theoretical covariance matrix (an Hermittian matrix‘BB R ‚R Ñ

Theoretical Model - Multi-dimensional Correlator

BÐ>Ñ ‘BB

i.e. an complex matrix (always Hermittian)‘ X?BB

Lœ Ð>ÑÞ Ð>Ñ ÐR ‚RÑ˜ ™B B

œ

B Ð>ÑÞB Ð>Ñ ß B Ð>ÑÞB Ð>Ñ ß ÞÞÞß B Ð>ÑÞB Ð>ÑB Ð>ÑÞB Ð>Ñ ß B Ð>ÑÞB Ð>Ñ ß ÞÞÞß B#Ð>ÑÞB Ð>Ñ

ÞÞÞß ÞÞÞß ÞÞÞß ÞÞÞ

Ô ×Ö ÙÖ ÙÕ Ø

e f e f e fe f e f e fX X XX X X

" " " # " R‡ ‡ ‡

# " # # R‡ ‡ ‡

X X Xe f e f e fB Ð>ÑÞB Ð>Ñ ß B Ð>ÑÞB Ð>Ñ ß ÞÞÞß B Ð>ÑÞB Ð>ÑR " R # R R‡ ‡ ‡



i.e.

‘ X?BB

Lœ Ð>ÑÞ Ð>Ñ˜ ™B B œ œX

Ú ÞÝ áÛ ßÝ áÜ àðóóóóñóóóóòa b a b’ ’Þ Ð>Ñ Þ Þ Ð>Ñ

œ Ð>Ñ

7

B

7n nÐ>Ñ Ð>Ñ L

œ Xe f’ ’ ’ ’Þ Ð>ÑÞ Ð>Ñ Þ Þ Ð>ÑÞ Þ Ð>Ñ Þ7 7 7 7L L L Ln n n nÐ>ÑÞ Ð>Ñ Ð>Ñ Ð>ÑL L

œ Þ Ð>ÑÞ Ð>Ñ Þ ’ ’X X˜ ™ ˜ ™7 7 L L n nÐ>ÑÞ Ð>ÑL

Þ Ð>ÑÞ Þ Ð>Ñ Þ

œ œ

’ ’

X X˜ ™ ˜ ™7 7n nÐ>Ñ Ð>ÑL

Q‚R R‚Q

L L

i.e. ‘BB œ Þ Þ ’ ‘ ’77L ‘nn (35)

with 2nd order statistics of unknown)‘77 œ Ð>ÑÞ Ð>Ñ Ð>Ñ? X˜ ™7 7 7L œ Ð

2nd order statistics of ‘nn œ Ð>ÑÞ Ð>Ñ œ Ð>Ñ? X˜ ™n n nL



Note that

‘77 œ Ð>ÑÞ Ð>Ñ? X˜ ™7 7 L

œ

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

èëëëëëëéëëëëëëêe f e f e fe f èëëëëëëéëëëëX X X

X

7 Ð>ÑÞ7 Ð>Ñ ß 7 Ð>ÑÞ7 Ð>Ñ ß ÞÞÞß 7 Ð>ÑÞ7 Ð>Ñ

T

7 Ð>ÑÞ7 Ð>Ñ ß

" " " # " Q‡ ‡ ‡

7

# "‡

œ?"

ëëêe f e f

e f e f e fèëëëëëëëéëëëëëëëêX X

X X X

7 Ð>ÑÞ7 Ð>Ñ ß ÞÞÞß 7#Ð>ÑÞ7 Ð>Ñ

T


7 Ð>ÑÞ7 Ð>Ñ ß 7 Ð>ÑÞ7 Ð>Ñ ß ÞÞÞß 7 Ð>ÑÞ7 Ð>Ñ

# # Q‡ ‡

7

Q " Q # Q Q‡ ‡ ‡

œ

œ

?

?

#

T7Q

œ an complex matrix (always Hermittian) - unknownÐQ ‚QÑ



and

‘nn œ Ð>ÑÞ Ð>Ñ? X˜ ™n n L

n n n n n n

œ

Ð>ÑÞ Ð>Ñ ß Ð>ÑÞ Ð>Ñ ß ÞÞÞß Ð>ÑÞ Ð>Ñ

! !Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

èëëëëëéëëëëëê èëëëëëéëëëëëê èëëëëëéëëëëëêe f e f e feX X X

5

X

" " " # " R‡ ‡ ‡

#œ œ œ?

n

f e f e fèëëëëëéëëëëëê èëëëëëéëëëëëê

e f e f

n n n n n n

n n n n

# " # # # R‡ ‡ ‡

#

R " R #‡ ‡

Ð>ÑÞ Ð>Ñ ß Ð>ÑÞ Ð>Ñ ß ÞÞÞß Ð>ÑÞ Ð>Ñ

!


Ð>ÑÞ Ð>Ñ ß Ð>ÑÞ Ð>Ñ

X X

5

X X

œ œ?

n

ß ÞÞÞß Ð>ÑÞ Ð>Ñèëëëëëëéëëëëëëêe fX

5

n nR R‡

#œ? n

(36)œ œ5 ˆ 5n n2 2

R an matrix with unknownÐR ‚RÑ



CASE-2: Consider that the signal is observed overBÐ>Ñ œ ’7 8Ð>Ñ Ð>Ñfinite observation interval equivalent to snapshots.P

These observations (snapshots) P > > >at times " # P, , ..., Ði.e. finite observationinterval) are denoted asÒ Ð Ñß Ð Ñß ÞÞÞß Ð ÑÓB B B> > > R ‚P" # P and represented by the complex matrix —

i.e. — œ? Ò Ð Ñß Ð Ñß ÞÞÞß Ð ÑÓB B B> > >" # P

œ Ò’ ’ ’Þ Ó7 8 7 8 7 8Ð> Ñ Ð> Ñß Þ Ð> Ñ Ð> Ñß ÞÞÞß Þ Ð> Ñ Ð> Ñ" " # # P P

œ Þ’ Œ

with (ÚÛÜ’ œ Ò ß ß ÞÞÞß ÓW W W" # Q ÐR ‚QÑ

Q ‚ PÑÐR ‚ PÑ

Œ

œ Ò Ð> Ñß Ð> Ñß ÞÞÞß Ð> ÑÓœ Ò Ð> Ñß Ð> Ñß ÞÞÞß Ð> ÑÓ

7 7 7" # P

" # Pn n n(37)

where the matrices and (as well as the dimension are unknown’, Œ Q Ñ



the 2nd order statitics of are given by the practical covariance matrix BÐ>Ñ ‘BB

Practical Model:

BÐ>Ñ ‘BB

‘BB œ " "P P !

6 "

P

=B BÐ> ÑÞ Ð> Ñ œ Þ6 6

L L— —

i.e. . .‘BB œ Þ œ Þ

œ

"P— — ’ ŒŒ ’

‘

L L L

77

ðñò ðñò" "P P

‘

. L

œ nn

(38)



Two Important Comments:1. In an array system the matrix theoretical or practical) contains all the‘BB Ð

geometrical information with respect to the about the various sources array reference point.

2. Procedure for generating random vectors (snapshots) with given 2ndP Border statistics (useful for computer simulation studies):Let such as D D D− G œR X˜ ™. L

Rˆ

‘ XBB œ ˜ ™B B. L

‘ „ƒ„ „ƒ ƒ „ „ƒ ƒ „BBL L Lœ œ œ

" " " "# # # #ˆR

œ

œ

„ƒ X ƒ „" "# #ðñò˜ ™D D. L

Rˆ

L

œ X „ƒ ƒ „š ›" "# #D D. L L

. i.e. œ Ö × œX „ƒ

„ƒ

B B BÅ

"#

"#

D

DL

↑White-Gaussian(0,1)



15. Revisiting the General Problem FormulationConsider an observed complex signal-vector modelled asÐ ÑR ‚ " BÐ>Ñfollows

( ) (39)BÐ>Ñ œ Þ ? ’ p 7 8Ð>Ñ Ð>Ñ

or, equivalently, its 2nd order statistics modelled as

(40) ‘ ’ ’BBLœ ‘ 5 ˆ77 R

#n

where


: signal-vector (unknown)

2nd or

ÚÝÝÝÝÝÝÝÝÛÝÝÝÝÝÝÝÝÜ

c d’ ’? ( )œ p œ œ R ‚Q: Ñß : Ñß ÞÞÞß : Ñ

ÐQ ‚ "Ñ

À ÐQ ‚QÑ

W W W" # Q

7Ð>Ñ

‘77 der statistics of (unknown)

: is an ( AWGN vector - power unknown

7Ð>Ñ

R ‚ "Ñ 8Ð>Ñ 582



with

the set (vector) of generic parameters ÚÝÝÝÝÝÛÝÝÝÝÝÜ

p œ (unknown) : ß : ß ÞÞÞß :" # Q

R

Q

œ

œ

known

(this is a system parameter)

(unknown this is a signal parameter - number of signals)(later this condition will be removed)with Q R

Note that the system used to observe can be represented byBÐ>Ña function known f{.} which maps an unknown real parameters to a:manifold vector WÐ:)

: )f{.}Ø WÐ:

Estimate etc. Qß: ß : ß ÞÞÞß : ß" # Q ‘ 577#, n ß



General Problem Formulation or

Condition: Q R

Estimate etc. Qß: ß : ß ÞÞÞß : ß" # Q ‘ 577#, n ß



16. The 'Detection' Problemì This is to determine the parameter Q

i.e. to determine the number of signals and thus the dimensionsof the vectors/matrices 7Ð>Ñß’, and ‘ Œ77

Ðe.g. how many emitting sources/transmitters are present in anarray environment i.e. ) to detect the presence of sourcesQ



16.1. Detection Criteria for infinite obervation interval ( = snapshots)P _

ì Based on we can form the matrix (representing theBÐ>Ñ ‘BB

statistics of :BÐ>ÑÑ

‘ ‘BB 77œ Þ Þ œ œ

’ ’L

‘signals

‘

5 ˆnn

n#

R

ì When the number of sources is smaller than the number ofQsystem dimensions (e.g. number of array-sensors) then theRdeterminant of the is equal to zero‘signals i.e. if ( )=0Q R Ê det ‘signals

This is due to the fact that the presence of an emitting sourceincreases the rank of the matrix by one.‘signalsi.e. ranke f‘signals œ Q Ê œ Qranke f‘BB 5 ˆn

#R



ì However, . . .

‘BB œ

œ œ

‘

„ „ „ „signals

A L # L

5 ˆ

5n

n

#R

where A

--

-œ

! ÞÞÞ ! ! ! ÞÞÞ !! ÞÞÞ ! ! ! ÞÞÞ !ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ! ! ÞÞÞ ! ! ÞÞÞ !! ! ÞÞÞ ! ! ÞÞÞ !! ! ÞÞÞ ! ! ÞÞÞ !ÞÞÞ ÞÞÞ ÞÞÞ

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

"

#

Q

!!

ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ! ! ÞÞÞ ! ! ! ÞÞÞ

ÐR ‚ RÑ

...!

. .Ê œ Ð Ñ

œ

‘BB „ „

ƒ

ðóóñóóòA 5 ˆn#

RL



‘BB œ

! ÞÞÞ ! ! ! ÞÞÞ !

! ÞÞÞ ! ! ! ÞÞÞ !Þ

„.

ðóóóóóóóóóóóóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóóóóóóóóóóóóò


-

-"

#

++

5

5

#

#n

nÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ

! ! ÞÞÞ ! ! ÞÞÞ !

! ! ÞÞÞ ! ! ÞÞÞ !

! ! ÞÞÞ ! ! ÞÞÞ !ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ

! ! ÞÞÞ ! ! ! ÞÞÞ

œ œ

-

A

Q+55

5

5

5 ˆ

#

#

#

#

#R

n

n

n

n

n

...

.

ƒ

„L

ì Note: another useful expression is

‘BB œ Þ Þ œ Ò ß ÓÞ Ò ß Ó„ ƒ „

L L „ „

ƒ= =

=„ „ƒ8 8

8” •

œ „ ƒ „= = =Þ Þ L L„ ƒ „8 8 8





ì This implies that the eigenvalues of the data covariance matrix are‘BB

related to the eigenvalues of the emitting signals covariance matrix ‘signalsas follows: eig eig3 3e f e f‘ ‘BB œ signals 5n

#

Now, since the smallest eigenvalue of is zero,‘signals

that is, ,eigmine f‘signals œ !with multiplicity , that meansR Q

eigmine f‘BB œ 5n#

also with multiplicity .R Q

ì Therefore, , the number of emitting sources can betheoretically Qdetermined by the of the covariance matrix of theeigenvalues ‘BB

received signal-vector , and more specifically by the followingBÐ>Ñexpression multiplicity of min. eigenvalue of )Q œ R Ð ‘BB (41)

N.B.: The above expression cannot be used in practice. Why?



16.2. Detection Criteria for a finite obervation interval ( snapshots)P

‘BB œ

! ÞÞÞ ! ! ! ÞÞÞ !

! ÞÞÞ ! ! ! ÞÞÞ !ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ

! ! ÞÞÞ ! ! ÞÞÞ !

! ! ÞÞÞ ! ! ÞÞÞ !

! ! ÞÞÞ

„.


-

-

-

"

#

Q

++

+

5

5

5

5

#

#

#Q

#Q"

1

2

! ! ÞÞÞ !

ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ

! ! ÞÞÞ ! ! ! ÞÞÞ

5

5

#Q#

#R

...

.„L

ì However, 5 5 5 5 5# # # # #" # Q Q" RÁ Á ÞÞÞ Á Á Á ÞÞÞÞ Á

but 5 5 5 5 5 5# # # # # #" # Q Q" Rn ¸ ¸ ¸ ÞÞÞ ¸ ¸ ¸ ÞÞÞÞ ¸

ì if is known thenQ i.e. the average of the smallest eigenvalues5s#

n œ R Q 5 5 5 5s# "

RQ# # #Q" Q# Rn œ ÞÞÞÞ a b (42)



1) AIC - Detection Criterion

AIC"

3œ"

R

3

" # $

" #

œ #P

.

. . .. ..

R RR "

#"

Î ÑÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÏ Ò

Ô × Ô × Ô ×Ö Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÕ Ø Õ Ø Õ Ø

# Î ÑÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÏ Ò

ln ln... ...3

1

R "

#"

.

. . .. .

.

...3

....ln

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

!3œ"

R

3

" # $

" #

"

an real vector # œ ÐR ‚ "Ñ

! #R" #R "ÞÞÞ ÞÞÞ

R # R #R " R "

Ô × Ô ×Ö Ù Ö ÙÖ Ù Ö ÙÖ Ù Ö ÙÖ Ù Ö ÙÕ Ø Õ Ø

(43)



2) Minimum Description Length (MDL) Detection Criterion

MDL"

3œ"

R

3

" # $

" #

œ P

.

. . .. ..

RR "R #ÞÞÞ#"

Î ÑÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÏ Ò

Ô × Ô × Ô ×Ö Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÕ Ø Õ Ø Õ Ø

# Î ÑÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÐ ÓÏ Ò

ln ln...

1

RR "R #ÞÞÞ#"

.

. . .. .

.

ln


!3œ"

R

3

" # $

" #

"

....

an real vector P œ ÐR ‚ "Ñ

! #R" #R "ÞÞÞ ÞÞÞ

R # R #R " R "

"# ln

Ô × Ô ×Ö Ù Ö ÙÖ Ù Ö ÙÖ Ù Ö ÙÖ Ù Ö ÙÕ Ø Õ Ø

(44)



Note :=3Ñ the of the vector or is minimumif first element AIC MDL then Q œ 0

if the of the vector or is minimumsecond element AIC MDLthen Q œ 1

if the of the vector or is minimumthird element AIC MDLthen Q œ 2

etc.

33Ñ ß 3 œ " R 3 (for to with denotes the -th. . . ÞÞ . Ñ3 " # R< < <eigenvalue of ‘BB

333 ß) M. Wax and T. Kailath "Reference: Detection of Signals by Information TheoreticCriteria" IEEE Trans. on ASSP, vol. ASSP-33, pp. 387-392, Apr. 1985ß



16.3. Detection Problem- Summary

"Ñ P œ _If i.e. Theoretical Covariance matrixÐ ‘ XBB œ ˜ ™B BÐ>Ñ Ð>Ñ. L

then • Q œ R Ðmultiplicity of min. eigenvalue of )‘BB

• noise power min. eigenvalue of 5 ‘8#

BBœ

2 If finiteÑ P œ

i.e. Practical Covariance matrixÐ ‘BB œ " "P P

=!6 "

P B BÐ> ÑÞ Ð> Ñ œ Þ Ñ6 6

L L— —

then • can be found using AIC, or MDL, criterionQ œ • the average of the smallest eigenvalues5s#

n œ R Q i.e 5 5 5 5s# "

RQ# # #Q" Q# Rn œ ÞÞÞÞ a b

Note: number of array elementsR œ



17. The 'Estimation' Problem

ì There are parameter estimation techniques,many

ì Currently the most powerful estimation techniques can beconsidered the Signal-Subspace type techniques

Signal-Subspace type techniques a class of techniques= superresolution



17.1. The ML-AlgorithmConsider an observed complex signal-vector modelled asÐ ÑR ‚ " BÐ>Ñfollows (45)BÐ>Ñ œ Þ? ’( )p 7 8Ð>Ñ Ð>Ñ

In this case the observation P = at times > > >" # P, ,... Ði.e. finite observationinterval) are defined as the complex matrix Ò Ð> Ñß Ð> Ñß ÞÞÞß Ð> ÑÓB B B" # P R ‚ P —

since the noise is modeled as a zero mean complex Gaussian random process,with a covariance matrix then the observed array signal has‘ 5 ˆ?

88 R#œ BÐ>Ñ

a mean vector and covariance matrix which are given as follows:

Xe fBÐ>Ñ œ Þ Ð> Ñ’a bp 7 6

X X X 5 ˆ˜ ™a b a be f e fB B B BÐ>Ñ Ð>Ñ Ð>Ñ Ð>Ñ Þ œH

n#

R



This implies that if there are observations, which are independent, then thePconditional probability density function (likelihood function - LF)

LF pdfB B " " P#œ Ð> Ñß Ð> Ñß ÞÞÞß Ð> Ñ ß? Œ ºB B B p Œ 5ß n

is as follows:

LF B 6" "

Þ#œ Þ Ð> Ñ Þ

P

6 œ "# Š ‹k k

1R deta b5 ˆ 5n n#

R#exp B ’a bp 7Ð> Ñ Ð Ñ6 46

By taking the ln of Equation-46 we have

ln lna b k kˆ ‰ "LFB 6#œ PÞRÞ Þ Ð> Ñ Þ

" P

6 œ "15

5nn

## 6B ’a bp 7Ð> Ñ Ð Ñ47

Let , i.e.5sn œ arg max ln5n

e fa bLFB



5 55 5

s œ Ð> Ñnn n

narg max ln Ÿˆ ‰ " k kPÞRÞ Þ Ð> Ñ Þ

" P

3 œ "1 #

# 6B 6#’a bp 7 Ð Ñ48

Then it is not difficult to see that

5s Ð> Ñ#

6n œ Þ Ð> Ñ Þ"

PÞR

P

6 œ "

" k kB 6#’a bp 7 Ð Ñ49

Substituting given by Equation 49 back into Equation 47 and then5sn

maximizing the result with respect to the signal parameters we have

a b k k ŸŠ ‹ a b Ÿ k ka b

pp

p

pp

s œ PÞRÞ Þ Ð> Ñ Þ Ð> Ñ Ð Ñß

s"

s

P

6 œ "

œ PÞRÞ Þ Ð> Ñ Þ Ð> Ñß

"

PÞR

P

3 œ "

, 50Œ 15 ’Œ 5

Œ’

QP#

# 6 6#

3 3#

arg max ln

arg max ln

nn

""

B 7

B 7



i.e. , 51a b k kœ a bpp

ps œ Ð> Ñ Þ Ð> Ñ Ð Ñß

P

3 œ "Œ ’

ŒQP 3 3#arg min ! B 7

However, by keeping constant the parameters Equation-51 becomespß

a b k k Ÿa bŒ ’ŒQP 3 3

#œ Ð> Ñ Þ Ð> Ñ Ð ÑP

3 œ "arg min " B 7p 52

Equation-52 has an analytical solution given by the following expression

Œs œ Š ‹’ ’ ’a b a b a bp p pL L"

— Ð Ñ53i.e.

7s Ð ÑÐ> Ñ œ3 Š ‹’ ’ ’a b a b a bp p pL L"

BÐ> Ñ3 54

that is, is given as a function of Œ p

Substituting given by Equation-54, back into Equation 51 we have7s Ð> Ñ3 ,



a b k k Ÿ" Ÿ" ¸ ¸ Ÿ" k k Ÿ"

p p

p

p

p

p

s œ Ð> Ñ P

3 œ "

œP

3 œ "

œP

3 œ "

œP

3 œ "œ

QP 3#

#

#

arg min

arg min

arg max

arg max

arg max

B

’

’

’

’

Þ Ð> Ñ

Þ Ð> Ñ

Þ Ð> Ñ

Ð> Ñ Þ Ð> Ñ

B

B

B

B B

3

¼3

3

3 3L

˜ ™ˆ ‰˜ ™ˆ ‰X<

X<

—L

L

—

— —

’

’

Þ

Þ Þœ

Ð Ñ

arg maxp

55



i.e.

p psQP

œ Ð Ñarg maxe fa bX< Þ’ ‘BB 56

where ’ ’ ’ ’’ œ Ð Ð Ð Ðp p p p). ) ) )ˆ ‰L L"

Equation 56 provides the ML estimator for the DOA's. This is a highly non-linear multivariable cost function with many local minimum and a globalmaximum point thats is equal to X< a bƒ=

IndeedX< X<a b Š ‹ ’ ’Þ‘BB œ Þ„ ƒ „Þ Þ œL

œ X< ’ÞÒ ß ÓÞ Ò ß Ó„ „ „ „ƒ ƒ= 8 = 8=

8

L” • œ X< Š ‹’ÞŠ ‹„ ƒ „ „ ƒ „= = 8 8= 8

L L



maxp e fa bÎ ÑÐ ÓÐ ÓÐ ÓÏ ÒïX< ’ ’ ’

Þ Þ‘BB œ X< s s„ ƒ „ „ ƒ „

„

= = 8 8

=

= 8L L

RßRœ

ðóóóñóóóò

œ X< Š ‹„ ƒ „= = =L

œ X<Î ÑÐ ÓÏ Òƒ „ „

ˆ

= ==L

Q

ï œ œ ÞÞÞ œX< a bƒ - - - -= " # Q 3

3œ"

Q!where ’ ’s Ðsœ? p

QP)



Summary of ML algorithm (assumption )Q œ known1) Estimate the set of parameters by maximizing the cost functionp

X<a b’Þ‘BB . That is

57psQP

œ Ð Ñarg maxp e fa bX< Þ’ ‘BB

2) Based on the estimated parameters evalute the manifold matrix p ps sQP QP

’ˆ ‰3) Weight the received signal-vector with Š ‹’ ’ ’ˆ ‰ ˆ ‰ ˆ ‰p p ps s s

QP QP QP

L L"

to estimate the unknown 'message'-signals. That is

587sÐ> Ñ3 œ Š ‹’ ’ ’ˆ ‰ ˆ ‰ ˆ ‰p p ps s sQP QP QP

L L"

BÐ> Ñ3 Ð Ñ

4) Estimate the noise power by evaluating the following expression

59 5s Ð> Ñ#

6n œ Þ Ð> Ñ ÞP

6 œ "

"PÞR 6

#! k kB ’a bp 7 Ð Ñ



17.2. Subspace-Type Algorithmsì In this type of algorithms the parameter is assumed knownQ

( and involves in some way, or another, two concepts:Q R Ñ

3) "manifold"the concept of the associated with thesystem/problem's characteristics in the case of arrays knownÐas "array manifold" . It is independent of the noisy observedÑsignal-vector and its properties.BÐ>ÑThis is a (e.g. a curve, surface etc) -non-linear subspaceembedded in an -dimensional observation spaceR

33) "signal-subspace"the concept of the associated with theobserved signal-vector and its propertiesBÐ>ÑThis is an of dimensionality equalunknown linear subspace Q R Q R - embedded in an -dimensional ( Ñ



ì−−

Solution = points system's manifold'signal subspace' of Q

Ð>Ñœ B

As a result the objective is firstly, from the data, to estimate thesignal subspace and then to to find its=/+<-2 the manifoldintersection with the estimated signal-subspace



17.2.1. The concept of the "Manifold"

ì we have seen that a system (e.g. an array system) maps one or more realparameters or ( to an vector , known as: :ß ;Ñ R ‚ "a b W WÐ: Ð:ß ;Ñ) or'manifold vector'

That is (for one unknown parameter per signal/source)

: Ð− − −e V e" R R )f{.}Ø W p or

or Ðfor two unknown parameters per signal)

( , ) f{.}

: ; Ñ Ð− − − −e e V e" " R RØ W p,q or

Note:1)This should be an ' ' mappingone-to-one2) the system/function is assumed to be f{.} known (i.e. system is 'calibrated', i.e. there are no modelling uncertainties in the system)



ì By recording s as a function ofthe locus of the manifold vectorthe parameter e.g. direction), a "continuum" (i.e. a: Ðgeometrical object lying such as a curve or surface) is formed in an -dimensional spaceR .

This (locus of manifold vectors i.e.geometrical objectWÐ a:p), ) is known as the system's manifold.

ì In an array system the manifold (array manifold) can becalculated (and stored) from only the knowledge of the locationsand directional characteristics of the sensors.



ì R Let be the manifold vector of a system of W( ): − −V eR R or dimensions (e.g. of an array of sensors) where is a genericR :system parameter.

This is a single-parameter vector function and as varies the:point will trace out a (see figure), embedded in anW( ): curve TR Ð Ñ-dimensional space .V eR R or

Array Manifold

S( ))

Origin

N-dim complex observation space This was expected, as vectorfunctions of one parameter areused to define space curvesÐalso known assingle-parameter manifoldsÑÞ



ì The curve ( ) is formallyT which is the locus of all manifold vectors WÐ:Ñ a:

defined as follows:

Array Manifold - T Hœ Ö ß a À − ×? WÐ:Ñ : :− ÐV eR R or Ñ :

where denotes the .H: parameter space

ì This curve is said to be a T regular parametrized differentialcurve if W

ÞÐ:Ñ :Á ß0R a

Ði.e. exists at all points on the manifold curve )tanget vector T

ì For more information see Chapter-2 of my book on Differential Geoemtry in ArrayProcessing



For a point on the (the locus of the manifold vectors: manifold curve T Wa b: ) the most important parameters are:

s=0

M i o d an f l l ine

Origin


p=0 SÐ Ñp

ì the , (the most basicarc length =feature of a curve)

ì the .rate of change of arc length =

ì the of the curveset of curvatures,3a b: for 1,2,... forming a matrix3 œknown as the Cartan Matrix ‚a b: Þ

‚

,, ,

,

,,

a bÔ ×Ö ÙÖ ÙÖ ÙÖ ÙÖ ÙÖ ÙÕ Ø

a ba b a ba ba ba b

= œ

! = ! â ! != ! = â ! !

! = ! â ! !ã ã ã ä ã ã! ! ! â ! =! ! ! â = !

"

" #

#

."

."



ì In a similar fashion there are (unknown) parameters if two ( , ): ;per signal is a then WÐ: ;Ñ, − ÐV eR R or Ñ two-parametermanifold vector (a vector function)and as varies the point will form a surface (see( , ) ( ): ; :ß ;W `figure), formally defined as followsArray Manifold - ( )` Hœ Ö ß a À − ×? WÐ: ;Ñ : : ;, ,q ,− Ð ÑV eR R or where denotes the .H parameter space

SÐ ß Ñ) 9


Origin

9

)

9œ!

9 1œ y#

9

ì This surface is the locus`

of all manifold vectorsWÐ: ;Ñ : ;, , a



S( )p,q


Origin

KG( )p,q

M


Origin

A B

S( )p,q

,g( )p,q

ì For a on the manifold surface the most important parameterspoint a b:ß ;

are: ˆ The manifold metric: († :ß ;Ñ ˆ The Christoffel matrices: =>Ð:ß ;Ñ ˆ The Gaussian curvature: OÐ:ß ;Ñì For a on the manifold surface, the parameters of interest are:curve

ˆ The arc length: = ˆ The geodesic curvature: ,1 Ð-?<@/ œ Êgeodesic ,1=0Ñ



17.2.2. The concept of the "Signal Subspace"

ì The first step is to utilize the observed (received) signal vector

(infinite observation interval)BÐ>Ñ œ a>’7 8Ð>Ñ Ð>Ñ or, over snapshots (finite observation interval)P

to estimate the " ".signal subspace

ì The " should have (in mostsignal subspace" dimensionalitycases) equal to known - or estimated) and the signal termQ Ð’.7Ð>Ñ belongs always to this subspace.

i.e. "Ñ œ Qdim( )signal subspace #Ñ −’7Ð>Ñ signal subspace



ì As the dimensionality of this subspace is and the number ofQcolumns of is equal to this’ Q Ð Ñremember ’ œ Ò ß ß ÞÞÞß ÓW W W" # Q

implies that"signal subspace" œ _[ ]’ with [ ]dima b_ ’ œ Q

That is, the signal subspace is spanned by the unknown Qmanifold vectors associated with the signals (one signal - oneQvector)

Origin

N-dim complex observation spaceL[ ]’The complement subspace to thesignal subspace is known as" "noise subspacei.e. " " noise subspace œ _[ ]’ ¼

dimwith [ ]a b_ ’ ¼ œ R Q



ì Signal-Subspace type techniques are based on thepartitioning observation space into

the andˆ Signal Subspace _[ ] ’

the .ˆ Noise Subspace _[ ]’ ¼

L[ ]’Origin

N-dim complex observation spaceL[ ]’

However, as the matrix ’remains unknown, the signalsubspace and consequentlythe noise subspace remainunknown.

Note: observation-space œ œ œ? _ _ _[ ] [ ] [ ]— ‘BB BÐ>Ñ



ì : This is achieved byEstimation of the two subspacesperforming an Eigenvector decomposition of the received datacovariance matrix ‘BB

i.e.

‘BB œ Þ Þ œ Ò ß ÓÞ Ò ß Ó„ ƒ „

L L„ „

ƒ= =

=„ „ƒ8 8

8” • (60)

‘BB œ

œ Þ

„ ƒ „= = =Þ Þ L L

#R

ðóñóò„ ƒ „8 8 8

5 ˆn

(61)

Remember: ‘BB œ Þ Þ Þ’ ’‘ 5 ˆ77 RL #

n

This implies that signal subspace œ _ _[ ] [ ]’ „œ =

and consequently noise subspace œ _ _ _[ ] [ ] [ ]’ „¼ ¼œ = œ „8



Summary:ì œ œ Signal subspace _ _[ ] [ ]„= Noise subspace „8

dim dima b a b_ _[ ] [ ]„= œ Q œ R Q„8

L[ ]„s

L[ ]„nOrigin


observationspace= =_ _[ ] [ ]‘ —BB

ì œ Note that: _ _[ ] [ ]„8 „=¼

although _ _[ ] [ ]„ ’ „ ’= =œ Á



17.2.3. Intersections of with the array manifold_[ ]„=

ì Both the and are embedded on the same -manifold _[ ]„= Rdimensional observation space

L[ ]„s

L[ ]’Origin


L[ ]’

Array Manifold

T




S( )pL[ ]„s

L[ ]’Origin

L[ ]’

Array Manifold

Tp

z( )p

p1 pi

pM

ì Therefore, the intersection of the manifold with will_[ ]„=

provide the end-points of the columns of the matrix ’i.e. it will provide the parameters : ß : ß ÞÞÞß :" # Q



ì Example:

S2

S1


Origin

S( )pp

L[ ]’

T

Note: W ß W ß I ß I ¼" # " # _[ ]„8

W ß W ß I ß I −" # " # _[ ]„=

_[ ]„= is a plane which intersects the array manifold in 2 pointsW ß W" #



ì How to estimate the intersections of with the array manifold_[ ]„= . Employ the following procedure:

ˆ : Let be a parameter value. Form the associated andW(:Ñthen project on to the subspace This will giveW(:Ñ _[ ]„8 Þus the vector

DÐ:Ñ œ Þ„8W(:Ñ

ˆ The norm-squared of can be written asDÐ:Ñ

0 (:Ñ œ Ð:Ñ Ð:Ñ œ Þ ÞD DL LW W( (:Ñ :ÑL„ „8 8

(62)

ˆ It is obvious that iff or0( 0:Ñ œ

::œ œœ

::"

#



ˆ Therefore, we search the array manifold, i.e. we evaluate theexpression (62), , and we select as our estimates the a: :'swhich satisfy

0( 0:Ñ œ ß i.e. W W( (:Ñ :ÑLÞ Þ Þ œ !

œ

œ

ðóóñóóòí

L„

„

„

„

8

8

8

8

Ê W W( (:Ñ :ÑLÞ Þ œ !„8

Ê Þ Þ Þ Þ ÞW W( (:Ñ :ÑL „ „ „ „8 88 8

Î ÑÏ ÒðñòL L

"

œ ˆR‚R

œ !

Ê Þ Þ a3W W( (:Ñ :ÑL „ „8 8Þ L œ ! (63)



ì Equation-63 is known as the Multiple Signal Classification(MUSIC) algorithm and it is a Signal-Subspace type technique.

ì Example of MUSIC used in conjunction with a Uniform LinearArray of 5 receiving elements.The array operates in the presence of emitting sources with3 unknownDOA's (30° ° (3 ° ° ( 0° °ß ! Ñß & ß ! Ñß * ß ! Ñ

0 20 40 60 80 100 120 140 160 180-5

0

5

10

15

20

25

30

35MuSIC spectrum (Theoretical) SNR=10dB

Azimuth Angle - degrees

dB



ì MUSIC Limitations: MUSIC breaks down if some incidentsignals are coherent, i.e. fully correlated, (e.g. multipath situationor 'smart' jamming case)

Then or, to be more precise, ._ _ _ _[ ] [ ] [ ] [ ]„ ’ „ ’= =Á −

Therefore the 'intersection' argument cannot be used.



e.g. same environment as before but the (30° ° & (3 ° ° ß ! Ñ & ß ! Ñ sources are coherent (fully correlated)

0 20 40 60 80 100 120 140 160 180-5

0

5

10

15

20

25

30

35MuSIC spectrum (coherent)


dB

ì b algorithms which can handle coherent signals.



17.2.4. Resolution thresholdì Consider two signals with parameters 'close together'

1st source 2nd source Commentsparametersignal-to-noise ratio

: : :" # "

#

œ ? ?: : = very smallSNR SNR1

Resolution threshold ( snapshots available):P

?: ‚ res-thr œ" # " "

=Ð:ÑÞ

v É Š ‹%

#"

"R

% %, È ÈSNR SNR1‚P ‚P#

whereÚÛÜ=Ð: Ñ œ :Þ

: œ

v v

v

1

,

= principal curvature

l lr sin: :

#

"

" #

ì Ê Note: Pp_ : p!? res-thr

ì (Reference: Chapter-8 of my book on "Differential Geometry in Array Processing")



17.3. Estimation of Signal Powers, Cross-correlation etc.

ì Firstly estimate the DOA's and noise power and then use theconcept of ' ' to estimate pseudo inverse ‘77

i.e.

step- : Based on , estimate and " ‘BB p 58#

step-2: form ’

step-3: ‘ 577 8#œ Þ Þ’ ’# #a b‘BB ˆR

L (64) where pseudo-inverse of’ ’ ’ ’ ’# œ ˆ ‰L L"

Þ Þ œ (65)



ì proof of Equation-64:

‘ ’ ‘ ’ 5 ˆBB 77 RL #

8œ Þ Þ ‘ 5 ˆ ’ ‘ ’BB R 778

# L œ Þ Þ

By pre&post multiplying both sides of the previous equationwith the pseudo inverse of we have’

’ ‘ 5 ˆ ’ ’ ’ ’ ’ ‘ ’ ’ ’ ’

’ ’

# #

# #

Þ Þ œ Þ Þ Þ Þ Þ Þ Þ Þ

œ œ

a b èëëëéëëëê èëëéëëêˆ ‰ ˆ ‰BB R 778# L L L LL " "

L

Ê Þ Þ œ’ ‘ 5 ˆ ’ ‘# #a bBB R 778# L

ì Note that:5 ‘8#

BB (min eigenvalue of or given by Equation-0)œ œ 893=/ :9A/<



18. The 'Reception' Problem

18.1. Array Pattern & Beamforming

ì If the array elements are weighted by complex-weights then thearray pattern provides the gain of the array as a function ofDOAse.g. if then ) )3 3 3 3

LÈ 1Ð Ñ œW A W (66)

where denotes the gain of the array for a signal arriving1Ð Ñ)3 from direction )3Thenß the function , , is known as the array pattern1Ð Ñ a3)3 (67)



ì : N.B. default pattern: 1Ð Ñ œ œ)3 3 3XR" W Wl l

"‡

i.e. i.e. no weights)A "œ R Ð

e.g. Array Pattern of a Uniform Linear Array of 5 elements( i.e. no weightsA "œ ß Ñ

0 20 40 60 80 100 120 140 160 180-20

-15

-10

-5

0

5

10initial pattern


gain

in d

B



ì Spatial Correction WeightThe array pattern a number of .has lobes

The one is called the while the largest remaining'main lobe' lobes are known as .'sidelobes'

beamwidth arcœ # ‚sinˆ ‰-1R.")!

. œ . œ Ê œ # ‚ Ñintersensor spacing (if beamwidth arc-1# R

# ")!sinˆ ‰To towards a specific (known)steer the main lobedirection a can be used:main

lobe'spatial correction weight' Amain

lobewhich should be equal to

rA A Wmain main main mainlobe lobe lobe lobe

= j =expŒ Êœ

X 5Ð: Ñ Ð: Ñ



ARRAY PATTERNfor arrays having 2 and 5 sensorsR œ ß$ß %

Mainlobe at 90°

< < < <BX X X X

B B Bœ Ò !Þ&ß !Þ&Ó œ Ò "ß !ß "Ó œ Ò "Þ&ß !Þ&ß !Þ&ß "Þ&Ó œ Ò # "ß !ß "ß #Ó- - - - - ,- A A A Aœ Ò"ß "Ó œ Ò"ß "ß "Ó œ Ò"ß "ß "ß "Ó œ Ò"ß "ß "ß "ß "ÓX X X X



ARRAY PATTERNfor arrays having 2 and 5 sensorsR œ ß$ß %

Mainlobe at 120°

< < < <BX X X X

B B Bœ Ò !Þ&ß !Þ&Ó œ Ò "ß !ß "Ó œ Ò "Þ&ß !Þ&ß !Þ&ß "Þ&Ó œ Ò # "ß !ß "ß #Ó- - - - - ,-

rA œ Ð W "#! ß ! Ð"#! ß ! Ñ° ° ° °Ñ œ expˆ ‰j œ

X 5

j œ expˆ ‰<BcosÐ"#! Ñ° Ðsimplified)



ì A is an array system whichbeamformerreceives a 'desired' signal and (according to a criterion) suppresses co-channelinterference and noise effects,by synthesizing an array pattern with high-gain towards the DOA of the desiredsignal and deep nulls towards the DOAs of the interfering signals (adaptive arrays).

ì Superresolution 'blind' beamformersCurrent state of art:



18.2. Some Popular Beamformers

ìWIENER-HOPF Beamformer:

A Wœ Þ‘BB"

./=3</.=318+6

(68)

ˆMaximizes the SNIR at the array output.ˆ It is optimum wrt SNIR criterionˆ It is a conventional beamformer (i.e. resolution is a function of

the SNRinÑˆ No need to know the DOAs of the interfering signalsˆ Ðplease try to prove Equation-68)



ìModified WIENER-HOPF Beamformer: A Wœ -Þ Þ‘8 N

"./=3</.=318+6

+ (69)

where a constant scalar- œ

ˆ comments similar to Wiener-Hopf

ˆ robust to 'pointing' errors (i.e. robust to errors associated with the direction of the desired signal)



ìA Superresolution Beamformer based on DOA estimation:

A Wœ Þ’N¼

./=3</.=318+6

(70)

where ’ ’œ Ò ß ÓW./=3</. N=318+6

ˆ Provides complete (asymptotically) interference cancellation.ˆ Maximizes the SIR at the array output.ˆ It is optimum wrt SIR criterionˆ It is a superresolution beamformer (i.e. resolution is not a

function of the SNRinÑˆ Needs an estimation algorithm to provide the DOAs of the

incident signals.



ìA Superresolution Beamformer not based on DOA estimationof interfering sources

A Wœ Þ„84./=3</.=318+6

(71)

where noise subspace of „ ‘8 8 N4œ +

Note: covariance matrix where the effects of the‘8 N+ œ desired signal have been removed

ˆ Provides complete (asymptotically) interference cancellationˆ Maximizes the SIR at the array output.ˆ It is optimum wrt SIR criterionˆ It is a superresolution beamformer (i.e. resolution is not a

function of the SNRinÑ



18.3. Examples of Array Patterns (Beamformers)ìConsider a uniform linear array of 5 elements operating in the

presence of one 'desired' source and two unknown interferences. 'desired' DOA : ( 0° ° knownˆ * ß ! Ñ œ DOAs of interfering sources: ˆ (30° ° Unknownß ! Ñ œ (35° ° Unknownß ! Ñ œ



ìWIENER-HOPF Beamformer (Equation 68):

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5W -H array pattern SNR=40dB


gain

in d

B

0 20 40 60 80 100 120 140 160 180-30

-25

-20

-15

-10

-5

0W -H array pattern SNR=10dB


gain

in d

B



ìSuperresolution Beamformer (Equation-71)

0 20 40 60 80 100 120 140 160 180-160

-140

-120

-100

-80

-60

-40

-20

0

20array pattern (90 deg) - Complete Interference Cancellation


gain

in d

B



ìSuperresolution Beamformer (Equation-70) .(all DOAs known)

a) if desired source=30° b) if desired source=35°

0 20 40 60 80 100 120 140 160 180-160

-140

-120

-100

-80

-60

-40

-20

0



gain

in d

B

0 20 40 60 80 100 120 140 160 180-160

-140

-120

-100

-80

-60

-40

-20

0



gain

in d

B



19. Performance Evaluationì Two polular performance evaluation criteria are: 1) SNIRout 2) Outage Probability

19.1. SNIR Criterionoutì The signal at the output of the beamformer can be expressed as CÐ>Ñ œ Ð>Ñ œ A AL LB a b’7Ð>Ñ >n( ) œ ALa bW"7 Ð>Ñ" ’N7N Ð>Ñ n( )>

where ’ œ Ò ÓW"ß ðóóñóóòW W# Q

N

ß ÞÞÞÞß

œ? ’ 7Ð>Ñ œ Ò ß 7 Ð>Ñß ÞÞÞß 7 Ð>ÑÓ7 Ð>Ñ" ðóóóóóñóóóóóò# Q

XN

X

œ? 7



ì À Power of CÐ>Ñ T œ CÐ>Ñ œC#Xe f

œ CÐ>Ñ CÐ>Ñ œ Ð>Ñ Ð>ÑX Xe f ˜ ™‡ LA ALB B

œ Ð>Ñ Ð>Ñ

œ

A AL ðóóóóñóóóóò˜ ™X

‘

B B L

BB

œ A AL

Î ÑÐ ÓÐ ÓÏ ÒðñòT

œ

" " "L

..

W W

? ‘

ðóñóò’ ‘ ’

‘

N NL

7 7N N

œ? JJ

î5 ˆ

‘

#R

88œ?

œ A AL a b‘.. ‘JJ ‘88

Ðassuming desired, interfs & noise are uncorrelated)



i.e. T œ C ðóñóò ðñòA AL..

.ß9?>

‘

œ T

A AL

Nß9?>

‘JJ

œ T

ðóñóòA AL88

8ß9?>

‘

œ T

œ

where T T T Þ.ß9?> " " " "

L L L"

#œ œ œ o/p desired term A W W A A Wˆ ‰

T ÞN ß9?> 34 3L L

4œ œo/p interf. term ! !3œ# 4œ#

Q Q

3 A W W A

T8ß9?># L8œ œo/p noise term 5 A A

Ê œSNIR9?>

œT T Þ.ß9?> " "

L #

TÞ

œ

N ß9?>34 3

L4L

L L

TÞ

8ß9?> # L8

ˆ ‰A W

ðóóóóóóóóñóóóóóóóóò! !3œ# 4œ#

Q Q

3

’ ‘ ’

A W W A

A A

N 7 7N N N

5 A A

(72)



19.2. Outage Probability Criterionì outage probability ( ) is defined as follows:OP

OP SNIRœ Pra bSNIR:<

or OP SIRœ Pra bSIR:<

ì It is a performance evaluation criterion.

ì An example of an array-CDMA system's Outage Probabilitywith =1, 8 and15 receiving elements is shown below:R



ì The figure above clearly shows that by employing an antenna array andusing, for instance, the beamformer of Equation 70 (complete interferencecancellation beamformer) at the base station, the system capacity isincreased. For example, for 0.001 outage probability, the system capacityper cell increases from 20 mobiles, for a single antenna case i.e. , toÐ R œ "Ñabout 40 and 80 mobiles for equal to 8 and 15, respectivelyR



20. Comments

ì An alternative equivalent expression for SNIR9?>

SNIR =9?>A A

A A

L

L

‘‘ ‘

dd

dda bBB

ì Some Applicationd of Beamformers in Communications:

1 analoque access methods FDMA (e.g. AMPS, TACS, NMT)2 digital access methods TDMA (.e.g GSM, IS136) CDMA3 duplex methods FDD, TDD



ì Main categories of beamformers:1) : there is a finite number of fixed array patternsswitched beamformerand the system chooses one of them to maximise signal strength (the onewith main lobe closer to the desired user/signal) and switches from one tobeam to another as the user/signal moves throughout the sector)2) : array patterns are adjustedadaptive beamformer (or adaptive array)automatically (main lobe extending towards a user/signal with a nulldirected towards a cochannel user/signal)

switched beamformer adaptive beamformer



user-1 user-2

Fully Adaptive Spatial Processing,

for two users operatingon the same channel in the same Cell



ì Beamforming Systems - main properties:

Properties Advantages1 signal gain better range/coverage(see figure below)2 Interference rejectiom increase capacity3 spatial diversity multipath rejection4 power efficiency reduced expense

Coverage patterns for switched beam and adaptive array antenna



21. VIVO Channels (MA Comm. Systems with Array Rx): Capacity with a beamformer at the receiver

G œ F "log#ˆ ‰SNIR9?>

(73)œ F "log#Š ‹T Þ" "L #ˆ ‰A W

A AL L’ ‘ ’N 7 7N N N 5# L8 ÞA A

based on Equ. 72)Ð

21.1. 'Unweighted' Beamformers

if A "œ R

then G œ F "log#Š ‹T Þ" "RL #ˆ ‰" W

" "RL L

R’ ‘ ’N 7 7N N N 5#8 ÞR



21.2. 'Steering Vector' Beamformers

if A Wœ ‚constant "

i.e. the main lobe towards the desiredsteer the main lobe(known) signal direction

then G œ F "log#Î ÑÏ ÒT Þ

œR

" "%

#çl lW

W W"L L

"’ ‘ ’N 7 7N N N 5#8 ÞR



21.3. 'Wiener- Hopf' Beamformers

if A Wœ Þ‘BB"

"

then

G œ F "log#Î ÑÐ ÓÏ Ò

T Þ" ""L #

" "L L

" "

ˆ ‰ a bW W

W W W W

‘

‘ ’ ‘ ’ ‘ ‘

‘

BB"

BB BB BB" L " #

44

ðóóñóóòN 7 7N N N

œ

5#8 Þ



21.4. 'Complete Interference Cancellation' Beamformers

if A Wœ Þ’N¼

"

then

G œ F "log#

Î ÑÐ ÓÐ ÓÐ ÓÐ ÓÏ Ò

T Þ

œ!

" "L"

#

" "L L

" "

Š ‹Š ‹

W W

W W W W

’

’ ’ ’

N

¼

N N N

¼ ¼ ¼

ðóóóóóóóóóóóñóóóóóóóóóóóòðóóñóóò’ ‘ ’

‘

N 7 7N N NL

44œ

5#8 Þ

Ê G œ F " log# T Þ" "L"

#

"L

"

Š ‹Š ‹W W

W W

’

’

N

¼

N

¼5#8 Þ



Ê G œ F " log#Œ T Þ" "L"Š ‹W W’N

¼

5#8

Ê G œ F " log#Š ‹T Þ Þ" " "a bl lW cos< 2

5#8

Ê G œ F " log#Š ‹R T Þ" "#cos <

5#8



22. MIMO Systems• Let us consider a communication system with multiple antennas at both the

transmitter and the receiver.

This is a multiple-element Tx multiple-element Rx (MEME) system.Another popular name is multiple-input multiple-output (MIMO) system.



• Equivalent Representation

Ì œ ß ß ÞÞÞß − GË‡ ˜ c d2 2 2" # R

R‚R

ME MEh11

h

1

2

N

1

2

NNN œ œ œ˜ ˜ ˜2 2 2" # R



• 24 œ 4 gain from the Tx-antenna to the ME-Rxth

234 œ 4 3gain from the Tx-antenna to the Rx-antennath th

• Received Signal-Vector:

B 7Ð>Ñ œ Ð>Ñ Ð>Ñ ÐR ‚RÑ‡ n

• If Rx is synchronised to the Tx then for the data symbol8>2

interval we have:

B 7Ò8Ó œ Ò8Ó Ò8Ó a8‡ n (74)



• Data observation matrix of symbols:P

[1] [2] [— œ? Ò ß ß ÞÞÞß PÓÓB B B œ Ò Þ Ó‡ ‡ ‡7 7 7Ò"Ó Ò"Óß Þ Ò#Ó Ò#Óß ÞÞÞß Þ ÒPÓ ÒPÓn n n

Ê œ Þ (75)— Œ ‡

with (œŒ œ Ò Ò"Óß Ò#Óß ÞÞÞß ÒPÓÓœ Ò ß ß ÞÞÞß P Ó

7 7 7 Q ‚PÑÐR ‚ PÑn n n[1] [2] [ ] (76)

where the matrices and (as well as the dimension ‡, Œ Q Ñ are unknown



• Second order statistics (covariance matrix) of BÐ>Ñ

‘ ‘ ‘

5 ˆ

BB 77 88L

#8 R

œ ÐR ‚RÑ

œ

‡ ‡å

(77)

• In a MIMO system it is assumed that the Matrix ‡ is known

• Capacity of MIMO Channel - General Expressions

GÎF œ log#Š ‹detdet

a ba b‘‘BB

nnbits/sec/Hz (78)

GÎF œ log det# R 77" LŠ ‹Š ‹ˆ ‘5n#‡ ‡ bits/sec/Hz (79)



• Capacity of MIMO Channel for independent parallel channels:

In this case: diagonal‘77

"

#

R

œ œ

T ! ÞÞÞ !! T ÞÞÞ !ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ! ! ÞÞÞ T

Ô ×Ö ÙÖ ÙÕ Ø

(80)

GÎF œ "log#4œ"

RT #Š ‹l l24

#4

#5n

GÎF œ "! Š ‹4œ"

R

#Tlog l l24

#4

#5nbits/sec/Hz (81)

Note: Total transmission power = Trace(T Ñ œ T‘77 44œ"

R!



23. MIMO Systems Expressed Geometrically

• Reciprocity Theorem

ˆ Antenna characteristics are independent of thedirection of energy flow.

The impedance & radiation pattern are the same whenthe antenna radiates a signal and when it receives it.

ˆ The Tx and Rx array patterns are the same.





• We have seen that

if in the "local area" of the Rx we have an array system of PA Relements sensors (antennas) with locationsÎ

rœ œ Ò ß ß ÞÞÞß Ó œ Ð$ ‚ RÑ< < <" # R ‘< ß < ß Ñ œ 7Ð> Ñ

4

4

ÞÞÞ

4

ÞÞÞ

4

3

1-

1-

1-

-

# X3 "

# X3 #

# X3 5

#

"



Š ‹Š ‹Š ‹Š ‹

exp

exp

exp

exp

-

-

-

1--? <

W

X3 R

3 3œ Ð ß Ñ˜

Ð>Ñ

) 9

+ n (83)





• Tx small scale displacement:

If, however, the Tx is displaced at a specific point (within its<mlocal area and the direction of the planewave propagation isP ÑE

described by the vector? ?3 3 3œ ß( )) 9

where ( ) ? ) 9 ) 9 ) 9 9ß œ Þ ß Þ ßc dcos cos sin cos sin X(84) a (3 1) real unit-vector pointing towardsœ ‚ the propagation direction ( , )) 9

then Equ. 101 becomes

+ BÐ>Ñ œ 7Ð> Ñ Ð>Ñ3- "WÐ ß Ñ) 93 3 expŠ ‹+4# X

31--? <m n (85)



• Tx Array System:Let us assumed that in the "local area" of the x there is anPA Xarray system of elements sensors (antennas)R ÎÐ R œ RÑwith having locations rœ œ Ò ß ß ÞÞÞß Ó œ Ð$ ‚ RÑ< < <" # R

‘< ß < ß 2

ma œ "ß #ß ÞÞÞß R

then we will consider the following two cases:

: Case-1 Each Tx array element transmits (via a demultiplexer) a different part of the Tx signal

: All Tx-array elements transmit a weighted versionCase-2 of the same signal.



23.1. MIMO Systems: Case-1• if each Tx array element transmits (via a demultiplexer) a different

part of the Tx signal then

+ B 7

7

Ð>Ñ œ Ð> Ñ Ð>Ñ

œ Ð> Ñ

"W3ðóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóò!7œ"

R

-

L

-

expŠ ‹ 4# X3

3

1--? <

W

m3

3

n

+ (86)B 7Ð>Ñ œ Ð> Ñ Ð>Ñ"W3WL

-33 n


(‘ " " 5 ˆ

‘

BB RL

77

L ‡ #œ

œ

W W3 3W W3 3ðóóóóóóóñóóóóóóóòÔ ×Ö ÙÖ ÙÕ ØT ß !ß ÞÞÞß !!ß T ß ÞÞÞß !ÞÞÞß ÞÞÞß ÞÞÞß ÞÞÞ!ß !ß ÞÞÞß T

"

#

R

n 87)

• Important Comment: Equs77, 80 & 87 = (88)Ê ‡ " W3WL

3



• bits/sec/HzCapacity GÎF œ log#Š ‹detdet

a ba b‘‘BB

nn

Ê GÎF œ log det# R 77"Š ‹Š ‹ˆ ‘5n#" "W W3 3W W

L L ‡3 3

Ê GÎF œ " log#œ"

T # Š ‹m

R l lW3#

#

"#m

n5

Ê GÎF œ " log#œ"

T # Š ‹m

R

R "#m

n5#

Ê GÎF œ " ! Š ‹mœ"

#T

R

log R "5

#

#m

nbits/sec/Hz 89Ð Ñ



23.2. MIMO Systems: Case-2• If all Tx array elements transmit the same signal (but weighted)

then received vector signal may be expressed as follows:BÐ>Ñ

+ BÐ>Ñ œ 7Ð> Ñ A Ð>Ñ

œ

3-

7œ"

R

7"W3ðóóóóóóóóóóóóñóóóóóóóóóóóóò! expŠ ‹ 4# X4

L

3

1--? <

W

m

A

n

Ê Ð>Ñ œ 7Ð> Ñ Ð>ÑB A3-

L"W3W3 + (n 90)

where A œ ÒA ß A ß ÞÞÞß A ß ÞÞÞß A Ó" # RX

m


(‘ " 5 ˆBB 7 R

## #

3œ T Š ‹WL

3 A W W3L

n 91)



• bits/sec/HzCapacity GÎF œ log#Š ‹detdet

a ba b‘‘BB

nn

GÎF œ log det# R"Œ Œ ˆ 5n#TmŠ ‹W

L

3 A#

#3" W W3L

Ê GÎF œ " log#

œ

TÎ ÑÐ ÓÏ Ò

æl lW3#

#

R

"##

mŠ ‹WL3 A

5nÐ Ñ92

and if A œ ßW3

then GÎF œ "log# TŠ ‹RR "#m

5n# bits/sec/Hz 93Ð Ñ



23.3. Summary

Ì

Ë‡ œ ß ß ÞÞÞß − G˜ c d2 2 2" # R

R‚R ME MEh11

h

1

2

N

1

2

NNN ‡ ´ "3W3W

L

3 œ œ œ˜ ˜ ˜2 2 2" # R



1) SISO Capacity: SNRlog#a b" bits/s/Hz

2) MIMO Capacity, Tx and Rx antennas, unknown Channel:R R SNRR " ‚log# R

RŠ ‹ bits/s/Hz

N.B.:general expressions No CSI: det log# R

" LŠ ‹Š ‹ˆ ‡‘ ‡ 58# 77

CSI: det max log‘

ˆ ‡‘ ‡77

77# R" LŠ ‹Š ‹ 58#

3) RX/Tx beamformer Tx and Rx antennas, known steering directions:R R SNR log#ˆ ‰" RR ‚ bits/s/Hz



Cellular System EvolutionAdvanced Communication Theory Compact Lecture Notes


24. Transmit Diversity• Provide diversity benefit to a mobile using base station

antenna array for frequency division duplexing (FDD)schemes. .Cost is shared among different users

• Order of diversity can be increased when used with otherconventional forms of diversity.

• Two types of transmit diversity techniques: Transmit diversity with feedback from receiverˆ

(close loop) Transmit diversity without feedback from receiverˆ (open loop)



24.1. Example of Transmit Diversity: Close Loop

• The transmitter transmits some pilot signals• The mobile (based on this pilot signals) estimates the Channel State Information• The mobile transmits the CSI to the BS uplink)Ð• The base station generates the weights and transmits data to the mobile



• UMTS 3GPP Standard: 2 Tx antennas

i.eÞˆ ß w w" # are adjusted such as is maximisedl<Ð>Ñl#

ˆ are adjusted based on the feedback information from the receiverw w" #ß

e.g.



24.2. Example of Transmit Diversity: Open Loopa)without spreader

b) with spreader



Receiver input := s s

= s s( multipaths)P

ÚÝÝÝÛÝÝÝÜr

r

" 4 " #4 "4œ"

P‡

# 4 # #4 #4œ"

P

"‡

!a b!a b

" "

" "

1 2

1

8

8

(flat fading, i.e. unresolvable paths)

Ê2 2 82 2 8œ r

r" " # "

‡

# # # #"‡

= s s= s s

1 21

where ; 2 ´ 2 ´1 1 2 2! !4œ" 4œ"

P P

4 4" "



(Equation-A)” • ” •” • ” •îrr" "

# #

"

#œ

= ß == ß =

1‡#

#‡"

22

88

´ 2or, equivalently,

<

= 8

œ œ 2 ß 22 ß 2” • ” • ” • ” •ðóóóñóóóòî î

rr" " "

# # #‡ ‡ ‡

1 #‡ ‡# "

´ ´ ´‡

= 8= 8

i.e. where < = 8

2

œ œ l2 l l2 l‡ ‡ ‡ ˆL # #" #ðóóóñóóóòa b2

œ l l#



decoder:

‡ ‡ ‡L L< = 88

œ î‡L

´ µ

decision varibales Ê G œ l l2 = 8# µ

i.e. l ll lK œ = 8

K œ = 8" " "

#

# ## ‡

#

2

2

µ

µ

Note:1) the receiver needs to know (estimate) the channel weights and but2 2" #

there is no need to send them back to the transmitter (i.e. open loop)2) and can be estimated by transmitting some pilot symbols as2 2" #

= =" and and then using Equation-A2



25. Arrayed CDMA Systemsì Space-Time Communications can be employed andin both TDMA/FDMA

CDMA type systems

ì In the signal is not spread and (mainly one or twoTDMA/FDMA only few- i.e. ) strong cochannel interferences (CCI) are present when theQ Rsystem employs channel reuse between cells. The array can be used to 'null'(remove/reduce) these few interferences.

ì In all active users use the same bandwidth and are separated byCDMAemploying different PN-codes to reduction/remove the MAI interferencefrom other user.

i.e. the array has to deal with a very large in a environment CDMAnumber of weak interferences ÐQ RÑ.





25.1. Decoupled Space-Time CDMA Receiverì Two representative examples of decoupled 'Space-Time CDMA Base

Station Receiver' architectures are shown below 1)

2)



ì Space-Time Matched Filter CDMA Receiver (Decoupled ST Rx)



ì Space-Time Rake Receiver (Decoupled ST Rx)Advanced Communication Theory Compact Lecture Notes


25.2. Integrated Space-Time CDMA Receiver Architecture

exp(-j F )π ctcarrier

Inner(Discretizerplus TDL)

Processor-3

SuperresolutionBeamformer

&Despreader

wx(t)

Nx1

Nx1

x[ ]n

2NNcx1Preprocessor

Processor-2

ChannelParameterEstimator

Gj{a [ ]}D n^x[ ]n

2NNcx1

Spatio-Temporal Array (STAR) Manifold Receiver



26. Modelling the array signal-vector at the input of a Space-Time CDMA Receiver

26.1. CDMA Transmitterì Consider an DS-CDMA (BPSK or QPSK) VIVOQ -user asynchronous

communication system with the transmitted signal of the userbaseband 3thmodelled as follows

7 Ð>Ñ œ Ò8Ó3 3

8œ_

_" a - Ð> 8X Ñ3ßTR -= (94)

with ,8X Ÿ > Ð8 "ÑX-= -=

where is the period of a channel period and is theX Ö Ò8Óß a8 − ×-= 3a a3th user's data sequence of symbols,

ì Examples: = ( 1, ) if ( 1) if a3Ò8Ó œ „ „ 4„

QPSKBPSK



ì In Equation 94, denotes one period of the PN-signal - Ð>Ñ3ßTR , Ð>Ñ3

associated with the th user,3

i.e. - Ð>Ñ œ -Ð> 5X Ñ3ßTR -"5œ!

R "

3

-

! Ò5Ó (95)

where is the corresponding PN sequence of sÖ Ò5Ó ! Ÿ 5 R ×!3 -, „ "(Gold or m-sequence) of period , with denoting the chipR œ X ÎX X- -= - -

interval i.e. the duration of the chip pulse waveform .ß -Ð>Ñ=rect >X-



26.2. ST-CDMA Channel

ì Let us assume that the transmitted signal of the user arrives at3>2

the reference point of an array CDMA receiver (e.g. a basestation CDMA receiver) via paths (multipaths).O3

ì Consider that the path of the user arrives at the array from4 3>2 >2

direction with channel propagation parameters and( ) ) 934 34ß "34

734 representing the complex path gain and path-delay,respectively.

ì Let us assume that the paths of the th user are arranged such thatO 33

.7 7 73" 3# 3OŸ Ÿ ÞÞÞ Ÿ3

ì Note that and represent the azimuth and elevation angles) 934 34

respectively.



ì Furthermore, the path coefficients model the effects of path"34

losses and shadowing, in addition to random phase shifts due toreflection; they also encompass the effects of the phase offsetbetween the modulating carrier at the transmitter and thedemodulating carrier at the receiver, as well as differences in thetransmitter powers.

ì The path delays are such that the delay spread is in the734 Xspread region of a channel symbol period .X-=



ì The continuous time baseband received signal-vector due to 3th

user is

!4œ"

O

34 3 34

3

WÐ ß Ñ) 934 34 " 77 Ð> Ñ (96)

In a more compact form, Equation-96 may be rewritten (SIVO)asi.e. ’ "3 3 3diagˆ ‰ a b7 > (SIVO)where’ V

" " " " V

3 3 3# 3OR‚O

3 3" 3 3OX O ‚"

3X O ‚"

œ ÒW W á W Ó −

œ Ò á Ó −

7 > œ − V

1

2

, , ,, , ,

3

3

33

3a b Ò7 > 7 > á 7 > Ó3 3" 3 3 3 3Oa b a b a b7 7 7, , ,2 3



ì Based on the above SIVO description, the received complexbaseband signal-vector at the antenna array d canue to usersQ be represented as:

B > œa b ! !3œ" 4œ"

Q O Ð>Ñ

34 3 34

3

W34Ð>Ñ" 7Ð>Ñ7 Ð> Ð>ÑÑ (97)

nœ 7 > >! ˆ ‰ a b a b3œ"

Q

3 3 3’ "diag (98)

œ ’ "diagˆ ‰ a b7 > >n( ) (99)where ’ ’ ’ ’œ Ò ß ß ÞÞÞß Ó1 2 Q



and n is a complex white Gaussian bandpass noise vector witha b>covariance matrix 5 ˆn

#R



• Time varying multipath channel. The receivedoverall BASEBAND signal at the antenna array is the due to all usersR Qsum of signals

B Ð>ÑÐ>Ñ œ function{ }W34

n

œ ! !3œ" 4œ"

Q O Ð>Ñ

34 3 34

3

W34Ð>Ñ Ñ Ð>Ñ" 7Ð>Ñ7 Ð> Ð>ÑÑ n



26.3. The STAR Manifold Vectorì Let us focused on the path of the user received by an array system of4 3th th

R elements sensors (antennas) Î in the "local area" of the RxPA

334- 34œ 7



if the locations of sensors (antennas) of the Rx-array are rœ œ Ò ß ß ÞÞÞß Ó œ Ð$ ‚ RÑ< < <" # R ‘< ß < ß Ñ œ 7 Ð> Ñ

4

4

ÞÞÞ

4

œ Ð ß˜

34 34-

# X34 "

# X34 #

# X34 R

34 34

3

1-

1-

1-

34

-

-

-

"

) 9



Š ‹Š ‹Š ‹

exp

exp

exp

Ñ

Ð>Ñ+ n (100)

+ B WÐ>Ñ œ 7 Ð> Ñ Ð>Ñ3 34 34-334 " n (101)



• For the path of the user the 4 3>2 >2 array manifold vector is: exp(W œ 434 Ò ß ß ÞÞÞß Ó Ñ< < <" # R 34

T k − VR‚"

where Ò ß ß ÞÞÞß Ó< < <" # R represents the array geometry and k34 is the wavenumber vector.

• By taking and into consideration, wethe PN-code multipath delay extend the concept of the array manifold vector to

S T ARPATIO- EMPORAL RAY (STAR) manifold vector,

defined as follows:

¡ ‰ œ34 34

j3œ −

˜ # R‚"W Œ 34 V ac (102)



ì This extention can been achieved by transforming, the array received signalvector to a discrete signal B > − B 8 −a b c dV VR‚" # R‚"a-

• at point : A BÐ>Ñ œ function{ }W34

nÐ>Ñ ÐR ‚ "Ñ

• at point :B B 8 œc d function{ }¡34

nÒ8] Ð #;R R ‚ "Ñ-



ì Thus, the array received signal-vector is discretised by aB >a bchip rate sampler.

comb comb combX X X- - -e f e f e fB Ð>Ñ œ 7 Ð> Ñ Ð>Ñ W

34" 734 3 34 n

œ W34" 734 34 ! !

8œ_

_

3 35œ!

R "

a Ò8Ó Ò5Ó-

! $Ð> 8X 5X Ñ-= -

n combX-e fÐ>Ñ

where

combX-Ö × œ7 Ð> Ñ3 7 734 34

! !8œ_

_

3 35œ!

R "

a Ò8Ó Ò5Ó-

! $Ð> 8X 5X Ñ-= -

has been used.



ì The discrete samples are then passed through a tapped-delay line(TDL) of length equal to .#a-

This is to ensure that one whole data symbol of the desireda3Ò8Óuser is captured within this interval.#a-

ì Note that, due to the lack of synchronisation, the tapped-delaylines contain contributions from the , the and theprevious nextcurrent data symbols.



Synchronous DS-CDMA

( -1) n th n th(n 1) + th

Tcs TcsTcs

Asynchronous DS-CDMA

( -1) n th n th(n 1) + th

Tcs TcsTcsTcs

Tcs Tcs

oj p34

Note: X œ R ÞX-= - -



ì In order to model such contributions due to lack ofsynchronisation and thus to further model the received signalvector, , we need the term B 8c d ‰j34œ3

whereˆ j34 is the discrete version of the path delay = ,734

334-

i.e. j34 œ ÎXi j7 a34 - -mod . (103)

ˆ œ3 − 3e# ‚" >2a- representing one period of the PN-sequence of the userpadded with zeros at the end, i.e.a-

œ ! ! !3 3 3 3 - RX

X

œ ! ß " ß âß R " ß !

3

” •ðóóóóóóóóóóóóóñóóóóóóóóóóóóóò ‘c d c d c d

user's PN-code

-

th

(104)

This corresponds to a zero path delay situation synchronous system .Ð Ñ



ì In addition, is a matrix defined as follows‰ a a# ‚ #- -

‰ œ œ

! ! â ! !" ! â ! !! " â ! !ã ã ä ã ã! ! â " !

! !

!

Ô ×Ö ÙÖ ÙÖ ÙÖ ÙÕ Ø

– — # "X

# " # "

a

a a

-

- -ˆ

(105)

having the property that every time the matrix (or ) operates‰ ‰X

on a column vector it down-shifts (or up-shifts) the elements ofthe vector by one.



ì For instance, h is a version of h down-shifted by elements,‰j j

while h is a version of h up-shifted by elements.ˆ ‰‰Xj

j

h= ; h= ; h=

hhh h

hh

h

hh

h

Ô ×Ö Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÖ Ù Ö Ù Ö ÙÕ Ø

Ô ×Ö ÙÖ Ù Ö ÙÖ Ù Ö ÙÖ ÙÕ Ø

Ô ×

Õ Ø

"

O

"

#

O$

$

%

O

2

3ã

!!!

ã

ã

!!

‰ ‰$ X #ˆ ‰



ì Based on the previous discussion the signal ( path of user)4 3th th

at the output of the TDL during the observation interval can8th

be modelled as follows:

BÒ8Ó œ "34W34Œ

Î ÑÐ ÓÐ ÓÏ Ò

aa 1

a 1

3 3j

3 3j Ñ

3 3j Ñ

Ò8Ó

Ò8 Ó -

Ò8 Ó -

Ò8Ó

‰ 34

34

34

-ˆ‰‰

X Ð

Ð

‰ a

a

-

-

n (106)

Ê BÒ8Ó œ a current3 3jÒ8Ó o"34 W34

Œ ‰ 34-

a 1 previous Ò8 Ó o3 3j Ñ

"34 W34Œ ˆ‰X Ð ‰ a- 34 -

a 1 next Ò8 Ó o3 3j Ñ"34 W34

Œ ‰Ð a- 34 -

noise Ò8Ó on(107)

where n is the sampled noise vector at the o/p c d8 of the TDL



Using the STAR Manifold vector ( W Ñ Œ - œ) ‰ ¡34

63

˜

34ˆ ‰34 (108)

for the path of the user, Equ.107 can be re-written as4 3>2 >2

follows: BÒ8Ó œ a3Ò8Ó"34 ¡

34

a 1 Ò8 Ó3 "34 èëëëëëëéëëëëëëêŠ ˆˆN Œ ‰X ‰ ‹a- ¡

¡

34

34ßœ previous

a 1 Ò8 Ó3 "34 èëëëëéëëëëêŠˆN Œ ‰a-‹¡

¡

34

34ßœ previous

Ò8Ón



ì Preprocessor: Sampler + TDL

ith user

carrier

x(t)

Nx1Nx1

x[ ]n

2qN Nc x1

Ts Ts Ts

Ts Ts Ts

Ts Ts Ts

x1[ ]n

x2[ ]n

xN[ ]n

Array

Ts TcsTDL length=2qNc

x t1( )

x t2( )

x tN( )

1st path2nd path

Ki-th path

Note: carrier ; oversampling factor ( is used for convenience)œ Ð 4# J >Ñ ; œ ; œ "exp 1 -





26.4. Some properties of the STAR Manifold Vector

½ ½¡ ¡ ¡

¡ ¡ ¡

¡ ¡ ¡

34 34 34

#

-L

34ß 34 34

#L

34ß 34ß 34

#

-L

œ RR œ !

œ R œ !

œ R R œ !

,previous

previous ,next

next previous ,next

½ ½½ ½

j

j Ñ

34

34(



ULAPNcode: goldweight-vector

R œ &R œ "&-

A = vector of 1s

gain

(degrees) delay () X Ñ-



ULA (intersensor spacing = /2)PNcode: goldweight-vector = STAR manifold vector ( =90 delay=7)

R œ &R œ "&

à

-

)-

A °

STAR manifold Array Pattern

gain

(degrees) delay () X Ñ-



Capacity of a Spatio-Temporal Link

PN-code: ; ULA:intersensor spaceing= /2R œ $" Ñ- -



26.5. ST CDMA RECEIVERì Preprocessor: Sampler + TDL

ith user

carrier

x(t)

Nx1Nx1

x[ ]n

2qN Nc x1

Ts Ts Ts

Ts Ts Ts

Ts Ts Ts

x1[ ]n

x2[ ]n

xN[ ]n

Array

Ts TcsTDL length=2qNc

x t1( )

x t2( )

x tN( )

1st path2nd path

Ki-th path

Note: carrier ; oversampling factor ( , say, for convenience)œ Ð 4# J >Ñ ; œ ; œ "exp 1 -



ì As shown in preprocessor's figure the received space-time signalvector is formed by concatenating the contents of theB 8c dtapped-delay lines of all the antennas,

i.e. B 8 œ B 8 ß B 8 ß ÞÞÞ ß B 8 −c d c d c d c d’ “" RX X X

X# R‚"

2 V a- (109)

where represents the contents of the tapped-delay line atB 85c dthe antenna associated with the data symbol period.5 8>2 >2



ì By taking into account all the multipaths of the users, theQreceived signal-vector B 8c d can be written, in a compact form, asfollows:

B 8 œ B 8 ß ÞÞÞ ß B 8 ß ÞÞÞ ß B 8c d c d c d c d’ “" 5 RX X X

X

= !ˆ ‰c d c d c d3œ"

Q

3 3 3 33a a a8 8 " 8 "‡ " ‡ " ‡ "3ß 3ß3 3prev next

n 8c d (110)

where ÚÛÜ

Š ˆŠ

‡ ‰ ‡

‡ ‰ ‡

3ß 3X

3ß 3

prev

next

œ Œ

œ Œ

ˆ

ˆ

N

N

‰ ‹‹a

a

-

-

(111)



Notice that by rearranging the terms in 110 , can be decoupled intoa b c dB 8four components, namely the desired (1st user, say), ISI, MAI and noisecomponents. That is,

B 8c d œ Ò8Ó

Ò8 Ó Ò8 Ó

ðóóñóóò

ðóóóóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóóóóò

ðóóóó

‡ "

‡ " ‡ "

" ""

" " " "" "

a

desired term

a 1 a 1

ISI ,prev ,next

óóóóóóóóóóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóò"ˆ ‰3œ#

Q

3 3 3 3 3 33 3 3‡ " ‡ " ‡ "a a 1 a 1

MAIn

Ò8Ó Ò8 Ó Ò8 Ó

Ò8Ó

,prev ,next

(112)



i.e.

B 8 œ 8 M 8 M 8 8c d c d c d c d c d‡ "" "a" ISI MAI n (113)

where


a

a a

" " "

" " " "

c dc d c d c dc d

8

M 8 œ 8 " 8 "

M 8

‡ "

‡ " ‡ "

‡

is the desired signal component

ISI prev next1 1ß ß

MAI œ !ˆ ‰c d c d3œ#

Q

3 3 33a a8 8 "‡ " 3ß 3ß3 3prev next" ‡ " a3c d8 "



ì This modelling is excellent for Signal-Subspace type Receiverarchitectures aiming at removing the M 8 M 8ISIc d c d & andMAIreduce the noise level.

This implies that the receiver initially estimates, over anobservation time, the ofspatio-temporal manifold parametersthe desired signal(s)

e.g., using the following 2-dim. 'STAR' MuSiC-type cost function - interval):Ð8 >2 0Ð ß Ñ œj ) "

(ðóóóóñóóóóò ðóóóóñóóóóòa b a b™ ‰ ™ ‰" " " "LW

W

W

W

Ð ÑŒ Ñ T Ò Ð ÑŒ

œ Ð ÑŒ œ Ð ÑŒ

) )

) )

j jœ œ

9 9j j

c d8 Ó

which are then employed to remove the MAI and ISI terms



STAR - Architecture

carrier

x(t)

Nx1

Nx1

x[ ]n

2NNcx1

Tc Tc Tc

Tc Tc Tc

Tc Tc Tc

x1[ ]n

x2[ ]n

xN[ ]n

ArrayTc TcsTDL length=2Nc

Z1,star

Preprocessor

SubspaceTracking

Construction ofSTAR manifold

vectors

WeightConstruction

2D MuSIC-typeSTAR Spectrum

z n1[ ]

w1

Inner Product

h1 j for all j

A B C D{a [ ]}1 n

( for all θ1 j , , j l1 j)

‹[n] [n]







26.6. A Representative Example of a Pre-Processor

• at point :E z" " " R

¼j j j jÒ8Ó œ BÒ8Ó œ ŒT Ò Ó ˆ ‚ where 1

• ‚ ‚ ‰ ‰ ‰1 1 1 1 11 2

j is formed from the matrix œ ß ß ÞÞß œ œ œR- ‘ by its removing columnjth



The following property for the 'desired' term ,is valid 3 œ "

‡ ¡

" "" "4

jj

œ!ß !ß ÞÞÞß ! ß ß ! ÞÞÞß !ß ! − 6 œ

!ß !ß ÞÞÞß ! ß ! ß

ÚÝÝÝÛÝÝÝÜ ‘èëéëêðóñóò

, if

j "

j

j "

V#R R‚O"4

- 3 (for a single path)

!, if , ÞÞÞß !ß ! − 6 Á a4‘ V#R R‚O"4

- 3 j

but not valid for and contributions which are 'transformed' ratherMAI ISIthan 'simplified'.

EXAMPLE:desired user = 6 pathswith its arriving with a delay equal to i.e. .2nd path Ð Ñ Ð 6 œ $Ñ4 œ # $Xc "#

• at point desired term = B: a [ ]" " "Ò8Ó 8‡ "Ò8Ó

• for at point : transformed desired termj œ $, E œ a " "#"#Ò8Ó ¡ "1$



A STAR Manifold Channel Estimator:

x[ ]n

Bank of ParallelPre-Processors

1l

z [ ]n1lfor 1 l=

z [ ]n1lfor 2 l=

z [ ]n1lfor l=Nc

for 2 l=

for 1 l=

for l=Nc

Noise-Subspace Estimation

Eigendecomp. Of zz‘ 12

Eigendecomp. Of zz‘ 1Nc

1-dim ‘STAR’ MuSICfor TOA=1Tc

1-dim ‘STAR’ MuSICfor TOA=2Tc

1-dim ‘STAR’ MuSICfor TOA= TcNc

Peak-search of 2-dim‘STAR’

spectrum

B

B

2 1N Nc ´

1l

1l

Eigendecomp. Of zz1‘ 1

• ‡"j jœdef" –ðóóóóóóóóñóóóóóóóóò ðóóóóóóóóñóóóóóóóóò “‡ ‡ ‡ ‡ ‡ ‡1 prev 1 1 next 2 prev 2 2 nextß ß ß ß

" " " " " "1 1 1 2 2 2ß ß ß ß

desired user 2nd user

, , etc



27. Some Results of a 'STAR'-family CDMA Receiver27.1. Example: Estimation and Reception

Environment:Receiver Array

User 1(Desired)

User 3(Interference)

Delay 711Direction )11Coefficient "11



User 2(Interference)




exp(-j F )π ctcarrier

Inner(Discretizerplus TDL)

Processor-3

SuperresolutionBeamformer

&Despreader

wx(t)

Nx1

Nx1

x[ ]n

2NNcx1Preprocessor

Processor-2

ChannelParameterEstimator

Gj{a [ ]}D n^x[ ]n

2NNcx1



User 1 (Desired) Path 1 Path 2 Path 3 Path 4 Path 5

Path Delay 1 9 17 21 27Path Direction 50 94 125 141 76Path Coefficient -0.10 + 0.26

ÐX ÑÐ Ñ

4

-9

-0.01 - 0.24 -0.31 - 0.02 -0.31 - 0.02 0.42 - 0.354 4 4 4

User (Interfer) Path 1 Path 2 Path 3 Path 4 Path 5


#

ÐX ÑÐ Ñ

4

-9

-0.41 - 0.74 -0.39 - 0.92 -0.91 - 0.12 0.76 - 0.004 4 4 4

User (Interfer) Path 1 Path 2 Path 3 Path 4 Path 5


$

ÐX ÑÐ Ñ

-9

7 -0.71 - 0.24 -0.11 - 0.01 -0.21 - 0.05 0.45 - 0.554 4 4 4 4



Surface and contour plots of MuSIC-type cost function shows that all 5path delays and directions are correctly estimated

0 5 10 15 20 25 30

40

60

80

100

120

140

Dire

ctio

n D

egre

es

Delay Tc



Diagram of Decision Variables ( :K Ñ4

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Re

Im

-2000 -1500 -1000 -500 0 500 1000 1500 2000-2000

-1500

-1000

-500

0

500

1000

1500

2000

Im

Re-8 -6 -4 -2 0 2 4 6 8

x 106

-8

-6

-4

-2

0

2

4

6

8x 106

Re

Im

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Re

Im

ST ST Decorrel. MU Rx Decorrel. MU Rx (Incomp) ST-RAKE SU Rx

STAR manifold Rx (Imperial College)



0 5 10 15 20 25 30 35 40 45 50-20

-10

0

10

20

30

40

50Average Output SNIR Against Number of Users

Number of Users

Aver

age

Out

put S

NIR

(in

dB)

DecorrProposed

Limited Decorr

ST-RAKE



27.2. Example: 'Near-Far' Resistance

0 10 20 30 40 50 60-60

-40

-20

0

20

40

60

NFR (in dB)

Average Output SNIR Against NFR

Aver

age

Out

put S

NIR

(in

dB)

Decorr

Proposed

Decorr (Limited Information)

ST-RAKE



27.3. Example-2: Tracking

System parameters

Base Station:Linear array of antennas (with 2 spacing)ñ Î5 -

co-channel DS-CDMA users, each with multipathsñ Q=8 3

short PN-code sequences with ñ R œ- 31

SNR of and SIR of ñ 20dB -40dB (i.e.near-far problem)



• Track desired users paths over . of1000 data symbols First paththe desired user during period disappears a b)!! Ÿ 8 Ÿ *!!

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

path 1path 2path 3

n-th symbol

TOA

(Tc

)

n-th symbol

DO

A (

Deg

rees

)

100 200 300 400 500 600 700 800 900 100060

80

100

120

140

160

180Path 1Path 2Path 3



0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

ActualEstimated

Path 3

Path 2

Path 1

Symbol

TOA

/ (Tc

)

100 200 300 400 500 600 700 800 900 1000

60

80

100

120

140

160

180ActualEstimated

Path 3

Path 2

Path 1

Symbol

DO

A /

(De

g)

60 80 100 120 140 160 1800

5

10

15

20

25

30

DOA / (Deg)

TOA

/ (T

c)

60 80 100 120 140 160 1800

5

10

15

20

25

30

DOA / (Deg)

TOA

/ (T

c)

for for 8508 œ $)& 8 œ



Main Properties of :STAR subspace-type receiver

ñ blind

ñ near-far resistant,

ñ superresolution capabilities

ñ The number of multipaths that can be resolved is not constrained by the number of array elements (antennas). Indeed, the desired user's STAR MuSIC-type spectrum (and contour diagram) for an array of 2 antennas operating in the presence of three CDMA users is shown below:

Þ



Specifications: 2 element array. Desired user's parameters = 8 paths with (TOA in , DOA in degrees)X-

as follows: (5, °), (6,6 °), (12,27°),(15,85°), "!! ! (18,45°),(24,30°),(27,118°),(29,105°)

2-Dimensional MuSIC Spectrum (with a 2-antenna array system)



28. Examples of Real Array SystemsA 2GHz Antenna Array of 48 elements



Owens Valley Radio Observatory Array



The New Mexico Very Large Array of 27 elements

(along railroad tracks - 35km)



A Large Circular Array



29. AN IMAGINARY ILLUSTRATIONì The figure below shows of the future handheld mobile setan imaginary illustration

with a Monolithic Microwave Integrated Circuit (MMIC) .antenna array

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COMPACT LECTURE NOTES on ADVANCED COMMUNICATION THEORY. 5 2005 - LN - TRANS... · IMPERIAL COLLEGE LONDON, DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING. COMPACT LECTURE NOTES