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Short Course @ISAP2010
in MACAO

Eigenvalues
gand Eigenvectors
gin Array Antennas
Optimization of Array Antennas forHigh Performance

Nobuyoshi Kikuma
Nagoya Institute of Technology,Japan

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Self-introduction| Nobuyoshi Kikuma
z Professorz Research interests:• adaptive and signal processing array• multipath propagation analysis• mobile and indoor wireless communication• wireless power transmissionz Hobby:• MUSIC (karaoke, etc).

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On this short course| Spatial signal processing technology usingarray antenna has been one of theimportant approaches for improving theperformance of communications andradars.
| ThisThi shorth t course expresses ththeoptimization of the array antenna for its
high performance in various applications.

4On this short course(cont’d)| Significance of eigenvalues and eigenvectorsof various matrices used in the array antenna
is explained.
| Key words:
z Gain of the array antennaz OptimumO tiweights
i ht off theth adaptive
d ti arrayz Array weights for direction-of-arrival (DOA)

estimation (e.g. MUSIC)
Ref.: D.K. Cheng, "Optimization techniques for antenna array," Proc. IEEE,vol.59, No.12, pp.1664-1674, Dec. 1971.

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Applications of radio waves

5Communications, Broadcasting, andSensors

Antennas are important components.

Radio Propagation in MobileCommunications

6• No line of sight• Reflection, diffraction andscattering (multipath)• Heavy fading
Amplitude (dB)

100-1 0-2 0-3 0f = 800MHzv = 60km/h-4 0-5 00

0.05

0 .1

0.15

0.2

0 .2 50 .3

0 .35

0 .4

0.45

0.5

Time (seconds)
Nagoya Institute of Technology3

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Example of antenna pattern for
multipath waves (1)Only one dominant wave is received.

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Example of antenna pattern for
multipath waves (2)All multipath waves are combined appropriately.

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9Radio Propagation in Cellular
Communication Systems• Effective utilization of frequency
qy resources• Co-channel interference (CCI)
BSInterferenceMT
Multipath waves

Same frequency

Nagoya Institute of Technology

10Techniques expected in antennasystems• Fading recovery• Suppressing or Canceling CCI

Control of directional pattern usingarray antenna:
Adaptive array signal processing(pattern optimization)
Nagoya Institute of Technology5

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Why array antenna is used?
| SNR increased byy in-phase combination ofarray-element signals| High angular resolution

with narrow mainlobe| Electronic scan ofmainlobe/null

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General configuration ofK-element array antenna
Transmitting moder(r,θ,φ)Ik

position vectorof observation
point

I1
IK
Directional pattern ofelement

Ik

Excitation current

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Radiation field from antenna
at origin (phase center)
Maxwell’s equationqspherical wave

Focus onθ –componentNo
r-component

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Approximation of radiation
field (e.g., θ-component)
#1

#2.
.
.

phasehdiff
difference ffrom origin
i iphase difference from origin

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Approximation of radiation field

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Radiation field of array antenna(Combined electric field)phase difference
spherical wave

due to element position

Directional functionof array antenna8

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Radiation field of array antenna(Cont’d)| If directional functions of all elements are

id ti l thenidentical,thPrinciple ofpattern multiplication: array factor: element pattern

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Directivity and its
optimization

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Directivity of array antenna
definition
impedance
power density

Identicalelements

Normalized power pattern ofantenna element

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Directivity of array antenna(Cont’d)
Vector notation: excitation current vector: array steering vector
Directivity

Inner product of above
two vectors
Ratio of two
Hermitian forms

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Maximization of directivity ofarray antenna
Generalized
Rayleigh quotient

What is w maximizing G?

Generalized eigenvalue problem:[W,Lambda]=eig(A,B);

Eigenvector corresponding tomaximum eigenvalue λM

22Maximization of directivity ofarray antenna (Cont’d) Example| Uniform linear array of isotropicelements with element spacing of d

| InI addition,
ddi iwhenh d = λ/2,
(Uniform excitation：Kronecker delta)

Number of elements112010/11/30

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Q factor & mainbeam radiation
efficiency of array antenna
Q factor

Mainbeam radiation efficiency

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Numerical Examples
6-element uniform linear array (ULA)

Broadside array with isotropic elements

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Numerical Examples (Cont’d)6-element uniform linear array (ULA)

Endfire array with isotropic elements

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Numerical Examples (Cont’d)6-element uniform linear array with d = λ/4
Directional patterns

wM =[1.11, -0.91, 0.80, 0.80, -0.91, 1.11] T wM =[-2.06+1.71j, 6.92-4.15j, -11.73+5.10j,
v0=[1, 1, 1, 1, 1, 1] T

12.28 - 3.59j, -7.98 + 1.23j, 2.68 ] T

v0=[1, -j, -1, j, 1, -j] T132010/11/30

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Adaptive array andits optimization

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Adaptive array model(receiving mode)
Vector notation ofinputs and weights

K-element array antenna

Output is expressed as
inner product of two vectors.

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Adaptive array model (Cont’d)| ArrayAoutputt t power：| Covariance matrix：

Hermitian form

Hermitian matrix

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Adaptive array model (Cont’d)| Output SINR：

at output
Large value means goodreceiving performance

Maximizationof SINR

*SINR: Signal-to-Interference-plus-Noise Ratio

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Adaptive array output (1)(linear array, no interference)
SNI
Input
t vector:t

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Adaptive array output (1) (Cont’d)(linear array, no interference)
Array output:

Input SNR

Rayleigh
quotient

Equal to mainbeam
radiation efficiency

Weight vector maximizing SINR:

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Adaptive array output (2)(linear array, L interferences)
Input vector:

SI

N

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Adaptive array output (2) (Cont’d)(linear array, L interferences)
Array output:

Generalized Rayleighy g qquotient

P0=Ps

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Adaptive array output (2) (Cont’d)(linear array, L interferences)
Quasi-Normalized
SINR

Generalized
Rayleigh quotient36

Directivity and Output SINR
Same form

D i d signal
Desiredil

Angular distributionof incident waves

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Typical criteria of adaptivearray¾ Maximum Signal-to-Noise Ration: MSN¾ Minimum Mean Square Error: MMSE38

MSN adaptive array| This adaptive array controls weights tomaximize the output SNR(SINR).
SNR(SINR)| A priori knowledge: DOA of desired signal
Cost function:
maximized

Angular distributionof incident waves

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MSN adaptive array (Cont’d)| Optimum weight vector:

MSN adaptive array (Cont’d)| Optimum weight vector w’opt maximizing

SINR

γ20

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MMSE adaptive array| Minimizing error e(t) which is adifference between reference signalr(t) and array output y(t):Cost function:minimized

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MMSE adaptive array (Cont’d)| Optimum weightvectortIn the case of one desired
signal incident

This weightThi
i ht isi equall tot thatth tof MSN adaptive array.
Maximizing output
SINR

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Example of 6-element
adaptive array (K = 6)

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Example of 6-element
adaptive array (K = 6) (Cont’d)Directional ppatterns
Adaptive(MMSE & MSN)
SINR = 27.66dB
SINR0 = 0.9983
Uniform
SINR = 15.31dB
SINR0 = 0.9714

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DOA estimation andarray optimization

Lineararray46

Beam-scan scheme¾ Beamformer

Maximization ofdirectivity in θ

Angular spectrum¾ Capon Beamformer

Constrained minimization

Angular spectrum

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Beam-scan scheme(Rayleigh quotient expression)¾ Beamformer¾ Capon Beamformer

Mainbeam radiation efficiencyy

SINR

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Problems of Beamformer

Wave 2 is received bysidelobe.

Closely spaced waves cannot
be separated.

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Advantage of Capon
Beamformer
Desired signal

Interference

Interference

Desired signal

Examples of directional patterns by optimum weights

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How angular resolution isenhanced furthermore?
Nulls are utilizedutilized.

Null-scan
NullscanschemeNulls are adaptively steered in therespective directions of incident waves.
Angular spectrum of null-scan scheme:
Inverse ofpower pattern25

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Null-scan scheme(Constrained minimization)¾ linearly constrained minimization (Linear prediction)¾ Quadratically constrained minimization

Pisarenko’s method, Min-Norm method,
MUSIC

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Null-scan scheme(Rayleigh quotient expression)¾ linearlyy constrained minimization ((Linear pprediction))¾ Quadratically constrained minimization(Pisarenko’s method)26

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53Eigenvectors of minimum
eigenvalues of covariance matrix
Eigenvalues:
Eigenvectors:

Noise power

Example of 6-element
ULA with element spacingof a half wavelength (L=3)(L 3)

3 eigenvectos
used individually
Pisarenko’s method

Some smoothing required

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Min-Norm method and MUSICMin--Norm method
Min

Signal subspace

Linear combinationof eigenvectors

Noise subspace

MUSICCombination ofpower patterns

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5 estimators
6-element ULA
3 waves0゜, 110゜, 160゜, 0.5

Noise power
0.01

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Summary and conclusions| Optimization of array antennas using

eigenvalues and eigenvectors
z Directivityz Adaptive arrays
z DOA estimation| Rayleigh quotient expressionz Link to eigenvalue problem
z Optimization of array antennas basedon Rayleigh quotient expression
Unified array antenna theory including
adaptive arrays and DOA estimation

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