Slides Array Eng

7/29/2019 Slides Array Eng

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Introduction to Array Processing

Olivier Besson

O. Besson (U. Toulouse-ISAE) Introduction to array processing 1 / 114
http://find/http://goback/


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Outline

1 Introduction

2

Signal model

3 Beamforming

4 Source localization



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Introduction Arrays of sensors

A toy example

y2(t) As(t t0 t)

y1(t) As(t t0)

s(t)

t depends on the direction of arrival of s(t) and on the relative

(known) positions of the antennas:if is known, one can obtain s(t): spatial filtering (beamforming)

if one can estimate t from y1(t) and y2(t), then follows: sourcelocalization.

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A toy example

For narrowband signals, a delay amounts to a phase shift. Hence

y1(t) = As(t) + n1(t)

y2(t) = As(t)ei + n2(t)

Let us estimate s(t) using a linear filter:

s(t) = w1y1(t)+w2y2(t) = As(t)

w1 + w2ei

+(w1n1(t) + w2n2(t))

The output signal to noise ratio (SNR) is given

SN R =|w1 + w2ei|2

|w1|2 + |w2|2|A|2Ps

Pn

and is maximal for w2 = w1ei, so that s(t) y1(t) + y2(t)e

-i.


I d i A f


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Array of sensors

PotentialitiesArray of sensors offer an additional dimension (space) which enables one,possibly in conjunction with temporal or frequency filtering, to performspatial filtering of signals:

1 source separation2 direction finding

Fields of application

1

radar, sonar (detection, target localization, anti-jamming)2 communications (system capacity improvement, enhanced signals

reception, spatial focusing of transmissions, interference mitigation)


I t d ti S t llit i ti
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Introduction Satellite communications

Satellite communications

Multi-beam coverage.

Available bandwidth distributed over space.

Improved interference mitigation.




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Multi-beam coverage




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Example of area coverage through multiple beams

0.0200.020.040.060.080.10.02

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u

v

Individual antenna beampatterm and beam footprint

Footprint of the spot

1

2 3

4

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7

Isolevel antenna gain




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Fixed beams, fixed bandwidth allocation

S p o t 1S p o t 2

3 c h a n n e l s / s p o t

F r e q u e n c y

S p o t 1

S p o t 2

3 c h a n n e l s / s p o t

F r e q u e n c y

3 fixed channels per sub-band pb: heterogeneous users distribution


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Fixed beams, variable bandwidth allocation

S p o t 1

S p o t 2

V a r i a b l e

allocation

p e r s p o t

F r e q u e n c y

S p o t 1

M a x i mum ga i n

at spot center

G a i n l o s s

about 3-4dB

o n s p o t e d g e s

U 1 U 2

heterogeneous users distribution pb: gain differences between users




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One beam per user

S p o t 1

S p o t 2

M a x i m u m g a i n

for all users

F r e q u e n c y

User of interest

( U O I )

C o - u s e r

( I n t e r f e r e n c e )

S a m e c h a n n e lr e - u s e d

I n t er f er ence due t o

co-user in sidelobe

F r e q u e n c y

F r e q u e n c y

T i m eU 3 , U 5 , U 8

U 1 , U 4 , U 7

U 2 , U 6

U 8

U 2

U 3

U 3

U 4

U 5

U 7 U 6

U 1




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SDMA (space-division multiple access)

J a m m e rI n t e r f e r e n c eU O I

F r e q u e n c y

F 1 F 3F 2


Introduction Radar detection


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Radar detection

A fundamental task of any radar system is to detect a target in the

presence of noise, clutter and interference.Consider an airborne radar aimed at detecting a ground movingtarget:




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Principle

The radar antenna sends a series of pulses and collects the variousback-scattered signals (clutter, noise and possibly target).

Within each pulse, the received signal is decomposed in short-timeintervals which correspond to scatterers located at a certain distancefrom the radar (range cells).

Cell under test

Adjacent cells

TR

Time


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Received data

When an array of sensors is used at receiver, the received data is

organized in a so-called datacube:

Range

Arrayelements

Pulses

samplesat range cell under test

(fast

time)

(slow time)

spa

ce

x

1

1

M

M

N

N


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Target space-time signature

If a target (with given direction of arrival and velocity) is present, the

transmitted waveform will undergo a phase shift from pulse to pulse (Doppler effect); a phase shift from antenna to antenna (propagation).

1

2

N

Space

TimeTR 2TR M TR

DOA shifts = 2

d sin

Doppler shiftD = 2

vTR

The main goal is to decide for the presence/absence of a signal withknown signature s(m, n) = ei2(nfs+mfd) among noise and clutter.


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Clutter space-time signature

For side-looking airborne radars, the spatial and Doppler frequencies of the

clutter are proportional:

Spatial frequency

Dopplerfrequency

Clutter spacetime signature

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Detection viewed as filtering

z = s + n

primary data

ws = f(s,z,Z)

secondary data

Z=z1 zK

wHs z

|.|2target

noise

Adaptive detection of a target in the presence of noise with unknown statistics.


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Optimal space-time filter response

Spatial frequency

Dopplerfrequency

Matched filter

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Output of the detector

Spatial frequency

Dopplerfrequency

Output of the detector

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Signal model Principle
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Arrays and waveforms

y

z

x

k

The array performs spatial sampling of a wavefront impinging fromdirection(, ).

Assumptions: homogeneous propagation medium, source in thefar-field of the array plane wavefront.


Signal model Multi-channel receiver
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Multi-channel receiver

HF

filter

thermalnoise

ADC

ADC

sin(ct)

cos(ct)

Im[yn(t)]

Re[yn(t)]

Lowpass filter

Lowpass filter

Im [yn(t)]

Re [yn(t)]

snapshoty(kTs)


Signal model Signals
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Signals (in the frequency domain)

X()

S()H() H()

c c


Signal model Signals received on the array
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Modeling

Source signal (narrowband)

x(t) = 2Re

s(t)eict

Re(t)ei(t)eict= (t)cos[ct + (t)]

(t) and (t) stand for amplitude and phase of s(t), and have slowtime-variations relative to fc.

Channel response

Receive channel number n has impulse response hn(t).


http://goforward/http://find/http://goback/


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Model of received signals

Signal received on n-th antenna

yn(t) = hn(t) x(t n) + nn(t)

where n is the propagation delay to n-th sensor.

In frequency domain :

Yn() = Hn()X()ein + Nn()

After demodulation ( + c) and lowpass filtering:

Yn() = Hn( + c)S()ei(+c)n + Nn( + c)

Hn(c)S()eicn + Nn( + c)


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Taking the inverse Fourier transformF1 Yn() = Hn(c)S()eicn + Nn( + c)

yn(t) Hn(c)s(t)eicn + nn(t)

The signal is then sampled (temporally) to obtain the so-called

snapshot y(k) = y1(kTs) y2(kTs) yN(kTs)T.Narrowband assumption: The propagation time across the arrays isnegligible compared to the inverse of the bandwidth:

D

c

1

B

L

c

1

B LB fc

Under this assumption, propagation delay phase shift:

s(t n) s(t) eicn .


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Snapshot

y1(kTs)y2(kTs)

...

yN(kTs)

=

H1(c)eic1

H2(c)ec2

...

HN(c)eicN

s(kTs) +

n1(kTs)n2(kTs)

...

nN(kTs)

y(k)N|1

= aN|1

s(k) + n(k)

a is the steering vector.n depends on the array geometry and the direction of arrival of thesource.

Hn(c) is the frequency response of n-th channel at c.


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Steering vector

y

z

x

k

n =1

c[xn cos cos + yn cos sin + zn sin ]

an(, ) = ei 2

[xn cos cos +yn cos sin +zn sin ]


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Uniform linear array (ULA)

Steering vector

1 2 3 N

d sin

d

a() =

1 ei2fs ei2(N1)fsT

; fs = fcd sin

c=

d

sin

Shannon spatial sampling theorem

|fs| 0.5 d

2


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Signals and covariance matrix

Signals

In the case of P sources

y(k) =P

p=1

a(p)sp(k) + n(k) =N|P

A()s(k)P|1

+ n(k)

Covariance matrix

The covariance matrix is defined as

R= Ey(k)yH(k)

R(n, ) = E {yn(k)y (k)} measures the spatial correlation between sensorsn and . For instance, in case of a ULA and a direction of arrival equal to0: R(n, ) = exp

i2 d (n )sin 0

.



M d l li i i
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Model limitations

y(k) = a()s(k) + n(k) is an idealized model of the signals received on

the array. It does not account for:a possibly non homogeneous propagation medium which results incoherence loss and wavefront distortions. This leads to amplitude andphase variations along the array, i.e.yn(k) = gn(k)e

in(k)an()s(k) + nn(k).

uncalibrated arrays, i.e., different amplitude and phase responses foreach channel.

wideband signals for which a time delay does not amount to a simplephase shift. In the frequency domain, one has

y(f) = af()s(f) + n(f) withaf() =

1 ei2f () ei2f(N1)()

T.

possibly colored reception noise, i.e. En(k)nH(k)

= C= 2I.


Beamforming Principle

S i l fil i
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Spatial filteringPrinciple: use a weighted combination of the sensors outputs in order topoint towards a looked direction.

y1(k)

w1y2(k)

w2yN(k)

wN

yF(k) =

Nn=1 w

nyn(k) as(k)

s(t) i(t)


Beamforming Array beampattern

ULA b
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ULA beampattern

Obtained as a simple sum of the sensors outputs:

g() = 1Ha()

=N1

n=0ei2n

d

sin

= ei(N1)d

sin sin

N d sin

sin

d sin

G() = |g()|2 = sin N d sin sin d sin 2



ULA b tt
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ULA beampattern

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20

30Beampattern of the uniform linear array

dB

Angle of arrival

N=6, d=0.5

N=6, d=2N=12, d=0.5

3dB 0.9

N d



Wi d i


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Windowing

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ULA beampattern with windowing

dB

Angle of arrival

Rect

Cheb 30dB

Cheb 50dB


Beamforming Spatial filtering

Principle


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Principle

We aim at pointing towards a given direction in order to enhancereception of the signals impinging from this direction, and possiblymitigate interference located at other directions.

Each sensor output is weighted by wn before summation:

yF(k) = wHy(k) =

w1 w

2 w

N

y(k) =

Nn=1

wnyn(k)

Assume we look for direction 0. Then, a simple way is to use

w = a(0)



Principle
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Principle

Source at 0

yF(k) = wHy(k) = aH(0)a(0)s(k)

=N1n=0

ei2d

n sin 0 e+i2d

n sin 0 s(k)

=N1n=0

s(k) = Ns(k)

so that the gain towards 0 is maximal and equal to N. The beamfomer

w = a(0)/(aH

(0)a(0)) is referred to as the conventional beamformer.

Principle

Basically, one compensates for the phase shift induced by propagationfrom direction 0 and then sum coherently.



Array beampattern with beamforming


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Array beampattern with beamforming

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Conventional beamformer

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Beamforming SNR improvement

SNR improvement
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SNR improvement

Before beamforming

y(k) = a0s(k) + n(k); SN Relem E|s(k)|2E {|nn(k)|2} = P2 .After beamforming

yF(k) = wHy(k) = wHa0s(k) + w

Hn(k)

SN Rarray =|wHa0|2

w2SN Relem

a02SN Relem = N SN Relem with equality ifw a0

White noise array gain

For any w such that wHa0 = 1, the white noise array gain isAw = w

2.


Beamforming SNR improvement

SNR improvement
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SNR improvement

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Signals before and after beamforming

Frequency

before CBF

after CBF


Beamforming Adaptive beamforming

Conventional beamforming versus adaptive beamforming


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Conventional beamforming versus adaptive beamforming

The conventional beamformer is optimal in white noise: it amounts tominimize wHw (the output power in white noise) under theconstraint wHa(0) = 1. Any other direction is deemed to beequivalent it does not take into account other signals present in

some directions.

Adaptive beamforming takes into account these other signals. It

consists in minimizing the output power EwHy(k)

2

while

maintaining a unit gain towards looked direction tends to place

nulls towards interfering signals.



Minimum Power Distortionless Response (MPDR)
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Minimum Power Distortionless Response (MPDR)

Principle

minwEwHy(k)2 = wHRw subject to wHa0 = 1

where

R= Ey(k)yH(k) stands for the signal plus interferencecovariance matrix;

a0 is the assumed steering vector of the signal of interest (SOI).

the constraint guarantees that the SOI passes the filter undistorted

(provided its DOA is a0) while the constraint aims at eliminatinginterference.



Reminder (Lagrangian)
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Reminder (Lagrangian)

Let x be the solution to the following constrained minimization problem

minx

f(x) subject to g(x) = 0 (1)

and let p = f(x). In order to solve (1), one writes the LagrangianL(x, ) = f(x) + g(x) and looks for x() = arg minx L(x, ). Let us denote

h() = minx L(x, ) = L(x(), ). Then

h() = minx

L(x, ) minx / g(x)=0

L(x, ) = p.

If there exists such that g(x()) = 0, then x() is the solution to (1).

Indeed, one has f(x()) = L(x(), ) = h() p

and hence h() = p. Observe that h() = max h().



MPDR
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MPDR

Solution

The Lagrangian is given by ( C

)

L(w, ) = wHRw + wHa0 1

+ aH0 w 1

=

w + R1a0

HR

w + R1a0

||2aH0 R1a0.Minimizing with respect to w leads to w = R1a0. Enforcing theconstraint, one finally obtains

wMPDR = R1

a0aH0 R

1a0(MPDR)

and the minimum power is

aH0 R

1a0

1

.



MPDR
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MPDR

Alternative proof

For any vector w such that wHa0 = 1, one has

1 =

wHa0

2

=wHR1/2R1/2a0

2

wHRw aH0 R1a0with equality if and only ifR1/2w are R1/2a0 co-linear. SincewHa0 = 1, it follows that

w = aH0 R1a01R1a0which concludes the proof.



SINR maximization
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Problem statement

The received signal in the presence of interference is given by

y(k) = ass(k) + yI(k) + n(k)

R= PsasaHs + C

where C stands for the interference plus noise covariance matrix andas is the actual SOI steering vector.

For a given beamformer w, the Signal to Interference plus NoiseRatio (SINR) can be written as

SINR(w) =Ps wHas2wHCw

Note that SINR(w) = SINR(w).



SINR maximization and MVDR


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Optimal beamformer

The optimal beamformer maximizes SINR while ensuring a unit gaintowards as:

minwwHCw subject to wHas = 1 (2)

Following the derivations on the previous slides

wopt =C1as

aHs C1as

; SINRopt = PsaHs C

1as.

Minimum Variance Distortionless Response (MVDR)

The MVDR solves (2) with a0 substituted for as, which leads to

wMVDR =C1a0

aH0 C1a0

(MVDR)



Generalized Sidelobe Canceler


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The MVDR (or MPDR) weight vector can be decomposed into acomponent along a0 and a component orthogonal to a0, i.e.,w = a0 + w:

1 The component along a0 ensures that the constraint is fulfilled since

wHa0 = aH0 a0 + wHa0 = aH0 a0 + 0 = aH0 a01 .2 The orthogonal component is arbitrary in the subspace (of

dimension N 1) orthogonal to a0.

one can minimize, in an unconstrained way, the power at the

output of the beamformer with respect to w.





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The MVDR beamformer can be implemented as

y(k)

a0aH0a0

d(k)

+

B

N|N 1

wa

N 1|1

z(k)+

d(k) wHa z(k)

with BHa0 = 0. The equivalent beamformer is

wgsc = wCBF Bwa

The SOI (as well as interference and noise) passes through the mainchannel but is canceled in the auxiliary channels: wa enables one toestimate, from z(k), the part of interferences contained in d(k) sincethe latter is correlated with z(k).



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The power at the output of the beamformer is given by

E

d(k) wHa z(k)

2

= E

|d(k)|2

wHa rdz r

Hdzwa + w

Ha Rzwa

= wa R1z rdzHRz wa R1z rdz+ E|d(k)|2

rHdzR

1z rdz

with rdz = E {d(k)z(k)} and Rz = Ez(k)zH(k).



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Minimizing the output power yields

wa = R1z rdz wgsc = wCBF BR

1z rdz

= wCBF BBHRyB

1BHRywCBF

= aH0 R1y a01R1y a0

where Ry = R in a MPDR scenario and Ry = C in a MVDRscenario.

The SINR is inversely proportional to the output power when

Ry = C, i.e.,

SINRgsc = PswHCBFCwCBF r

HdzR

1z rdz1



Minimisation of the mean-square error (MSE)
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Assume we have a reference signal s(k) (e.g. pilot signal). Then, onemay try to minimize the MSE:

E|wHy(k) s(k)|2

= wHRyw w

Hrys rHysw + Ps

where rys = E {y(k)s(k)

}.The solution is given by

w = R1y rys

Ifrys = Psas then w = PsR1as, which is exactly the MPDRbeamformer (without requiring knowledge ofas).



Example: SINR in the case of one interference
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y(k) = ass(k) + ajsj (k) + n(k)C= Pjaja

Hj +

2I

It is straightforward to show that when IN R =Pj2 1

SINRCBF Ps

2 1

g IN R; SINRMV DR P

s

2N(1 g)

with g = cos2 (as,aj) = |aHs aj |

2/(aHs as)(aH

j aj ).

RemarksWith CBF, the SINR decreases when Pj increases while it isindependent of Pj with adaptive beamforming.

The SINR decreases when aj as (g 1).



Comparison CBF-MVDR


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Comparison CBFMVDR

dB

Angle of arrival

CBF

MVDR



Interpretation of the MVDR
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Assuming J interferences

C=J

j=1

PjajaH

j + 2I=

Jn=1

n +

2unu

Hn +

2N

n=J+1

unuHn

so that the MVDR beamformer can be rewritten as

wMVDR =

2

a0

Jn=1

nn + 2

uHn a0un

The MVDR beamformer amounts to subtract from the CBF a linearcombination of the J principal eigenvectors ofC, and the latter spanthe interference subspace.



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CBF

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1st eigenbeam

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0MVDR



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CBF

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1st eigenbeam

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Number of eigenbeams

dB

Signal to interference and noise ratio

SINR versus number of eigen-beams (J = 2)



MVDR versus MPDR
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The SINR is not modified when using the MPDR beamformer if

minw wHC+ PsasaHs w subject to wHa0 = 1 (MPDR)

minwwHCw subject to wHa0 = 1 (MVDR)

minwwHCw subject to wHas = 1 (opt)

which is true only when the 2 following conditions are satisfied:

1 the assumed steering vector a0 coincides with the actual steeringvector as: in practice, uncalibrated arrays or a pointing error lead toa0 = as;

2 the covariance matrix R is known: in practice, one needs to estimateit which results in estimation errors RR.

= It ensues that degradation compared to SINRopt is unavoidable inpractice, and it can be quite different between MPDR and MVDR.



Influence of a steering vector error
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We assume that the SOI steering vector is a0 while it is actually as.

The SINR obtained with wMVDR =aH0 C

1a01

C1a0 becomes

SINRMVDR =

Ps wHMVDRas

2

wHMVDRCwMVDR = Ps aH0 C

1as2

aH0 C1a0

= SINRopt

aH0 C1as2(aH0 C

1a0)(aHs C1as)

= SINRopt cos2 as,a0;C1 SINRopt



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The MPDR beamformer can be written as

wMPDR =R1a0

aH0 R1a0

; R= PsasaHs + C

Its SINR is decreased compared to that of the MVDR, viz

SINRMPDR =SINRMVDR

1 +

2SINRopt + SINR2opt

sin2as,a0;C

1

SINRMVDR.

The degradation is all the more important that Ps increases.



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dB

Angle of arrival

0

s=2

opt

MVDR

MPDR



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20 15 10 5 0 5 10 15 20100

80

60

40

20

0

20SINR loss

dB

Pointing error

MVDR

MPDR



Case of an uncalibrated array
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Let us consider an uncalibrated array with actual steering vector

[a()]n = (1 + gn)einei 2 (xn+xn)sin ei 2 yn cos For any beamformer w, the resulting beampattern B() = w

H

a() isrelated to the nominal beampattern B() = wHa() throughE| B()|2 = |B()|2 exp (2 + 2)

+ w2 (1 + 2g ) exp

(2 + 2)The term proportional to w2 leads to sidelobe level increase.



Influence of a finite number of snapshots
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In practice, K snapshots are available

y(k) = ass(k) + yI(k) + n(k); k = 1, , K

The covariance matrices are thus estimated as

R= 1K

Kk=1

y(k)yH(k); C= 1N

Kk=1

[yI(k) + n(k)] [yI(k) + n(k)]H

from which one can compute the corresponding beamformers

wsmiMPDR =R

1a0

aH0 R1a0

; wsmiMVDR =C

1a0

aH0 C1a0



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10 20 30 40 50 60 70 80 90 1008

6

4

2

0

2

4

6

8

10

Number of snapshots

dB

SINR versus K

optimum

MVDR

MPDR

CBF



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80 60 40 20 0 20 40 60 8050

40

30

20

10

0

10Beampatterns CBFMPDRMVDR

dB

Angle of arrival

N=10, K=20

MVDR

MPDR

CBF



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Rate of convergence

In order to achieve the optimal SINR up to 3dB:

MVDR :

K 2N

MPDR :K (N 1) SINRopt



Robustness issues
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Estimation of covariance matrices leads to a significant SINR loss(especially for the MPDR beamformer) due to

the interference being less eliminated a sidelobe level increase which results in a higher white noise gain.

In case of uncalibrated arrays, steering vector errors are all the moreemphasized that the white noise gain is low (or w2 large).

Idea: restrain w2, or equivalently enforce a minimal white noisegain in order to robustify the MPDR beamformer.



White noise gain and finite number of snapshots
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10 20 30 40 50 60 70 80 90 1006

4

2

0

2

4

6

8

10

Number of snapshots

dB

White noise gain versus K

MVDR

MPDR


Beamforming Robust adaptive beamforming

Diagonal loading
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Principle

One tries to solve

minwwHRw subject to wHa0 = 1 and w

2 =

SolutionThe Lagrangian is given by (with C and R)

L(w, , ) = wHRw + wHa0 1

+ aH0 w 1

+ wHw

= w + R+ I1 a0HR+ I w + R+ I1 a0 ||2aH0

R+ I

1a0.



Diagonal loading

Solution
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Solution

The solution thus takes the form wMPDR-DL = R+ I1 a0. SincewHMPDR-DLa0 = 1, it follows that

wMPDR-DL = R+ I

1a0

aH0R+ I1 a0

and is selected such that wMPDR-DL2 = .

Choice of the loading level

Solving for wMPDR-DL2 = is complex one usually chooses the

loading level in a heuristic way.



Diagonal loading : adaptivity vs robustness


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MPDR CBF

Aw

NCBF

MPDR

Diagonal loading



An interpretation of diagonal loading and the choice of
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The array beampattern with the true covariance matrix is given by

g() =

2

aH0 a()

Jn=1

nn + 2

aH0 un

uHn a()

The array beampattern with an estimated covariance matrix becomes

gsmi() =

min

aH0 a()

Nn=1

n

n + min

aH0 un

uHn a()

Degradation is due toJ+1 =

J+2 =

N =

min.

Replacing Rby R+ I enables one to equalize the eigenvalues,provided that 2 and < J.



Diagonal loading and finite sample errors
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6

4

2

0

2

4

6

8

10

Number of snapshots

dB

SINR versus K Diagonal loading

optimum

MPDR

MPDRDL=5dB

MPDRDL=10dB



Diagonal loading and finite sample errors
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50

40

30

20

10

0

10

Angle of arrival

dB

Beampattern of MPDR with diagonal loading

N=10, K=20

MPDR

MPDRDL=5dB

MPDRDL=10dB



Influence of the loading level
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20 15 10 5 0 5 10 15 20 25 302

0

2

4

6

8

10

Diagonal loading level (relative to 2)

dB

SINR versus diagonal loading level

K=20

K=200

20 15 10 5 0 5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

Diagonal loading level (relative to 2)

dB

White noise gain versus diagonal loading level

K=20

K=200



Diagonal loading and uncalibrated arrays
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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.16

4

2

0

2

4

6

8

10

Value of

dB

Diagonal loading for uncalibrated array

MPDR

MPDRDL=5dB

MPDRDL=10dB



Linearly constrained beamforming
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To mitigate pointing errors, one can resort to constraints, i.e. solvethe problem

minwHCw subject to ZHw = d

whose solution is w = C1ZZHC1Z

1d.

One can use a unit gain constraint around the presumed DOA or asmoothness constraint:

Z=

a(0) a(0 + 1) a(0 + L)

d =

1 1 1

T

Z= a(0) a() 0 La()L 0 d = 1 0 0T


Beamforming Partially adaptive beamforming

Partially adaptive beamforming
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Objective

Decrease computational cost and improve rate of convergence (recall thatto achieve the optimal SINR up to 3dB, one needs at least K = 2N snapshots).

Principle

Project the snapshots in a lower dimensional subspace and performbeamforming in this subspace.

Interpretation

The (columns of) matrices T and U can be viewed as beams pointedtowards interference (and possibly the SOI) prior to filtering them(beamspace filtering).

O. Besson (U. Toulouse-ISAE) Introduction to array processing 81 / 114 Beamforming Partially adaptive beamforming

Derivation of the partially adaptive beamformer
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Direct formNew snapshots after transformation z(k) = THy(k) whosecovariance matrix is Rz = T

HRyT.

Minimization of the output power

minwwHRzw subject to w

HTHa0 = 1 (DF)The solution is given by

wdf = R1z T

H

a0 weq = T THRyT1 THa0.

O. Besson (U. Toulouse-ISAE) Introduction to array processing 82 / 114 Beamforming Partially adaptive beamforming

Derivation of the partially adaptive beamformer
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GSC form

New snapshots after transformation z(k) = UHz(k) = UHBHy(k)whose covariance matrix is Rz = U

HRzU.

Minimization of the output power

minwa Ed(k)wHa z(k)2 (GSC)The solution is given by

wa = R1

z

rdz = UHRzU1 UHrdzwgsc = wCBF BUR

1z rdz .

O Besson (U Toulouse-ISAE) Introduction to array processing 83 / 114 Beamforming Partially adaptive beamforming

Selection of matrices T and U
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Fixed transformations

For instance using subarrays or spatial filtering, i.e.

T = a(1) a(2) a(R)U= BHa(1) a(2) a(R)

Require some prior knowledge about the interference DOA in orderfor them to pass through the beams.


Selection of matrices T and U

Adaptive transformations
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Adaptive transformations

Matrices T or U depend on the snaspshots. For example, in GSC form,if

Rz =N1n=1

nunuHn ; 1 2 N1

one can choose the R principal eigenvectors ofRz (Principal Component), i.e.

U=

u1 uR

wgsc-pc = wCBF BU

1UHrdz

where = diag {1, , R}. the R eigenvectors which contribute most to increasing the SINR

(Cross Spectral Metric).


Partially adaptive beamforming: examples


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6

4

2

0

2

4

6

8

10

Number of snapshots

d

B

SINR of partially adaptive beamformers

optimum

MPDR

MPDRPA [20 15]

MPDRPA [30 20]

PC


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3

4

5

6

7

8

9

10

Number of snapshots

d

B

SINR of PC and CSM beamformers

R=J=2

optimum

MVDR

PC

CSM


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1 2 3 4 5 6 7 8 95

0

5

10

Rank of the transformation

d

B

SINR of PC and CSM beamformers

K=20

optimum

MVDR

PC

CSM

O Besson (U Toulouse-ISAE) Introduction to array processing 88 / 114 Source localization Non parametric methods

Beamforming for direction finding purposes

Principle
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The idea is to form a beam for each angle and to evaluate the power at

the output of the beamformer: the largest peaks provide the directions ofarrival. It amounts to evaluate E

|yF(k)|2

= EwH()y(k)2 where

w() is some beamformer pointing towards , for instance

E|yF(k)|2 = aH()Ra() (CBF)=

1

aH()R1a()(Capon)

In practice, R is estimated as

R=1

K

Kk=1

y(k)yH(k)

O Besson (U Toulouse-ISAE) Introduction to array processing 89 / 114 Source localization Non parametric methods

Comparison CBF-Capon (low resolution scenario)
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15

10

5

0

5

10

Angle of arrival

d

B

Comparison CBFCAPON

=[30, 10, 20]

3dB

5.1

CBFCAPON

O Besson (U Toulouse ISAE) Introduction to array processing 90 / 114 Source localization Non parametric methods

Comparison CBF-Capon (high resolution scenario)
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15

10

5

0

5

10

Angle of arrival

d

B

Comparison CBFCAPON

=[30, 10, 15]

3dB

5.1

CBFCAPON

O Besson (U Toulouse ISAE) Introduction to array processing 91 / 114 Source localization Parametric methods for DOA estimation

Model-based methods


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PrincipleBased on the model

y(k) = A()s(k) + n(k)

where = 1 2 PT,A() =

a(1) a(2) a(P)

s(k) =

s1(k) s2(k) sP(k)

T

and a() stands for the steering vector.


Classes of methods
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Maximum Likelihood methods are based on maximizing the likelihoodfunction, which amounts to finding the unknown parameters whichmake the observed data the more likely.

Subspace-based methods rely on the fact that the signal subspacecoincides with the subspace spanned by the principal eigenvectors of

R. Moreover, the latter is orthogonal to the subspace spanned by theminor eigenvectors. These two algebraic properties are exploited fordirection finding.

Covariance matching relies on a model R() for the covariancematrix and looks for the model parameters which minimize thedistance between R() and the sample covariance matrix R.


Maximum Likelihood Estimation
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The MLE consists in finding the parameter vector which maximizesthe probability density function (pdf) p(Y;) of the snapshotsY= y(1) y(2) y(K), where is the model parametervector.Multi-dimensional optimization problem (usually).


Stochastic MLE


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Assume that s(k) is Gaussian distributed with E {s(k)} =0

,Es(k)sH(k) = S(k, k) and Es(k)sT(k) = 0.The pdf of the snapshots is thus given by

y(k) CN0,R= A()SAH() + 2I .

The joint pdf can be written as

p(Y;) =K

k=11

N|R|ey

H(k)R1y(k).


Stochastic MLE

Th ML i i b i d
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The ML estimate is obtained as

= arg min,S,2

logp(Y;)

= arg min,S,2

log |R|+ Tr

R1R

Closed-form solutions for 2 and S can be obtained so that thelikelihood function is concentrated, yielding a minimization over theangles only:

sto

= arg min log A()S()AH() + 2()I


Deterministe MLE

The signal waveforms are assumed deterministic so that


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The signal waveforms are assumed deterministic so that

y(k) CNA()s(k), 2I.The MLE is now given by

= arg min

,S

,2

N Klog 2 + 2K

k=1 y(k)A()s(k)2 .

The likelihood function can be concentrated with respect to S and2, and finally

det

= arg min

TrPA()R .

For a single source det = arg max1

NaH()Ra() CBF.

O B ss (U T l s ISAE) I t d ti t ssi 97 / 114 Source localization Parametric methods for DOA estimation

Subspace methods

Eigenvalue decomposition of the covariance matrix

If P signals are present one has
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If P signals are present, one has

R= A(0)SAH(0) +

2I=P

p=1

pepeH

p + 2I

=

P

p=1 p + 2 epeHp + 2N

p=P+1

epe

H

p = Es

sE

H

s +

2

EnE

H

n

Eigenvalues of the covariance matrix

Signal subspace

Noise subspace

2

O B (U T l ISAE) I t d ti t i 98 / 114 Source localization Parametric methods for DOA estimation

Subspace methods

Signal and noise subspaces
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Signal and noise subspaces

Since

REn = 2En = A(0)SA

H(0)En + 2En

AH(0)En = 0

we have

NAH(0)

= R{En} = R{Es}

= R{A(0)}

R{Es} = R{A(0)} .

The signal subspace is spanned by Es: it is thus orthogonal to En.


MUSIC

The signal steering vectors are orthogonal to En
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eHn a(p) = 0 aH(p)EnE

Hn a(p) = 0

One looks for the P largest maxima of

VMUSIC() =

1

aH()EnEHn a()

For a ULA, one can either compute the P roots (root-MUSIC) of

VMUSIC(z) = aT

(z1

)En

E

H

n a(z)

closest to the unit circle, where a(z) =

1 z zN1T

.


Subspace Fitting
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Since R{Es} = R{A(

0)}, there exists a full-rank matrix T

(P P) such that

Es = A(0)T .

The idea is to look for the DOA which minimize the error between thesubspaces spanned by E

sand A() :

, T = arg min,T

Es A()T2W

= arg min,T

TrEs A()TWEs A()TH

.

O B (U T l ISAE) I d i i 101 / 114 Source localization Parametric methods for DOA estimation

Subspace Fitting

There exists a closed form solution for T and finally
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There exists a closed-form solution for T and finally

SSF

= arg min

TrPA()EsWE

Hs

.

Alternative: use the fact that

R{En} = NAH(0) EHn A(0) = 0and estimate the angles as

NSF

= arg min EHn A()2U .

O B (U T l ISAE) I d i i 102 / 114 Source localization Parametric methods for DOA estimation

ESPRIT
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We assume that the array is composed of 2 sub-arrays which arerelated to by a known displacement. Then

A2 = A1 = A1

eic(1)

eic(2)

. . .. . .

eic(P)

.


Source localization Parametric methods for DOA estimation

ESPRIT
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The signals received on the two sub-arrays can be written as

y1(k) = A1s(k) + n1(k)

y2(k) = A1s(k) + n2(k).

Let

z(k) =

y1(k)y2(k)

=

A1A1

s(k) +

n1(k)n2(k)

= As(k) + n(k)

and Rz = Ez(k)zH(k) be the covariance matrix ofz(k).



ESPRIT
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IfRz = EssEHs + EnnE

Hn then

Es = AT

E1E2

=

A1A1

T

E1 = A1T et E2 = A1T

E2 = E1T1T

E2 = E1.

The eigenvalues ofare eic(p)P

p=1.



Low-resolution scenario

25Eigenvalues of the covariance matrix
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0 2 4 6 8 10 12 14 16 18 205

0

5

10

15

20

25

dB

R theory

R estimated



Low-resolution scenario
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100 80 60 40 20 0 20 40 60 80 10015

10

5

0

5

10

15

20 Comparison CBFMUSIC

Angle of arrival

dB

=[30, 10, 20]

3dB

5.1

CBF

MUSIC

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Roots

rootMUSIC

arbitrary in R(Un)



High-resolution scenario

30Eigenvalues of the covariance matrix
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0

5

10

15

20

25

dB

R theory

R estimated



High-resolution scenario


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10

5

0

5

10

15

20 Comparison CBFMUSIC

Angle of arrival

dB

=[30, 10, 13]

3dB

5.1

CBF

MUSIC

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Roots

rootMUSIC

arbitrary in R(Un)



Covariance matching

The covariance matrix is given by R( P ) = R ( P ) + Q()


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The covariance matrix is given by R(,P,) = Rs(,P) + Q()

r = vec(R) = ()P+ =()

P

().

The parameters are estimated by minimizing the error between R and

its estimate R :

, = argmin[r ()]W1 [r ()] .

The criterion can be concentrated with respect to : minimization

with respect to only.



Covariance matching
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In case of independent Gaussian distributed snaphots,Wopt = R

T R and covariance matching estimates areasymptotically (i.e. when K) equivalent to ML estimates.

In contrast to MLE, no need for assumptions on the pdf, only anassumption on R. The criterion is usually simpler to minimize.

Covariance matching can be used with full-rank covariance matrix Rswhile subspace methods require the latter to be rank deficient.



Synthesis
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Hypotheses Algorithm Perf. Problems

ML pdf Min. optimal Computational cost

COMET R Min. optimal Computational costMUSIC R EVD optimal Coherent signals


Conclusions

Conclusions

Array processing thanks to additional degrees of freedom enables
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Array processing, thanks to additional degrees of freedom, enablesone to perform spatial filtering of signals.

Adaptive beamforming, possibly with reduced-rank transformations,enables one to achieve high SINR with a fast rate of convergence inadverse conditions (interference, noise).

Robustness issues are of utmost importance in practical systems, andshould be given a careful attention.

Non-parametric direction finding methods are simple and robust butmay suffer from a lack of resolution.

Parametric methods offer high resolution at the price of robustness.


References

References
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1 H.L. Van Trees, Optimum Array Processing, John Wiley, 2002

2 D.G. Manolakis, V.K. Ingle et S.M. Kogon, Statistical and Adaptive Signal Processing, ch.11, McGraw-Hill, 2000

3 Y. Hua, A.B. Gershman et Q. Cheng (Editeurs), High-Resolution and Robust Signal

Processing, Marcel Dekker, 2004

4 J.R. Guerci, Space-Time Adaptive Processing, Artech House, 2003

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Slides Array Eng

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