Spatial Array Processing - The University of Texas at Austin

Spatial Array Processing

Signal and Image Processing Seminar

Murat Torlak

,

Telecommunications & Information Sys. Eng. The University of Texas at Austin

1

Introduction

 A sensor array is a group of sensors located at spatially separated points

 Sensor array processing focuses on data collected at the sensors to carry out a given estimation task

 Application Areas – Radar – Sonar – Seismic exploration – Anti-jamming communications – YES! Wireless communications

2

Problem Statement s1 (t) s2 (t) θ1

θ2

∆

x1 (t)

x2 (t)

x3 (t)

x4 (t)

Find 1. Number of sources

2. Their direction-of-arrivals (DOAs)

3. Signal Waveforms

3

x5 (t)

x6 (t)

Assumptions

 Isotropic and nondispersive medium – Uniform propagation in all directions

 Far-Field – Radius of propogation

>> size of array

– Plane wave propogation

 Zero mean white noise and signal, uncorrelated  No coupling and perfect calibration

4

Antenna Array

Source θ

X1

X

X4

X3

2

∆

X5

 Array Response Vector–Far-Field Assumption Narrowband - Delay Phase Shift =) Assumption

a() = [1; ej2fc 4 sin =c ; : : : ; ej2fc 44 sin =c ]T  Single Source Case =) x(t) 2 x1 (t) 6 6 x2 (t) 6 6 6 . 6 . 6 . 4

xM (t)

3 2 7 66 7 7 66 7 = 6 7 7 6 7 5 64

s1 (t) s1 (t ,  ) . . .

s1 (t , (M , 1) )

3 2 7 6 7 6 7 6 7 6  7 6 7 6 7 5 6 4

1

e,j 2fc  . . .

e,j 2fc (M ,1)

where  = 4 sin 1 =c.

5

3 7 7 7 7 s1 (t) = a(1 )s1 (t) 7 7 7 5

General Model

 By superposition, for d signals, x(t)

= =

a(1 )s1 (t) +


+ a(d )sd (t) d X a(k )sk (t) k=1

 Noise x(t)

= =

d X

a(k )sk (t) + n(t) k=1 AS(t) + n(t)

where

A = [a(1 ); : : : ; a(d )] and

S(t) = [s1 (t); : : : ; sd (t)]T :

6

Low-Resolution Approach:Beamforming

 Basic Idea d X

d

X ( i , 1)( j 2 f 4 sin  =c ) c k xi (t) = =e sk (t) = sk (t)ejwk (i,1) k=1 k=1 where wk

= 24 sin(k )=c and i = 1; : : : ; M .

 Use DFT (or FFT) to find the frequencies fwk g F F w


= [

(

1)

2 66 6 F(wM )] = 666 64

1

1

ejw1

ejw2

. . .

. . .

ej (M ,1)w1

ej (M ,1)w2







.

jF (xi (t))j = jF x(t)j2

N X 1  x(t)j2 B (wi ) = j F N t=1

7

.






 Look for the peaks in  To smooth out noise

.

1

ejwM . . .

ej (M ,1)wM

3 77 77 77 75

Beamforming Algorithm

 Algorithm

Rx =

PN

 (t) x ( t ) x t=1 2. Calculate B (wi ) = F (wi )Rx F(wi )

1. Estimate

1 N

B (wi ) for all possible wi ’s. Calculate k , i = 1; : : : ; d.

3. Find peaks of 4.

 Advantage - Simple and easy to understand

 Disadvantage - Low resolution

8

Number of Sources

 Detection of number of signals for d < M , Rx

= =

where

x(t) = As(t) + n(t) E fx(t)x (t)g = A E fs(t)s (t)g A + E fn(t)n (t)g | {z } | {z } Rs

 + I A R A s n |{z} |{z} |{z}

n2 I

2

M d dd dM

n2 is the noise power.

 No noise and rank of Rs is d

Rx = ARs A will be f1 ; : : : ; d ; 0; : : : ; 0g: Real positive eigenvalues because Rx is real, Hermition-symmetric

– Eigenvalues of

–

– rank

d

 Check the rank of Rx or its nonzero eigenvalues to detect the number of signals

 Noise eigenvalues are shifted by n2

f1 + n2 ; : : : ; d + n2 ; n2 ; : : : ; n2 g:

where 1 > : : : > d and  >>

0

 Detect the number of principal (distinct) eigenvalues 9

MUSIC

 Subspace decomposition by performing eigenvalue decomposition

M

X  2 Rx = ARs A + n I = k ek ek k=1

ek is the eigenvector of the k eigenvalue  spanfAg = spanfe1; : : : ; edg = spanfEs g  Check which a()  spanfEsg or PA a() or P?Aa(), where PA is a projection matrix  Search for all possible  such that where

jP? a()j2 = 0 or M () = A

1 PAa() = 1

 After EVD of Rx

P?A = I , EsEs = EnEn where the noise eigenvector matrix

En = [ed+1; : : : ; eM ]

10

Root-MUSIC

 For a true , ej2fc 4 sin =c is a root of P (z ) =

M X

k=d+1

[1; z; : : : ; z M ,1 ]T ek ek [1; z ,1 ; : : : ; z ,(M ,1) ]:

 After eigenvalue decomposition, - Obtain fek gdk=1 - Form p(z ) - Obtain 2M , 2 roots by rooting p(z ) - Pick d roots lying on the unit circle - Solve for fk g

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Estimation of Signal Parameters via Rotationally Invariant Techniques (ESPRIT)

 Decompose a uniform linear array of M sensors into two subarrays with M , 1 sensors  Note the shift invariance property 2 66 66 (2) a () = 6 64

ejw ej 2w . . .

ej (M ,1)w

3 2 77 66 77 66 77 = 66 5 4

1

ejw . . .

ej (M ,1)w

3 77 77 jw (1) jw 77 e = a e 5

 General form relating subarray (1) to subarray (2) 2 3 jw e 6 7 1

A(2) = A(1) 664

..

.

ejwd

7 = A(1): 7 5

  contains sufficient information of fk g 12

ESPRIT

 spanfEsg = spanfAg and Es = AT - T is a d  d nonsingular unitary matrix -

T comes from a Grahm-Schmit orthogonalization

of

Ab in

Rx

EssEs + En En AH RsA + n2 I (1)  E(2) s = A(2) T and Es = A(1) T =

Es (2) = A(2) T = A(1) T = Es (1)T,1 T

 Multiply both sides by the pseudo inverse of E(1) s

E(1)# s Es (2) = (E(1) E(1) ),1 E(1) E(1) T,1 T = T,1 T where # means the pseudo-inverse

A# = (AsH A),1AsH  Eigenvalues of T,1T are those of . 13

Superresolution Algorithms

1. Calculate

Rx =

1 N

PN

 (k) x ( k ) x k=1

2. Perform eigenvalue decomposition 3. Based on the distribution of

fk g, determine d

4. Use your favorite diraction-of-arrival estimation algorithm: (a) MUSIC: Find the peaks of

180

- Find

M () for  from 0 to

f^k gdk=1 corresponding the d peaks of

M (
).

(b) Root-MUSIC: Root the polynomial

p(z )

d roots that are closest to the unit rk c . circle frk gdk=1 and ^k = sin,1 2f c


- Pick the

(c) ESPRIT: Find the eigenvalues of

fk g k c - ^k = sin,1 2f c4

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(2) E(1)# s Es ,

Signal Waveform Estimation

 Given A, recover s(t) from x(t).  Deterministic Method

wk such that wk ? a(i ); i 6= k; wk 6? a(k )

– No noise case: find

 A# can do the job

A#x(t) = A#As(t) = s(t)  With noise, n(t)

A#x(t) = s(t) + A#n(t) ) increased noise

– Disadvantage =

15

Stocastic Approach

 Find wk to minimize 2g =  Rk wk min E fj w x ( t ) j min w k k a ( )w =1 a ( )w =1 k

k

k

k

 Use the Langrange method

min E fjwk x(t)j2 g , min wk Rk wk + 2(a (k )wk , 1) ;wk a (k )wk =1

 Differentiating it, we obtain

Rx wk = a(k ); orwk = R,x 1 a(k )

.

 Since a (k )wk = a(k )R,x 1 a(k ) = 1,  Then  = a (k )R,x 1 a(k )  Capon’s Beamformer

wk = R,x 1a(k )=(a (k )R,x 1a(k )) 16

Subspace Framework for Sinusoid Detection

 x(t) =

d P k=1

k e(k +j!k )t

 Let us select a window of M , i.e., x t ; : : : ; x(t , M + 1)]T

xt

( ) = [ ( )

 Then xt

( )

=

=

2 66 66 66 64

3 2 7 6 7 6 d 7 X6 7 6 = 7 6 7 6 7 5 k=1 6 4

x(t) x(t , 1) . . .

x(t , M + 1)

2 66 d X 66 66 k=1 64 |

1

e,(k +j!k ) . . .

e(k +j!k )(,M +1) {z

a k (

=

d X k=1

k e(k +j!k )t k e(k +j!k )(t,1) . . .

k e(k +j!k )(t,M +1)

3 7 7 7 7 k e(k +j!k )t 7 7 | {z } 7 5 s (t) }

k

)

a k sk t As t ; (

)

( ) =

( )

where M is the window size, d the number of sinusoids, and

k

= ek +j!k .

17

3 77 77 77 75

Subspace Framework for Sinusoid Detection

 Therefore, the subspace methods can be applied to find fk + j!k g  Recall x(t) =

d X k=1

k e(k +j!k )t

 Then finding f k g is a simple least squares problem.

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Wireless Communications co

-c

ha

nn

el

in

te

rf

er

en

ce

Multipaths th

t ec

Pa

r

Di

Cellular Telephony Office Building

Residential Area

Personal Communications Services (PCS)

Outdoors To Networks

Di

re

ath ct P e r i D

ct

th

th

pa

i lt

Mu

Wireless LAN

 Increasing Demand for Wireless Services  Unique Problems compared to Wired communications 19

Pa

Problems in Wireless Communications

 Scarce Radio Spectrum and Co-channel Interference 1 4

2 3 1 4

1 2

3 1

 Multipath Multipath

Direct P

ath

Mu

ltip

ath

Base Station

Time

Desired Signal

Reflected Signal

 Coverage/Range 20

Smart Antenna Systems

 Employ more than one antenna element and exploit the spatial dimension in signal processing to improve some system operating parameter(s): - Capacity, Quality, Coverage, and Cost. User One

User Two

Multiple RF Module

Advanced Signal Processing Algorithms

Conventional Communication Module

21

Experimental Validation of Smart Uplink Algorithm

 Comparison of constellation before (upper) and after imaginary axis

smart uplink processing (middle and lower)

imaginary axis

real axis Antenna Output

imaginary axis

real axis Equalized Signal 1

real axis Equalized Signal 2

22

Selective Transmission Using DOAs

 Beamforming results for two sources separated by 20 Power Spectrum

1 0.8 0.6 0.4 0.2 0

0.5

1 1.5 Frequency [Hz], User #1

2

1 1.5 Frequency [Hz], User #2

2

4

x 10

Power Spectrum

1 0.8 0.6 0.4 0.2 0

0.5

23

4

x 10

Selective Transmission Using DOAs

 Beamforming results for two sources separated by 3

Power Spectrum

1 0.8 0.6 0.4 0.2 0

0.5

1 1.5 Frequency [Hz], User #1

2

1 1.5 Frequency [Hz], User #2

2

4

x 10

Power Spectrum

1 0.8 0.6 0.4 0.2 0

0.5

24

4

x 10

Future Directions

 Adapt the theoretical methods to fit the particular demands in specific applications – Smart Antennas – Synthetic aperture radar – Underwater acoustic imaging – Chemical sensor arrays

 Bridge the gap between theoretical methods and real-time applications

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