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)

θ 1 θ 2

∆ x (t) x (t) x (t) x (t) x1 (t) 2 3 x4 (t) 5 6

Find

1. Number of sources

2. Their direction-of-arrivals (DOAs)

3. Signal Waveforms

3 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

X X X X12X 3 45∆

Array Response Vector–Far-Field Assumption

Narrowband

- Delay = Phase Shift

Assumption

j2f 4 sin =c j2f 44 sin =c T

c c

a = [1;e ;::: ;e ]

= xt

Single Source Case

2 3 3 3 2 2

x t s t 1

1 1

6 7 7 7 6 6

j 2f

6 7 7 7 6 6

x t s t e

2 1

6 7 7 7 6 6

s t = a s t =

6 7 7 7 6 6

1 1

. . . 1

6 7 7 7 6

. 6 . .

6 7 7 7 6 6

4 5 5 5 4

. 4 . .

j 2f M 1

x t s t M 1 e

= 4 sin =c

where 1 .

General Model d

By superposition, for signals,

xt = a s t+ + a s t

1 1

d d

= a s t

k k

k=1

Noise

xt = a s t+ nt

k k

k =1

= ASt+ nt

where

A = [a ;::: ;a ]

1 d

and

St = [s t;::: ;s t] :

1 d

6 Low-Resolution Approach:Beamforming

Basic Idea

d d

X X

i1j 2f 4 sin =c jw i1

k k

x t = = e s t = s te

k k

k =1 k =1

w = 2 4 sin =c i = 1;::: ;M

where k and .

fw g

Use DFT (or FFT) to ﬁnd the frequencies k

2 3

1 1 1

6 7

jw jw

6 7

1 2 M

e e e

6 7

F = [Fw Fw ] =

6 7 1

M . . . . 6

. . . . 7 6

. 7 4

. . . 5

jM1w

jM 1w j M 1w

1 2 M

e e e

Look for the peaks in

jF x tj = jF xtj i

To smooth out noise

jF xtj B w =

t=1

7 Beamforming Algorithm

Algorithm

R = xtx t

1. Estimate x

t=1

B w = F w R Fw

i x i

2. Calculate i

B w w i

3. Find peaks of i for all possible ’s.

i = 1;::: ;d

4. Calculate k , .

Advantage

- Simple and easy to understand

Disadvantage

- Low resolution

Number of Sources

d < M

Detection of number of signals for ,

xt = Ast+ nt

R = Efxtx tg = AEfsts tgA + E fntn tg

| {z } | {z }

I = A R A +

|{z} |{z}

|{z}

M d dM

dd 2

where is the noise power.

R d

No noise and rank of s is

R = AR A s

– Eigenvalues of x will be

f ;::: ; ;0; ::: ;0g:

d R

–Real positive eigenvalues because x is real, Hermition-symmetric

– rank d

Check the rank of x or its nonzero eigenvalues to

detect the number of signals

Noise eigenvalues are shifted by

2 2 2 2

f + ;::: ; + ; ;::: ; g:

n n n n

> ::: > >> 0 1

where d and

Detect the number of principal (distinct) eigenvalues

9 MUSIC

Subspace decomposition by performing eigenvalue

decomposition

I = e e R = AR A +

x s

k k

k =1

e k

where k is the eigenvector of the eigenvalue

spanfAg = spanfe ;::: ;e g = spanfE g

1 d s

a spanfE g P a A

Check which s or or

P a P

, where A is a projection matrix

Search for all possible such that

? 2

jP a j = 0 or M = = 1

P a

After EVD of x

P = I E E = E E

s n

A s n

where the noise eigenvector matrix

E = [e ;::: ;e ]

n d+1 M

Root-MUSIC

j 2f 4 sin =c

For a true , is a root of

M1 T 1 M1

P z = [1;z;::: ;z ] e e [1;z ;::: ;z ]:

k =d+1

After eigenvalue decomposition,

fe g

- Obtain k

k =1

-Formp

M 2 pz - Obtain 2 roots by rooting

-Pickd roots lying on the unit circle

f g

- Solve for k

11 Estimation of Signal Parameters via

Rotationally Invariant Techniques (ESPRIT) M

Decompose a uniform linear array of sensors into

1 two subarrays with M sensors

Note the shift invariance property

2 3 3 2

1 e

6 7 7 6

jw j2w

e e

6 7 7 6

jw 1 jw 2

6 7 7 6

= e = a e a =

6 7 7 6 . .

. . 6 7 7

6 . .

4 5 5 4

jM 1w jM 1w

e e

General form relating subarray (1) to subarray (2)

2 3

6 7

6 7 1

2 .

= A1: = A

A . 7

6 .

4 5

f g

contains sufﬁcient information of k

ESPRIT

spanfE g = spanfAg E = AT s

s and

d d - T is a nonsingular unitary matrix

- T comes from a Grahm-Schmit orthogonalization

of Ab in

R = E E + E E

x s s n

s n

H 2

A R A + I

2 1

E = A T E = A T s

s and

2 1 1

E 2 = A T = A T = E 1T T

s s

Multiply both sides by the pseudo inverse of s

1 1 1 1 1 1 1

E E 2 = E E E E T T = T T

s s

where means the pseudo-inverse

sH 1 sH

A = A A A

T Eigenvalues of T are those of .

Superresolution Algorithms

R = xk x k

1. Calculate x

k =1 N

2. Perform eigenvalue decomposition

f g d

3. Based on the distribution of k , determine

4. Use your favorite diraction-of-arrival estimation

algorithm:

(a) MUSIC: Find the peaks of M for from to

180

f g d

- Find k corresponding the peaks of

k =1

M .

z (b) Root-MUSIC: Root the polynomial p

- Pick the d roots that are closest to the unit

r c

fr g = sin k

circle k and .

k =1

2 1

E E s

f g

= sin

- k

2f 4 c

Signal Waveform Estimation

A st xt Given , recover from .

Deterministic Method w

– No noise case: ﬁnd k such that

w ? a ;i 6= k; w 6? a

k i k k

can do the job

A xt = A Ast = st

With noise,

A xt = st + A nt – Disadvantage = increased noise

Stocastic Approach

Find k to minimize

min E fjw xtj g = min w R w

k k k

a w =1 a w =1

k k k k

Use the Langrange method

min E fjw xtj g , min w R w + 2a w 1

k k k k k

a w =1

k k

Differentiating it, we obtain

R w = a ;orw = R a

k k k k x

a w = a R a = 1

k k k

Since k , x

Then

= a R a

k k x

Capon’s Beamformer

1 1

w = R a =a R a

k k k k

x x

16 Subspace Framework for Sinusoid

Detection

+j! t

k k

xt = e

k =1 M

Let us select a window of , i.e.,

xt = [xt;: : : ;xt M + 1]

Then

2 3 3 2

+j! t

k k

xt e

6 7 7 6

+j! t1

6 7 7 6

k k

xt 1 e

d k

6 7 7 6

xt = =

6 7 7

. 6 .

6 7 7

. 6 .

6 7 7 6

k=1

4 5 5

. 4 .

+j! tM +1

k k

xt M +1

3 2

7 6

+j!

7 6

k k

7 6

+j! t

k k

e =

7 6

. k

7 6

{z }

. |

7 6

k =1 5

4 .

s t

+j! M +1

k k

{z } |

a s t = Ast; =

k k

k =1 d

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

+j!

k k

= e

k .

17 Subspace Framework for Sinusoid Detection

Therefore, the subspace methods can be applied to

f + j! g k ﬁnd k

Recall

+j! t

k k

xt = e

k =1

f g

Then ﬁnding k is a simple least squares problem.

18 Wireless Communications

co-channel interference

Multipaths

Direct Path

Cellular Telephony

Office Building

Residential Area

Personal Communications Outdoors Services (PCS)

To Networks

Direct Path

Multipath Wireless LAN

Increasing Demand for Wireless Services

Unique Problems compared to Wired communications

19 Problems in Wireless Communications

Scarce Radio Spectrum and Co-channel Interference

1 2 4 3 1

4 1 3 2 1

Multipath

Multipath Direct Path Multipath

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 smart uplink processing (middle and lower)

real axis Antenna Output

real axis Equalized Signal 1 imaginary axis imaginary axis imaginary axis real axis Equalized Signal 2

22 Selective Transmission Using DOAs

Beamforming results for two sources separated by

0.8

0.6

0.4

Power Spectrum 0.2

0 0.5 1 1.5 2 Frequency [Hz], User #1 4 x 10 1

0.8

0.6

0.4

Power Spectrum 0.2

0 0.5 1 1.5 2 Frequency [Hz], User #2 4 x 10

23 Selective Transmission Using DOAs

Beamforming results for two sources separated by

0.8

0.6

0.4

Power Spectrum 0.2

0 0.5 1 1.5 2 Frequency [Hz], User #1 4 x 10 1

0.8

0.6

0.4

Power Spectrum 0.2

0 0.5 1 1.5 2 Frequency [Hz], User #2 4 x 10

24 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