This document summarizes spatial array processing and signal estimation techniques. It discusses how sensor arrays can be used to estimate the number and directions of arrival of multiple signal sources. Key techniques include beamforming, MUSIC, root-MUSIC, and ESPRIT algorithms. These algorithms use eigendecomposition and subspace projections to estimate signal parameters from sensor array data with high resolution. The document also covers stochastic approaches to estimate unknown signal waveforms after determining the signal parameters.

1. Spatial Array Processing
Signal and Image Processing Seminar
Murat Torlak
,
Telecommunications & Information Sys. Eng.
The University of Texas at Austin
1

2. 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

3. Problem Statement
s1 (t)
s2 (t)
θ1
θ2
∆
x1 (t) x2 (t) x3 (t) x4 (t) x5 (t) x6 (t)
Find
1. Number of sources
2. Their direction-of-arrivals (DOAs)
3. Signal Waveforms
3

4. 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

5. Antenna Array
Source
θ
X1 X
2 X3 X4 ∆ X5
Array Response Vector–Far-Field Assumption
Narrowband
- Delay = Phase Shift
Assumption
a

6. = 1; ej 2fc 4 sin

7. =c ; : : : ; ej 2fc 44 sin

8. =c T
Single Source Case = xt
2 3 2 3 2 3
x t s1 t 1
6 x1 t
6 2
6
7 6
7 6
7 6 s1 t ,
7 6
7 6
7 6 e,j 2fc 7
7
7
6
6 7=6
7 6 76
7 6 7 s t = a

9. s t
7 1
6
6
.
. 7 6
7 6
.
. 7 6
7 6
.
. 7
7
1 1
4 5 4 5 4 5
e,j 2fc M ,1
. . .
xM t s1 t , M , 1
where = 4 sin

10. 1 =c.
5

11. General Model
By superposition, for d signals,
xt = a

12. 1 s1 t + + a

13. d sd t
Xd
= a

14. k sk t
k=1
Noise
d
X
xt = a

15. k sk t + nt
k=1
= ASt + nt
where
A = a

16. 1 ; : : : ; a

17. d
and
St = s1 t; : : : ; sd t T :
6

18. Low-Resolution Approach:Beamforming
Basic Idea
d
X d
xi t = =e i,1j 2fc 4 sin

19. k =c sk t = X sk tejwk i,1
k=1 k=1
where wk = 24 sin

20. k =c and i = 1; : : : ; M .
Use DFT (or FFT) to find the frequencies fwk g
2 3
6
1 1 1
7
6
6 ejw1 ejw2 ejwM 7
7
F F w
= 1 FwM = 6
6
6 . . . .
7
7
7
6
4
. . .
.
. 7
5
ej M ,1w1 ej M ,1w2 ej M ,1wM
. . .
Look for the peaks in
jF xi tj = jF xtj2
To smooth out noise
N
1 X jF xtj2
B wi =
N t=1
7

21. Beamforming Algorithm
Algorithm
PN
1. Estimate Rx = 1
Nt=1 xtx t
2. Calculate B wi = F wi Rx Fwi
3. Find peaks of B wi for all possible wi ’s.
4. Calculate

22. k , i = 1; : : : ; d.
Advantage
- Simple and easy to understand
Disadvantage
- Low resolution
8

23. Number of Sources
Detection of number of signals for d M,
xt = Ast + nt
Rx = E fxtx tg = A E fsts tg A + E fntn tg
| z | z
Rs nI
2
= A |Rzs |Az
| z
+ nI
2
M d dd dM
2
wheren is the noise power.
No noise and rank of Rs is d
– Eigenvalues of Rx = ARs A will be
f1 ; : : : ; d ; 0; : : : ; 0g:
– Real positive eigenvalues because Rx is real, Hermition-symmetric
– rank d
Check the rank of Rx or its nonzero eigenvalues to
detect the number of signals
2
Noise eigenvalues are shifted by n
f1 + 2 2 2 2
n ; : : : ; d + n ; n ; : : : ; n g:
where 1 ::: d and 0
Detect the number of principal (distinct) eigenvalues
9

24. MUSIC
Subspace decomposition by performing eigenvalue
decomposition
M
+ n I = X k ek e
Rx = ARs A 2 k
k=1
whereek is the eigenvector of the k eigenvalue
spanfAg = spanfe1 ; : : : ; ed g = spanfEs g
Check which a

25. spanfEs g or PA a

26. or
P? a

27. , where PA is a projection matrix
A
Search for all possible

28. such that
1
jP? a

29. j2 = 0 or M

30. =
A PAa

31. = 1
After EVD of Rx
P? = I , EsE = EnE
A s n
where the noise eigenvector matrix
En = ed+1; : : : ; eM
10

32. Root-MUSIC
For a true , j 2fc 4 sin

33. =c is a root of

34. e
M
X
P z = 1; z; : : : ; z M ,1 T ek e 1; z ,1 ; : : : ; z ,M ,1 :
k
k=d+1
After eigenvalue decomposition,
- Obtain fek gd=1
k
- Form pz
- Obtain 2M , 2 roots by rooting pz
- Pick d roots lying on the unit circle
- Solve for f

35. k g
11

36. 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 3 2 3
ejw 1
6
6 7 6
7 6 7
7
ej 2w ejw
a 2

37. = 6
6 7 6
7=6 7 jw 1 jw
7e = a e
6
6 .
. 7 6
7 6 .
. 7
7
4 . 5 4 . 5
ej M ,1w ej M ,1w
General form relating subarray (1) to subarray (2)
2 3
6 ejw 1
7
A2 = A1 6
6
4
..
.
7 = A1:
7
5
ejwd
contains sufficient information of f

38. k g
12

39. ESPRIT
spanfEs g = spanfAg and Es = AT
- T is a d d nonsingular unitary matrix
- T comes from a Grahm-Schmit orthogonalization
of Ab in
RxEssE + En E
= s n
AH RsA + n I
2
E2 = A2T and E1 = A1T
s s
Es 2 = A2 T = A1 T = Es 1T,1 T
Multiply both sides by the pseudo inverse of E1
s
E1Es 2 = E1 E1,1 E1E1 T,1 T = T,1 T
s
where means the pseudo-inverse
A = AsH A,1AsH
Eigenvalues of T,1T are those of .
13

40. Superresolution Algorithms
PN
1. Calculate Rx = 1
N k=1 xkx k
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 M

41. for

42. from 0 to
180
- Find ^ k
f

43. k gd=1 corresponding the d peaks of
M .
(b) Root-MUSIC: Root the polynomial pz
- Pick the d roots that are closest to the unit
circle frk gd=1 and

44. k = sin,1 2fc .
k
^ rk c
(c) ESPRIT: Find the eigenvalues of E1E2,
s s
f kg
-

45. k = sin,1 2fcc4
^ k
14

46. Signal Waveform Estimation
Given A, recover st from xt.
Deterministic Method
– No noise case: find wk such that
wk ? a

47. i; i 6= k; wk 6? a

48. k
A can do the job
Axt = AAst = st
With noise, nt
Axt = st + Ant
– Disadvantage = increased noise
15

49. Stocastic Approach
Find wk to minimize
min E fjwk xtj2 g = min wk Rk wk
a

50. k wk =1 a

51. k wk =1
Use the Langrange method
min E fjwk xtj2 g , ;w wk Rk wk + 2a

52. k wk , 1
min
a

53. k wk =1 k
Differentiating it, we obtain
Rx wk = a

54. k ; orwk = R,1 a

55. k
x
.
Since a

56. k wk = a

57. k R,1a

58. k = 1,
x
Then
= a

59. k R,1 a

60. k
x
Capon’s Beamformer
wk = R,1a

61. k =a

62. k R,1a

63. k
x x
16

64. Subspace Framework for Sinusoid
Detection
P
d
xt = k e k +j!k t
k=1
Let us select a window of M , i.e.,
xt = xt; : : : ; xt , M + 1 T
Then
2 3 2 3
6
xt
7 6 k e +j!j!t,1
k+ k t
7
6
6 xt , 1 7
7 X6
d 6 k e k k 7
7
xt
=
6
6
6 .
7=
7 6
7 k=1 6
6 .
7
7
7
6
4
. 7
5 6
4
. 7
5
k e k +j!k t,M +1
. .
xt , M + 1
2 3
1
d 6
X6
6 e, k +j!k 7
7
7
6
6 7 e k +j!k t
7 k
k=1 6 7|
= .
6 . 7 z
4 5 sk t
e k +j!k ,M +1
.
| z
a k
d
X
= a k sk t As t ;
=
k=1
where M is the window size, d the number of sinusoids, and
k = e k +j!k .
17

65. Subspace Framework for Sinusoid
Detection
Therefore, the subspace methods can be applied to
find f k + j!k g
Recall
d
X
xt = ke k k
+j! t
k=1
Then finding f k g is a simple least squares problem.
18

66. Wireless Communications
co
-c
ha
nn
el
in
te
rf
er
en
ce
Multipaths
th
Pa
t
r ec
Di
Cellular Telephony
Office Building
Residential Area
Personal Communications
Outdoors
Services (PCS)
To Networks
Di
th re
t Pa ct
D irec Pa
th
th
ipa
lt
Mu
Wireless LAN
Increasing Demand for Wireless Services
Unique Problems compared to Wired
communications
19

67. Problems in Wireless Communications
Scarce Radio Spectrum and Co-channel
Interference
1
2 4
3
1
4 1
3 2
1
Multipath
Multipath
Direct P
ath
Mu
ltip
ath
Base
Station
Time
Desired Signal Reflected Signal
Coverage/Range
20

68. 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

69. Experimental Validation of Smart Uplink
Algorithm
Comparison of constellation before (upper) and after
smart uplink processing (middle and lower)
imaginary axis
real axis
Antenna Output
imaginary axis
real axis
Equalized Signal 1
imaginary axis
real axis
Equalized Signal 2
22

70. Selective Transmission Using DOAs
Beamforming results for two sources separated by
20
1
Power Spectrum
0.8
0.6
0.4
0.2
0
0.5 1 1.5 2
Frequency [Hz], User #1 4
x 10
1
Power Spectrum
0.8
0.6
0.4
0.2
0
0.5 1 1.5 2
Frequency [Hz], User #2 4
x 10
23

71. Selective Transmission Using DOAs
Beamforming results for two sources separated by
3
1
Power Spectrum
0.8
0.6
0.4
0.2
0
0.5 1 1.5 2
Frequency [Hz], User #1 4
x 10
1
Power Spectrum
0.8
0.6
0.4
0.2
0
0.5 1 1.5 2
Frequency [Hz], User #2 4
x 10
24

72. 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
25