Embed Size (px)
Citation preview
1
Deconvolution of Sources in Aeroacoustic Images from Phased Microphone Arrays Using Linear Programming Robert P. Dougherty OptiNav, Inc. Rakesh C. Ramachandran and Ganesh Raman Illinois Institute of Technology
Presentation for the 19th AIAA/CEAS Aeroacoustics Conference, May 2013. See the AIAA web site for the paper, AIAA-2013-2210.
2
Outline • Motivation
• Theory
• Speaker test
• Aeroacoustic test
• Convergence of DAMAS
• Line souces
• Conclusions and recommendations
3
Motivation
4
Prefer small arrays, except for resolution
J.F. Piet,, U. Michel, and P. Bönhing, “Location of the acoustic sources of the A340 with a large phased microphone array during flight tests” 8’th AIAA/CEAS Aeroacoustics Conference, 1998.
8m array 4m array
5
Theory
6
! ! !!! !!
!! !! !! ! !!
!
!!!
Complex, narrowband, slowly varying quantities
array data
CSM ! ! !!! ! !!!!!!!!
!!! (incoherent assumption)
!! ! !! ! where
often ! ! !!"#!
(nonnegative)
7
Beamform map
!! ! !!!!! ! !! !!!!!!!!!!
!
!!!!!!
! !!!!!!!!!!!! !!!
!
!!!!! !!!!!!!
!
!!!!!
where
!!!! ! !!!!! !
(nonnegative)
! ! !"
8
x1 x2 ...
xM
x7
x43 ...
b1 b2
bM
b7
b43 ...
Source distribution Beamform map
PSF
x =
x1x2...xM
!
"
########
$
%
&&&&&&&&
b =
b1b2...bM
!
"
########
$
%
&&&&&&&&
b = Ax
A
9
Deconvolution problem
b = Ax
b A xGiven and find
Typical idea:
!"#"$"%&! !"! !! !!!! !!!!! ! !
10
All terms nonnegative
!! ! !!!!!!!
!
!!!!!
11
Better formulation
!"#$%$&'!! ! !!"#$%&'!!"!!" ! !!! ! !!!
!!! ! !!!!
!
!!!!!! ! !!! !!
12
Bonus for LP
Can handle self noise by adding a constant column to A
13
Speaker test
14
0.35 m aperture 3 m distance
0.2 m source separation
NA = sin tan!1 0.1753
"
#$
%
&'= 0.058
Rayleigh limit = 17.9 kHz
Sparrow limit = 13.8 kHz
Speaker test setup
15
16
17
18
19
20
21
22
LP starts to work
23
24
25
26
DAMAS starts to work
27
28
29
30
31
32
33
34
35
36
37
38
CLEAN-SC starts to work
39
40
Sparrow Limit
41
42
43
44
45
46
47
48
49
50
51
DAMAS and LP stopped
52
53
54
!"#! "$%$& '( )'*$+,&)%-./!0123/456 7897 :9; 89< 7=9:%>?/!0123/456 =@9= =: =: AB
Frequency ranges in speaker test
Rayleigh limit = 17.9 kHz
Sparrow limit = 13.8 kHz
55
Aeroacoustic test
56
δ1 δ3
δ4
δ2
101mm
z = 0.3m D = 0.35m NA = 0.5
57
Hole pair
Hole separation !, mm
Observed LP minimum
frequency for separation, Hz
Rayleigh radius, mm
Sparrow limit, mm
Linear Programming Improvement vs. Rayleigh
Linear Programming Improvement vs. Sparrow
1 35.2 4728 87.7 67.5 2.5 1.9 2 26.5 7099 58.4 45.0 2.2 1.7 3 17.6 8944 46.4 35.7 2.6 2.0 4 8.8 15,976 26.0 20.0 2.9 2.3
Linear Programming Resolution Results
58
59
60
61
62
Hole pair
Hole separation !, mm
Frequency Hz
Number of source points
LP time, sec.
Number of DAMAS iterations
DAMAS time, sec.
1 35.2 4,728 2,177 9.9 60,000 390 2 26.5 7,099 4,694 182 60,000 1,776 3 17.6 8,944 7,558 840 120,000 9,120 4 8.8 15,976 24,703 5,880 120,000 90,000
Timing Results
63
DAMAS convergence
64
65
Line sources
66
Incoherent L- source
67
DAMAS 10k iterations 200k iterations
68
LP
69
LP Anti-dot
70
Compare
DAMAS 2M iterations* LP Antidot**
*Changed from presentation, which showed DAMAS after 200k iterations **4 iterations of the antidot procedure applied
71
Conclusions • Deconvolution problem is a linear programming problem • Efficient and unambiguous solution by simplex method • Simple aeroacoustic test technique • DAMAS and LP results very similar • Both DAMAS and LP have superresolution (factor of 2 vs. Sparrow limit) • LP is an order of magnitude faster than DAMAS • LP has a definite stopping condition and always converges • Can use LP to start DAMAS for better speed (but why use DAMAS?) • Simple way to account for self noise in LP (DD not currently used) • CLEAN-SC has better dynamic range at high frequency (when PSF poor)
Recommendations • Generalized inverse for coherent sources • LP for incoherent sources with accurate PSF • CLEAN-SC when PSF is poorly known • TIDY for broadband • Jury still out concerning incoherent line sources
