deconvolution of sources in aeroacoustic images from ... · presentation for the 19th aiaa/ceas...

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