NASA Technical Reports Server (NTRS) 20060020679: Extension of DAMAS Phased Array Processing for Spatial Coherence Determination (DAMAS-C)

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by  NASA Technical Reports Server (NTRS)

12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, May 8-10, 2006 AIAA-2006-2654

Extension of DAMAS Phased Array Processing
for Spatial Coherence Determination (DAMAS-C)

* †
Thomas F. Brooks and William M. Humphreys, Jr.
NASA Langley Research Center, Hampton, VA. 23681

The present study reports a new development of the DAMAS microphone phased array processing
methodology that allows the determination and separation of coherent and incoherent noise source
distributions. In 2004, a Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS) was
developed which decoupled the array design and processing influence from the noise being measured, using a
simple and robust algorithm. In 2005, three-dimensional applications of DAMAS were examined. DAMAS
has been shown to render an unambiguous quantitative determination of acoustic source position and
strength. However, an underlying premise of DAMAS, as well as that of classical array beamforming
methodology, is that the noise regions under study are distributions of statistically independent sources. The
present development, called DAMAS-C, extends the basic approach to include coherence definition between
noise sources. The solutions incorporate cross-beamforming array measurements over the survey region.
While the resulting inverse problem can be large and the iteration solution computationally demanding, it
solves problems no other technique can approach. DAMAS-C is validated using noise source simulations and
is applied to airframe flap noise test results.

SYMBOLS

a shear layer refraction amplitude correction for e
m mn
A DAMAS-C matrix with A components
C nn,n (cid:1) n(cid:1)
0 0
A reciprocal influence of cross-beamforming characteristics between grid points
nn,n (cid:1) n(cid:1)
0 0
B array half-power “beamwidth” of 3 dB down from beam peak maximum
c speed of sound in medium in the absence of mean flow
0
CSM cross spectral matrix
(cid:1)2 coherence between sources at n and n
nn 0
0
DR diagonal removal of G in array processing
e steering vector for array for focus at grid point n
n
e component of e for microphone m
mn n
f frequency
(cid:1)f frequency bandwidth resolution of spectra
G cross-spectrum between P and P
mm (cid:1) m m (cid:1)
G matrix (CSM) of cross-spectrum elements G
mm (cid:1)
H height of chosen scan plane
i iteration number
m microphone identity number in array
m(cid:1) same as m, but independently varied

*
Senior Research Scientist, Aeroacoustics Branch, Fellow AIAA.
†
Senior Research Scientist, Aeroacoustics Branch, Senior Member AIAA.

1
American Institute of Aeronautics and Astronautics
m total number of microphones in array
0
n grid point number on scan plane(s)
n (cid:1), n ,n (cid:1) same as n but independently varied
0 0
M wind tunnel test Mach number
N total number of grid points over scan plane(s)
P Fourier Transform of pressure time history at microphonem
m
QFF Quiet Flow Facility
Q idealized P for modeled source at n for quiescent acoustic medium
n m
r distance r for m equal the center microphone of array
c m
r retarded coordinate distance from focus point to m, (=(cid:1) c )
m m 0
SADA Small Aperture Directional Array
STD standard (or classical) array processing
T complex conjugate transpose (superscript)
(cid:1) propagation time from grid point to microphone m
m
w frequency dependent shading (or weighting) for m
m
W shading matrix of w terms
m
W width of scan plane
(cid:1)x widthwise spacing of grid points
X matrix of X terms
C nn
0
X (auto) spectrum of “noise source” at n , with levels defined with respect to array position, (=Q(cid:1) Q )
nn 0 n n
0 0 0 0
X cross-spectrum between sources at n and n, (=Q(cid:1) Q )
nn 0 n n
0 0
(cid:1)y heightwise spacing of grid points
Y matrix of Y terms
C nn
0
Y beamform power response of array at focus location n , Y of Ref. 1
nn 0 n
0 0
Y cross-beamform power response between locations n and n
nn 0
0

I. INTRODUCTION

The Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS) method1 represents a
breakthrough in phased array processing technology that can remove array-dependent beamforming characteristics
from output presentations. Whereas traditional beamforming produces an output that is dependent on array
geometry, size, source distance, and frequency, DAMAS can remove these dependencies to render an accurate and
explicit definition of acoustic source strength and location. Applications of DAMAS shown in Ref. 1 demonstrated
its ability to provide quantitative airframe noise definition that was previously unattainable. The DAMAS algorithm
is an iterative non-negative least squares solver of a linear system of equations (defining the DAMAS inverse
problem) that is formulated based on the array response to distributions of modeled noise sources1,2. Three-
dimensional applications were examined in Ref. 3. Dougherty4 proposed two extensions to DAMAS. These were
successfully applied to distributed source simulations and wake vortex turbulence decay noise data from flight tests.
One extension termed DAMAS 2 offers dramatic speedups of the iteration calculations and adds solution
regularization by a low pass filter. The other extension, DAMAS 3, does these and additionally reduces the required
number of iterations. Both extensions in Ref. 4 restrict the point-spread function (array beamform lobe
characteristics) to a translationally-invariant convolution form. This can be a serious limitation in spatial generality,
but this is addressed through spatial transforms. DAMAS1 does not have such limitation concerns, although it is
slower computationally. More recently, Dougherty applied the DAMAS methodology with success to turbofan
engine duct mode measurement analysis5.

The underlying premise of classical beamform processing methodology (as does DAMAS) is the assumption
that the source regions under study are distributions of statistically independent noise sources. This assumption is the
basis for source power integration methodology, for example see Ref. 6. Classical beamforming produces
insufficient information to determine how (and even if) source regions may be correlated. Normally one may suspect
coherence may be present when the beamformed maps are “peculiar”. Horne, et al.7 studied the effects of source
coherence on array response for simple distributions of noise sources. It was found that distributed, coherent sources
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will radiate multi-lobed interference patterns with integrated source levels depending on array size and frequency.
An expression for computing two-point cross beamforming over the array measurement scan plane was given and
was suggested as a possibly useful spatial coherence diagnostic. Oerlemans and Sijtsma8 employed such an
expression to perform coherence analyses between beamforming locations for reflections and aeroacoustic sources.
The array outputs for simulation sources well matched experimental data for known sources. Still, from a source
definition standpoint, the array beamforming coherence analysis approach is indeterminate for unknown source
distributions. It was suggested7 that this cross beamforming coherence approach, combined with other knowledge,
offers potential for the determination of coherence length of distributed sources and for the identification of
sidelobes and mirror sources. A possible alternate approach to determine coherent source distributions using a
microphone array is alluded to in Ref. 9. This would be a generalization of an array technique known as matched
field processing. It would find combinations of point sources in the region that would coherently sum to the array
cross-spectral matrix (CSM) eigenvectors. At present, there are no developed methods reported.

DAMAS, along with classical beamforming processing, employs the statistically independent (incoherent) noise
source distribution assumption. It can thus produce inaccurate and distorted results in the presence of coherent
sources. The original intent of the present study was to investigate the effect of source coherence on DAMAS
processing and to develop methods to interpret and minimize error. It became apparent that it would be difficult to
develop systematic diagnostic methods because of the complexity that coherence can cause in standard
beamforming and DAMAS. During the study, a break came which offered the opportunity to generalize DAMAS to
account for coherence, rather than just trying to identify and correct for it. In developing simulations of coherent
sources in the noise field under study by DAMAS, it became apparent that an equation form similar to that of
DAMAS can be rendered when coherence is allowed between sources and one employs cross-beamforming over the
source region. This meant that if coherences between sources could be regarded as independent variables, then the
cross-products between noise contributions of the sources are also independent variables. A DAMAS type linear
system and solution should then be appropriate to determine a coherent as well as an incoherent distribution of
sources. This paper presents the DAMAS-C (Coherence) development and some sample simulation and
experimental applications.

II. DAMAS-C ANALYSIS

Beamforming and Cross-beamforming. Figure 1 shows a distribution of microphones comprising a phased
array used to survey (or scan) noise regions. A scan plane with N grid points is shown which cuts through a noise
source region under study. Additional scan planes or individual grid points can also be placed over the source region
when performing three-dimensional (3D) beamforming. For classical or standard (STD) array beamforming, the
output power spectrum (or response) of the array is
obtained from (using the terminology from Ref. 1)

tangent
Y = eTnGen (1) at ray
n m2 crossing
0
n = N
where matrices are signified by bold letters. This is Eq. n actual ray path
(6) from Ref. 1, where here Yn =Y(e), when focused at ray path for no flow m
grid point n. The Cross-Spectral Matrix (CSM) is G, c
where
Phased Array

of Microphones
G=(cid:1)(cid:5)(cid:5) GM11 GG1222 L G1Mm0 (cid:6)(cid:6)(cid:3) (2) Flow MSLaheyeaeanrr
(cid:5) M O M (cid:6) Scan Plane(s)
(cid:5) (cid:6) n = 1 o ver Noise
(cid:2)G m 1 Gm m (cid:4) Source Region
0 0 0

where m is the total number of microphones in the Figure 1. Illustration of open test configuration where
0 the microphone array is placed outside the flow region
array. The steering “vector” matrix with respect to a
containing the noise source scan plane region.

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survey grid point at location n is 10 Yn,n 90 Yn,n n=1
o o o o 90
5
en=col[e1 e2 L em ] (3)
0 0 99 BAUTO 91 94
95
and the component for each microphone m is
n) -5
e =a rm exp(j2(cid:1)f(cid:2) ) (4) on(i -10 9910 9190 90 90
m m rc m locati 10 Yno,n no=113 95 Yno,n no=202 90
Y 5
In Ref. 1, Eqs. (2), (3), and (4) correspond to the same
equation numbers above. Figure 1 represents the test B
0 96 99 CROSS 92 95
condition of an open jet test section with a noise region
under study in the jet flow and the microphone array on -5
the outside with a shear layer in between. The steering 91 91
“vector” e terms account for the mean amplitude a -10 95 95 90 90
m m -10 0 10 -10 0 10
and phase changes due to the convection and refraction Xlocation(in)
through the shear layer to each microphone m. The time
delay from a grid point to a microphone m is(cid:1) . In Figure 2. Output dB level contours over scan planes of
m
(auto) Beamforming, Y and cross beamforming
retarded coordinates, the distance from a point on the nn
0 0
scan plane to the microphone m is r and the distance Y between grid points at n and at n. Point source
m n0n 0
to the reference center microphone is r . located at n=113.
c

n=N
o
For the present analysis, we employ a cross- Plane
n=n=N
beamform product o
n=5
o
eT Ge Plane
n n
Y = 0 (5)
n0n m02 no=1
Plane
that is a beamform cross-spectrum (or cross-response) of
the array between focused locations of grid points at
n=n and at another n. Equation (5) becomes Eq. (1)
0
when n =n. For the steering “vectors” of Eq. (5),
0

en =col[e1n e2n L em n ] (6)
0 0 0 0 0 n=n=1
o
and
Figure 3. Illustration of stack of individual n planes
0
en=col[e1n e2n L em n] (7) defining the Yn0n survey and Xn0n solution space.
0

Figure 2, in the top left frame, shows a standard (auto) beamform dB contour map of Y for a simulated point
nn
0 0
source in the center of a 15x15 point grid scan plane (the focus points n range from 1 to N=225). The array used
0
is that of the SADA reported in Ref. 1 and detailed subsequently. Whereas in Ref. 1, all presentations were of
beamforming and DAMAS solutions over the scan plane of N points, the present paper often presents results over
individual n planes with grid points n=1,2,3,...,N. Figures 2 present cross-beamform maps for the specific
0
n =1, 113, and 202 planes, respectively. The simulated source location is at n=113. In these planes, the respective
0
n grid locations are indicated by an open symbol. (Noted for subsequent use is that a length measure of the half-
0
power beamwidth B is shown for Y and for a cross-beamform case Y .) Figure 3 illustrates all N of the n
nn nn 0
0 0 0
planes. Each n plane contains all cross-beamform responses Y over n=1,2,3,...,N – which includes its standard
0 nn
0
beamform response Y at n .
nn 0
0 0
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DAMAS-C Inverse Problem Definition. From Ref. 1, Eq. (10), the pressure transform P of microphone m is
m
related to a modeled source located at position n in the source field.

P =Q e (cid:1)1 (8)
mn n mn

where here, we have used the terminology P =Q e (cid:1)1 rather than P =Q e (cid:1)1 from Ref. 1. In Eq. (8), Q
mn n mn m:n n m:n n
represents the pressure transform that P would be if flow convection and shear layer refraction did not affect
mn
transmission of the noise to microphone m and if m were at a distance of r from n rather than r . The e (cid:1)1 term
c m mn
is, from Ref. 1, those things that are postulated to affect the signal in the radiation transmission to render P (this is
mn
a way to back out Q information from P ). For a single source located at n, Eqs. (11) and (12) of Ref. 1 gives
n m
relations for P (cid:2)P and G .
mn m (cid:1)n nmod

In the present analysis, it is desired to obtain a more general distribution for the CSM than that of a distribution
of uncorrelated acoustic sources at different n, as given in Ref. 1. From Eq. (8), the cross-spectrum between
microphones m and m (cid:1) for a distribution of sources over all N grid points is

P *P =(cid:1)(cid:1)(Q e(cid:2)1 )*(Q e(cid:2)1 ) (9)
m m (cid:3) n(cid:3) mn(cid:3) n(cid:3) m (cid:3)n (cid:3)
0 0
n(cid:3) n(cid:3)
0

This reflects the fact that the acoustic pressure perceived at microphone m due to the sources at n (cid:1) and n (cid:1) are
0
generally different than that perceived at microphone m (cid:1) for these same sources. Note here that n , n, n (cid:1) , and n (cid:1)
0 0
are distinguished from each other only in that they are varied separately.

As in Ref. 1, the G terms of the CSM are proportional to the corresponding P *P terms. One has
mm (cid:1) m m (cid:1)

G =(cid:1)(cid:1)X (e(cid:2)1 )*e(cid:2)1 (10)
mm (cid:3) n(cid:3) n(cid:3) mn(cid:3) m (cid:3)n (cid:3)
0 0
n(cid:3) n(cid:3)
0

where X =Q(cid:2) Q . Upon accounting for FFT data ensemble processing parameters1, X represents the mean-
n(cid:1) n (cid:1) n(cid:1) n(cid:1) n(cid:1) n
0 0 0
square cross-spectral pressure per bandwidth, due to the coherent portion between the sources at n (cid:1) , and n (cid:1) , at each
0
microphone m (or m (cid:1) ) normalized in level for a microphone at r =r . As will be seen, the determination of X
m c n (cid:1) n(cid:1)
0
is the primary objective of DAMAS-C.

Note, that if the sources at n (cid:1) , and n (cid:1) each radiate noise in a statistically independent way then X =0. In
0 n(cid:1) n(cid:1)
0
particular, if X =0 when n (cid:2) (cid:1)n (cid:2) , then the result from Ref. 1 is attained,
n (cid:1) n (cid:1) 0
0

N
G =(cid:1)X (e(cid:2)1)*e(cid:2)1 (11)
mm (cid:3) nn mn m (cid:3)n
n

where G are components of G from Eq. (13) of Ref. 1. Here, X and e(cid:1)1 equal X and e(cid:1)1 , respectively,
mm (cid:1) mod nn mn n m:n
of Ref. 1.

For our present coherent source case, the presently modeled CSM is G with components G given by
modC mm (cid:1)
Eq. (10). Employing this in Eq. (5),

e TG e
n modC n
(Y ) = 0 (12)
n0n mod m2
0
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eT (cid:1)(cid:1)X [ ] e (cid:1)(cid:1)(eT [ ] e )X
n0 n(cid:2) 0 n(cid:2) n(cid:2) 0 n(cid:2) n n0 n(cid:2) 0 n(cid:2) n n(cid:2) 0 n (cid:2)
n(cid:2) n(cid:2) n(cid:2) n(cid:2)
(Y ) = 0 = 0 (13)
n0n mod m2 m2
0 0

where the bracketed term is

(cid:3)(cid:7) (e1(cid:1)n1(cid:2) 0 )*e1(cid:1)n1(cid:2) (e1(cid:1)n1(cid:2) 0 )*e2(cid:1)n1(cid:2) L (e1(cid:1)n1(cid:2) 0 )*em(cid:1)10n(cid:2) (cid:5)(cid:8)
[ ] =(cid:7) (e2(cid:1)n1(cid:2) 0 )*e1(cid:1)n1(cid:2) (e2(cid:1)n1(cid:2) 0 )*e2(cid:1)n1(cid:2) M (cid:8) (14)
n(cid:2) n(cid:2) (cid:7) (cid:8)
0 M O M
(cid:7) (cid:8)
(cid:4)(cid:7)( em(cid:1)1n(cid:2) )*e1(cid:1)n1(cid:2) L L (em(cid:1)1n(cid:2) )*em(cid:1)1n(cid:2) (cid:6)(cid:8)
0 0 0 0 0

To be explicit, one can look at the terms of Eq. (13), with n (cid:2) =1,2,(cid:1)(cid:1),N and n (cid:2) =1,2,(cid:1)(cid:1),N
0

m2Y =eT [ ] e X +eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X (15)
0 n0n n0 11 n 11 n0 12 n 12 n0 21 n 21 n0 N1 n N1 n0 NN n NN

And for n =1,2,(cid:1)(cid:1),N and n=1,2,(cid:1)(cid:1),N
0

m2Y =eT[ ] e X +eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X +(cid:1)(cid:1)+eT[ ] eX +(cid:1)(cid:1)+eT[ ] e X
0 11 1 11 1 11 1 12 1 12 1 21 1 21 1 N1 N1 1 NN 1 NN

m2Y =eT[ ] e X +eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X
0 12 1 11 2 11 1 12 2 12 1 21 2 21 1 N1 2 N1 1 NN 2 NN
M
m2Y =eT[ ] e X +eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X +(cid:1)(cid:1)+eT[ ] e X (16)
0 21 2 11 1 11 2 12 1 12 2 21 1 21 2 N1 1 N1 2 NN 1 NN
M
m2Y =eT [ ] e X +eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X
0 N1 N 11 1 11 N 12 1 12 N 21 1 21 N N1 1 N1 N NN 1 NN
M
m2Y =eT [ ] e X +eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X +(cid:1)(cid:1)+eT [ ] e X
0 NN N 11 N 11 N 12 N 12 N 21 N 21 N N1 N N1 N NN N NN

One notes that Eq. (16) can be written as

Y =A X (17)
C C C

that is the same inverse problem form as DAMAS, Eq. (18) of Ref. 1. Here, however, Y and X each have N2
C C
complex-number components, rather than the N real-number components of Ref. 1, that is

Y , with n n=11,12,(cid:1)(cid:1),1N,21,22,(cid:1)(cid:1),2N,31,32,(cid:1)(cid:1),NN
nn 0
0
and X , with n n=11,12,(cid:1)(cid:1),1N,21,22,(cid:1)(cid:1),2N,31,32,(cid:1)(cid:1),NN (18)
nn 0
0

And A has N4 complex-number components, rather than the previous N2 real-number components, which are
C

A =(eT [ ] e )/m2 (19)
n0n,n(cid:1) 0 n(cid:1) n0 n(cid:1) 0 n(cid:1) n 0

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with [ ] being defined by Eq. (14) and the order within A defined by
n(cid:1) n(cid:1) C
0

(cid:2) A11,11 A11,12 A11,13 L A11,21 A11,22 L A11,NN (cid:4)
(cid:6) (cid:7)
A A A
(cid:6) 12,11 12,12 12,13 (cid:7)
(cid:6) A A A (cid:7)
13,11 13,12 13,13
(cid:6) (cid:7)
A =(cid:6) M O M (cid:7) (20)
C (cid:6) A21,11 A21,12 O (cid:7)
(cid:6) (cid:7)
(cid:6) A22,11 A22,12 O (cid:7)
(cid:6) M ANN(cid:1)1,NN(cid:1)1 ANN(cid:1)1,NN(cid:7)
(cid:6) (cid:7)
(cid:3) ANN,11 ANN,12 L ANN,NN(cid:1)1 ANN,NN (cid:5)

The above equations contain terms that are complex conjugates of one another. These are

Y =Y(cid:1) (21)
nn nn
0 0

and

X =X(cid:1) (22)
nn nn
0 0

Relationships are also noted for A . For the diagonal terms of Eq. (20), where one has n n=n (cid:1) n (cid:1) ,
C 0 0

A =A =1 (23)
nn,n(cid:1) n(cid:1) nn,nn
0 0 0 0

Also, for all terms, because of Eqs. (21) and (22), it can be shown that

A =A(cid:2) (24)
nn,n(cid:1) n(cid:1) nn ,n(cid:1) n (cid:1)
0 0 0 0

whether n =n or n (cid:1)n and whether n (cid:1) =n (cid:1) or n (cid:2) (cid:1)n (cid:2) . The terms are in general complex, but when n =n and
0 0 0 0 0
n (cid:1) =n (cid:1) , A =A is real. (Note that the DAMAS problem of Ref. 1 is recovered if X =0 when n (cid:1)n
0 nn,n(cid:1) n(cid:1) n n ,n(cid:1) n(cid:1) nn 0
0 0 0 0 0 0 0
and one only considers the beamform power spectrum Y =Y rather than the cross-spectrum Y . Then A
nn n nn nn,n(cid:1) n(cid:1)
0 0 0 0 0
becomes A of Ref. 1.)
n

The above relationships between terms, Eqs. (21)-(24), mean that the DAMAS-C problem contains N(N+1)/2,
rather than N2, potentially independent equations and unknowns. In fact, the Y rows in Eq. (16) are complex
nn
0
conjugates of other Y rows. This means that for Eq. (17), the number of components can be reduced by almost
nn
0
half by removing rows. Then for Y =A X , the components can be defined as Y , X , and A with the
C C C nn n(cid:1) n(cid:1) nn,n (cid:1) n(cid:1)
0 0 0 0
indices n n and n (cid:1) n (cid:1) following the pattern,
0 0

n n=11,12,13,..,1N,22,23,24,..,2N,33,34,..,NN (25)
0
=(n =1,2,3,..,N)(n=n ,n +1,n +2,..,N)
0 0 0 0

and

n (cid:1) n (cid:1) =11,12,13,..,1N,21,22,23,..,2N,31,32,..,NN (26)
0
=(n (cid:1) =1,2,3,..,N)(n (cid:1) =1,2,3,..,N)
0

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Therefore, using Eqs, (25) and (26), the DAMAS-C problem, Eq. (17), is reduced in size from that indicated by Eqs.
(18)-(20). Y then has N(N+1)/2 terms, X has N2 terms (N(N+1)/2 are independent), and A has
C C C
N3(N+1)/2 terms (N2(N2+1)/2 are independent).

Modified Beamforming. Note that the inverse problem, Eq. (17), for X can be defined in terms of Y and
C C
A being generated by shaded standard, diagonal removal (DR), and shaded DR beamforming – in parallel with
C
that done in Ref. 1. That is, for shaded standard beamforming,

eT WGWTe
n n
Y = 0 (27)
n0n (cid:2) m (cid:4) 2
0
(cid:6) (cid:1)w (cid:7)
(cid:6) m(cid:7)
(cid:3) m=1 (cid:5)

and, correspondingly, A is defined by components
C

eT W[ ] WTe
A = n0 n(cid:2) 0 n(cid:2) n (28)
n0n,n(cid:2) 0 n(cid:2) (cid:3) m (cid:5) 2
0
(cid:7) (cid:1)w (cid:8)
(cid:7) m(cid:8)
(cid:4) m=1 (cid:6)

For diagonal removal, DR, beamforming

eT G e
Y = n0 diag=0 n (29)
n0n m 2(cid:1)m
0 0

and, correspondingly, A is defined by components
C

eT ([ ] ) e
A = n0 n(cid:2) 0 n(cid:2) diag=0 n (30)
n0n,n(cid:2) 0 n(cid:2) m 2(cid:1)m
0 0

For shaded DR beamforming

eT WG WTe
Y = n0 diag=0 n (31)
n0n (cid:3) m (cid:5) 2 (cid:3) m (cid:5)
0 0
(cid:7) (cid:1)w (cid:8) (cid:2)(cid:7) (cid:1)w (cid:8)
(cid:7) m(cid:8) (cid:7) m(cid:8)
(cid:4) m=1 (cid:6) (cid:4) m=1 (cid:6)

and, correspondingly, A is defined by components
C

eT W([ ] ) WTe
A = n0 n(cid:3) 0 n(cid:3) diag=0 n (32)
n0n,n(cid:3) 0 n(cid:3) (cid:4) m (cid:6) 2 (cid:4) m (cid:6)
0 0
(cid:8) (cid:1)w (cid:9) (cid:2)(cid:8) (cid:1)w (cid:9)
(cid:8) m(cid:9) (cid:8) m(cid:9)
(cid:5) m=1 (cid:7) (cid:5) m=1 (cid:7)

Note that for all the above types of special beamforming processing, the relationships Eqs. (21)-(26) are equally
valid.
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American Institute of Aeronautics and Astronautics
DAMAS-C Inverse Problem Solution. As in Eq. (22) of Ref. 1, a single linear equation component of the
present Eq. (17) is

Y =A X +A X +(cid:1)(cid:1)+A X +A X +A X +(cid:1)(cid:1)+A X (33)
nn nn,11 11 n n,12 12 nn,1N 1N nn,21 21 nn,22 22 n n,NN NN
0 0 0 0 0 0 0

N N
Y =(cid:1) (cid:1)A X (34)
nn nn,n(cid:2) n(cid:2) n(cid:2) n(cid:2)
0 0 0 0
n(cid:2) 0 =1 n(cid:2) =1

With A =1 from Eq. (23), this is rearranged to give
nn,nn
0 0

n0 n(cid:2)1 N N
X =Y (cid:2)[(cid:1) (cid:1)A X + (cid:1) (cid:1)A X ] (35)
nn nn nn,n(cid:3) n(cid:3) n(cid:3) n(cid:3) nn,n(cid:3) n(cid:3) n(cid:3) n(cid:3)
0 0 0 0 0 0 0 0
n (cid:3)0 =1 n(cid:3) =1 n(cid:3) 0 =n0 n(cid:3) =n+1

Because of the counting pattern of n (cid:1) n (cid:1) from Eq. (26), exceptions are noted in the upper and lower sum limits in Eq.
0
(35). If X =X , then the upper limits on the first set of sums, i.e. n and n(cid:1)1, are replaced by n (cid:1)1 and N,
nn n 1 0 0
0 0
respectively, and the lower limits on the second set of sums remain as they are. But, if X =X , then although
nn n N
0 0
the upper limits on the first set of sums remain as they are, the lower limits on the second set of sums, i.e. n and
0
n+1 are replaced by n +1 and 1.
0

Equation (35) is used in an iteration algorithm to obtain the source distribution strengths X (or X ) for all
nn nn
0 0
n and cross-strengths X for all combinations of n and n indicated by Eq. (25), as per the following equation.
nn 0
0

N N
X(i)=Y (cid:2)[0+(cid:1) (cid:1)A X(i(cid:2)1)]
11 11 11,n(cid:3) 0 n(cid:3) n(cid:3) 0 n(cid:3)
n(cid:3) 0 =1 n(cid:3) =1+1

n0 n(cid:2)1 N N
X(i) =Y (cid:2)[(cid:1) (cid:1)A X(i) + (cid:1) (cid:1)A X(i(cid:2)1)] (36)
n0n n0n n0n,n(cid:3) 0 n(cid:3) n(cid:3) 0 n(cid:3) n0n,n(cid:3) 0 n(cid:3) n(cid:3) 0 n(cid:3)
n(cid:3) 0 =1 n(cid:3) =1 n(cid:3) 0 =n0 n(cid:3) =n+1

N N(cid:2)1
X(i) =Y (cid:2)[(cid:1) (cid:1)A X(i) +0]
NN NN NN,n(cid:3) 0 n(cid:3) n(cid:3) 0 n(cid:3)
n(cid:3) 0 =1 n(cid:3) =1

Here, the first and last term of the ith iteration, (i), for X is shown for clarity. Equation (36) represents a similar
nn
0
form as that given in Eq. (24) of Ref. 1, except that these terms are complex and that the problem size for the same
number of N grid points is expanded. In the iterations, the same sum limit exceptions mentioned for Eq. (35) apply.

The iteration path is consistent with a progression through a stack of n =1,2,3,...,N solution maps (planes).
0
Figure 3 illustrates the N planes, where within each, the counting sequence starts at n=1 in the left bottom corner
and increases vertically along each column. The uppermost point in the last column to the right is the n=N grid
point. The last three frames (planes) of Fig. 2 show Y magnitude levels for the particular n planes shown. (The
nn 0
0
Y map of Fig. 2 is not a n plane, but one made up of points Y from each n plane.) For Y terms, as well as
nn 0 nn 0 nn
0 0 0 0 0
for X , with only N+1(cid:1)n terms calculated for each n plane, the values in the plots for the remaining n (cid:1)1
nn 0 0 0
0
grid points are determined from the complex conjugate relationships. The Y terms are defined from Eqs. (5), (27),
nn
0
(29), or (31). The A terms are determined from Eq. (19), (28), (30), or (32). For each (i) iteration for the X
nn,n (cid:1) n(cid:1) nn
0 0 0
terms, the values of X(i) replace the previous iteration X(i(cid:1)1) values. Therefore, whereas Eq. (17) with the
nn nn
0 0
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American Institute of Aeronautics and Astronautics
appropriate A defines the DAMAS-C inverse problem, Eq. (36) is the inverse problem iterative solution, with the
C
appropriate constraints described below. 10

Physical Interpretations and Positivity Constraints. In Ref. 1, a positivity constraint was used in the
iterations for X . This made the inverse DAMAS problem for X sufficiently deterministic to force unique
n n
solutions. The constraint is actually physically necessary in that X represents auto-spectral pressure-squared
n
(positive) amplitude for the sources at grid points n. In DAMAS-C, the value of X is Re(X )+Im(X ).
nn nn nn
0 0 0
When n=n , X =X is real and positive – equivalent to the DAMAS X . Thus for each iteration calculation
0 nn nn n
0 0 0
for Eq. (36), Im(X ) is set to zero and Re(X ) is set to zero if it is not already positive. This is equivalent to
nn nn
0 0 0 0
the positivity constraint X of Ref. 1.
n

When n (cid:1)n, X could be in any of four complex quadrants. In terms of a complex coherence definition
0 nn
0

X =(cid:1) X X (37)
nn nn nn nn
0 0 0 0

where the coherence (cid:1)2 = (cid:1)2 = X2 /X X = X(cid:1) X /X X (38)
nn nn nn nn nn nn nn nn nn
0 0 0 0 0 0 0 0 0

(cid:1) = (cid:1)2 exp(i(cid:2) ) (39)
n n nn n n
0 0 0

where (cid:1) is the cross-spectral phase or phase between coherent portions of source strength at point n with respect
nn
0
to n . This of course follows from X =X(cid:1) . Physically, (cid:1) is interpreted here as the coherence factor between
0 nn nn n n
0 0 0
the sources at n and n . It can be related to noise emission from unsteady aerodynamic related regions over
0
radiating surfaces (0(cid:1)(cid:2) (cid:1)1, for n and n being nearby radiating locations) or reflections, such as from image
nn 0
0
regions in the presence of tunnel sidewalls, ((cid:2) (cid:1)1 for n being the source and n being the reflection image).
nn 0
0

An appropriate constraint based on the above equations is that X (cid:1) X X be enforced in the iterations.
nn nn nn
0 0 0
This is not done in the coding for the present paper. Instead, X is regarded as an independent variable in the same
nn
0
manner as X and X . Also, with regard to the generality of X , there is a question concerning the rank of
nn nn nn
0 0 0
the DAMAS-C inverse problem, and thus the practicality of solving for X with arbitrary phase. If the inverse
nn
0
problem, Eq. (17), is of low rank (more unknowns than independent equations) this would mean that the problem is
indeterminate for a unrestricted solution space. That would suggest that the solution space should be limited, such
that was used for DAMAS with its positivity constraint. For the present paper, DAMAS-C applications are limited
to sources that have only in-phase coherence, that is (cid:1) =0 in Eq. (39), as a constraint. In this n (cid:1)n case, after
n n 0
0
each iteration calculation (as is done above for n =n), Im(X ) is set to zero and Re(X ) is set to zero if it is
0 n n nn
0 0
not already positive.

Problem Reduction by Zoning. The size of the matrices of Y =A X can be quite large and prohibitive for
C C C
many evaluation regions required in practical DAMAS-C applications. The storage of these large numbers of terms,
combined with the iteration procedure requirements, require the use of methods that can further reduce the problem
size wherever the physical problem permits. A method was developed called zoning, which is employed to restrict
the possible solutions to anticipated or realizable conditions of the noise source evaluation region under study. The
evaluation region can be composed of a number of grid point zones, each with assumed coherence criteria. The
criteria can be uniform over the zones or functionally dependent on, for example, the point-to-point distance and
frequency.

For the present paper, the source evaluation region (which can be contiguous or not) is composed of multiple
non-congruent zones A and B containing grid points (n) and (n) , respectively. The zone A is taken as a region
A B
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