What is Beamforming?

Abstract

Beamforming is an imaging technique that has found many applications in aeroacoustics, and continues to evolve to meet greater challenges. It has elements in common with other methods such as nearfield acoustic holography, but its strength is distributed, broadband, incoherent sources at arbitrary distance from the array. The formulation of the classical technique in the frequency domain is simple and lends itself to many types of analysis. A derivation is given here that leads to an expression for the variance of the beamform map when the integration time is finite and not all of the elements of the cross spectral matrix are included.

Full text

1
WHAT IS BEAMFORMING ?
Robert P. Dougherty
OptiNav, Inc.
10900 NE 8TH ST, Suite 900, Bellevue, WA, USA
ABSTRACT
Beamforming is an imaging technique that has found many applications in aeroacoustics,
and continues to evolve to meet greater challenges. It has elements in common with other
methods such as nearfield acoustic holography, but its strength is distributed, broadband,
incoherent sources at arbitrary distance from the array. The formulation of the classical
technique in the frequency domain is simple and lends itself to many types of analysis. A
derivation is given here that leads to an expression for the variance of the beamform map
when the integration time is finite and not all of the elements of the cross spectral matrix are
included.
BeBeC-2008-01

2nd Berlin Beamforming Conference
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NOMENCLATURE
Aj
r
ξ
(
)
= Point Spread Function (PSF) for a source at
r
ξ
jwhen beamforming to
r
ξ
b = beamform map
C = Cross Spectral Matrix (CSM)
C = trimmed CSM, e.g., after diagonal deletion
E(x) = Expectation value of x
e−i
ω
t = Fast phase factor divided out of all complex pressure quantities
g = steering vector
g
n = an element of a steering vector
j, k = beamform grid or source point indices
m, n = microphone indices
M = number of acoustic sources
N = number of array microphones
NI = number of blocks=TΔ
ν
NS = number of elements of S
p
n = complex narrowband unsteady pressure measured at microphone n
qr
ξ
j
(
)
= complex narrowband time history of a source at
r
ξ
j
R
n = complex self noise at microphone n
rn = microphone self noise power
(
)
2
n
RE=
S = the set of microphone pairs included in the cross spectral matrix for beamforming
s
j = power of source j =
(
)
jj qqE *
T = integration time
u = steered array data
w = weight vector
r
x
n = location of microphone n
Δt = block length =1
Δ
ν
=
T
NI
α
= weight vector normalization coefficient
Δ
ν
= analysis bandwidth
r
ξ
= point in the beamform grid
r
ξ
j = location of source j
f = time average f=1
NI
ftime block l
()
l
=
1
NI
∑
′
z = Hermitian conjugate (complex conjugate transpose) of z

2nd Berlin Beamforming Conference
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1 INTRODUCTION
Aeroacoustic beamforming is a method for processing microphone array data to produce
images that represent the distribution of the acoustic source strength. It is an imaging
technique that applies to continuous or discrete source distributions. The distance from the
source region to the array is not restricted. The resolution is governed by the same Rayleigh
formulas that govern diffraction-limited optics. Superresolution algorithms that can
potentially locate sources to the theoretical limit of the Cramer-Rao bound have been defined,
but are restricted in applicability. The aeroacoustic application requires the array to operate
over a very wide frequency range compared with electromagnetic beamforming. Grating lobes
are prevented by applying sparse, wideband microphone arrangements. These come with a
drawback of a number of sidelobes in addition the sidelobes related to the overall aperture
shape. Efforts to determine component spectra for subregions of the beamforming grid or to
image sources far below the highest source in level must be able to compensate for the
sidelobes. Classical beamforming gives the best results for incoherent, broadband, source
distributions. Airframe noise measurement provides an excellent match to the strengths of the
technique. Another strong selling point is the ability of beamforming to locate rogue sources;
sources that are not expected and would potentially contaminate the results of conventional
microphone measurements. Numerous extensions to the technique, in addition to
superesolution beamforming have been developed and are continuing to appear. Methods
have been developed for dealing with uniform and nonuniform flow effects, reverberant
environments, linearly and nonlinearly moving sources, pressurized wind tunnels, for fusing
optical and acoustic images, and for maintaining constant resolution over a range of
frequencies. Deconvolution techniques attempt to extract the true source distribution by
removing some of the artifacts introduced by the array. Notable examples include the
DAMAS and CLEAN-SC deconvolution techniques. Current research includes finding ways
to accurately represent extended, coherent, source distributions, beamform in complex, small
environments such as turbofan engine nacelles and automobile cabins, and expand the
beamforming space to include independent parameters in addition to frequency and spatial
coordinates, as well as multipole source distributions. New array designs include multiple
sensor modes. A guide star method has been developed to remove effects of turbulent
decorrelation, but this remains a big challenge in beamforming. Instrumentation and data
management are also continuing issues in beamforming. Since the results improve with the
number of channels, the budget for microphones and data acquisition systems is often a
limiting factor. A number of references can be found in the short review article [1] and the
book [2].
This paper presents the classical beamforming algorithm in the frequency domain using an
extension of the compact notation in [3] while filling in a few of the details. The derivation is
different from the one given in [3], emphasizing an intuitive imaging process rather than an
optimization problem. Reference [3] continues from classical beamforming to discuss several
important deconvolution algorithms. The only ambitious goal in this paper is to derive a
formula for the variance of the beamforming result for the case of finite integration time and a
partial (“trimmed” in the terminology of [3]) cross spectral matrix.

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2 PROBLEM FORMULATION
2.1 Source-receiver model
Consider an array of N microphones and a beamform grid (Fig. 1.) The Green’s function
for grid point
r
ξ
and microphone n is
gn
r
ξ
(
)
. An example is
(
)
n
xik
nxeg n
v
v
r
v
v
−= −
ξξ
ξ
/.
The model for the pressure is pis
p=qjg
r
ξ
j
()
j=1
M
∑+R, (1)
whereqjis the time history of source j, and
R
is the microphone self (flow) noise. The
pressure is recorded for a time T and divided into NI (conceptually non-overlapping) blocks
of length Δ
t
. An FFT is applied to each block, giving an analysis bandwidth of Δ
ν
=1
Δ
t
.
There are NI data vectors, p. Each source j has power
s
j and NI source time history values,
qj, that enter into the model. These are assumed to be zero-mean, random, and mutually
incoherent: cov(qj
*,qk)
=
sj
δ
jk . (2)
The self noise components,
R
n, have power rn
=
var( Rn
(
)
, n = 1,…, N, and are assumed to
mutually incoherent and uncorrelated with the acoustic sources.
Fig. 1. A beamform grid and a phased array of microphones
2.2 Beam steering
The array is steered to
r
ξ
by forming the NI complex numbers (
α
is determined below)
ur
ξ
()
=
α
′
g
r
ξ
()
p=
α
′
g
n*
r
ξ
()
pn
n
=
1
N
∑, (3)
The function
ur
ξ
(
)
is intended to be similar to the source time history for the point
r
ξ
:

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ur
ξ
()
=
α
qj′
g r
ξ
()
gv
ξ
j
()
j=1
M
∑+′
g r
ξ
()
R
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
. (4)
The array is designed so that
g
r
ξ
(
)
varies strongly with
r
ξ
. Ideally ′
g
r
ξ
(
)
gr
ξ
j
(
)
will have peak
at
r
ξ
=r
ξ
j since the inner product of a vector with itself is gives a maximum. For source k:
ur
ξ
k
()
=
α
qkgr
ξ
k
()
2+qj′
g r
ξ
k
()
gv
ξ
j
()
j≠k
∑+′
g r
ξ
k
()
R
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
. (5)
2.3 Source strength maps images
The average power of Eq. 3 is
br
ξ
()
=ur
ξ
()
2=
α
2′
g
r
ξ
(
)
p2=
α
2′
g
r
ξ
(
)p′
p g
r
ξ
(
)=
α
2′
g
r
ξ
(
)
Cg
r
ξ
(
)
, (6)
where the last form introduces the array Cross Spectral Matrix (CSM)
C
=
p
′
p . (7)
Substituting Eq. 1 into Eq. 6 for a single source and no self noise gives
sj=br
ξ
j
()
=
α
2sj′
g
r
ξ
(
)
g
r
ξ
j
(
)′
g
r
ξ
j
(
)
g
r
ξ
j
(
)
. (8)
Solving for
α
gives
α
=1
′
g g
()
2=1
gm
2gn
2
m,n
∑, (9)
Defining the array weight vector by
w
r
ξ
(
)=
α
g
r
ξ
(
)
, Eq. 6, can be rewritten
br
ξ
()
=′
w r
ξ
()
Cw r
ξ
()
. (Classical beamforming expression) (10)
It is often the case that some of the elements of the CSM do more harm than good in
beamforming. As show below, for example, the diagonal elements simply add a noise floor to
the beamform map [2]. Also, certain elements are deleted when using a cross-shaped array
[4]. The “trimmed” CSM, C [3] has elementsS
=
(m,n)Cmn is not set to 0
{
}
. This gives
br
ξ
()
=′
w r
ξ
()
C w
r
ξ
(
)
,
α
=1
gm
2gn
2
m,n
()
∈S
∑. (11)
3 ANALYSIS
3.1 Expectation value of the beamform map
Using the statistical assumptions, the expectation value of the beamform map becomes
Ebr
ξ
()
[]
=Aj
r
ξ
()
j=1
M
∑sj+wm
*
r
ξ
()
wn
r
ξ
()
m,n
()
∈S
∑
δ
mn rn, (11)

2nd Berlin Beamforming Conference
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where the array point spread function for a source at
r
ξ
j is given by
Aj
r
ξ
()
=wm
*
r
ξ
(
)gm
r
ξ
j
(
)gn
*
r
ξ
j
(
)wn
r
ξ
(
)
m,n
()
∈S
∑,Aj
r
ξ
j
(
)=1. (12)
3.2 Variance
Deleting the diagonal elements of the CSM completely removes the microphone self noise
from the expectation value of the beamform map. To choose the integration time, suppose that
the cross-terms between the acoustic source and the self noise can be neglected relative to the
cross terms between self noise at different microphones. Then
br
ξ
()
=qj
2Aj
r
ξ
j
(
)+wm
*
r
ξ
(
)wn
r
ξ
(
)
(m,n)∈S
∑
RmRn
*. (13)
Assuming the acoustic source is Gaussian broadband noise and that the data blocks are
independent, manipulation of Eq. 13 gives
var qj
2
⎛
⎝
⎜ ⎞
⎠
⎟ =2 var qj
(
)
[
]2
NI
=2sj
2
NI
and (14)
var wm
*r
ξ
()
wn
r
ξ
()
(m,n)∈S
∑RmRn
*
⎡
⎣
⎢
⎤
⎦
⎥ =NS
NI
w2r2 (15)
where NS is the number of elements of S and w and r are the magnitudes of weight vector
elements and the self noise power, respectively (assumed uniform over the array).
In terms of the pressure at the array, the variance of the beamforming peak simplifies to
var pbp
2
(
)=2p2
NI
+r2
NI
. (16)
4 SUMMARY
Beamforming is a powerful, flexible, and continuously evolving measurement technique in
aeroacoustics. A derivation of the classical formulation has been given, including formulas
giving the variance of the result in the practical case of finite integration time.
REFERENCES
[1] R. P. Dougherty, “Noise source imaging by beamforming,” SAE 2008-36-0518, 2008.
[2] T.J. Mueller, ed., Aeroacoustic Testing, Springer-Verlag, 2002.
J. Mueller, ed., Aeroacoustic Testing, Springer-Verlag, 2002.
[3] P. Sijtsma, “CLEAN Based on Spatial Source Coherence,” AIAA 2007-3436, 2007.
[4] J.-F. Piet and J. Elias, “Airframe Noise Source Localization Using a Microphone Array,
AIAA 97-1643, 1997