A hybrid spatial/spectral technique is presented below as an alternative to in-tegration of the beamform map. The experience base for this technique is much less extensive than that of integration. However, the new method offers the promise of being more stable and accurate than integration.
Relationship between Beamforming Weight Vectors and Cross Spectral Matrix Eigenvectors
Substituting Equation 2.27 into Equation 2.31 gives
f
--> -->tA
=
C (xs) CII
(xs) q (xs) d3 Xs + diag(h1, ... , hN) where(2.58) (2.59) Suppose that the noise source distribution is composed of M distinct sources at location xSj ' . . . , xSM ' In addition, assume that the channel noise strengths are equal: hi = h, i = I, ... ,N3• Then (2.58) becomes
3 There may be reason to believe that the channel noise strengths are unequal. For exam-ple, the array may be composed of two types of microphones, 1 and 2, where type 1 mi-crophones are more sensitive to boundary layer noise than type 2 mimi-crophones. Another possibility is that the array is installed in such a way that the boundary layer varies sig-nificantly from one portion of the array to another. In such cases, eigenvalue analysis de-scribed here will not operate correctly unless the CSM is re-scaled in such a way that the transformed channel noise strengths are equal. For each microphone n, a scale factor
in
is determined so that the values ofnhn are all equal. The scale factors are applied by mul-tiplying each element of the CSM, Aij, byJ,k
To preserve the overall array sensitivity, the scale factors should be chosen so that the trace of A is not changed, i. e,J, n -- N 1 ( A;; Ajj )-112
;:-1
~ A;;In the usual case that the boundary layer noise is much louder than the acoustic noise from the model, it is possible to derive the scale factors by using the total noise at each mi-crophone as an approximation of the channel noise. In this case, scale factors become
in=
L. N A;;
~,n=l, ... ,N NAnn
Dead microphones should be identified and excluded from this process to avoid dividing by zero.
M --7 --7 M --7
A= L C(xs)Ct(xs)q(xs) +hI= L w(xsJwt(xs)IIC(xs)Wq(xs) +hI
i= 1 i= 1 (2.60)
where I is the Nx N identity matrix and Equation 2.34 has been used to define the steering vectors.
Consider the source at xs, for some 1 ::;; j::;; M. Using Equation 2.60, the prod-uct A w(xs.) ] is seen to be ]
(2.61) Since the array is designed to be able to distinguish between different sources, i. e., to have small values of
for it:. j, the sum on the right hand side of Equation 2.61 should be small. In this case, Equation 2.61 shows that w(xs) is nearly an eigenvector of A with
eigenvalue ]
--7 2
IIC(xs)11
q(xs) + h.The Array-Centric Definition of Sources
This suggests the possibility of estimating the source strengths and locations by simply computing the eigenvector-eigenvalue decomposition of A. Before this can be done rigorously, it is necessary to reinterpret the problem slightly.
One indication that this is necessary is the fact that the number of sources, M, may exceed the number of available eigenvectors, N. In other words, there are many possible source distributions that could account for a given set of array data. The reinterpretation consists of defining the orthogonal components of the source distribution, as far as the array is concerned, to be the eigenvector-eigenvalue pairs for the CSM. Each eigenvector-eigenvalue, Ai' is identified with a source strength qi according to
Ai
= ~
+ h, i=
1, ... N (2.62)r
where a reference distance, r, has been introduced for dimensional consis-tency. It would be reasonable to define r as the distance from the center of the model to the center of the array. It is also reasonable to take h as the smallest eigenvalue of A.
Equation 2.62 applies to the case that the channel noise terms, hn , are equal for n
=
1 to N, and the diagonal elements of the CSM have not been deleted. Asnoted, the rows and columns of the CSM may need to be scaled to produce equal channel noise levels.
The eigenvector,
Vi
corresponding to each eigenvalue reflects the combined effects of the source location and structure, as well as propagation effects from the source to the array. In this view, the source is not necessarily a point source, but, instead, is a particular pattern of mechanical and fluid motion that gives rise to a certain eigenvector of the CSM. Each source is temporally incoherent with the other N-l sources. If desired, a singular value decompo-sition can be applied to the narrowband array data to isolate the time histories of the sources.Classification
Because the sources are not necessarily point monopoles, the eigenvectors may not be parallel to any of the array weight vectors used in beamforming, w(x). However, it is reasonable to expect that it will be possible to assign the eigenvalue-eigenvector pairs to particular regions of the wind tunnel and the model. The reason is the expectation that sources that are temporally coherent will also be spatially compact (Exceptions such as tones in resonant cavities and turbomachinery tone noise should be classified by mode rather than spa-tiallocation). The classification method consists of computing array weight vectors for a grid of points inside the volume V as well as at other locations where noise sources are expected. For each eigenvector,
Vi'
the inner product ofVi
with all of the array weight vectors is computed, and the largest result is identified. If the array weight vector giving the largest inner product cor-responds to a point within V and if the inner product exceeds a threshold, say 0.1, then the eigenvalue is diminished in proportion to the inner product and the result is assigned to V. When all of the eigenvalues have been processed, the sum represents the total noise strength for V.Benefits
The eigenvalue method should be much more robust than the integration method since a fixed sum of noise strengths, the trace of the CSM, is simply be-ing allocated to different parts of the wind tunnel. If the threshold is set higher than SLmax , then it is unlikely that sidelobes will have any influence on the re-sults, since no eigenvectors will be mistakenly matched with sidelobes. If the threshold is set too high or important grid points are omitted, then some eigenvectors will not be paired with any source region, and this portion of the total source strength will be lost from the analysis. This type of error will be evident during the calculation, and does not have the potential to make the final results for one region highly sensitive to the noise produced in another region.
Coherent Sources and Virtual Microphones
An extension of the eigenvalue method is to attempt to determine coherent source distributions that individually reproduce the CSM eigenvectors. Sup-pose that an eigenvalue has been assigned to a particular region on the basis that its eigenvector is somewhat parallel with the array weight vector associ-ated with a grid point within the region. The interpretation is that a distinctive pattern of airflow within the region has given rise to a coherent distribution of sources. Together, these sources have produced a portion of the radiation field that somewhat resembles the radiation from a point source at one of the grid locations. The analytical goal is to find a combination of point sources in the region that coherently sum to the eigenvector. Once determined, they can be taken as a model of the actual source distribution and used to predict the ra-diation at locations other than the array. This is a generalization of an array technique known as matched field processing.
Let
Vi
be a particular eigenvector, and let }I\, ... , WK be the weight vectors cor-responding to the volume V to whichVi
has been assigned. The number of grid points, K, may be smaller than, equal to, or larger than the number of micro-phones, N. The goal is to find complex coefficients, CI"'" CK that give the least squares solution of minimum norm to the problem(2.63) This can be solved in a standard way, using either the singular value decom-position of Wor the generalized inverse of
wtw,
where W is the NxK matrix with columns WI"'" WK' Once the coefficients have been determined, a radia-tion model similar to Equaradia-tion 2.26 can be applied to estimate one component of the narrowband signal that would have been measured by a virtual micro-phone at any location Xvirtual:_~ K -->
eigen componenti (u (Xvirtual, t))
=
-'I qiL
Cj C(Xvirtual, Xj)j=1 (2.64)
Here the time dependent, complex, narrowband source strength has been taken as a pure tone at a level determined by Equation 2.62. As noted previ-ously, the true time dependence of this source can be obtained from the sin-gular value decomposition of the narrowband array data. In principle, this process could be carried out for all of the eigenvectors, giving an approxima-tion to a complete mathematical reconstrucapproxima-tion of the sound field at an arbi-trary point. Alternatively, the eigenvectors identified with certain regions could be included and others excluded, in order to simulate the effect of vir-tual configuration changes.
The accuracy of the reconstruction will depend on the number of micro-phones in two ways. First, the effective number of independent sources is
lim-ited to N because this is the maximum number of eigenvalues. Second, the complexity of the radiation pattern that can be modeled for each source is lim-ited because the eigenvector containing the information that describes it is composed of only N complex values. In a typical wind tunnel test of an air-frame noise model, there are fewer than 60 strong, independent sources, and each of these is compact enough that its radiation pattern should be well de-scribed by 60 complex parameters. Thus, it seems probable that the method described here can be successful with an array of 60 microphones, at least over an angular sector that is not too much larger than the array. The limits of the method for this type of test, as well as others such as jet noise, turbomachin-ery noise, and helicopter noise have not been determined.
Array Calibration Using a Speaker
Beamforming to a point x involves computing the complex propagation vec-tors for the paths from x to the microphone locations. A successful result de-pends on a match between this computed array vector and the signals actually measured at the microphones. At high frequency, there are many ways that small differences between the theoretical array and the actual one can intro-duce unexpected phase shifts. For example, if a microphone is installed in a different location than expected, then there will be a phase error equal to the inner product of the wavenumber and the difference between the real and the-oretical microphone locations. At a frequency of 30 kHz, a microphone only has to move 3 mm to produce a 90° phase error. If these differences are not taken into account, the computed microphone weight vector will not match the measured data and the beamforming levels for real sources will be suppressed in relation to the levels for sidelobes.
Setup
One way to adapt the theory to the real array is to perform a speaker calibra-tion. A small speaker is used as an approximation to a point source. This is placed in front of the array, near the location of the model. The calibration is performed with the wind off. In the case of a hardwall wind tunnel, a tempo-rary anechoic enclosure must be constructed around the array and the speaker to eliminate reflections from the wind tunnel walls and other objects. Such an enclosure is called a "house of foam" after Tallman's House of Foam, the store in Palo Alto, California that supplied the mattress material for the first one.
The speaker is driven with a broadband signal generator and array data is collected. For each analysis frequency, a cross spectral matrix is computed and the eigenvalue decomposition is formed. There should be one dominant eigen-value and the others should be lower by at least a factor of 10. If this is not the case, it means speaker was not loud enough or the analysis bandwidth was too
wide for the speaker-array geometry. Decreasing the bandwidth will increase TB and reduce decorrelation effects due to differing path lengths between the speaker and the microphones.
The Diagonal Calibration Matrix
The eigenvector corresponding to the dominant eigenv~lue of the calibration CSM, Veal' is taken as the correct expression of C(xs)/IIC(xs)II, where Xs is the speaker location. The theoretical (normalized) propagation vector is
(2.65)
A diagonal calibration matrix Deal is defined by
(2.66) and used to correct the data cross spectral matrices according to Aeorr = D!al A Deal' Equivalently, Deal can be applied to the theoretical array weight vectors be-fore beamforming to produce weorr
=
Deal W. The beamforming expression (Equa-tion 2.29) is accordingly modified towt
(Xb) Aeorr W(Xb) or wtorr (xb)A Weorr(Xb), both of which reduce tow
t (Xb) DJalA Deal W(Xb)' It can be verified that applica-tion of this formula to the calibraapplica-tion data gives the leading eigenvalue, AI' The Effect of Speaker Calibration for Microphone Position ErrorsThe speaker calibration should remove any systematic differences between the microphones that are related to the construction of the microphones or the data acquisition system. Effects of errors in microphone position are only par-tially corrected by this method. To study the usefulness of the technique for correcting phase errors, it is appropriate to focus on the phase of a single mi-crophone. The amplitude factors in expressions like Equation 2.64 will not change significantly with a small microphone movement. Amplitude effects are neglected in the following analysis for simplicity.
Suppose that a microphone is theoretically located at the origin and actu-ally to be found at x', where r' =
Ix'i
is a small position error. Suppose that a speaker calibration is performed with the speaker at Xs' The phase factor of the theoretical propagation coefficient during the calibration is exp(ikI
Xs - 0 I).The actual phase factor during the calibration is exp(iklxs - x'l).
Neglect-ing amplitude effects, the calibration factor for this microphone is the ratio exp (ik
I
Xs - x' I)/exp(ikI
Xs - 0 I) . Now suppose that the calibrated array is used in an idealized test4 with a point source at location Xd. The theoretical propa-gation factor expected in this case is exp(ikI
Xd - 0 I) and the actual data will be proportional to exp(ikI
xd - x' I). Taking the point of view that the calibration is being used to correct the theoretical array weighting factor for the point Xd, this corrected weight will beeiklx,-x'i
W corr = D w cal = eikl x, _ 0 I eiklxrol (2.67)
The microphone weight factor that would produce perfect beamforming at Xd is, of course W exact = exp(iklxd - x'l). The net phase error is the ratio
WeOrT Wexaet
eiklx, -x'i eiklxrol
- - - = eik[lx, -x' I-ix,1 + IXd I-ix, - x'11
eiklx,-ol eiklx,-x'i (2.68)
This result can be simplified using an approximation that is valid for small
1x'1/llxll:
(Jackson, 1975)(2.69) Here nx is the unit vector in the direction of x.
With the approximation in Equation 2.69, the phase error with speaker cal-ibration reduces to
WeorT . I
- - "" e1kx . [nd - ns] (2.70)
Wexaet
where nd and ns are unit vectors pointing from the theoretical microphone po-sition toward the data source and calibration speaker locations, respectively.
The corrected phase is perfect if the data source lies in the same direction as the calibration speaker from the microphone. If the data source is in a differ-ent direction, then the error increases in proportion to the difference in direc-tions, projected onto the microphone displacement vector. Thus, microphone displacements toward or away from the source cause the most harm, and their effects are not corrected by the speaker calibration procedure to the extent that the sources to be measured lie in different directions than the calibration speaker did.
4 This idealized test refers to one in which the gas conditions are the same as those during the calibration, the single source is compact, and there are no convection effects, reflec-tions, or microphone boundary layer noise mechanisms. This can be realized in practice by simply moving the speaker within the house of foam!
For comparison, suppose the calibration step is omitted. Then the phase er-ror corresponding to Equation 2.70 becomes
W . ,
- - ' " e1kx ·nd
wexact (2.71)
The effect of the speaker calibration as far as errors in microphone placement are concerned, is to diminish nd in Equation 2.71 by replacing it with (na-nJ in Equation 2.70.
Array Level Calibration
One of the more difficult aspects of acoustic testing is establishing the sensi-tivities of the microphones. In the case of a phased array, the task of perform-ing level calibrations for the dozens or hundreds of microphones in the array may seem daunting. Speaker calibration can provide a practical shortcut. Use of the diagonal calibration matrix, as discussed above, corrects for differences between the sensitivities of the microphones in the array. Another step can be taken to establish the overall sensitivity of the array.
During the speaker calibration, a single microphone with known sensitivity is used to measure the level spectrum of the speaker at a position near the cen-ter of the array. This microphone can be an individual element of the array, or it can be a separate transducer. If a separate microphone is used, care must be taken to avoid interference between direct radiation from the speaker and re-flections from the array surface. In either case, a level standard, such as pis-tonphone, is employed to calibrate the single microphone.
For each analysis frequency, the measured level from the speaker is com-pared with the leading eigenvalue of the corrected cross spectral matrix, 71.[' In computing AI> the diagonal corrections (Equation 2.66) are applied. If the array microphones had been individually calibrated, then 71.[ would be higher than the single microphone result by a factor of N, as discussed in connection with Equation 2.30. The ratio of the single microphone spectrum to A[tN de-fines a factor that can be taken as the overall array sensitivity. If high-quality microphones with reasonably flat frequency-response curves are used, then it is best to determine a single, average, factor to be applied over the entire frequency range. Attempts to apply the level calibration on a frequency-by-frequency basis can introduce spurious oscillations related to the spectral characteristics of the speaker and residual reverberation within the house of foam.
Conclusions
Acoustic phased arrays are well matched to the wind tunnel setting. They can extend the capability of an aerodynamic wind tunnel into the realm of
acoustic testing without any need to redesign the facility. The characteristics of the array and the processing technique render it almost immune from the difficulties confronting conventional acoustic testing in a wind tunnel: bound-ary layer noise and reverberation.
As an imaging system, an array is a powerful tool for source location. The determination of quantitative spectra by the obvious method of integrating the beamform map is possible, but care is required to avoid errors that can be caused by sidelobe contamination. Eigenvalue classification, a hybrid scheme combining spectral and spatial analysis of the array data may provide a more robust method for source measurements than integration. An extension of this approach to distributed coherent sources may allow computation of directiv-ity patterns, within limits.
As an imaging system, an array is a powerful tool for source location. The determination of quantitative spectra by the obvious method of integrating the beamform map is possible, but care is required to avoid errors that can be caused by sidelobe contamination. Eigenvalue classification, a hybrid scheme combining spectral and spatial analysis of the array data may provide a more robust method for source measurements than integration. An extension of this approach to distributed coherent sources may allow computation of directiv-ity patterns, within limits.