Consider the point spread function for fixed x and varying xs. For Xs near x, the function has a large peak. As Xs moves away from x, p (x, xs) decreases to negligibly small values, and remains small until Xs approaches a sidelobe point. Each sidelobe consists of another peak that is similar to the central peak, but substantially distant from it and lower in level by a factor of SLmax or
less. Let p(x, xs) be partitioned into two functions P (x, xs) = Po (x, xs) + PI (x, xs), where Po(x, xs) is the central peak. To be concrete, letpo (x, xs) = p(x, xs) for Xs close enough to x that p(x, xs) > 0.1, and Po(x, xs)
=
0 otherwise. In addition, let space be partitioned according to 9t3 = Qo U QI' where Qo is a subset of V which has significant source strength. It is assumed that V is somewhat larger than Qo so that the central peak cannot extend from Qo to a point outside V.Specifically, it is assumed that if x E Qo and x ~ V, then Po (x, xs) = O. With these assumptions, Equation 2.54 can be expanded as
h(V) = f f Po(x,xs)
IIC(x)
Wq(X)d3Xd3XSVQo
+f f PI(x,xs)IIC(x)Wq(x)d3Xd3XS+PV
v 913 (2.55)
The integral over V in the first term can be extended to all space by the as-sumption that the central peak does not extend from Qo to V. Since the psf de-pends mostly on x -Xs, especially over a small region, the integrals in the first term can be performed separately, giving
(2.57) Here xcenter is a point within V.
Rejecting Sidelobes with a Threshold
The first term on the right of Equation 2.56 is the intended result,I(V); the sec-ond and third terms are errors. The third term can be eliminated by deleting the diagonal of the CSM. The second term represents the effect of sidelobes. A con-tribution to this integral occurs when a point Xs within V is connected to a side-lobe at x by PI (x, xs) and a source, usually not part of the source of interest, is pre-sent at x. The contribution is suppressed by the SLmax, which is typically 0.1 or less. However, it can happen that the interfering sources are stronger than the source of interest and/or several sidelobes from distinct sources overlap.
The risk of multiple, small, sidelobes overwhelming the integral of the beam-form map can be reduced by excluding points from the integral for which b is smaller than a certain threshold. For example, suppose that a scan of the entire wind tunnel shows that the highest beamforming result is bmax• This means that no sidelobe from any source can produce a beamforming peak that is higher in level than SLmax bmax. If this is set as a threshold for points to include in the in-tegral, then the sidelobe contamination of the result should be greatly reduced.
It will not be eliminated because it remains possible for sidelobes to exceed the threshold by overlaying other sidelobes or real sources.
Caution #1: The Sidelobes Are Controlled by the Loudest Source in the Wind Tunnel
The use of a threshold in this way has at least three points that require caution on the part of the analyst. At high frequency, it is often impractical to scan the entire wind tunnel with a dense grid to be sure that the highest level has been found.
The usual approach is to scan the model and presume that it contains the loudest noise source. If the highest source is missed, then the threshold will be too low, and the integrated results may be too high because sidelobes are included. This situation should be evident from examination of the beamform maps, since the sidelobes will not usually correspond to reasonable source locations.
Caution #2: The Threshold Excludes Some Real Noise
The second difficulty is that the threshold will exclude some legitimate noise from the integral. If the interfering level is higher that the source of interest by a factor of 1/SLmax or more, then the integral will be zero! Often, the objective of the test is to measure small changes in the noise from a portion of the model when the configuration is changed. If some extraneous factor causes the inter-fering source to vary between configurations, then the threshold will change, possibly giving misleading results. Even if the threshold does not change, the subtraction process may cause the integrated results to overstate the sensitiv-ity of the noise to the model changes.
Caution #3: Array Performance May Be less than Optimal
Finally, determination of the proper threshold depends on knowing the actual sidelobe levels in relation to the beamforming peak. Errors in computing the microphone weight vectors will typically reduce the central peak more than the sidelobes. This may cause the threshold to be set too low. Many factors in the data analysis have the potential to introduce small errors that erode the ac-curacy of the weight vectors. Examples include errors in microphone position, Mach number, temperature, and the flow description. At high frequency, prop-agation through a turbulent shear layer can disturb the sound in a way that seems to reduce the beamforming peak levels in relation to the sidelobes. If the sidelobe level is too high, then integration will not be effective.
Successful Integration
Despite the need to handle the details correctly, integration can be an effective technique. Care must be taken in the computation of the weight factors. For example, speaker calibration can be used to compensate for errors in micro-phone positioning. It is important to examine the beamform maps so that con-taminated results are not accepted uncritically. In designing an array for
inte-gration, it may be advantageous to use a reasonably small array and sacrifice some beamwidth performance in order to push the sidelobes away from the central peak.