Instrumentation for Noise Measurement and Analysis
3.15 SOUND SOURCE LOCALISATION
3.15.4 Beamforming
Φk(x,y,z) ' kz
ρcke&j(kxx%kyy%kz(zp&zm)) (3.66)
where SNR is the signal-to-noise ratio for the measured data and d = zm - zp is the distance between the sound source and the measurement plane.
The particle velocity is obtained using the same procedure and same equations as above, except that Equation (3.57) is replaced with:
This measurement technique is accurate in terms of quantifying the sound intensity as a function of location on the noise emitting structure and it has the same resolution and frequency range as NAH. No particle velocity measurement is needed.
Res ' 1.22 L
Dλ (3.67)
Table 3.3 Beamforming array properties
Array Array Number of Distance Frequency Backward
name size mics. from source (m) range (Hz) attenuationa
Star 3×2m arms 36 3–300 100–7,000 -21 dB
Ring 0.75 m dia 48 0.7–3 400–20,000 0 dB
Ring 0.35 m dia 32 0.35–1.5 400–20,000 0 dB
Ring 1.4 m dia 72 2.5–20 250–20,000 0 dB
Cube 0.35m across 32 0.3–1.5 1,000–10,000 -20 dB
Sphere 0.35 m dia 48 0.3–1.5 1,000–10,000 -20 dB
aThis is how much a wave is attenuated if it arrives from behind the array.
When a spherical array is used inside an irregular enclosure such as a car passenger compartment, it is necessary to use a CAD model of the interior of the enclosure and accurately position the beamforming array within it. Then the focus plane of the array can be adjusted in software for each direction to which the array is steered.
Beamformers have the disadvantage of poor spatial resolution of noise source locations, especially at low frequencies. For a beamforming array of largest dimension, D, and distance from the source, L, the resolution (or accuracy with which a source can be located) is given by:
For acceptable results, the array should be sufficiently far from the source that it does not subtend an angle greater than 30E in order to cover the entire source. In general, the distance of the array from the sound source should be at least the same as the array diameter, but no greater if at all possible. A big advantage of the beamforming technique is that it can image distant sources as well as moving sources. It is also possible to get quantitative measures of the sound power radiated by the source (Hald, 2005). One disadvantage of beamforming compared to NAH is that it is not possible to distinguish between sound radiated directly by the source and sound reflected from the source (as the measurements are made in the far field of the source). Also, for planar arrays, one cannot distinguish between sound coming from in front of or behind the array.
Beamforming can give erroneous results in some situations. For example, if the array is not focussed at the source distance, the source location will not be clear and sharp - it can look quite fuzzy. If two sources are at different distances from the array, it is possible that neither will be identified.
Beamforming array design is also important as there is a trade off between depth of focus of the array and its dynamic range. The spiral array has the greatest dynamic range (up to 15 dB) but a very small depth of focus whereas the ring array only has a dynamic range of 6 dB but a large depth of focus, allowing the array to focus on noise
p(n ,t) ' j
L R'1
wRpR(t&∆R(n )) (3.68)
∆R ' n ·r R
c (3.69)
sources at differing distances and not requiring such precision in the estimate of the distance of the noise source from the array. The dynamic range is greatest for broadband noise sources and least for low frequency and tonal sources.
3.15.4.1 Summary of the Underlying Theory
Beamforming theory is complicated so only a brief summary will be presented here.
For more details, the reader is referred to Christensen and Hald (2004) and Johnson and Dudgeon (1993). There are two types of beamforming: infinite- focus distance and finite-focus distance. For the former, plane waves are assumed and for the latter, spherical waves are assumed to originate from the focal point of the array.
In essence, infinite-focus beamforming in the context of interest here is the process of summing the signals from an array of microphones and applying different delays to the signals from each microphone so that sound coming from a particular direction causes a maximum summed microphone response and sound coming from other directions causes no response. Of course in practice, sound from any direction will still cause some response but the principle of operation is that these responses will be well below the main response due to sound coming from the direction of interest.
It is also possible to scale the beamformer output so that a quantitative measure of the active sound intensity at the surface of the noise radiator can be made (Hald, 2005).
Consider a planar array made up of L microphones at locations (xR, yR , R = 1,...., L) in the x - y plane. If the measured pressure signals, pR are individually delayed and then summed, the output of the array is:
where wR is the weighting coefficient applied to pressure signal pR,and its function is to reduce the importance of the signals coming from the array edges, which in turn reduces the amplitudes of side lobes in the array response. Side lobes are peaks in the array response in directions other than the design direction and serve to reduce the dynamic range of the beamformer. The quantity n in Equation (3.68) is the unit vector in the direction of maximum sensitivity of the array and the time delays ∆R are chosen to maximise the array sensitivity in direction, n . This is done by delaying the signals associated with a plane wave arriving from direction, n , so that they are aligned in time before being summed. The time delay, ∆R, is the dot product of the unit vector, n , and the vector, rR = (xR, yR) divided by the speed of sound, c. That is:
If the analysis is done in the frequency domain, the beamformer output at angular frequency, ω, is:
P(n ,ω) ' j
L R'1
wRPR(ω)e&jω∆R(n) ' j
I R'1
wRPR(ω)e&jk ·rR (3.70a,b)
∆R ' *r *&*r &r i*
c (3.71)
where k = -kn is the wave number vector of a plane wave incident from the direction, n , which is the direction in which the array is focussed. More detailed analysis of various aspects affecting the beamformer performance are discussed by Christensen and Hald (2004) and Johnson and Dudgeon (1993).
Finite-focus beamforming using a spherical wave assumption to locate the direction of a source and its strength at a particular distance from the array (array focal point) follows a similar but slightly more complex analysis than outlined above for infinite-focus beamforming. For the array to focus on a point source at a finite distance, the various microphone delays should align in time, the signals of a spherical wave radiated from the focus point. Equation (3.70a) still applies but the delay, ∆R, is defined as:
where r is the vector location of the source from an origin point in the same plane as the array, r R is the vector location of microphone, R, in the array with respect to the same origin and |r - r R| is the scalar distance of microphone R, from the source location.
More complex beamforming analyses applicable to aero-acoustic problems, where the array is close to the source and there is a mean flow involved, are discussed by Brooks and Humphreys (2006).