1. An apparatus for remotely locating and quantifying- sound sources comprising: an array of acoustic vector probes; said probes in said array connected to a multi-channel data acquisition system for rapid conversion of analog signals to digital form and for temporary data storage; said multi-channel system providing input to a digital signal processor programmed to compute the sound-intensity and sound- velocity vectors and sound pressure at each of said probes in said array; said processor connected to a device for outputting the results of the computations; and each of said acoustic vector probes comprises four microphones spatially positioned in three dimensional space.
2. The invention as in claim 1 wherein a least-squares triangulation formula determines the spatial coordinates of a sound source from a set of directions of the said sound intensity vectors in said array pointing to said source.
3. The invention as in claim 1 wherein said array is used to determine the sound power flow incident on said array from said sound source.
4. The invention as in claim 1 wherein said array surrounds a sound source and measures the sound-intensity vector simultaneously at all points in the array to determine the sound power of the source and investigate its acoustical properties.
5. The invention as in claim 1 wherein said array concurrently locates and tracks a moving source and measures the sound-power flow from said source incident on said array.
6. The invention as in claim 1 wherein the sound pressure measured by said probes in said array is phased to create a sensitivity beam capable of scanning a sound source.
7. The invention as in claim 6 wherein the said sound-intensity measurements made concurrently with said sound pressure by said probes in said array are used to determine the location of the part of the source within said beam using a least-squares triangulation formula; and to determine the sound power transmitted from said part within said beam.
8. The invention as in claim 1 wherein the quantities measured by said array are used to improve the input for computational methods of investigating sound sources.
9. The invention as in claim 1 wherein said output device enables an operator to use said acoustic apparatus interactively to locate and quantify sound sources, by filtering said data into any desired frequency bands and by adjusting the position of said array.
10. A method, for finding the location of a sound source using an array of acoustic vector probes, said method comprising the steps of: conduct a preliminary investigation to determine the frequency distribution of points in azimuth-elevation plots for each of said probes in said array;
find the concentrations of said points in said azimuth-elevation plots that indicate the presence of sound sources and select a single concentration common to said probes in said array indicating a particular source; for said single concentration find an average azimuth-elevation direction for each of said probes in said array and determine the unit vectors corresponding to each said direction; and from the known spatial coordinates of said probes in said array and the said unit vectors associated with each of said probes pointing towards said particular source, use a least-squares triangulation formula to determine the approximate spatial coordinates of said particular sound source.
11. An apparatus for remotely locating and quantifying sound sources as defined in claim 1 further comprising: each of said vector probes comprises four microphones spatially positioned at the vertices of a regular tetrahedron to determine sound intensity vectors.
12. An apparatus as defined in claim 11 further comprising: each of said acoustic vector probes having two microphones facing a first direction and two microphones facing an opposite direction.
13. An apparatus as defined in claim 1 further comprising: each of said acoustic vector probes having two microphones facing a first direction and two microphones facing an opposite direction.
14. An apparatus for remotely locating and quantifying sound sources comprising: an array of acoustic vector probes;
said probes in said array connected to a multi-channel data acquisition system for rapid conversion of analog signals to digital form and for temporary data storage; said multi-channel system providing input to a digital signal processor programmed to compute the sound-intensity and sound-velocity vectors and sound pressure at each of said probes in said array; and said processor connected to a device for outputting the results of the computations; and each of said acoustic vector probes having two microphones facing a first direction and two microphones facing an opposite direction.
SOUND SOURCE LOCATION AND QUANTIFICATION USING ARRAYS OF VECTOR PROBES
The invention relates to methods and means of remotely locating and quantifying sound sources, using arrays of recently-developed acoustic vector probes (AVPs).
BACKGROUND OF THE INVENTION
Acoustic Vector Probes
Recently a patent application was filed for a new acoustic instrument, the acoustic vector probe (AVP).
1. R. Hickling 2003, "Acoustic Measurement Method and Apparatus",
Patent Application to the United States Patent and Trademark Office,
Number 10/396541, Filing Date 03/25/2003.
The technical information contained in this application is hereby incorporated herein by reference.
An AVP consists of a tetrahedral arrangement of four small microphones that simultaneously measures at a point in air the three fundamental quantities of acoustics, namely the sound-intensity and sound-velocity vectors, and sound pressure. Sound intensity is the time average of sound power flow per unit area expressed in spectral form. The time dependence of sound intensity is determined by taking a series of averages over short intervals. AVPs are more accurate, more compact and less expensive than previous instruments for measuring sound intensity. Nested AVPs can be used to make accurate measurements over a broader frequency range than previous instruments. A normalization and
calibration procedure described by Hickling (Ref.1) ensures that the probe is omnidirectional and accurate.
The sound-intensity vector determines the direction of a sound source.
Because it is expressed as a fast Fourier transform (FFT), it also provides information about the frequency characteristics of the source, enabling the AVP to distinguish one source from another. Sources can also be distinguished by when they occur in time.
Subsequently a continuation-in-part was submitted describing the use of an array of AVPs to detect buried objects 2. R. Hickling, 2003, "Method and Apparatus for Acoustic Detection of
Buried Objects", Patent Application to the United States Patent and Trademark Office, Number 10/658076, Filing Date 09/09/2003. This subsequently issued on March las US 6,862,252.
The technical information contained in this application is hereby incorporated herein by reference. It describes how the compactness and inexpensiveness of
AVPs make them suitable for forming an array. It also indicates that modern digital signal processing permits simultaneous measurement at all the AVPs in the array.
Previous methods of locating and quantifying sound sources Previous methods of locating and quantifying sound sources use arrays of single microphones. These have been summarized by
3. M. Batel, M. Marroquin, J. HaId, J. J. Christensen, A. P. Schuhmacher and T. G. Nielsen, 2003, "Noise Source Location Techniques - Simple to Advanced Applications", Sound & Vibration, March issue, 24-38. The methods can be listed briefly as follows:
(a) Sound pressure mapping. This method consists of sound pressure measurements at different locations on the surface of a source. It is unsatisfactory because pressure measurements do not measure sound power flow at the surface.
(b) Sound intensity and selective intensity. Here a two-microphone probe is used to measure the component of sound intensity at points perpendicular to the surface
of a source. It can be used to rank the sound power of different components of the source and to sum these to obtain the total radiated sound power. However the measurements usually have to be made by hand and it is not easy to stand next to a sound source, such as an engine, for extended periods and perform careful, tedious measurements. Another disadvantage is the clumsy face-to-face microphone arrangement with a U-shaped holder that is currently used as a two- microphone probe. There is therefore a need to make measurements remote from the source, using methods with less emphasis on manual work and based more on improved measurement techniques and computation, as follows. (c) Near-field acoustic holography. This method measures sound pressure at an array of individual microphones remote from the source and computes the sound field from this data. The computed field is then used to determine how the source radiates sound. However the computations involve assumptions and approximations that can introduce misrepresentations and inaccuracy. Planar measurements of sound pressure are used to determine the components of sound velocity and sound intensity perpendicular to these planes.
(d) Non-stationary acoustic holography. This is a development of near-field acoustic holography for a non steady source.
(e) Beam forming. This method uses a phased array of sound-pressure microphones to form a beam that can scan the surface of a source to obtain the relative amounts of sound emitted by different parts of the source. Beam forming is a well-known technique. The side lobes of the primary beam cause error but methods developed by Batel et al can reduce this effect. The sound power transmitted by the source within the beam cannot be determined accurately. (f) Inverse boundary element methods. These provide additional mathematical methods for modeling the sound radiated by a source. Triangulation and Other Locating Techniques
A mathematical technique for locating sound sources using AVPs was published previously by
4. R. Hickling and A. P Morgan, 1996, "Locating sound sources with vector sound-intensity probes using polynomial continuation", Journ. Acoust. Soc. Amer 100(1), 49-56.
However the polynomial-continuation method has difficulty in dealing with measurement error and the finite size of a source. It is therefore necessary to use triangulation methods. Triangulation is a well-known concept. Systems that use triangulation have been described in texts such as
5. M. S. Grewal, A. P. Andrews and L. R. Weill, 2001,
"Global Positioning Systems, Inertial Navigation and Integration", John Wiley & Sons Inc.
6. Loran-C User Handbook, 1990, Department of Transportation, US Coast Guard, Commandant Instruction Ml 6562.3 Washington DC.
These systems are based on radio waves and not on sound. Generally they consist of one receiver and several transmitters, whereas source location with an array of AVPs involves a number of receivers and generally one transmitter.
Measuring Sound Power with an Array of Sound-Intensity Probes enclosing a Source
There are standard procedures for measuring the sound power of a source using an array of two-microphone sound-intensity probes enclosing the source: 7. ISO 9614- 1 : 1993 (E), "Acoustics - Determination of Sound Power
Levels of Noise Sources using Sound Intensity, Part I Measurement at
Discrete Points", International Organization for Standardization, Geneva,
8. ANSI S12-12-1992., "Engineering Methods for Determination of Sound Power Levels using Sound Intensity", American National
Standards Institute, New York.
Two-microphone probes measure one component of the sound-intensity vector and are aligned perpendicularly to the measurement surface enclosing the source. Generally, such probes are expensive, making it impractical to use them in
sufficient numbers to make simultaneous measurements at all points in- the enclosing array.
The sound power of a source moving in water was determined using a four-hydrophone probe. 9. W. Wei and R. Hickling, 1995, "Measuring the Sound Power of a
Moving Source", Journ. Acoust. Soc. Amer., 97(1), 116-120. Here it was assumed that the source moves along a known straight path and that its sound power can be determined by integrating over an imaginary infinite cylinder enclosing the path. The four-hydrophone probe is less compact, and is not as accurate as an AVP.
What is needed and desired is a new approach for locating and quantifying sound sources using an array of AVPs that
(a) replaces the previous use of single microphones and two-microphone probes. (b) uses a least-squares triangulation formula to determine the spatial coordinates of a source from the set of directions determined by an array of AVPs. (c) determines the sound power of a source using simultaneous sound-intensity measurements by an array of AVPs enclosing the source. (d) forms a beam using sound-pressure measurents made simultaneously with the sound-intensity measurements to determine the spatial coordinates of the part of an object highlighted by the beam and the sound power radiated within the beam
(e) distinguishes between sources with different frequency characteristics and locates these sources
Further objects and advantages of this invention will become apparent from a consideration of the following description and drawings.
SUMMARY OF THE INVENTION
The present invention includes and utilizes arrays of acoustic vector probes (AVPs) to remotely locate and quantify sound sources. AVPs are small, rugged and inexpensive and can easily be arranged into an array linked to a signal processing system.
An important part of the invention is a least-squares triangulation formula for locating sound sources that makes allowance for measurement error and the finite size of the source. A brief proof of the formula is given in an appendix at the end of the specifications. The need for the least-squares formula can be understood by considering the example of a single source with two AVPs at different positions in space. The source can be assumed to be roughly where the vector directions from the AVPs intersect. However because of experimental error and the finite size of the source the directions determined by the AVPs will generally not intersect. Hence the source has to be located using a least-squares fit. An array of AVPs determines a set of directions to a source and also determines the sound- power flow from the source to each AVP, which can be integrated to obtain the total sound power flow at the array either from the direction of the source, or perpendicular to the array, or from some other direction.
An array can enclose a source either totally or combined with rigid surfaces. Integration over the array of the component of the intensity vector perpendicular to the array determines the sound power of the source. Previously two-microphone probes were used. However such probes are clumsy and expensive, preventing their use in sufficient numbers to make simultaneous measurements at all points in the enclosing array. AVPs, on the other hand, are compact and inexpensive and can be used to make simultaneous measurements over the entire array. This greatly speeds up the sound power measurement, and also makes it possible to investigate the sound power of a non-steady source. It
also makes it possible to measure the sound power of a source in a noisy environment, provided the background noise is not overwhelming compared to that of the source. In addition, the techniques of this invention can be used to investigate the acoustical characteristics of the source. These improvements make it possible to use AVPs for quality control in manufacturing. When the source is moving or cannot be enclosed, an array can still locate and track the source and determine the sound power radiated at the array.
An additional feature of the invention is that a beam can be formed by phasing the sound pressure measured by an array of AVPs. The triangulation formula can then be used to determine the location of the region highlighted by the beam. Since sound intensity is measured concurrently with sound pressure, the intensity can be integrated over the cross-section of the beam to determine sound power flow in the beam. This is accomplished more accurately than previously. Since the sound-intensity vector is expressed as a digital Fourier transform its direction can be plotted as a function of frequency, in the form of a distribution of points in an azimuth-elevation plot. This can then be used to separate sources with different frequency characteristics and to locate such sources in space.
BRIEF DESCRIPTION OF THE DRAWINGS
In the drawings:
FIG. 1 is a block diagram showing a sound source, an array of acoustic vector probes (AVPs), a multi-channel data-acquisition system for rapid analog to digital conversion and temporary data storage, a signal processor, and a display unit.
FIG. 2 is a perspective view of an AVP forming a part of the invention.
FIG. 3 is a cubic lattice diagram showing the geometry of the tetrahedral arrangement of microphones in the AVP and the relation of the microphones to the system of Cartesian coordinates used in making measurements at the origin M.
FIG. 4 shows the coordinate system for determining the direction of a sound- intensity vector in azimuth and elevation, relative to the coordinate system of the AVP. FIG. 5 illustrates a set of directions towards a sound source from an array of AVPs, as used in the least-squares triangulation formula to determine the location of the source.
FIG. 6 shows the geometry of the least-squares triangulation formula in locating a sound source with two AVPs. The formula places the source at the midpoint of the normal between the directions towards the source from the AVPs
FIG. 7 depicts arrays of AVPs surrounding a source for measuring the sound power of the source: (a) a hemispherical array on a rigid base and (b) an array of known arbitrary shape adjacent to rigid surfaces.
FIG. 8 depicts examples of arrays of AVPs where it is not possible to surround the source with the array: (a) investigating the sound from an engine compartment; and (b) investigating the sound of an aircraft flyover.
FIG. 9 depicts the phasing of sound pressure measurements by an array of AVPs to form a beam.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
FIG. 1 is a block diagram illustrating the apparatus for source location and quantification in the present invention. Block 100 represents a sound source. Block 200 represents an array of AVPs. Block 300 represents a multi-channel data-acquisition system for rapid analog to digital conversion of the signals from the array, and for data storage, prior to input to the digital signal processor represented by block 400. The processor computes the three components of the sound-intensity vector and the sound pressure at each AVP in the array and interprets the data, displaying the results on an output device 500 such as a monitor screen. The sound-intensity vector is used to detect, position and quantify sound sources. The sound-pressure measurements can be phased to form a beam. In FIG. 2 numeral 40 generally indicates an AVP formed in accordance with the invention. Probe 40 includes a fixture 42 being an annular member formed as a ring with a central opening 46. Protruding from the ring are four support tubes for the microphones, parallel to the axis of the ring, two on one side of the ring pointing in one direction and two on the reverse side pointing in the opposite direction. These tubes are spaced around the ring at ninety degree intervals at openings in the ring at 48, 50, 52 and 54, and centered on an annular centerline 56 having a diameter d. The pair of tubes 58 on one side of the ring is attached to the ring at diametrically opposite openings 48 and 52, and the pair of tubes 60 on the reverse side of the ring is attached to the ring at diametrically opposite openings 50 and 54. The outer ends of the support tubes 58, 60 are each a distance d/ (2 y/2 ) from the central base plane 64 of the ring. Within the ends of the two support tubes 58 are located microphones 1, 2 and within the ends of the two support tubes 60 are located microphones 3 and 4. Microphones 1 through 4 are located at the vertices of an imaginary regular tetrahedron. The advantages of the structure in FIG. 2 are: (a) the microphones are symmetric on the two opposite
sides of the base ring so that they detect sound equally from both directions; (b) the measurement point M is well defined; (c) the procedure for normalizing and calibrating in Ref 1 can be applied easily. Since the dimensions of the probe are required to be much less than the wavelengths being measured, the effect of diffraction will be insignificant. Suitable microphones are type FG-3329-PO7 manufactured by Rnowles Electronics which have a diameter and length of about 2.6 mm and whose sensitivity is about 22 mV/Pa and relatively flat to about 5kHz. Other microphones could also be used.
In FIG. 3, the geometric placement of the four microphones in the tetrahedral arrangement is shown inserted within an imaginary cubic lattice 70 having 6 faces with midpoints 12, 13, 14, 23, 24, 34. Lines through the midpoints of the opposite faces of the lattice pass through an origin M, which is the measurement point, and form X, Y and Z axes of the cubic lattice 64. The dashed lines between the microphones form diagonals across the faces of the cubic lattice, representing the edges of the regular tetrahedron of length d with midpoints 12, 13, 14, 23, 24 and 34. These lines form hypotenuse lines for the respective faces of the cubic lattice 64 so that the edges of the sides of the lattice have dimension d/ λ /2 .
At the microphones 1, 2, 3 and 4 at the vertices of the regular tetrahedron in FIG 2, the corresponding sound pressures pi, p2, p3 and p4 are measured and digitized. The discrete Fourier transforms (DFTs) of the sound pressures are then computed, normalized and calibrated using the transfer-function procedure described by Hickling in Ref.l, giving the modified transforms Fpl(f), Fp2(f), Fp3(f) and Fp4(f) at the discrete points f = f . , i=l,.n. For simplicity, the frequency dependence (f) will be dropped. Finite difference approximations (derived from Taylor series expansions) are then used to obtain the DFTs of the sound pressures at the six midpoints of the edges of the regular tetrahedron at 12, 13, 14, 23, 24 and 34 in FIG. 3, giving respectively Fpl2 = (Fpl+Fp2)/2 Fρl3 = (Fpl+Fp3)/2 FpU = (Fpl+JFp4)/2
Fp23 = (Fp2+Fp3)/2 Fp24 = (Fp2+Fp4)/2 iφ34 = (Fp3+Fp4)/2. (1) These approximations are accurate to the second order, i. e. order (kd) 2 /4, provided. kd/2<l (2) where k = 2JI/X, λ being the wavelength.
The components of the sound-intensity vector at the measurement point M are determined from the sound pressure DFTs in Equation (1), using the cross- spectral formulation for sound intensity described by Hickling (Ref 1). The components are FIX = Im CS[Fp24, Fpl3]/(p2πf(d/ -Jϊ ))
FTY = Im CS[iφ23, Fpl4]/(p2πf(d/V2 ))
FYL = Im CS[Fpl2, Fp34]/(p2πf(d/V2 )) (3) where hn is the imaginary part and CS is the cross spectrum of the sound pressures at the midpoints of the opposite edges of the imaginary regular tetrahedron in FIG. 3, and p is the density of the fluid medium, which is approximately 1.3 kg/m 3 for air. The amplitude of the sound-intensity vector is given by
FlA = ^I[FDC 2 + FTY 2 + FlZ 2 ] (4)
Sound intensity is expressed in SI units of watts per meter squared per hertz. . The direction of a sound source can be expressed in terms of the horizontal (azimuth) angle θ. and the vertical (elevation) angle φ. The combination of these two angles specifies the direction to the source, as shown in FIG. 4. The vector probe points in the direction of the Z-axis in FIG. 3 and the Y-axis is vertical. The angles θ and φ are determined from the relations
θ = arccos(iTZ/iTXZ) (5) and φ = arcsin(FIY/iTA) (6)
where FIXZ = V (FIX 2 + FIZ 2 ) and the other terms come from Equations (3) and (4). The angles θ and φ are functions of frequency. They can be represented over the frequency range by a set of discrete points in an elevation-azimuth (or vertical- horizontal) plot, relative to the direction of the probe. The DFT of the sound-intensity vector determined by a probe provides a set of angles in azimuth and elevation θ ,• and φ i for each point f = f ; , i =1, n in the frequency range of the DFT, together with a corresponding set of amplitudes of the sound intensity vector w ,• = FϊA(f . ) from
Equation (4). Azimuth-elevation plots generally show a scatter of points as a function of frequency and it is necessary to interpret this scatter both in terms of sources that may be present and in terms of the acoustic environment. Usually an azimuth- elevation plot shows a concentration of points in the direction of a source and a preliminary study has to be made to find such concentrations. For a particular concentration, a single representative direction in the azimuth and elevation angles θ and φ is determined for each probe, using a techniques based on weighted averages m m θ = ∑ w,.θ,./W φ = ∑ w,.φ,./W (7) ι=l (=1 where W = ∑™ w,. and the range i = 1 to m covers the concentration of points in the azimuth-elevation plot. If more than one source is indicated, by frequency content or from other characteristic such as timing, the direction to each of the sources can be determined using the same averaging procedure. After finding a set of averaged directions the next step is to apply the least- squares triangulation formula derived in the Appendix to determine the spatial coordinates of the source. Vectors and matrices are indicated in bold type. As shown in FIG. 5, it is assumed that there is an array of n AVPs with coordinate vectors Pu P 2 , ..., p n ,and corresponding unit vectors u j , u 2 , , u n pointing towards the source at the location 100 or P s . The least-squares formula then determines the coordinate vector q, of P s . From the Appendix the formula is
q [I - U 1
-Uf] P,. (8)
U 2 = (sin θ 2 cos φ 2 , sin φ 2 , cos φ 2 cos θ 2 ). For n=2 Equation (2) becomes q ^ pi - uX - U 2 U 2 T' [(1 - U 1 Uf) P 1 H- (I - U 2 U^ p 2 ] (9) It can be shown that the geometric form of this equation positions the source at the midpoint of the normal connecting the directions from P x and P 2 , as depicted in FIG. 6.
After using Equation (8) to locate the sound source, the next step is to determine sound-power flow using the sound intensity vector measured by the AVPs. Usually this is performed for stationary (steady-state) sources but it can also be applied to quasi-stationary sources, i. e. to sources whose sound varies slowly compared to the rate of signal processing. Additionally it can be applied to impulsive sound.
An important application is determining the sound power of a source using an array enclosing the source 100. Generally the array 200 is combined with rigid surfaces 220, as shown in FIG. 7. Previously measurements were made with two- microphone intensity probes aligned perpendicularly to the array. Such probes are bulky and expensive and it is impractical to have a sufficient number of them to make measurements simultaneously at all points in the enclosing array. However replacing these probes with AVPs makes simultaneous measurements possible. Determining the sound power of the source then becomes much more rapid. Also, it becomes possible to measure the sound power of a quasi-stationary source. In addition use can be made of the triangulation formula to locate components of the source with a distinctive feature, such as a resonance. Such improved procedures can be used for quality control.
In FIG. 8 examples are shown where it would not be practical to surround the source 100 with an array 200. FIG. 8 (a) shows an array used to investigate the sound from beneath the hood of a car. Here the least-squares triangulation formula can be used to locate sources that have distinctive features. FIG 8 (b) shows an array used to investigate the flyover of an aircraft. The array concurrently locates and tracks the aircraft and measures its sound-power. A single AVP could be used for tracking and measuring sound power but an array provides statistically more reliable data. A sound-absorbing backing can prevent interference from sound reflected by the ground or by the supporting base.
Sound pressure measured by an array of AVPs can be phased to form a beam 250 with side lobe 275, as shown in FIG. 9. This beam can scan different parts of a source. Since the AVPs in the array measure sound-intensity concurrently with sound-pressure, the information can be used to determine the location of the part of the source highlighted by the beam, using the triangulation formula. Also the information can be used to measure sound-power flow from the highlighted part, which is obtained by integrating sound intensity over the cross- section of the beam. This is determined more accurately than previously.
Finally replacing individual sound-pressure microphones in the array with AVPs will greatly improve the data input for computational methods of locating and quantifying sound sources.
Derivation of Formula for Determining the Position of a Sound Source from Directions Provided by an Array of AVPs.
In the derivation, bold-face characters represent 3-dimensional vectors and matrices and the superscript T represents a vector transpose.
Given n AVPs at locations represented by the vectors p x , p 2 , ...., p n ,with corresponding unit vectors U 15 U 2 , , u n pointing towards a sound source at an unknown location q. The least-squares estimate of q can then be determined from the given data using the formula
where I is the 3 x 3 identity matrix. If δ ,. is the perpendicular (shortest distance) from q to the ray from p, in the direction of u . then Equation (Al) locates q such that σ " =1 δ 2 is a minimum. Proof
The displacement vector q - p , can be resolved into a component u,uf (q - p ; ) in the direction of u ; and a component (q - p .) - u ( uf (q - p .) perpendicular to u . , which can be rewritten as (I - u . uf )(q - p . ). This latter component equals δ t in length i. e. δ? =(q-p,) r (I-u,.uf) r α-u,.uf)(q-p / ) which is then
,ur)'(q-p,) because (I-u,.uf) r
(I - u . u f ) being symmetric and idempotent.
For brevity let A 1 - = (I - u ; u f ). Then summing over the n AVPs yields,
5 pfA lPi
Unless the vectors u are all parallel, A is positive definite, so that making q = r globally minimizes ∑"_ δ ) . This proves Equation (Al).
10 While the invention has been described by reference to certain preferred embodiments, it should be understood that numerous changes could be made within the spirit and scope of the inventive concepts described. Accordingly it is intended that the invention not be limited to the disclosed embodiments, but that it have the Ml scope permitted by the language of the following claims.
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