Statistical errors in acoustic intensity measurements

Statistical errors in acoustic intensity measurements

Journal of Sound and Vibration (1981) 75(4), 519-526 STATISTICAL ERRORS IN ACOUSTIC INTENSITY MEASUREMENTS A. F. SEYBERT DepartmentofMechanicalEng...

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Journal of Sound and Vibration (1981) 75(4), 519-526

STATISTICAL

ERRORS IN ACOUSTIC

INTENSITY

MEASUREMENTS A. F. SEYBERT DepartmentofMechanicalEngineering,

UniversityofKentucky,

Lexington, Kentucky 40506, U.S.A.

(Received 8 July 1980, and in revised form 29 September 1980)

A bivariate stochastic process is used to evaluate sources of error in acoustic intensity measurements. Three acoustic intensity estimators are examined for both bias and random errors. All three acoustic intensity estimators are shown to be biased by the presence of a second uncorrelated acoustic source. It is further shown that the presence of phase errors introduces additional bias which may limit the accuracy at low frequency. An expression is derived for the normalized standard error of the acoustic intensity estimates. The normalized standard error is a strong function of the coherence function and the phase angle spectrum, with maximum error occurring at low frequency and low coherence. Further interpretation of the normalized standard error is given by using two monopole acoustic sources as components in the bivariate process.



1. INTRODUCTION

In several recent publications [l-5], various procedures for estimating acoustic intensity by utilizing near field microphones and/or vibration transducers have been described. While the reported experimental results appear encouraging, bias and random errors inherent in the intensity estimation procedure may restrict the accuracy of these techniques under certain experimental conditions. The purpose of this paper is to point out potential sources of error and suggest possible alternatives. Additional sources of error due to the methodology of performing the intensity measurement have been adequately described elsewhere [3].

2. DESCRIPTION

OF THE TECHNIQUE

The acoustic intensity I(f) may be expressed [l, 2,4] as an approximation finite difference

based cn a

I(f) = Im G&)/27&d,

(1)

where Im Gr~(f) is the imaginary component of the one-sided cross-spectrum between two microphones, A is the distance between the microphones, p is the mass density of the medium, andf is frequency. The microphones lie on a line normal to a hypothetical surface enclosing the source region. The cross-spectrum has real and imaginary components Gr2(f) = C&f) + jQrZ(f) and equation (1) may be written as 1(f) = Q12(f)/27+A.

(2)

In this paper errors that result from using either equations (1) or (2) to estimate the intensity of a single acoustic source will be considered. Errors introduced in the estimation of acoustic intensity by a second, uncorrelated acoustic source may be evaluated by 519 0022-460X/81/080519+08

$02.00/O

@ 1981 Academic

Press Inc. (London)

Limited

520

A. Bivoriate

F. SEYBERT

stochost;c

idealized with

process ,

phase

measurement mismatch

3-F

Source 1

Source

2

Figure 1. Schematic representation

of idealized acoustic intensity measurement.

considering the bivariate stochastic process shown in Figure 1. Source 1 is the source being measured while source 2 is a second (contamination) source. The quantities hii, h12, etc., are linear systems between each source and the measurement points 1 and 2. The quantities pl, pi, etc., are the acoustic pressures produced at the measurement points by each source acting individually. Because the microphone separation distance A is on the order of centimeters [l-3], the true cross-spectrum Giz(f) will have a very small imaginary component: Q,,(f) = ]G&)]sin &#). For a simple acoustic source with the microphones aligned in the direction of sound propagation &(f) = kA, where k is the acoustic wavenumber. Therefore, even small phase errors in measuring the acoustic pressure may substantially bias an estimate of I(f). These phase errors are primarily due to instrumentation phase response and, to a lesser extent, uncertainty in the value of A. To evaluate the effect of instrumentation phase errors on the estimation of acoustic intensity, an idealized measurement process has been included in the model in Figure 1. In Figure 1 &(f) and &(fl are phase shifts between the true acoustic pressures at points 1 and 2 and the measured aCOUStiC pressures y, and yb. Because p1 and pz cannot be measured directly, ,an estimate of G&) must be used in equation (1). This estimate is the cross-spectrum Gab(f) between the measured acoustic pressures yn and yb. An estimate f(f) of the acoustic intensity is given by f(f) = Im &(f)/27rfpA. This estimate is assumed to be obtained by Fourier transformation G,,(f) = (2/T) Y:

(f)yb(f),

(3) of y, and yb, so that (4)

where Yz is the conjugate Fourier transform of y,, Yb(f) is the Fourier transform of yb, and T is the length of the raw data recoid. The bar over the product Yz (f) Yb(f) in equation (4) indicates that the quantity Gab(f) is a smoothed cross-spectrum estimate obtained, for example, by averaging nd individual raw estimates of the cross-spectrum: i.e.

In addition to the estimate given in equation (3), the following estimates of acoustic intensity may be used: f(f)=Im

i(f)=Im[(~nb(f)+~Sob(f))/21/2~~~A. (56) [~~b(f)e~b(n1"'/2~~~Arrfpd,

The estimate in equation (5) has been used by Chung [2,3] and is based on the geometric mean of the cross-spectra &,(fl and &z(f), where r%,(f) is obtained after interchanging the measurement system between 1 and a with the measurement system between 2 and b

INTENSITY

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521

ERRORS

in Figure 1. The estimate in equation (6) is based on the arithmetic mean of the same two spectra. 3. BIAS

Each of the estimates in equations normalized bias error &b,

ERRORS

(3), (5) and (6) may be evaluated

by defining a

&b= (E[j] - I)/1

(7)

where 1 is the acoustic intensity given by equation (1). From Figure 1 it is apparent that

Y,(f)=[Pl(f)+P;(f)le’~a’f’,

yb(f)

=

[f%(f)

+

Pi

are the Fourier transforms where PI(~), Pi VI, &(f) and Pi(f) respectively. Combining equations (4) and (8) yields

(f)l

ej’h(i)?

(8)

of pl, pi, p2 and pi, (9)

~ab(f)=[Gl*(f)+G~z(f)le”~,

where G;*(f) is the cross-spectrum between the acoustic pressures pi and pi and AC#J= &(f)-&~~(f). Cross-spectra between p1 and pi and between p2 and p; do not appear in equation (9) becaus: of the assumption that sources 1 and 2 are uncorrelated. Equation (9) shows that Gab(f) is a function of A4 rather than of &(f) or &(f) individually. In a typical measurement Aq5 is quite small because most acoustical instrumentation of like manufacture is reasonably well phase-matched. Figure 2 shows the phase spectrum of Gab(f) and &z(f), respectively, for a loudspeaker source of band-limited Gaussian noise with A = 15.9 mm. Each measurement channel consisted of a one-quarter inch microphone, a preamplifier, and an amplifier. Although the instrumentation was of like manufacture, no attempt was made to phase-match the two measurement channels. Figure 2 is quite typical, and for most measurements eiA’ = 1 + jA4.

‘O’

I

-40

I

I

2k Frequency

Figure

2. Phase spectra

I

lk

0 of typical

microphone

I

I 3k

(Hz)

cross-spectra.

-,

Phase of C?.*(f); - - -, phase of b”,,(f).

522

A. F. SEYBERT

The normalized bias error for each of the estimates in equations (3), (5) and (6) may be found by combining these equations with equations (7) and (9). For the estimate in equation (3), E,, = I’/I + A4 cot 412 + (I’/I)Ac#J cot &;z,

(10)

for the estimate in equation (5), &b= 1’11,

(11)

&b-II/I.

(12)

and for the estimate in equation (6),

In equations (10) through (12) I’ is the acoustic intensity of source 2 estimated by using equation (1) with Gr2(f) replaced by G;*(f), +r2(f) is the phase spectrum of Gr2(f) and +i2(f) is the phase spectrum of Gi2(f). The third term in equation (10) shows that the presence of both phase mismatch and a second source compounds the normalized bias error. Either of the estimates in equations (5) and (6) will yield acceptable results if the effect of secondary acoustic sources can be minimized. The estimate in equation (5) yields slightly better results if AC#J is not small, but the estimate in equation (6) is easier to evaluate for spectrum analyzers having limited post-processing capability. In an ideal environment with a single source, the normalized bias errors in equations (10) through (12) become, respectively, Eb

=

A4

cot

&b

412,

=

0,

Eb

=O.

(13-15)

Further interpretation of equation (13) can be provided by noting that, for a simple source with kA cc 1, cb = Ac$/kA. Thus, for a given A, phase mismatch establishes a low frequency limit below which the intensity estimate may not be acceptable. It is important to note that the bias error in equations (10) through (12) is present even though the two acoustic sources were assumed uncorrelated. Additional bias error terms would be produced for correlated sources. A time delay between two signals will cause a negative bias error in the cross-spectrum [6]. This bias error is transferred to other quantities computed from the cross-spectrum such as transfer functions, coherence functions or acoustic intensity estimates. However, for T,,/T <<1, where r. is the time delay and T is the raw data record length, the bias error is negligible. This inequality is fulfilled in practice where A is small.

4. RANDOM ERRORS In addition to bias errors, the estimates in equations (3), (5) and (6) contain random (variance) errors. Consider for example, equation (3) which may be written as f(f) =

@ab(f)l

sin

(16)

&b(f)/2d’A,

are estimates of the magnitude and phase of the cross-spectrum i&b(!?i and &b(f) between ya and yb. To a first order approximation the variance of the intensity in equation (16) is [7]

where

var@(f)] =

@f/d&b\)* +a(a~/lale,bI)(ai/a~,,)

var

{i&b(f)i}+

@i/$ab>2 cov{@ab(f)&df)h

var

{&b(f)}

(17)

where the derivatives are evaluated at the true values ]Gar,(f)] = (Gnb(f)] and &b(f) = &,(f). Expressions for the variance and covariance of the smoothed magnitude and phase

INTENSITY

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523

ERRORS

estimators may be found in references [8] and [9]: they are var {l&,(f)]]=

(18)

lGa&V2/~&b(f),

(19920) ~%(f)l/b&(f), cov{I~ab(f)l,~ab(f)}~O,

var{&b(f))=[l-

where y&(f) is the true coherence between y0 and yb. The derivation of equation (20) is given in the Appendix. For small phase mismatch (small time delay), IGab( and yfb(f) (18) and (19). Using equations may be replaced by IG&)] and yTz(f) in equations (17)-(20) yields the normalized standard error F (1) as &=JvarIi(f))ll(f)

= (n~)-“*~llrT~(f)+cot2

412

(l- Y:2(fwY:2(fP2.

(21)

The normalized standard error of the estimates in equat&ns (5) and (6) is identical to equation (21). As calculated by using equation (21), E(I)Jnd is plotted as a function of ~5,~ and yf2 in Figure 3.

I

O-O

0

IO

I

I

I

20

30

40

50

Phase angle (degrees)

Figure

3. Normalized

random

error of acoustic

intensity

estimate.

For a fixed microphone separation d, the phase angle cSi2 is approximately a linear function of frequency (see Figure 2). Equation (21) and Figure 3 show that for small phase angles (low frequency) the second term in equation (21) dominates the normalized standard error. If, in addition, the coherence is low (which is usually the case at low frequency where background noise is present), the normalized standard error will be quite large. However, for large phase (hi h frequency) the normalized standard error approaches the limiting value of l/ SB n dy2i2. At high frequency the normalized standard error can usually be made sufficiently small by appropriate selection of It& The information required to use equations (10) and (21) is normally available at the time a measurement is conducted. As an example consider a situation where I’= 0.11, c5r2= c5i2 = 0.03 rad, d~$ = 0.01 rad , yf2 = O-6 and nd = 50. From equation (10) the bias error is 0.47 (1.7 dB) and the random error (equation (21)) is 8 (I^)= 2.73 (5.7 dB).

524

A. F. SEYEERT

Equations (10) and (21) are useful for euafuating the results of an experiment but, without further information, are not very useful for designing the experiment. Equations (10) and (21) do not give insight in the selection of the position of the microphones relative to source 1, or in the selection of A. While the final selection of these parameters may be made by trial-and-error during measurement, some general guidelines may be developed by considering the bivariate stochastic process in Figure 1. The coherence function can be written as &(f)

= IGl*(f)l*lGll(f)G22(f),

(22)

where Gil(f) and &2(f) are the auto-spectra of the acoustic pressures at measurement points 1 and 2 respectively. For a general bivariate process (with omission, for brevity, of the frequency f) IGi212= G: ]Hii121Hi~12+2GlG2 Re WTIHI~H~IH& I+ G: IH22121H2~12, GiiG22=

G:l~,~l21~~2l2+G~G2~lH~~l2l~22l2+IH2,l2lH,2l2~+G:IH2212lH2~~2,

(23) (24)

where Gi and G2 are the auto-spectra of sources 1 and 2, respectively, and where Hii is the Fourier transform of hii, etc. An examination of equations (22)-(24) shows that yT2 = 1 if G2=0,

(25)

or if Hii = HI2

and

H22 = H2i.

(26)

If source 1 is a monopole acoustic source with source strength qi(t), the acoustic pressure p1 at measurement point 1 a distance rll from source 1 is pi(~ii, t)= A ql(t) exp (--jkrll)/rll, where A is a constant. The auto-spectrum of the acoustic pressure at measurement point 1 is ]Hll(f)]2Gl(f)r where Gi(f) is the auto-spectrum of the source strength and Hii =A exp (-jkrii)/ rll. If source 2 is also a monopole, then, by using equations (22) through (24), one finds y:z = [l + 2a/3S cos 2kA + (c~pS)~]/[l+ a(@ + S2) + (a@)*],

(27)

where (Y= G2/G1, /3 = r11/r21, S = r12/r2*, rzl is the distance from source 2 to measurement point 1, etc. For simplicity the measurement points are restricted to be on a line between the sources (rz2 = r21 -A, rll = r12 - A). From equation (27) it can be seen that equation (26) implies A = 0 for the limiting case of y:2 = 1. Equation (21) may be misinterpreted because it appears that ~(1) can be reduced by increasing A. However, equation (27) shows that y & is also a function of A, even though cos 2kA 2: 1 for kA K 1. Figure 4 shows equation (27) plotted against R’/R for several values of A/R where R is the distance between the sources and R’ = rll + A/2. The normalized standard error obtained by using equation (27) with equation (21) is also plotted in Figure 4. Figure 4 shows that s(i) increases slightly as A is increased. Similar observations have been made by Fahy [lo].

5. CONCLUSIONS As discussed in references [2] and [3], and as shown here, instrument phase mismatch will bias acoustic intensity measurements. This bias error may be large at low frequency where Ac$ is not negligible when compared to the true phase spectrum 4i2. At high frequency where Acf~/c$12 -K1 the bias error due to phase mismatch is small.

INTENSITY MEASUREMENT ERRORS

01

I

0

0. I

I

0.2

525

I

I

0.3

0.4

I

0.5

R’/R

Figure 4. Coherence and normalized random error of acoustic intensity for G2/GI values of A/R. -, Coherence; ---, normalizedrandom error.

= 1 and kR = 0.5. for

various

Secondary acoustic sources will bias the acoustic intensity even when these sources are with the primary source. The presence of secondary sources reduces the coherence between the measured acoustic pressures and increases the random error of the acoustic intensity estimates. Random error may be very large at low frequency, or where the coherence is low, but is not usually significant at high frequency. The use of the acoustic intensity method at low frequency requires a compromise between acceptable random and bias errors in the selection of A. Random error decreases with decreasing A, but uncorrected phase mismatch will cause the bias error to increase. The results in this paper show these effects as adequately as is possible, when using idealized models, but the applicability of the acoustic intensity method will ultimately be decided in the laboratory. uncorrelated

REFERENCES 1. F. J. FAHY 1977 Journal of the Acoustical Society of America 62, 1057-1059. Measurement of acoustic intensity using the cross-spectral density of two-microphone signals. 2. J. Y. CHUNG 1978 Journal oftheAcoustica1 Society ofAmerica 64, 1613-1616. Cross spectral method of measuring acoustic intensity without error caused by instrument phase mismatch. 3. J. Y. CHUNG, J. POPE and D. A. FELDMAIER 1979 Proceedings, Diesel Engine Noise Conference, Detroit, Michigan, 26 February-2 March 1979, SAE P-80, 353-364. Application of acoustic intensity measurement to engine noise evaluation. 0. K. 0. PETTERSON 1979 Journal of Sound and Vibration 66, 626-629. A procedure for determining the sound intensity distribution close to a vibrating surface. R. J. ALFREDSON 1980Journalof Sound and Vibration 70,181-186. The direct measurement of acoustic energy in transient sound fields. A. F. SEYBERT and J. F. HAMILTON 1978 Journalof Sound and Vibration 60,1-9. Time delay bias errors in estimating frequency response and coherence functions. A. PAPOULIS 1965 Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill Book Company. See section 7-3.

526

A. F. SEYBERT

See 8. G. M. JENKINS and D. G. WAITS 1968 Spectral Analysis. San Francisco: Holden-Day. section 9.2. 9. J. S. BENDAT 1978 Journal of Sound and Vibration 59, 40.5421. Statistical errors in measurements of coherence functions and input/output quantities. 10. F. J. FAHY 1977 Noise Control Engineering 9, 155-162. A technique for measuring sound intensity with a sound level meter.

APPENDIX

wen without proof in reference [8]. The The result that cov {16,,(f)], &b(f)}=0 1s ’ g’ proof folloys, where the frequency f has been dropped for simplicity. The smoothed estimates IGab] and &, may be represented in terms of &, and dab, the smoothed estimates of the real and imaginary components of Gab : IGab/ = (&

dab = arctan(QJeahb).

+ &b)1’2,

(AI, A2)

If these estimates have negligible bias error E{&b} = C,, and E{Q’,b}= Qab. Let S&, and SQab be perturbations about the true values: i.e., & = C,b f s&,

Qab = Qab + S&b,

(A3)

where E[S&,] = 0 and E[SQi,b] = 0. A Taylor series expansion of equations (Al) and (A2) about the expected values yields leabI”IGabI+(Cabs~~~+Qabs~‘ab)lIGabI, &lb =

‘#‘a6 +

(cabdab

-

(A4)

Qab~~ab)/l66

i2,

WI

where the higher order terms have been neglected. The covariance is COV {I&I,

dab}

=

E{(@abI

-E[~&bI])(&b

J3&xdI= E{I&bI&ab}lGabk#%zb. (A61

-

The first term in equation (A6) may be evaluated by multiplying equations (A4) and (AS) and taking expected values: E{@,&nb}”

]Ga&‘ab +

+[(&b

-

c’,b)/lGabi31

(QabCab/lGabi3hr

cov

{&b}-var

{dab,

CabI

(A7)

{dab>).

From reference [8] COV {dab, tab}

=

QabCnb/nd,

var

{kb)-var

{dab>

=

cc’,6

-

&b)/Q.

w3)

Combining equations (A6)-(A8) gives cov {I&&], &b}= 0, which is equation (20) in this paper.