Statistical errors in the estimation of time-averaged acoustic energy density using the two-microphone method

Transcription

1 Statistical errors in the estimation of time-averaged acoustic energy density using the two-microphone method Short title: Statistical errors in acoustic energy density Justin Ghan, Ben Cazzolato, and Scott Snyder Department of Mechanical Engineering, University of Adelaide, SA 5005, Australia. Address all correspondence to this author. Typeset by REVTEX

2 Abstract Time-averaged acoustic energy density can be estimated using the auto-spectra and crossspectrum between two closely spaced microphones. In this paper, an analysis of the random errors that arise using two microphone measurement is undertaken. An expression for the normalised random error of the time-averaged acoustic energy density estimate is derived. This expression is verified numerically. The lower and upper bounds of the normalised random error are derived. It is shown that the normalised random error of the estimate is not a strong function of the sound field properties, and is chiefly dependent on the number of records averaged. PACS numbers: e, Qv

3 I. INTRODUCTION Acoustic energy density is defined by the sum of the acoustic potential energy density and the acoustic kinetic energy density at a point. It has been shown [] that energy density provides a significantly better estimate of the total acoustic energy within an enclosure than does the acoustic potential energy (estimated by microphones. Subsequently, several authors have found energy density to be an effective sensor for active noise control applications, as it measures the total energy at a point, and generally outperforms microphones [ 6]. The estimation of potential energy density and kinetic energy density in the frequency domain and the associated statistical errors have been discussed by Elko [7]. An expression for the estimation of time-averaged total acoustic energy density in the frequency domain has been derived [8]. In this paper, the statistical errors associated with this estimate are analysed. An expression for the normalised random error of the estimate is derived. The method here is similar to that used to derive the normalised random error in the estimation of sound intensity [9]. A Simulink model will be used to simulate the estimation of the time-averaged acoustic energy density. The resulting measurements will be analysed to verify the derived expression for the normalised random error. It will be shown that the normalised random error does not depend heavily upon the nature of the sound field being measured. In fact, it is bounded between two functions dependent only on n d, the number of records averaged to produce the smooth spectra estimates used in the expression for the time-averaged acoustic energy density estimate. As a result, the averaging time required to attain a desired level of accuracy can be chosen without knowledge of any of the specifics of the experiment. II. ANALYTICAL DERIVATION OF THE RANDOM ERROR The instantaneous acoustic energy density, E D (t, at a point is defined as the sum of the acoustic potential energy density, U(t, and the acoustic kinetic energy density, T (t, 3

4 at that point, given by [0] U(t = p (t ρc, ( T (t = ρv (t, ( E D (t = U(t + T (t = p (t ρc + ρv (t, (3 where p(t and v(t are the instantaneous pressure and particle velocity magnitude, respectively, at that point, c is the speed of sound, and ρ is the mean density of the fluid. It has been shown [8] that, using the two-microphone measurement method with a microphone separation distance of h, the time-averaged acoustic energy density is approximated by E D (ω ( 8ρc + (G 8ρω h (ω + G (ω ( + 8ρc 8ρω h (Re [G (ω], (4 where G (ω and G (ω are the single-sided auto-spectra, and G (ω is the singlesided cross-spectrum, of the two microphone pressure signals. If the cross-spectrum is separated into real and imaginary components, G (ω = C (ω + jq (ω, equation (4 can be rewritten as E D (ω ( 8ρc + (G 8ρω h (ω + G (ω ( + 8ρc 8ρω h An estimate of the time-averaged acoustic energy density is given by ( E ˆ D (ω 8ρc + (Ĝ(ω Ĝ(ω + 8ρω h + ( 8ρc 8ρω h (C (ω. (5 ( Ĉ(ω, (6 where Ĝ(ω, Ĝ (ω, and Ĉ(ω are smooth estimates of the auto-spectra and the real part of the cross-spectrum, obtained by averaging n d statistically independent raw estimates. Estimates of time-averaged potential energy density and time-averaged kinetic energy density are given by U(ω ˆ (Ĝ (ω + 8ρc Ĝ(ω + Ĉ(ω, (7 T ˆ (ω (Ĝ (ω + 8ρω h Ĝ(ω Ĉ(ω. (8 4

5 Elko [7] derived the normalised random error of these estimates. While the derivations were based upon incorrect expressions for the estimates, the method and results remain valid: It can be shown that the variance of the estimate in (6 is, to a first order approximation, ε (Û(ω ε (ˆT (ω nd, (9 nd. (0 { var ED ˆ (ω ( ˆ E D Ĝ ( ˆ E D Ĉ ( ˆ E D ( var {Ĝ (ω ˆ E D + var {Ĝ (ω Ĝ Ĝ ( ˆ E D Ĝ ( ˆ E D Ĝ var {Ĉ (ω ( ˆ E D Ĝ ( ˆ E D Ĉ ( ˆ E D Ĉ cov {Ĝ (ω, Ĝ(ω cov {Ĝ (ω, Ĉ(ω cov {Ĝ (ω, Ĉ(ω, ( where the derivatives are evaluated at the true values Ĝ(ω = G (ω, Ĝ(ω = G (ω, and Ĉ(ω = C (ω. Assuming stationary Gaussian random signals, the relevant variances and covariances are [] var {Ĝ (ω var {Ĝ (ω var {Ĉ (ω cov {Ĝ (ω, Ĝ(ω cov {Ĝ (ω, Ĉ(ω cov {Ĝ (ω, Ĉ(ω = G (ω n d, ( = G (ω n d, (3 = G (ωg (ω + C (ω Q (ω n d, (4 = G (ω n d, (5 = G (ωc (ω n d, (6 = G (ωc (ω n d. (7 5

6 Therefore, { ( var ED ˆ (ω 8ρc + G (ω ( + 8ρω h n d 8ρc + G (ω 8ρω h n d ( +4 8ρc G (ωg (ω + C (ω Q (ω 8ρω h n d ( + 8ρc + G (ω 8ρω h n ( ( d +4 8ρc + 8ρω h 8ρc G (ωc (ω 8ρω h n ( ( d +4 8ρc + 8ρω h 8ρc G (ωc (ω. (8 8ρω h n d Noting that [ ED (ω ] ( 8ρc + ( G 8ρω h (ω + G (ω + G (ωg (ω ( + 8ρc ( 4C 8ρω h (ω ( ( + 8ρc + 8ρω h 8ρc 8ρω h (G (ω + G (ω (C (ω, (9 it can be seen that [ { var ED ˆ (ω [ED (ω ] ( + n d 8ρc ( G(ωG (ω C 8ρω h (ω Q (ω ( + 8ρc + ( G(ω G ] (ωg (ω. (0 8ρω h Since C (ω + Q (ω = G (ω = γ (ωg (ωg (ω, where γ (ω is the coherence between the two microphone pressure signals, { var ED ˆ (ω [ [ED (ω ] ( ( 8 n d 8ρc 8ρω h ( γ(ω ] G (ωg (ω. ( is Therefore, the normalised random error of the estimate E ˆ D (ω, defined as { ( var ED ˆ (ω ε ED ˆ (ω =, ( E D (ω ( ε ED ˆ (ω 8 nd ( ( ( γ 8ρc 8ρω h (ω G (ωg (ω [ ED (ω ]. (3 6

7 Since Re [G (ω] = G (ω cos φ (ω = γ (ω G (ωg (ω cos φ (ω, where φ (ω is the phase angle between the two microphone pressure signals and γ (ω is the positive square root of γ (ω, equation (4 can be rewritten as ( E D (ω 8ρc + (G 8ρω h (ω + G (ω ( + 8ρc ( γ (ω G (ωg (ω cos φ (ω. (4 8ρω h Substituting equation (4 into equation (3 and multiplying the numerator and denominator of the fraction by (8ρω h /4G (ωg (ω yields ( ε ED ˆ (ω (kh [ ( γ (ω ], (5 nd [((kh + X + ((kh γ (ω cos φ (ω] where the wave number is given by k = ω, and c ( X = G (ω G (ω + G (ω G (ω. (6 If the auto-spectra of the two microphone pressure signals are approximately equal, G (ω G (ω, then X, so ( ε ED ˆ (ω ] (kh [ ( γ (ω. (7 nd [((kh + + ((kh γ (ω cos φ (ω] The normalised random error for this case is plotted in Figure, assuming that the phase between the two signals is solely due to the propagation delay so that φ = kh. It is important to note that, while the bias errors that occur when using the two-microphone method are affected by spatial aliasing, the random errors are not. III. SIMULINK SIMULATION A Simulink model (see Figure was created to simulate the two-microphone method for measuring acoustic energy density. The two microphone pressure signals were calculated by simulating two points in a free space pressure field generated by a white noise point source. The transfer function from a monopole point source to a sensor at distance r is G(s = r r e c s, (8 where c is the speed of sound. These transfer functions were implemented using a second order Padé approximation in series with a gain. In order to vary the coherence between 7

8 the two pressure readings, an amount of incoherent noise was introduced to one of the signals. To compute a single estimate, a frequency spectrum of each of the microphone pressure readings was obtained by performing a 5 point FFT (see Figure 3. The auto-spectra and the cross-spectrum were then calculated, and equation (4 was applied to produce an acoustic energy density spectrum estimate with n d = (see Figure 3. A number, N, of acoustic energy density spectrum estimates were calculated, and the standard deviation of each spectral line was divided by the mean value to produce an estimate of the normalised random error. The microphone pressure readings, p and p, were also sampled in the time domain. Each time signal was divided into overlapping 5 point sections, each of which was windowed. The FFTs of these windowed sections were averaged to obtain spectra estimates, which were then used to calculate the auto-spectra, cross-spectrum, coherence and phase angle. Equation (5 was applied to calculate the normalised random error. Comparisons of the results between the two methods for estimating the normalised random error for various values of N are shown in Figure 4. As the number, N, of acoustic energy density estimates used in the average increases, the estimate of the normalised random error converges to that predicted by the derived equation (5. This would seem to confirm the validity of the expression for the normalised random error of the acoustic energy density estimate. IV. BOUNDS OF THE RANDOM ERROR Since the fraction in equation (5 is always non-negative, the normalised random error is bounded above by ( ε ED ˆ (ω, (9 nd and equality occurs when γ (ω =. The denominator of the fraction in equation (5 can be rearranged as [ ((kh + X + ((kh γ (ω cos φ (ω ] = [ (X + γ (ω cos φ (ω(kh + (X γ (ω cos φ (ω ]. (30 Since the arithmetic mean of (X + γ (ω cos φ (ω(kh and (X γ (ω cos φ (ω 8

9 is greater than or equal to their geometric mean, [ (X + γ (ω cos φ (ω(kh + (X γ (ω cos φ (ω ] 4(X γ (ω cos φ (ω(kh. (3 Since cos φ (ω, 4(X γ (ω cos φ (ω(kh 4(X γ (ω(kh. (3 Equation (30 and inequalities (3 and (3 yield [ ((kh + X + ((kh γ (ω cos φ (ω ] 4(X γ(ω(kh. (33 Therefore, γ(ω (X γ (ω (kh ( γ(ω [((kh + X + ((kh γ (ω cos φ (ω], (34 so that the normalised random error is bounded below by ( ε ED ˆ (ω [ ] γ (ω. (35 nd (X γ (ω Equality occurs in inequality (3 when (X + γ (ω cos φ (ω(kh = (X γ (ω cos φ (ω, and equality occurs in inequality (3 when cos φ (ω =. Therefore, equality occurs in inequality (35 either when φ (ω = 0 and (X + γ (ω(kh = (X γ (ω, or when φ (ω = π and (X γ (ω(kh = (X + γ (ω. Inequalities (9 and (35 together yield [ X γ (ω ] ( ε ED ˆ (ω nd X γ(ω. (36 nd When G (ω = G (ω, X = so that the lower bound in (36 becomes / n d. The definition of X (6 shows that X always, since the arithmetic mean of G (ω/g (ω and G (ω/g (ω is greater than or equal to their geometric mean. Therefore, X γ (ω X γ (ω = γ (ω (X γ (ω γ (ω ( γ (ω =. (37 This shows that the lower bound for the normalised random error is minimum at / n d when G (ω = G (ω. Indeed, ( ε ED ˆ (ω. (38 nd nd 9

10 V. DISCUSSION The expression (36 shows that the normalised random error of time-averaged acoustic energy density spectrum estimates is always bounded within a narrow tolerance. As the upper bound in (9 is dependent only on n d, the number of records averaged, it can be ensured that the normalised random error is no greater than a specified tolerance by an appropriate choice of n d, without the need to know any other information about the measurement. Thus it is not in fact necessary to first estimate the coherence or phase angle. The lower bound in (38 shows that the averaging time resulting from this approach will exceed the minimum necessary averaging time at most by a factor of. VI. CONCLUSIONS An expression for the normalised random error of the time-averaged acoustic energy density estimates has been derived. A simulation in Simulink was used to validate the expression numerically. The estimates of the normalised random error resulting from the simulated measurements agree with the error predicted by the expression. Because the normalised random error is only heavily dependent on n d, an experiment can be designed to achieve a desired normalised random error without measuring the local properties of the sound field first. 0

11 [] R.K. Cook and P.A. Schade. New method for the measurement of the total energy density of sound waves. In Proceedings of Inter-Noise 74, pages 0 06, 974. [] S.D. Sommerfeldt and P.J. Nashif. A comparision of control strategies for minimising the sound field in enclosures. In Proceedings of Noise-Con 9, pages , 99. [3] P.J. Nashif and S.D. Sommerfeldt. An active control strategy for minimising the energy density in enclosures. In Proceedings of Inter Noise 9, pages , 99. [4] S.D. Sommerfeldt and J. Parkins. Active control of energy density in three dimensional enclosures. Journal of the Acoustical Society of America, 95(5:989, 994. [5] W. Shen and J.Q. Sun. A study of shell interior noise control. SPIE, 304:8 88, 997. [6] Y.C. Park and S.D. Sommerfeldt. Global attenuation of broadband noise fields using energy density control. Journal of the Acoustical Society of America, 0(: , 997. [7] G.W. Elko. Frequency domain estimation of the complex acoustic intensity and acoustic energy density. PhD thesis, The Pennsylvania State University, 984. [8] J. Ghan, B.S. Cazzolato, and S.D. Snyder. Expression for the estimation of time-averaged acoustic energy density using the two-microphone method. Accepted for publication in the Journal of the Acoustical Society of America, 003. [9] A.F. Seybert. Statistical errors in acoustic intensity measurements. Journal of Sound and Vibration, 75(4:59 56, 98. [0] F. Fahy. Sound Intensity. E&FN Spon, London, nd edition, 995. [] J.S. Bendat and A.G. Piersol. Random Data - Analysis and Measurement Procedures. John Wiley & Sons, New York, nd edition, 986.

12 List of Figures Normalised random error of the acoustic energy density estimate (n d =, for h = 50 mm, c = 343 m s. It is assumed that G (ω G (ω and φ = kh. When the microphone separation distance exceeds λ, half the wavelength, the bias errors become very large, so the random error is unimportant. In this case, this occurs at frequencies above 3430 Hz The simulated sound field. The microphones (separation distance, h = 50 mm are placed at distances r = 5.00 m and r = 5.05 m from the noise sources. The primary noise source produces continuous white noise of.0 Pa at m (94 db re 0 µpa. The secondary (contaminating noise source produces continuous white noise of 0.00 Pa at m (54 db re 0 µpa. Speed of sound, c = 343 m s, and density of air, ρ =. kg m 3. The Energy Density Calculation sub-model is shown in Figure Model for the calculation of the time-averaged acoustic energy density estimate using the frequency domain expression in terms of the auto-spectra and cross-spectrum of the two pressure readings Estimates of the normalised random error of the acoustic energy density spectrum estimates, obtained from the standard deviation of measurements, compared with the calculated normalised random error as predicted by the derived theoretical expression. Results are shown for (a 0, (b 00, and (c 000 acoustic energy density spectrum estimates

13 .05 γ = Normalised Random Error γ = 0.4 γ = 0.8 γ = γ = γ = Frequency (Hz Figure : Normalised random error of the acoustic energy density estimate (n d =, for h = 50 mm, c = 343 m s. It is assumed that G (ω G (ω and φ = kh. When the microphone separation distance exceeds λ, half the wavelength, the bias errors become very large, so the random error is unimportant. In this case, this occurs at frequencies above 3430 Hz. 3

14 p Secondary Noise Source Primary Noise Source num(s den(s Transfer Fcn num(s den(s Transfer Fcn s (s+*pi*0 Mic Response HighPass Filter 0 Hz s (s+*pi*0 Mic Response HighPass Filter 0 Hz To Workspace p p p ED Energy Density Calculation To Workspace ED To Workspace Figure : The simulated sound field. The microphones (separation distance, h = 50 mm are placed at distances r = 5.00 m and r = 5.05 m from the noise sources. The primary noise source produces continuous white noise of.0 Pa at m (94 db re 0 µpa. The secondary (contaminating noise source produces continuous white noise of 0.00 Pa at m (54 db re 0 µpa. Speed of sound, c = 343 m s, and density of air, ρ =. kg m 3. The Energy Density Calculation sub-model is shown in Figure 3. 4

15 p p FFT 5 point FFT FFT 5 point FFT u Conjugate u Conjugate Product Product Product 3 Spp Re(u Complex to Real Spp Gain 6 4 Gain 3 Gain 7 Gpp Re{*Gpp Gpp wsq /(8*rho*c^ Gain 4 /(8*rho*h^ Product 6 Gain 5 ED /w^ Figure 3: Model for the calculation of the time-averaged acoustic energy density estimate using the frequency domain expression in terms of the auto-spectra and cross-spectrum of the two pressure readings. 5

16 .6 (a N = 0.4. Normalised Random Error Measured Calculated Frequency (Hz.6.4 (b N = 00. Normalised Random Error Measured Calculated Frequency (Hz 6

17 .6 (c N = Normalised Random Error Measured Calculated Frequency (Hz Figure 4: Estimates of the normalised random error of the acoustic energy density spectrum estimates, obtained from the standard deviation of measurements, compared with the calculated normalised random error as predicted by the derived theoretical expression. Results are shown for (a 0, (b 00, and (c 000 acoustic energy density spectrum estimates. 7