- Email: [email protected]
Development of a multi-microphone calibrator
Applied Acoustics 70 (2009) 790–798
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Development of a multi-microphone calibrator Jonathan R. Oldham a, Jason D. Sagers a, Jonathan D. Blotter a,*, Scott D. Sommerfeldt b, Timothy W. Leishman c, Kent L. Gee d a
Department of Mechanical Engineering, 435 CTB, Brigham Young University, Provo, UT 84602, United States Department of Physics and Astronomy, N283 ESC, Brigham Young University, Provo, UT 84602, United States c Department of Physics and Astronomy, N247 ESC, Brigham Young University, Provo, UT 84602, United States d Department of Physics and Astronomy, N319 ESC, Brigham Young University, Provo, UT 84602, United States b
a r t i c l e
i n f o
Article history: Received 4 April 2008 Received in revised form 11 September 2008 Accepted 15 September 2008 Available online 8 November 2008 Keywords: Calibration Energy density Multi-microphone probes
a b s t r a c t This paper presents the theory, design, and validation of a microphone calibrator used to simultaneously calibrate the amplitudes of multiple microphones on a single probe. The probe uses four 6 mm diameter electret microphones to acquire the data needed to compute acoustic energy density. This probe has prompted the need for simultaneous, multi-microphone amplitude calibration. The calibration process simultaneously subject each microphone on the probe to the same known acoustic pressure using four equal-length, small-diameter tubes connected to a single excitation source. A reference microphone connected to a fifth tube is used to calibrate the microphones. Test results show that the calibrator can calibrate each probe microphone within ±0.5 dB up to 2 kHz, and within ±1 dB up to 4.9 Hz with a confidence level of 95%. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Pressure measurements made by microphones are one of the most common acoustic field measurements. One reason for this is that pressure measurements can be used to calculate many other acoustic quantities such as acoustic intensity, acoustic sound power, acoustic energy density, and particle velocity. Single microphones, small microphone groups, and various types and sizes of microphone arrays are often used to gather acoustic field data. Accurate and frequent calibration is essential when using microphones to acquire data. As a result, calibration for systems with multiple microphones can be a tedious task. The primary objective of this paper is to present and validate a multi-microphone calibrator. Although the specific calibrator shown in this paper was developed for an acoustic energy density probe, it should be noted that the concepts in this paper could be extended to calibration of other types of probes or microphone arrays. The calibrator presented in this paper was developed for use with a multi-microphone probe. The probe is used to calculate acoustic energy density and has a usable bandwidth of 20 Hz– 2 kHz. One of the design objectives for the calibrator was to simultaneously calibrate the amplitude of each microphone on the probe within ±0.5 dB (re: 20 lPa) over the usable bandwidth. The probe
* Corresponding author. Tel.: +1 801 422 7820; fax: +1 801 422 0516. E-mail addresses: [email protected] (J.R. Oldham), [email protected] (J.D. Sagers), [email protected] (J.D. Blotter), [email protected] (S.D. Sommerfeldt), [email protected] (T.W. Leishman), [email protected] (K.L. Gee). 0003-682X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2008.09.013
microphones are calibrated relative to a high-quality reference microphone with a known calibration curve. Calibrating the probe to within ±0.5 dB would classify the probe as a class 1 probe, and within ±1 dB would classify the probe as a class 2 probe [1]. The primary objective of this paper was to calibrate the amplitudes of each of the probe microphones. Although phase calibration is not part of the current calibrator, results are presented that show phase calibration may be possible with this calibrator. It is further noted that the precision phase-matching is not as critical when energy density is the quantity of interest. However, if intensity measurements are desired from this particular probe, a phase calibration for the condenser microphones would also be required. The effectiveness of the calibrator was validated in three different stages. The first stage validated the splitter, where the volume velocity of the source tube is divided equally into five different paths. The second stage validated the receiver tubes, where identical pressures must exist at the end of each tube. The third stage validated the entire calibrator, where a multi-microphone probe was used in conjunction with the calibrator. The three stages were all validated at several discrete frequencies, and the final stage was further validated over a continuous range of frequencies. 2. Acoustic path concept 2.1. Theory The objective of this research was to develop a calibrator that would simultaneously calibrate multiple microphones using a
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single excitation source. In order to do this, each microphone on the probe must be exposed to a known acoustic pressure at the same time. A design with a single excitation source and multiple identical acoustic paths from the source to the microphone locations was developed. The theory behind this concept was derived from the conceptual drawing shown in Fig. 1. A loudspeaker driver was used as the excitation source and was located at the left end of the driver tube. The length of the driver tube allowed for evanescent modes to decay, leaving only a plane wave at the right end of the tube [2]. This allowed for the assumption of a uniform volume velocity U (units of m3/s) at the right end of the driver tube. The splitter was inserted at the right end of the driver tube in order to equally divide the volume velocity into each of the five receiver tubes. As long as the total acoustic impedance of each path, ZA (units of N s/m5), was the same for each of the receiver tubes, the amplitude of the acoustic pressure would then be the same
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at each of the microphones [3]. The acoustic impedance of each path can be assumed to be the same as long as the total length, cross-sectional dimensions, and termination impedance of each path are identical. 2.2. Analogous circuit model To more fully understand and validate the experimental results, analogous circuit techniques were used to model the system shown in Fig. 1. The electrical and mechanical properties of the loudspeaker driver are easily represented in an analogous circuit (this will be discussed later). The acoustic waveguides consisting of the driver tube, splitter ports, receiver tubes, and inserts can each be modeled below the cutoff frequency, fc, using a wellknown acoustic impedance circuit. An acoustic impedance circuit for a finite-length one-dimensional waveguide with an arbitrary source S at one end and an arbitrary termination T at the other is shown in Fig. 2. The impedance circuit parameters in Fig. 2, labeled ^ ð0Þ and P ^ ð1Þ are the acousU1 and U2, are the volume velocities and P tic pressures at the source and the termination respectively. The impedances ZA1 and ZA2 are the acoustic impedances relating to the particular tube and are defined in Eqs. (1) and (2), respectively [4]
Z A1 ¼ j
qo ~c S
Z A2 ¼ j
tan
qo ~c S
~ kl 2
!
~ cscðklÞ
ð1Þ
ð2Þ
In Eqs. (1) and (2), S is the cross-sectional area of the tube, j is pffiffiffiffiffiffiffi 1, qo is the ambient fluid density, ~c is the lossy acoustic phase ~ is the lossy acoustic wave number defined as k ~ ¼ k ja p speed, k (where ap is the total thermoviscous absorption coefficient, which is a sum of the classical absorption coefficient and the absorption coefficient for wall losses), and l is the length of the tube. The lossy acoustic phase speed and the lossy acoustic wave number account for the losses in the tubes. This approach assumes that the fluid boundary layer thickness is much less than the tube radius. The calibrator uses a total of sixteen circular waveguides (one for the driver tube, five for the splitter, five for the receiver tubes, and five for the inserts) to channel the sound from the excitation source to the microphones. Each of these tube sections has a different diameter and length, but the circuit shown in Fig. 2 can be used to model all of the waveguides. The complexity of the analogous circuit model was reduced by combining all of the acoustic waveguide circuits (except for the driver tube) into a Thevenin equivalent impedance. This was accomplished in two steps. First, three circuits representing the splitter port, receiver tube, and insert were combined in series for each of the five paths. Second, the five path circuits were combined in parallel. The result was a single acoustic impedance value that represented all of the elements to the right of the driver tube.
Fig. 1. Calibrator path concept drawing.
Fig. 2. An acoustic impedance circuit for a finite-length one-dimensional waveguide with a source at one end and an arbitrary termination at the other.
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Each of the five calibrator paths is terminated by a microphone. In this model it was assumed that the impedance of a microphone was much larger than the impedance of the waveguides, and therefore the termination impedance of each path was assumed to be infinite in the analogous circuit. It should also be noted that effective acoustic lengths were added to each of the tubes in the five paths. The change in cross-sectional area between tubes creates a discontinuity inductance that can be interpreted physically as an increase in the length of the tube [5]. This change in length must be accounted for in order to obtain accurate predictions from the analogous circuit model. To further increase the accuracy of the calibrator model, the actual loudspeaker driver was modeled and included in the circuit. This produced a more accurate model of the calibrator than if the excitation source was assumed to be a constant volume velocity source. The driver model combines the acoustic, electrical, and mechanical elements of the driver by converting the electrical and mechanical components into the acoustic impedance domain. These elements are shown in Fig. 3, along with the rest of the analogous circuit model for the calibrator. All of the elements from different parts of the calibrator have been transformed into the acoustical impedance domain. In the circuit of Fig. 3, there are five acoustic elements, two electrical elements, and three mechanical elements. The acoustic elements Zrad and ZATh represent the radiation impedance of the back of the driver and the Thevenin acoustic impedance of all of the calibrator waveguides, respectively. The electrical circuit shows a source ð^eg Þ which is the complex voltage supplied to the loudspeaker terminals. The electrical circuit also contains an electrical impedance representation of the voice coil which is defined as [6]
Z E ¼ RE þ jxL1 þ
jxL2 R2 : R2 þ jxL2
ð3Þ
The mechanical circuit consists of three elements, the mechanical resistance of the suspension (RMS), the mechanical compliance of the suspension (CMS), and the actual moving mass of the driver diaphragm (MMD). The mechanical resistance of the suspension RMS and the moving mass of the driver diaphragm (MMD) are calculated from measured Thiele-Small parameters using Eq. (4) and Eq. (5), respectively
M MD
8 ¼ MMS qo a3 3
ð4Þ
1 2pF s C MS Q MS
ð5Þ
RMS ¼
In Eq. (4), a is the radius of the driver. In Eq. (5), Fs is the ‘‘free air” resonance frequency of the driver, and QMS is the quality factor of the driver at Fs considering mechanical losses of the moving system only [7].
2.2.1. Measured driver parameters The driver parameters used to model the driver were measured using the actual calibrator driver. The driver was suspended in an anechoic chamber. Using a maximum-length sequence system analyzer (MLSSA) with the speaker parameter option (SPO) the driver element values were measured. The MLSSA SPO determines the driver parameters by analyzing the complex impedance of the driver over the frequency range of interest. All of the complex impedance points are used in the analysis of the driver parameters. Driver impedance is complex and can be thought of as a curve in 3 D space where the x-axis is the real part of the impedance, the y-axis is the imaginary part and the z-axis is the frequency. The measured parameters are valid to use in the model because the MLSSA accounts for the real, imaginary, and frequency dependence of each driver parameter. The parameters were measured ten different times and the average of all ten runs for each parameter was used in the model. The added-mass method was used to measure the parameters. The first five parameter measurements were made using an added mass of 1 g and the second five were made using 2 g. The averaged results of all of the parameters used in the model are shown in Table 1. 2.2.2. Model results The model was evaluated numerically to predict the frequency response function (FRF) at the end of one of the paths. The results were plotted along with the measured FRF as measured at the end of one of the paths. These plots are shown in Fig. 4. As can be seen, the model accounts for every resonance up to the cutoff frequency of the first cross-mode. The model predicts all but one of the resonances to occur within 25 Hz of the corresponding measured resonance. The smallest difference was within 5 Hz. The largest error in a predicted resonance and the measured resonance occurred in the prediction of the resonance near 4.5 kHz, and was 70 Hz. The model still predicts several resonances to be higher than the actual measured resonances. This is caused by the path tube radii not
Table 1 Measured loudspeaker parameters used in circuit model. Parameter
Mean
Std dev.
Units
SD (effective driver diameter) Fs (free-air resonance) QMS (mechanical quality factor) MMD (mechanical mass) CMS (mechanical compliance) Bl (force factor) RE (voice-coil resistance) L1 (voice-coil inductance) L2 (voice-coil inductance) R2 (voice-coil resistance)
1.30E03 268.03 6.56 0.00214 0.000164 2.72 6.24 0.0000892 0.00013 7.44
N/A 1.066 0.0579 6.15E05 5.64E06 0.0416 0.00202 4.22E07 1.71E12 0.0198
m2 Hz
Fig. 3. The entire calibrator circuit model in the acoustic domain.
kg m/N Tesla-m X H H X
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Fig. 4. Frequency response function of the calibrator analogous circuit model and the actual calibrator.
being large compared to the fluid flow boundary thickness. If the losses are increased in the model these effects are reduced, but the model also loses accuracy. The model may also be improved by including the acoustic impedance of the microphones (instead of assuming infinite impedances), and by modeling the radiation impedance of the rear of the driver more accurately. The model was intended to predict the overall trend and number of resonances of the FRF; therefore, for this work the model proved to be sufficient. 3. Probe and calibrator hardware This section presents the hardware design of the microphone probe and the calibrator developed in this research. 3.1. The probe The probe used in this research to validate the calibrator is spherically shaped with four pressure transducers mounted in the sphere. The microphones are 6 mm, Primo EM123 electret microphones with an input impedance of 1.6 kX. The sphere is 50.8 mm in diameter. A hollow shaft is used to connect the sphere to the digital signal processor (DSP) housing. Wires pass through the shaft to connect the microphones to the DSP. The z-axis, or the axis which runs down the length of the shaft, is considered the natural axis of the sphere. One sensor microphone is mounted on the sphere opposite the shaft and shares the same z-axis as the shaft. This microphone is known as the pole microphone (see Fig. 5). The other three microphones are located 68.75off the z-axis as measured from the pole microphone. These three side microphones are spaced equally around the sphere at 120increments. All four microphones are mounted perpendicular to and flush with the surface of the sphere at their locations. In order for the probe to accurately calculate energy density, each microphone must be accurately calibrated. Since the probe is intended to measure between 20 Hz and 2.0 kHz it needs to be calibrated over this entire range. 3.2. The calibrator A photograph of the entire calibrator is shown in Fig. 6. The driver is used to excite the driver tube with tonal or broadband sound. Below the fc of the driver tube, only plane waves [8] exist at the splitter. As a result of the plane waves, the splitter evenly splits
Fig. 5. Acoustic energy density probe.
the volume velocity into five paths. The receiver tubes attach to the splitter, and guide the sound to the four probe microphones and the reference microphone. The left half of the calibrator clamp aligns the receiver tubes with probe microphones. The two halves of the calibrator clamp are connected with three latches. The calibrator surrounds the entire probe and provides a consistent seal at each microphone. The individual components of the calibrator are described in detail in the following sections. 3.2.1. The excitation source The loudspeaker driver used in the calibrator is a HiVi A2S fullrange loudspeaker. The driver response shown in Fig. 7 is relatively flat over all frequencies of interest in calibrating the probe. This characteristic makes the driver useful in a frequency band calibration application. 3.2.2. Driver tube The driver tube was designed to act like a waveguide and to separate the source from the splitter. Waveguides can be used to remove higher order modes that are induced by a source. The driver tube was chosen to be a circular waveguide with a radius of 20.0 cm. This radius was designed large enough to cover the diaphragm of the driver, and small enough to keep the fc above the
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Fig. 6. Photograph of the entire calibrator.
Deg.
dB spl 90
180
80
135
70
90
60
45
50
0
40
-45
30
-90
20
-135
10
-180
10
Freq
50
100
500
1k
Hz
5k
10k
40k
Fig. 7. HiVi A2S full frequency loudspeaker frequency response.
usable bandwidth of the probe. The resulting fc of the first higher order mode of the driver tube is 5.0 kHz. The length of the driver tube was calculated to be about 0.13 m. This length corresponded with a first higher order mode attenuation of 95 dB when the driver tube is driven at 2.0 kHz (the upper frequency limit of the probe). All other higher order modes would attenuate 95 dB in less spatial distance than the first one. Since only plane waves propagate in the driver tube, the pressure will be uniform at each port of the splitter. 3.2.3. Splitter The splitter was designed and built to divide the volume velocity into equal separate paths as shown in Fig. 1. The splitter is equipped with five holes (four for the probe microphones and one for a reference microphone) located symmetrically about its center as shown in Fig. 8. On the ‘‘front” side of the splitter, each hole is equipped with a port that is designed to connect to a receiver tube. The ‘‘back” side of the splitter is also shown in Fig. 8 and illustrates how the driver tube connects to the splitter. 3.2.4. Receiver tubes Flexible PVC tubes with an outside diameter of 11.9 mm were added to the end of each port of the splitter. The receiver tubes were chosen to have an inside diameter of 6.35 mm and are 23.5 cm in length. These tubes connect to the ports on the splitter at one end and the inserts of the calibrator clamp on the other. The receiver tubes were carefully constructed so that they were identi-
Fig. 8. Splitter (CAD model).
cal in length and diameter. The physical similarities are essential to ensure the acoustic path (and thus the acoustic impedance) is the same for each. The receiver tubes needed to be flexible to allow them to have the same length and yet reach all of the microphones as shown in Fig. 6. 3.2.5. Calibrator clamp The purpose of the calibrator clamp is to align the end of each receiver tube with a probe microphone. The calibrator clamp was designed using two halves with spherical cavities that come together and enclose the probe. The top half of the calibrator clamp
J.R. Oldham et al. / Applied Acoustics 70 (2009) 790–798
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is equipped with a groove that allows the probe shaft to pass through to its center. The bottom half of the calibrator clamp where the probe microphones are located, is equipped with four ports. Each port lines up with a microphone on the probe. Each port also has an insert that is fit with a rubber o-ring to seal the interface between the probe and the bottom half of the calibrator clamp. The inserts connect to the receiver tubes and are held in place by a bolt that has a hole in the center to allow the insert to pass through it. The bolts screw into threaded holes in the bottom half of the calibrator clamp. A CAD model assembly drawing of the calibrator clamp in relation to the probe is shown in Fig. 9. 4. Experimental results The effectiveness of the calibrator was validated in two stages. The first stage validated that the receiver tubes are acoustically the same. In the second stage, the full calibrator was validated. Both stages were validated at several discrete frequencies. The entire calibrator was then validated over a continuous range of frequencies. The results of these validation steps are presented in the following sections. 4.1. Receiver tube validation The receiver tubes were tested similar to the splitter port test described above. All five tubes were connected to the splitter and one tube was tested at a time. The same microphone was used to test all the tubes and all of the tubes except the one being tested were terminated with a microphone and adapter as if they had been mounted in the calibrator. Again, 20 data sets were initially used to compute a mean but it was noted that no change occurred after using only 5 data sets. Testing was done at frequencies of 250 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1.0 kHz, 1.5 kHz, and 2.0 kHz. The 2.0 kHz upper limit was chosen based on the operational limit of the probe. At each of the above frequencies, the pressure was measured at the end of each receiver tube, and the largest difference between any of the five receiver tubes was calculated. This difference was used to calculate the error at each of the frequencies listed above. The error was plotted at each of the measured frequencies. The results of this test are shown in Fig. 10. The goal was to limit the error to less than
Fig. 10. Receiver tube validation results.
±0.5 dB (a class 1 probe), and as can be seen from Fig. 10, this was accomplished at these discrete frequencies. In order to reach each microphone on the probe, the receiver tubes had to be bent slightly. The error introduced as a result of bending the tubes was determined by measuring the pressure at the end of each tube with the tube straight and then bent at 30°, 60°, and 90°. The pressure was measured at the end of each tube in each position five different times. A statistical test was used to determine if bending the tubes had any effect on the measured pressure. The null hypothesis was that the bending of the tubes did not cause any change in the acoustical path of the receiver tubes. A single factor analysis of the variance was performed on the five runs at each orientation with a = 0.05. The analysis produced a P-value of 0.57. Since the P-value is greater than the a value of 0.05 the null hypothesis was not rejected [9]. With a = 0.05 and a P-value that is larger, the probability that the variance in the data was just noise is 95%. This analysis provided confidence that the receiver tubes could be bent to align with the probe and still remain acoustically the same.
Fig. 9. Calibrator clamp (CAD model).
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Fig. 13. FRF of the five receiver tubes.
Fig. 11. Relative phase calibration results between the five calibrator tubes.
The phase difference in the tubes was also measured using a two microphone calibration technique. The tubes were terminated in the calibrator box and phase matched precision microphones
were used to acquire the relative phase between the five tubes in the calibrator. The results for all five tubes are shown in Fig. 11. It should be noted that the phase results are relatively flat over the majority of the frequency range. These curves could be used to calibrate the phase of one probe microphone relative to another.
Fig. 12. Calibration error at various frequencies and 114 dB.
J.R. Oldham et al. / Applied Acoustics 70 (2009) 790–798
4.2. Full calibrator validation This full calibrator validation test was performed to show the overall uncertainty bounds in the entire calibrator, to show the insensitivity to the probe orientation in the calibrator (i.e. that all paths in the probe are similar), and to show repeatability of the measurements. The calibrator driver was excited with a 114-dB (as measured with the 6 mm, GRAS, reference microphone) harmonic wave at 250 Hz, 500 Hz, 1.0 kHz, 2.0 kHz, and 3.0 kHz. The probe was inserted into the calibrator housing and a one second time signal
797
was recorded by each microphone on the probe and the reference microphone. The probe was taken out of the calibrator housing and rotated so that the three side microphones on the probe were each connected to a different receiver tube. The pole microphone and the reference microphone remained connected to the same receiver tubes. Another time signal was recorded and the rotation was repeated. The data for each of the three measurement configurations were normalized about the first position. The error at the second and third positions relative to the first position was calculated. This entire process was repeated for the five frequencies mentioned above. It should be noted the physical limitations of
Fig. 14. The error in receiver tube 1 with a confidence interval.
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the driver reduced the amplitude of the harmonic wave at 2 kHz and 3 kHz to 110 dB. The error plots for each of the five frequencies can be seen in Fig. 12. The error at 250 Hz, 500 Hz, 1.0 kHz, and 2.0 kHz were all less than the 0.5 dB. The error at 3.0 kHz was slightly greater than 0.5 dB. The results indicate that the calibrator is capable of calibrating the probe at 250 Hz, 500 Hz, 1.0 kHz, and 2.0 kHz to the class 1 probe standard. The results at 3.0 kHz indicate that the calibrator is capable of calibrating the probe to the class 2 probe standards at 3.0 kHz. The calibrator was further tested using white noise as the signal input. These measurements were made with the probe in the calibrator just as it would be during calibration. The FRF was calculated using 20 magnitude averages. This number of averages corresponded with a coherence greater than 0.99. The FRF was measured at the end of each receiver tube five times and the average of each receiver tube was calculated. The FRF of each of the receiver tubes is shown in Fig. 13. The FRF results from each receiver tube were averaged to obtain an overall mean. To calculate the error at each receiver tube, the overall mean was subtracted from each of the five FRF measurements. The resulting errors at each receiver tube were then averaged to obtain an average receiver tube error (l). The standard deviation of each of the five error calculations for each receiver tube was calculated (r). Using the mean and standard deviation of the error, an interval at each receiver tube was calculated using Eq. (6)
Interval ¼ l 2r
ð6Þ
The interval corresponds to a confidence of 95.45% that the error will fall in the limits of the interval. The interval for each receiver tube was plotted along with the mean error l and the upper and lower control limits for a class 1 probe and a class 2 probe. These plots are shown in Fig. 14. In Fig. 14 the upper and lower limits of the confidence interval are illustrated using the down pointing and up pointing triangle, respectively. The upper and lower control
limits for a class 1 probe, and a class 2 probe are illustrated by a dashed and dotted lines respectively. 5. Conclusions It has been shown that the calibrator presented in this paper can accurately calibrate four microphones within ±0.5 dB up to 2.0 kHz, and within ±1 dB up to 4.9 kHz. The calibrator is designed for a specific probe, but the concept could easily be applied to use on other probes or arrays. The concept could also be extended to work with higher frequencies by decreasing the size of the driver tube which would increase the cutoff frequency of the first higher order mode and would increase the bandwidth of the calibrator. With a higher cutoff frequency, the driver tube could limit the propagation to only plane waves. As long as only plane waves are present at the splitter the calibrator error will be minimal. Using this calibrator to calibrate this probe would result in a class I probe up to 2.0 kHz and a class II probe up to 5.0 kHz. References [1] Harris Cyril M. Handbook of acoustical measurements and noise control. New York: McGraw-Hill; 1991. [2] Pierce AD. Acoustics: an introduction to its physical principles and applications. New York: Acoustical Society of America; 1989. [3] Beranek LL. Acoustics. New York: (Acoustical Society of America; 1996. [4] Leishman TW. Active control of sound transmission through partitions composed of discretely controlled modules (Ph.D. dissertation. PA: The Pennsylvania State University, University Park; 2000). [5] Karal FC. The analogous acoustical impedance for discontinuities and constrictions of circular cross section. J Acoust Soc Am 1953;25:327–34. [6] Rife DD. MLSSA SPO speaker parameter option. Reference manual version 4WI Rev. 8. DRA Laboratories; 2005. [7] Leishman TW. Physics 562 class notes: applied acoustics. Brigham Young University, Dept. of Physics and Astronomy; 2005. [8] Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of acoustics. 4th ed. New York: Wiley; 2000. [9] Lawson J, Erjavec J. Modern statistics for engineering and quality improvement. California: Duxbury; 2001.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Development of a multi-microphone calibrator Jonathan R. Oldham a, Jason D. Sagers a, Jonathan D. Blotter a,*, Scott D. Sommerfeldt b, Timothy W. Leishman c, Kent L. Gee d a
Department of Mechanical Engineering, 435 CTB, Brigham Young University, Provo, UT 84602, United States Department of Physics and Astronomy, N283 ESC, Brigham Young University, Provo, UT 84602, United States c Department of Physics and Astronomy, N247 ESC, Brigham Young University, Provo, UT 84602, United States d Department of Physics and Astronomy, N319 ESC, Brigham Young University, Provo, UT 84602, United States b
a r t i c l e
i n f o
Article history: Received 4 April 2008 Received in revised form 11 September 2008 Accepted 15 September 2008 Available online 8 November 2008 Keywords: Calibration Energy density Multi-microphone probes
a b s t r a c t This paper presents the theory, design, and validation of a microphone calibrator used to simultaneously calibrate the amplitudes of multiple microphones on a single probe. The probe uses four 6 mm diameter electret microphones to acquire the data needed to compute acoustic energy density. This probe has prompted the need for simultaneous, multi-microphone amplitude calibration. The calibration process simultaneously subject each microphone on the probe to the same known acoustic pressure using four equal-length, small-diameter tubes connected to a single excitation source. A reference microphone connected to a fifth tube is used to calibrate the microphones. Test results show that the calibrator can calibrate each probe microphone within ±0.5 dB up to 2 kHz, and within ±1 dB up to 4.9 Hz with a confidence level of 95%. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Pressure measurements made by microphones are one of the most common acoustic field measurements. One reason for this is that pressure measurements can be used to calculate many other acoustic quantities such as acoustic intensity, acoustic sound power, acoustic energy density, and particle velocity. Single microphones, small microphone groups, and various types and sizes of microphone arrays are often used to gather acoustic field data. Accurate and frequent calibration is essential when using microphones to acquire data. As a result, calibration for systems with multiple microphones can be a tedious task. The primary objective of this paper is to present and validate a multi-microphone calibrator. Although the specific calibrator shown in this paper was developed for an acoustic energy density probe, it should be noted that the concepts in this paper could be extended to calibration of other types of probes or microphone arrays. The calibrator presented in this paper was developed for use with a multi-microphone probe. The probe is used to calculate acoustic energy density and has a usable bandwidth of 20 Hz– 2 kHz. One of the design objectives for the calibrator was to simultaneously calibrate the amplitude of each microphone on the probe within ±0.5 dB (re: 20 lPa) over the usable bandwidth. The probe
* Corresponding author. Tel.: +1 801 422 7820; fax: +1 801 422 0516. E-mail addresses: [email protected] (J.R. Oldham), [email protected] (J.D. Sagers), [email protected] (J.D. Blotter), [email protected] (S.D. Sommerfeldt), [email protected] (T.W. Leishman), [email protected] (K.L. Gee). 0003-682X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2008.09.013
microphones are calibrated relative to a high-quality reference microphone with a known calibration curve. Calibrating the probe to within ±0.5 dB would classify the probe as a class 1 probe, and within ±1 dB would classify the probe as a class 2 probe [1]. The primary objective of this paper was to calibrate the amplitudes of each of the probe microphones. Although phase calibration is not part of the current calibrator, results are presented that show phase calibration may be possible with this calibrator. It is further noted that the precision phase-matching is not as critical when energy density is the quantity of interest. However, if intensity measurements are desired from this particular probe, a phase calibration for the condenser microphones would also be required. The effectiveness of the calibrator was validated in three different stages. The first stage validated the splitter, where the volume velocity of the source tube is divided equally into five different paths. The second stage validated the receiver tubes, where identical pressures must exist at the end of each tube. The third stage validated the entire calibrator, where a multi-microphone probe was used in conjunction with the calibrator. The three stages were all validated at several discrete frequencies, and the final stage was further validated over a continuous range of frequencies. 2. Acoustic path concept 2.1. Theory The objective of this research was to develop a calibrator that would simultaneously calibrate multiple microphones using a
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single excitation source. In order to do this, each microphone on the probe must be exposed to a known acoustic pressure at the same time. A design with a single excitation source and multiple identical acoustic paths from the source to the microphone locations was developed. The theory behind this concept was derived from the conceptual drawing shown in Fig. 1. A loudspeaker driver was used as the excitation source and was located at the left end of the driver tube. The length of the driver tube allowed for evanescent modes to decay, leaving only a plane wave at the right end of the tube [2]. This allowed for the assumption of a uniform volume velocity U (units of m3/s) at the right end of the driver tube. The splitter was inserted at the right end of the driver tube in order to equally divide the volume velocity into each of the five receiver tubes. As long as the total acoustic impedance of each path, ZA (units of N s/m5), was the same for each of the receiver tubes, the amplitude of the acoustic pressure would then be the same
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at each of the microphones [3]. The acoustic impedance of each path can be assumed to be the same as long as the total length, cross-sectional dimensions, and termination impedance of each path are identical. 2.2. Analogous circuit model To more fully understand and validate the experimental results, analogous circuit techniques were used to model the system shown in Fig. 1. The electrical and mechanical properties of the loudspeaker driver are easily represented in an analogous circuit (this will be discussed later). The acoustic waveguides consisting of the driver tube, splitter ports, receiver tubes, and inserts can each be modeled below the cutoff frequency, fc, using a wellknown acoustic impedance circuit. An acoustic impedance circuit for a finite-length one-dimensional waveguide with an arbitrary source S at one end and an arbitrary termination T at the other is shown in Fig. 2. The impedance circuit parameters in Fig. 2, labeled ^ ð0Þ and P ^ ð1Þ are the acousU1 and U2, are the volume velocities and P tic pressures at the source and the termination respectively. The impedances ZA1 and ZA2 are the acoustic impedances relating to the particular tube and are defined in Eqs. (1) and (2), respectively [4]
Z A1 ¼ j
qo ~c S
Z A2 ¼ j
tan
qo ~c S
~ kl 2
!
~ cscðklÞ
ð1Þ
ð2Þ
In Eqs. (1) and (2), S is the cross-sectional area of the tube, j is pffiffiffiffiffiffiffi 1, qo is the ambient fluid density, ~c is the lossy acoustic phase ~ is the lossy acoustic wave number defined as k ~ ¼ k ja p speed, k (where ap is the total thermoviscous absorption coefficient, which is a sum of the classical absorption coefficient and the absorption coefficient for wall losses), and l is the length of the tube. The lossy acoustic phase speed and the lossy acoustic wave number account for the losses in the tubes. This approach assumes that the fluid boundary layer thickness is much less than the tube radius. The calibrator uses a total of sixteen circular waveguides (one for the driver tube, five for the splitter, five for the receiver tubes, and five for the inserts) to channel the sound from the excitation source to the microphones. Each of these tube sections has a different diameter and length, but the circuit shown in Fig. 2 can be used to model all of the waveguides. The complexity of the analogous circuit model was reduced by combining all of the acoustic waveguide circuits (except for the driver tube) into a Thevenin equivalent impedance. This was accomplished in two steps. First, three circuits representing the splitter port, receiver tube, and insert were combined in series for each of the five paths. Second, the five path circuits were combined in parallel. The result was a single acoustic impedance value that represented all of the elements to the right of the driver tube.
Fig. 1. Calibrator path concept drawing.
Fig. 2. An acoustic impedance circuit for a finite-length one-dimensional waveguide with a source at one end and an arbitrary termination at the other.
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Each of the five calibrator paths is terminated by a microphone. In this model it was assumed that the impedance of a microphone was much larger than the impedance of the waveguides, and therefore the termination impedance of each path was assumed to be infinite in the analogous circuit. It should also be noted that effective acoustic lengths were added to each of the tubes in the five paths. The change in cross-sectional area between tubes creates a discontinuity inductance that can be interpreted physically as an increase in the length of the tube [5]. This change in length must be accounted for in order to obtain accurate predictions from the analogous circuit model. To further increase the accuracy of the calibrator model, the actual loudspeaker driver was modeled and included in the circuit. This produced a more accurate model of the calibrator than if the excitation source was assumed to be a constant volume velocity source. The driver model combines the acoustic, electrical, and mechanical elements of the driver by converting the electrical and mechanical components into the acoustic impedance domain. These elements are shown in Fig. 3, along with the rest of the analogous circuit model for the calibrator. All of the elements from different parts of the calibrator have been transformed into the acoustical impedance domain. In the circuit of Fig. 3, there are five acoustic elements, two electrical elements, and three mechanical elements. The acoustic elements Zrad and ZATh represent the radiation impedance of the back of the driver and the Thevenin acoustic impedance of all of the calibrator waveguides, respectively. The electrical circuit shows a source ð^eg Þ which is the complex voltage supplied to the loudspeaker terminals. The electrical circuit also contains an electrical impedance representation of the voice coil which is defined as [6]
Z E ¼ RE þ jxL1 þ
jxL2 R2 : R2 þ jxL2
ð3Þ
The mechanical circuit consists of three elements, the mechanical resistance of the suspension (RMS), the mechanical compliance of the suspension (CMS), and the actual moving mass of the driver diaphragm (MMD). The mechanical resistance of the suspension RMS and the moving mass of the driver diaphragm (MMD) are calculated from measured Thiele-Small parameters using Eq. (4) and Eq. (5), respectively
M MD
8 ¼ MMS qo a3 3
ð4Þ
1 2pF s C MS Q MS
ð5Þ
RMS ¼
In Eq. (4), a is the radius of the driver. In Eq. (5), Fs is the ‘‘free air” resonance frequency of the driver, and QMS is the quality factor of the driver at Fs considering mechanical losses of the moving system only [7].
2.2.1. Measured driver parameters The driver parameters used to model the driver were measured using the actual calibrator driver. The driver was suspended in an anechoic chamber. Using a maximum-length sequence system analyzer (MLSSA) with the speaker parameter option (SPO) the driver element values were measured. The MLSSA SPO determines the driver parameters by analyzing the complex impedance of the driver over the frequency range of interest. All of the complex impedance points are used in the analysis of the driver parameters. Driver impedance is complex and can be thought of as a curve in 3 D space where the x-axis is the real part of the impedance, the y-axis is the imaginary part and the z-axis is the frequency. The measured parameters are valid to use in the model because the MLSSA accounts for the real, imaginary, and frequency dependence of each driver parameter. The parameters were measured ten different times and the average of all ten runs for each parameter was used in the model. The added-mass method was used to measure the parameters. The first five parameter measurements were made using an added mass of 1 g and the second five were made using 2 g. The averaged results of all of the parameters used in the model are shown in Table 1. 2.2.2. Model results The model was evaluated numerically to predict the frequency response function (FRF) at the end of one of the paths. The results were plotted along with the measured FRF as measured at the end of one of the paths. These plots are shown in Fig. 4. As can be seen, the model accounts for every resonance up to the cutoff frequency of the first cross-mode. The model predicts all but one of the resonances to occur within 25 Hz of the corresponding measured resonance. The smallest difference was within 5 Hz. The largest error in a predicted resonance and the measured resonance occurred in the prediction of the resonance near 4.5 kHz, and was 70 Hz. The model still predicts several resonances to be higher than the actual measured resonances. This is caused by the path tube radii not
Table 1 Measured loudspeaker parameters used in circuit model. Parameter
Mean
Std dev.
Units
SD (effective driver diameter) Fs (free-air resonance) QMS (mechanical quality factor) MMD (mechanical mass) CMS (mechanical compliance) Bl (force factor) RE (voice-coil resistance) L1 (voice-coil inductance) L2 (voice-coil inductance) R2 (voice-coil resistance)
1.30E03 268.03 6.56 0.00214 0.000164 2.72 6.24 0.0000892 0.00013 7.44
N/A 1.066 0.0579 6.15E05 5.64E06 0.0416 0.00202 4.22E07 1.71E12 0.0198
m2 Hz
Fig. 3. The entire calibrator circuit model in the acoustic domain.
kg m/N Tesla-m X H H X
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Fig. 4. Frequency response function of the calibrator analogous circuit model and the actual calibrator.
being large compared to the fluid flow boundary thickness. If the losses are increased in the model these effects are reduced, but the model also loses accuracy. The model may also be improved by including the acoustic impedance of the microphones (instead of assuming infinite impedances), and by modeling the radiation impedance of the rear of the driver more accurately. The model was intended to predict the overall trend and number of resonances of the FRF; therefore, for this work the model proved to be sufficient. 3. Probe and calibrator hardware This section presents the hardware design of the microphone probe and the calibrator developed in this research. 3.1. The probe The probe used in this research to validate the calibrator is spherically shaped with four pressure transducers mounted in the sphere. The microphones are 6 mm, Primo EM123 electret microphones with an input impedance of 1.6 kX. The sphere is 50.8 mm in diameter. A hollow shaft is used to connect the sphere to the digital signal processor (DSP) housing. Wires pass through the shaft to connect the microphones to the DSP. The z-axis, or the axis which runs down the length of the shaft, is considered the natural axis of the sphere. One sensor microphone is mounted on the sphere opposite the shaft and shares the same z-axis as the shaft. This microphone is known as the pole microphone (see Fig. 5). The other three microphones are located 68.75off the z-axis as measured from the pole microphone. These three side microphones are spaced equally around the sphere at 120increments. All four microphones are mounted perpendicular to and flush with the surface of the sphere at their locations. In order for the probe to accurately calculate energy density, each microphone must be accurately calibrated. Since the probe is intended to measure between 20 Hz and 2.0 kHz it needs to be calibrated over this entire range. 3.2. The calibrator A photograph of the entire calibrator is shown in Fig. 6. The driver is used to excite the driver tube with tonal or broadband sound. Below the fc of the driver tube, only plane waves [8] exist at the splitter. As a result of the plane waves, the splitter evenly splits
Fig. 5. Acoustic energy density probe.
the volume velocity into five paths. The receiver tubes attach to the splitter, and guide the sound to the four probe microphones and the reference microphone. The left half of the calibrator clamp aligns the receiver tubes with probe microphones. The two halves of the calibrator clamp are connected with three latches. The calibrator surrounds the entire probe and provides a consistent seal at each microphone. The individual components of the calibrator are described in detail in the following sections. 3.2.1. The excitation source The loudspeaker driver used in the calibrator is a HiVi A2S fullrange loudspeaker. The driver response shown in Fig. 7 is relatively flat over all frequencies of interest in calibrating the probe. This characteristic makes the driver useful in a frequency band calibration application. 3.2.2. Driver tube The driver tube was designed to act like a waveguide and to separate the source from the splitter. Waveguides can be used to remove higher order modes that are induced by a source. The driver tube was chosen to be a circular waveguide with a radius of 20.0 cm. This radius was designed large enough to cover the diaphragm of the driver, and small enough to keep the fc above the
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Fig. 6. Photograph of the entire calibrator.
Deg.
dB spl 90
180
80
135
70
90
60
45
50
0
40
-45
30
-90
20
-135
10
-180
10
Freq
50
100
500
1k
Hz
5k
10k
40k
Fig. 7. HiVi A2S full frequency loudspeaker frequency response.
usable bandwidth of the probe. The resulting fc of the first higher order mode of the driver tube is 5.0 kHz. The length of the driver tube was calculated to be about 0.13 m. This length corresponded with a first higher order mode attenuation of 95 dB when the driver tube is driven at 2.0 kHz (the upper frequency limit of the probe). All other higher order modes would attenuate 95 dB in less spatial distance than the first one. Since only plane waves propagate in the driver tube, the pressure will be uniform at each port of the splitter. 3.2.3. Splitter The splitter was designed and built to divide the volume velocity into equal separate paths as shown in Fig. 1. The splitter is equipped with five holes (four for the probe microphones and one for a reference microphone) located symmetrically about its center as shown in Fig. 8. On the ‘‘front” side of the splitter, each hole is equipped with a port that is designed to connect to a receiver tube. The ‘‘back” side of the splitter is also shown in Fig. 8 and illustrates how the driver tube connects to the splitter. 3.2.4. Receiver tubes Flexible PVC tubes with an outside diameter of 11.9 mm were added to the end of each port of the splitter. The receiver tubes were chosen to have an inside diameter of 6.35 mm and are 23.5 cm in length. These tubes connect to the ports on the splitter at one end and the inserts of the calibrator clamp on the other. The receiver tubes were carefully constructed so that they were identi-
Fig. 8. Splitter (CAD model).
cal in length and diameter. The physical similarities are essential to ensure the acoustic path (and thus the acoustic impedance) is the same for each. The receiver tubes needed to be flexible to allow them to have the same length and yet reach all of the microphones as shown in Fig. 6. 3.2.5. Calibrator clamp The purpose of the calibrator clamp is to align the end of each receiver tube with a probe microphone. The calibrator clamp was designed using two halves with spherical cavities that come together and enclose the probe. The top half of the calibrator clamp
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is equipped with a groove that allows the probe shaft to pass through to its center. The bottom half of the calibrator clamp where the probe microphones are located, is equipped with four ports. Each port lines up with a microphone on the probe. Each port also has an insert that is fit with a rubber o-ring to seal the interface between the probe and the bottom half of the calibrator clamp. The inserts connect to the receiver tubes and are held in place by a bolt that has a hole in the center to allow the insert to pass through it. The bolts screw into threaded holes in the bottom half of the calibrator clamp. A CAD model assembly drawing of the calibrator clamp in relation to the probe is shown in Fig. 9. 4. Experimental results The effectiveness of the calibrator was validated in two stages. The first stage validated that the receiver tubes are acoustically the same. In the second stage, the full calibrator was validated. Both stages were validated at several discrete frequencies. The entire calibrator was then validated over a continuous range of frequencies. The results of these validation steps are presented in the following sections. 4.1. Receiver tube validation The receiver tubes were tested similar to the splitter port test described above. All five tubes were connected to the splitter and one tube was tested at a time. The same microphone was used to test all the tubes and all of the tubes except the one being tested were terminated with a microphone and adapter as if they had been mounted in the calibrator. Again, 20 data sets were initially used to compute a mean but it was noted that no change occurred after using only 5 data sets. Testing was done at frequencies of 250 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1.0 kHz, 1.5 kHz, and 2.0 kHz. The 2.0 kHz upper limit was chosen based on the operational limit of the probe. At each of the above frequencies, the pressure was measured at the end of each receiver tube, and the largest difference between any of the five receiver tubes was calculated. This difference was used to calculate the error at each of the frequencies listed above. The error was plotted at each of the measured frequencies. The results of this test are shown in Fig. 10. The goal was to limit the error to less than
Fig. 10. Receiver tube validation results.
±0.5 dB (a class 1 probe), and as can be seen from Fig. 10, this was accomplished at these discrete frequencies. In order to reach each microphone on the probe, the receiver tubes had to be bent slightly. The error introduced as a result of bending the tubes was determined by measuring the pressure at the end of each tube with the tube straight and then bent at 30°, 60°, and 90°. The pressure was measured at the end of each tube in each position five different times. A statistical test was used to determine if bending the tubes had any effect on the measured pressure. The null hypothesis was that the bending of the tubes did not cause any change in the acoustical path of the receiver tubes. A single factor analysis of the variance was performed on the five runs at each orientation with a = 0.05. The analysis produced a P-value of 0.57. Since the P-value is greater than the a value of 0.05 the null hypothesis was not rejected [9]. With a = 0.05 and a P-value that is larger, the probability that the variance in the data was just noise is 95%. This analysis provided confidence that the receiver tubes could be bent to align with the probe and still remain acoustically the same.
Fig. 9. Calibrator clamp (CAD model).
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Fig. 13. FRF of the five receiver tubes.
Fig. 11. Relative phase calibration results between the five calibrator tubes.
The phase difference in the tubes was also measured using a two microphone calibration technique. The tubes were terminated in the calibrator box and phase matched precision microphones
were used to acquire the relative phase between the five tubes in the calibrator. The results for all five tubes are shown in Fig. 11. It should be noted that the phase results are relatively flat over the majority of the frequency range. These curves could be used to calibrate the phase of one probe microphone relative to another.
Fig. 12. Calibration error at various frequencies and 114 dB.
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4.2. Full calibrator validation This full calibrator validation test was performed to show the overall uncertainty bounds in the entire calibrator, to show the insensitivity to the probe orientation in the calibrator (i.e. that all paths in the probe are similar), and to show repeatability of the measurements. The calibrator driver was excited with a 114-dB (as measured with the 6 mm, GRAS, reference microphone) harmonic wave at 250 Hz, 500 Hz, 1.0 kHz, 2.0 kHz, and 3.0 kHz. The probe was inserted into the calibrator housing and a one second time signal
797
was recorded by each microphone on the probe and the reference microphone. The probe was taken out of the calibrator housing and rotated so that the three side microphones on the probe were each connected to a different receiver tube. The pole microphone and the reference microphone remained connected to the same receiver tubes. Another time signal was recorded and the rotation was repeated. The data for each of the three measurement configurations were normalized about the first position. The error at the second and third positions relative to the first position was calculated. This entire process was repeated for the five frequencies mentioned above. It should be noted the physical limitations of
Fig. 14. The error in receiver tube 1 with a confidence interval.
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the driver reduced the amplitude of the harmonic wave at 2 kHz and 3 kHz to 110 dB. The error plots for each of the five frequencies can be seen in Fig. 12. The error at 250 Hz, 500 Hz, 1.0 kHz, and 2.0 kHz were all less than the 0.5 dB. The error at 3.0 kHz was slightly greater than 0.5 dB. The results indicate that the calibrator is capable of calibrating the probe at 250 Hz, 500 Hz, 1.0 kHz, and 2.0 kHz to the class 1 probe standard. The results at 3.0 kHz indicate that the calibrator is capable of calibrating the probe to the class 2 probe standards at 3.0 kHz. The calibrator was further tested using white noise as the signal input. These measurements were made with the probe in the calibrator just as it would be during calibration. The FRF was calculated using 20 magnitude averages. This number of averages corresponded with a coherence greater than 0.99. The FRF was measured at the end of each receiver tube five times and the average of each receiver tube was calculated. The FRF of each of the receiver tubes is shown in Fig. 13. The FRF results from each receiver tube were averaged to obtain an overall mean. To calculate the error at each receiver tube, the overall mean was subtracted from each of the five FRF measurements. The resulting errors at each receiver tube were then averaged to obtain an average receiver tube error (l). The standard deviation of each of the five error calculations for each receiver tube was calculated (r). Using the mean and standard deviation of the error, an interval at each receiver tube was calculated using Eq. (6)
Interval ¼ l 2r
ð6Þ
The interval corresponds to a confidence of 95.45% that the error will fall in the limits of the interval. The interval for each receiver tube was plotted along with the mean error l and the upper and lower control limits for a class 1 probe and a class 2 probe. These plots are shown in Fig. 14. In Fig. 14 the upper and lower limits of the confidence interval are illustrated using the down pointing and up pointing triangle, respectively. The upper and lower control
limits for a class 1 probe, and a class 2 probe are illustrated by a dashed and dotted lines respectively. 5. Conclusions It has been shown that the calibrator presented in this paper can accurately calibrate four microphones within ±0.5 dB up to 2.0 kHz, and within ±1 dB up to 4.9 kHz. The calibrator is designed for a specific probe, but the concept could easily be applied to use on other probes or arrays. The concept could also be extended to work with higher frequencies by decreasing the size of the driver tube which would increase the cutoff frequency of the first higher order mode and would increase the bandwidth of the calibrator. With a higher cutoff frequency, the driver tube could limit the propagation to only plane waves. As long as only plane waves are present at the splitter the calibrator error will be minimal. Using this calibrator to calibrate this probe would result in a class I probe up to 2.0 kHz and a class II probe up to 5.0 kHz. References [1] Harris Cyril M. Handbook of acoustical measurements and noise control. New York: McGraw-Hill; 1991. [2] Pierce AD. Acoustics: an introduction to its physical principles and applications. New York: Acoustical Society of America; 1989. [3] Beranek LL. Acoustics. New York: (Acoustical Society of America; 1996. [4] Leishman TW. Active control of sound transmission through partitions composed of discretely controlled modules (Ph.D. dissertation. PA: The Pennsylvania State University, University Park; 2000). [5] Karal FC. The analogous acoustical impedance for discontinuities and constrictions of circular cross section. J Acoust Soc Am 1953;25:327–34. [6] Rife DD. MLSSA SPO speaker parameter option. Reference manual version 4WI Rev. 8. DRA Laboratories; 2005. [7] Leishman TW. Physics 562 class notes: applied acoustics. Brigham Young University, Dept. of Physics and Astronomy; 2005. [8] Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of acoustics. 4th ed. New York: Wiley; 2000. [9] Lawson J, Erjavec J. Modern statistics for engineering and quality improvement. California: Duxbury; 2001.