A Comparison of Impedance Measurements Using One and Two Microphones

Measurements of acoustic input impedance of wind instruments using two different approaches are presented. In the first approach, commonly referred to as the two-microphone transfer function method, a tube is connected to the instrument and excited with broad-band noise. Signals recorded at microphone pairs placed along the tube are then analyzed to estimate the instrument input impedance. A calibration step is described, wherein the position of each microphone pair is determined from the measurement of a rigid termination. The second technique, a novel variant of pulse reflectometry, makes use of a long tube with a single microphone located at its midpoint. Using a long-duration broad-band stimulus, the impulse response is measured for the tube, first with a rigid termination, and then with the system to be characterized attached. The system reflectance, and therefore its impedance, is found by comparing the first reflection from the tube end for both measurements. The design of the impedance probes and the data sampling and analysis procedures are presented. Measurements obtained using the two techniques are compared for various acoustic systems, including an alto saxophone neck and fabricated conical objects. The results show good agreement between the methods. Advantages of the one-microphone technique include ease of use and robustness to noise, while the two-microphone approach can provide a better high-frequency response for long objects. INTRODUCTION The measurement of acoustic impedance has been the subject of much research since the beginning of the last century and a great number of publications have been written on the subject. Benade and Ibis (1987) and Dalmont (2001) provide a good background on the historical origins and development of these techniques. Since the 1980s, two measurement techniques have become widely used: the two-microphone transfer function (TMTF) technique and pulse reflectometry. The use of two microphones located along an acoustic transmission line to evaluate the impedance of an object dates back to the early 19th century (see Beranek, 1988). The twomicrophone transfer function technique introduced by Seybert and Ross (1977) made use of a broad-band source signal and Fourier analysis to evaluate the impedance over the entire spectrum in one measurement. It has also been described by Chung and Blaser (1980a,b). Pulse reflectometry originated from geophysical studies of the earth’s crust but, throughout the 1970s and 1980s, it was applied to the study of the vocal tract (see Fredberg et al., 1980) and to musical instruments. The novel approach reported here is based on the same principle as pulse reflectometry but achieves an improved signal-to-noise ratio (SNR) by using wide-band signals of ISMA 2007 A Comparison of Impedance Measurements long duration, such as swept sines. For the purposes of this paper, we shall refer to this technique as “impulse reflectometry” (IR). The objective of this paper is to compare impedance measurements obtained with both techniques in order to identify and characterize possible discrepancies between the two, as well as to better assess the accuracy of the results and the importance of measurement errors. In the context of musical acoustics, we are mainly interested in the magnitudes and frequencies of the maxima and minima of strongly resonant bodies. We first detail the experimental setup, calibration procedures, and signal analysis methods for both techniques. We then present impedance measurement results for three objects: an alto saxophone neck, a short carbon fiber cone, and a long carbon fiber cone coupled with the neck. We conclude with a comparison of the advantages and disadvantages of both techniques. THE TWO-MICROPHONE TRANSFER FUNCTION TECHNIQUE x = 0 x2 Source P+(x, f ) P−(x, f ) Zin x1 s Impedance measurement tube Measured object Microphone 2 Microphone 1 Figure 1. Diagram of the two-microphone measurement apparatus. In the two-microphone transfer function technique, the impedance of an object is evaluated from the measurement of the transfer function between two microphones located at different positions along a waveguide connected to that object. A horn driver emits a broad-band signal, such as white noise, in the waveguide over a time duration adequate to reduce variance in the results, as computed with a modified average periodogram. This technique is based on the mathematical theory of one-dimensional planar pressure wave propagation in a cylindrical duct. Such waves, including attenuation, can be described by the equation P (x, f) = P+(x, f) + P−(x, f) = Ae −Γx +Be, (1) where A and B are the complex frequency-dependent amplitudes of the progressive and regressive traveling-wave components. The propagation parameter is defined as Γ = α + iω/vφ, where α is the attenuation and vφ the phase velocity. Estimation of this parameter has been described by Pierce (1989). It can be approximated by Γ = iω/c+ (1 + i)α, where α ∝ √f by a constant that depends on air properties. From these equations, it can be shown (Lefebvre, 2006) that the impedance Z̄in of an object located at x = 0 (see Fig. 1) is given by Z̄in = Z Zc = H12 sinh(Γx1)− sinh(Γx2) H12 cosh(Γx1)− cosh(Γx2) , (2) where H12 is the transfer function between the two microphones and Zc is the characteristic impedance. This approach is based on one-dimensional wave propagation and thus, it is limited in frequency to the first higher-order mode that occurs at f = 1.84c/(2πr), where r is the cylinder radius and c is the speed of sound. For our measurement system, the cutoff frequency is approximately 16.5 kHz (r = 0.006 meters). The TMTF technique is also incapable of providing results ISMA 2007 A Comparison of Impedance Measurements Microphone Pair Distance Frequency Range (Hz) 1 and 2 3 cm 575 4600 1 and 3 12 cm 290 1150 1 and 4 36 cm 95 380 Table 1. microphone pairs use in our measurement apparatus at critical frequencies where the two pressure signals become linearly dependent, which equates to half-wavelengths that are an integer multiple of the microphone spacing: fc = mc/2s,m = 1, 2, ..., N. (3) The consequence is that we need several pairs of microphones to cover a sufficient frequency range for musical instrument characterization. To achieve a frequency range of 100 – 5000 Hz, we use four microphones. Table 1 indicates the microphone distances and valid frequency ranges. Final impedance results are realized by concatenating impedances from three microphone pairs. Prior to the measurement, a relative calibration of microphones pairs is performed, as described by Seybert and Ross (1977) and Krishnappa (1981), in order to eliminate frequency response differences between them. This calibration is made using a special apparatus such that the four microphones are located at the same reference plane and exposed to a broadband noise signal. The microphone positions used in Eq. (2) can also be fine-tuned with a measurement obtained when the plane at x = 0 is rigidly terminated. For this condition, the transfer function between two microphones is given by (see Lefebvre, 2006): H12 = cosh(Γx2) cosh(Γx1) . (4) The attenuation parameter, α, which will be higher than predicted if the tube inner surface is not completely smooth, can also be calibrated from the magnitudes of these maxima and minima. We evaluate the transfer function H12 between the recorded signals at the two microphones with the total least square formulation, which reduces the impact of noise (see P.R. White, 2006): H12 = C12 · Sp2p2 − Sp1p1 + √ (Sp1p1 − Sp2p2) + 4|Sp1p2 |2 2Sp2p1 (5) where Sp1p1 is the auto-correlated spectral density of the first microphone signal, Sp1p2 is the cross-correlated spectral density between microphones 1 and 2, etc. C12 is the calibration function previously measured. IMPULSE REFLECTOMETRY x = 0 L2 L1 Adaptor Object Microphone Source Figure 2. Setup for the one-microphone measurement system. The impulse reflectometry (IR) technique uses a setup with a single microphone and a calculation based on two measurements. The apparatus consists of a horn driver connected to a long probe tube and a microphone located near its midpoint, as illustrated in Fig. 2. After ISMA 2007 A Comparison of Impedance Measurements performing a measurement with the probe tube rigidly terminated, the object to be measured is attached to the end of the probe and another measurement is made. In contrast to traditional pulse reflectometry techniques, a long duration source signal, such as a swept sine, is used to make the measurements. The benefit is that a lot of energy can be supplied to the system, increasing the measurement signal-to-noise ratio. In the measurements reported here, a logarithmically swept sine was used. The impulse response of the system fitted alternately with a rigid termination and with the object of interest is measured by deconvolving the recorded signal, y(t), from the input or source signal x(t):