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Page 517 ï~~Forward-Going Wave Extraction in Acoustic Tubes Tim Stilson Center for Computer Research in Music and Acoustics (CCRMA) Stanford University stilti~ccrma.stanford.edu ABSTRACT: A simple estimator is developed to extract the forward-going traveling pressure wave from physical measurements of pressure in a tube, which consist of the sum of the forward-going wave and a reverse-going wave. The estimator is then analyzed and limits of performance are discussed. Finally, a few extensions are discussed and analyzed which overcome some limitations. 1 Introduction The solution to the wave equation in simple acoustic waveguides such as the cylindrical tube prescribes the superposition of two waves traveling in opposite directions down the waveguide (Morse). Any physical measurement of the waves in the tube will always return the sum of the two waves. Sometimes, it is desired to be able to extract just one of these waves from the measurements. For example, to measure bell reflection functions. The two-sensor method is a well known technique for handling two-way propagation in tubes, and has been explored deeply (Spiekermann et al.), (Gibiat et al.) and (Abom et al), but always in directly working the problem of measuring reflection functions and other properties of the tube. In fact these methods never seem to actually estimate the forward-going wave (or the backwardgoing wave), instead they directly calculate higher level functions. In this paper, we will derive estimators for the forward-going wave using the two-sensor method. We will assume a tube oriented horizontally, with the two (pressure) waves labeled 'R(x - ct)' and 'L(x + ct)', (for 'Right'- and 'Left'-going), where x is position in the tube, and c is the wave velocity. The absense of DC flow (and turbulence) is assumed, along with negligible attenuation between sensors. 2 The Two-Sensor Estimator If we at first assume perfect noiseless measurements, and let the space- and time-separation of the sensors be: Ax = x2- x1, At = Â~. We use the fact that the right and left going waves propagate to get the following relations between the measurements and waves at the sensor locations: si (t) = R(xi - ct) + L(xi + ct) R(x - vt) *L(x + Vt) = -- v) ) + + +vt S2 (t) = R(x2 - ct) + L(x2 + Ct) R(x2 - c(t + At)) = R(xi - ct) S1,&4 L(x2 + c(t - At)) = L(xi + ct) At 24t A2t) We solve this set of equations to get: 2At R(x2 - Vt) = R(X2 - v(t - 2At)) + si(t - At) - s2(t - 2At) Figure 1: Two-Sensor EstiL(x1 + vt) = L(xl + v(t- 2At)) + s2(t- At) - s (t - 2At) mator [CMC PROCEEDINGS 199551 517
Page 518 ï~~[R() -j21n f~t [ 1 e2 7fAt irS(s)1 L (s)- 1- ej4"fAt --e-j2rfAt 1 s2(s)2 J This matrix equation is quite useful for analysis of the system, and extends well to more complicated systems, such as those with more sensors. This generalizes well to discrete-time systems, simply by substituting z = e e. Note, however, that the delays in the estimator become integer-length, thus restricting the choice of sensor spacings ax to f-c 2.1 Effects of Noise Noise is inevitable in the measurements si and s2, so the sensor equation is rewritten as: [i ()]r 1 e-i2lr f~t 1[R2 (s) 1 z (S)] s2(s) j e-j2irfot 1 J1[Li(s) + 2(s) If we assume that the noises are zero-mean, then the same estimation as before should work, but let's see how it is affected by the noise: if we plug the above equation for Si and s2 into the estimator and assume the noises are independent, we get the following estimator performance equation: 102 R2(s) R= +2(S) 1 [113(8) K Where v3 and v4 are simply another set of noises obtained from the addition of vl and v2. We see that the noise gain has poles at DC and at multiples of (DC and t in discrete-time, using one-sample 100 _ sensor spacing). Thus, although the transfer function from R1 to R 0 o2 is flat, the estimation SNR could get rather low at the pole locations, Figure 2: Condition Number thus making the estimation quite inaccurate, for D TS two-sensor system This can also be seen by looking at the condition number (s) of the sensor matrix vs. frequency (Figure 2). One interpretation of is the "effective-singularity" of the matrix ( -+ 0c = Singular). Since the condition number approaches oc at DC and L, we can say that those frequencies, the sensor matrix is essntially singular, so the individual waves are not estimable. We can use the condition number as a quick check to see if a given sensor configuration is good for estimating the travelling waves. FIR Estimators: If the noise gain at DC and 2 is too disturbing, one may try an FIR estimator, simply by removing the recursion from the estimator, which removes the poles in the transfer functions (we also removed the extra delay, which became unnecessary). This gives an estimator whose performance is: 1(s)1 - L )4() Technically, the SNR is the same as before (the zeroes reduce the signal the same amount as the noise was boosted before), but in some cases this may be preferred (ease of analysis, for example). The zero at Â~L: These zeroes (at 4 and multiples) come from the fact that these wavelengths fall unobservably between the sensors, and the subtraction in the estimator (ft1 (k) = Sj (k) - s2(k - 1) in DTS) causes these frequencies to cancel at the sensors. One possible solution to this problem is to use multiple sensors. The idea being that the zeroes of the various pairs of sensors would not all 518 8ICM C PROCEEDINGS 1995
Page 519 ï~~land at the same frequencies, so that the estimation from one pair could be used to "fill-in" where another is at a zero. Unfortunately, in discrete-time all the possible sensor separations are n cT8, giving zeros at multiples of L for each sensor pair. Thus all sensor spacings will have a zero at - 22" Oversampling, however, can be used to get a sensor separation that doesn't have a zero at the original 4. For example, we could double the sampling rate and place the sensors at xk, xk+2, and xk+3 according to the new sampling rate. The first two sensors are separated by one sample in the original sampling rate, and the second pair are spaced by one half-sample in the original sampling rate, so their first zero is effectively at f or.i2 These sensors could be combined so that the zero at f or /2 is "filled-in", and then the estimate resampled back to f s8.,o. The condition number of this method's sensor matrix is shown in Figure 3 for the frequency range of interest corresponding to the larger sensor spacing, which for this derivation is one sample at fso.,o. Thus the zero at f8oi, /2 has disappeared (or, for a nonFIR estimator, this means that the noise gain at that frequency has become no longer a problem). S Sl vao8 2 S3 102 0 100 0 fs.aig 2 Figure 3: K for oversampled three-sensor system The zero at DC: No combination of sensors can fill-in the zero at DC. In fact, it is difficult to define the right-going wave as being distinct from the left-going wave at DC (in any finite-length tube), which reinforces the implication that DC (and near-DC) is unobservable. All estimation transfer functions will thus fall to zero at DC. The important question is how close to DC the estimator can get without breaking down. 4-sample 1-sample There are a few ways to approach this question. We can extend the spacing spacing multiple-sensor system and add sensor pairs that have less-severe 2 attenuation at low frequencies. For example, a pair with a very wide spacing will have quite a few zeros in the range of interest, f / but will have regions of little attenuation much closer to DC than Figure 4: Filling-in near DC the 1-sample sensor spacing (see Figure 4). Multiple such sensor-pair estimates can be combined to get a better response close to DC. Note that a similar result can be had near DC by subsampling a closely-spaced pair of sensors. This can also be viewed as an FIR filter design problem: one of choosing the weightings of a regularly-spaced set of sensors to get a desired estimation frequency response, under the added condition that the transfer function from the unwanted wave be attenuated as far as possible. 2.2 Effects of Sensor Misplacement All of the preceding derivations assumed that the estimator could implement delays precisely equal to the wave propagation delays between the sensors. In the discrete-time estimator case, this may be particularly difficult, since the time delays are fixed, so the sensor placement must be rather precise. Here, we analyze the effects of sensor misplacement. Say the sensor spacing is Ax = vAtl and the delay used in a twosensor FIR estimator is At2, let the delays be unequal, but quite similar (At2 - At1 = E << At1, At2). If we look at RFIR2,u.(t), we see that L(xi + vt) is no longer fully canceled, we get: RFIR2mte (t) = RFR2(t) + [L(xi + vt) - L(xi + vt - ye)] - RFIR2(f) + 2e-21rfE/2 sin(2irf E)Lx, (f) R(x-v) L(x+vt) 2 weightin weighting 0.1%: -44dB 1%: -24dB 10%: -4.2dB Figure 5: Gain of L at 4L vs. sensor placement error (-) sensr paceenterrr Dig I CMC PROCEEDINGS 199551 519
Page 520 ï~~Note: if it were possible to fine-tune the sampling rate, then misplacement would not be a problem, since the sampling rate could be chosen to get delays that match the wave-propagation delays. 2.3 Effects of Sensor Spread Similar to the sensor misplacement problem is the fact that most pressure sensors do not pick up just the pressure at a single point. Instead, they have some area over which they pick up pressure, effectively giving the 'average' of the pressure over the area. It is possible that this averaging could cause an effect similar to misplacement in the estimator. If we model the spreading as a space convolution along the length of the tube, we get s1 (t) = R(xi - vt) *Sp (x) + L(xi + Vt) *Sp2 (x), and a similar function for s2(t). Luckily, if we assume that the sensors have the same spread functions, and that these spreads are symmetric about the pickup point, (both of which are fair assumptions), the linearity of convolution gives the following result for a two-sensor FIR estimator (After we note that in the case of traveling wave measurements, space convolution is equivalent to time convolution): R2FIRs.p(t) = R2FIa(t)* vSp(vt). This gives a simple filtering of the signal, which, given the typical size of the spread function, only affects the highest of frequencies. Thus we find that sensor spread doesn't become a problem for identical, symmetric-pattern sensors. 3 Conclusions The sensor matrix can be inverted to give a simple forward-wave estimator, but it breaks down at DC and 2 (L' in the simple discrete-time setup). Multiple sensors (together with oversampling for discrete-time estimators) can reduce the problem at the higher frequency, and can push the break-down point closer to DC. The condition-number of the sensor matrix can be used to gauge the effectiveness of a sensor configuration. If the sensors are incorrectly placed, or if the estimator implements incorrect delays, the unwanted wave will leak into the estimator output, at a gain of approximately 20 dB per order-of-magnitude placement error. Finally, sensor spread produces no leakage if the sensors have identical spread functions and these functions are symmetric. References [Abom et al.] Mats Abom and Hans Boden. Error analysis of two-microphone measurements in ducts with flow. Journal of the Acoustical Society of America (June 1988) vol.83, no.6, p. 2429-38. [Gibiat et al.] V. Gibiat and F. Laloe. Acoustical impedance measurements by the two-microphonethree-calibration (TMTC) method. Journal of the Acoustical Society of America (Dec. 1990) vol.88, no.6, p. 2533-45. [Morse] Philip M. Morse Vibration and Sound Acoustical Society of America, 1981 [Spiekermann et al.] Charles E. Spiekermann and Clark J. Radcliffe. Stripping one-dimensional acoustic pressure response into propagating- and standing-wave components. Journal of the Acoustical Society of America (Oct. 1988) vol.84, no.4, p. 1542-8. This material is based upon work supported under a National Science Foundation Graduate Fellowship. 520 0ICMC PROCEEDINGS 1995