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Noninvasive Model Independent Noise Control with Adaptive Feedback Cancellation

Noninvasive Model Independent Noise Control with Adaptive Feedback Cancellation Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 863603, 7 pages doi:10.1155/2008/863603 Research Article Noninvasive Model Independent Noise Control with Adaptive Feedback Cancellation Jing Yuan Department of Mechanical Engineering, Faculty of Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Correspondence should be addressed to Jing Yuan, mmjyuan@polyu.edu.hk Received 12 November 2007; Accepted 2 February 2008 Recommended by Marek Pawelczyk An active noise control (ANC) system is model dependent/independent if its controller transfer function is dependent/independent on initial estimates of path models in a sound field. Since parameters of path models in a sound field will change when boundary conditions of the sound field change, model-independent ANC systems (MIANC) are able to tolerate variations of boundary conditions in sound fields and more reliable than model-dependent counterparts. A possible way to implement MIANC systems is online path modeling. Many such systems require invasive probing signals (persistent excitations) to obtain accurate estimates of path models. In this study, a noninvasive MIANC system is proposed. It uses online path estimates to cancel feedback, recover reference signal, and optimize a stable controller in the minimum H norm sense, without any forms of persistent excitations. Theoretical analysis and experimental results are presented to demonstrate the stable control performance of the proposed system. Copyright © 2008 Jing Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION changes to path transfer functions P(z)and S(z). Since a model-dependent ANC system only remembers initial path estimates P(z)and S(z), variation of P(z)and S(z)may Most active noise control (ANC) systems are model depen- cause mismatch with initial estimates P(z)and S(z)tode- dent. Let P(z)and S(z) denote estimates of primary and sec- gradeANC performance. In casesofseveremismatchbe- ondary path transfer functions P(z)and S(z). Either S(z) tween path transfer functions and their initial estimates, a or both P(z)and S(z) must be obtained by initial system model-dependent ANC system may generate constructive in- identification for model-dependent ANC systems. Controller stead of destructive interference, or even become unstable. transfer function C(z) of a model-dependent ANC system is Model-independent ANC (MIANC) systems depend on either designed by minimizing P(z)+S(z)C(z),oradapted online path modeling or invariant properties of sound fields with the aid of S(z)[1, 2]. If estimates P(z)and S(z)contain to update or design controllers [6–8]. These systems avoid too much error, a model-dependent ANC system may gen- initial path modeling and are adaptive to variations of en- erate constructive instead of destructive interference. This is vironmental or boundary conditions of sound fields. Many mathematically equivalent to P(z)+ S(z)C(z) > P(z) adaptive MIANC systems require invasive persistent excita- even if P(z)+ S(z)C(z) is minimized. If phase error in tions to obtain accurate path estimates and ensure closed- S(z) exceeds 90 in some frequency, an ANC system adapted loop stability [6, 7, 9, 10]. Noninvasive MIANC systems are by the filtered-X least mean square (FXLMS) algorithm may able to ensure closed-loop stability without persistent ex- become unstable [3–5]. An operator of a model-dependent citations, which are possible by a recently developed algo- ANC system must have the knowledge and skill to obtain ac- rithm, known as orthogonal adaptation [11, 12], if the pri- curate estimates of path models by initial system identifica- mary noise signal is directly available as the reference signal. tion for each individual application. In many real applications, the primary noise signal is not During the operation of an ANC system, changes of en- necessarily available and the reference signal must be recov- vironmental or boundary conditions may cause significant ered from the sound field [1, 2]. When an ANC system is 2 Advances in Acoustics and Vibration 1 p p u Z Z p s cos(kd) jZ sin(kd) p p p 1 2 3 sin(kd) d cos(kd) MIANC system (a) Figure 1: Configuration of the proposed MIANC system. p p 1 2 cos(kd) jZ sin(kd) sin(kd) cos(kd) active, a measured signal is a linear combination of primary and secondary signals. Feedback of ANC signal in the mea- surement is mathematically modeled by a feedback transfer (b) function F (z) from the controller to the reference sensor. Ac- curate estimation of F (z)isasimportant as accurate estima- 1 p u Z p p tion of P(z)or S(z)[9, 13]. A complete noninvasive MIANC cos(kd) jZ sin(kd) (CNMIANC) system must be able to suppress the noise sig- sin(kd) Z cos(kd) nal without injecting probing signals for online modeling of j P(z), S(z), and F (z). Most available methods for adaptive feedback cancellation require persistent excitations [9, 13]. (c) In this study, a new method is presented for adaptive feed- Figure 2: (a) Acoustical two-port circuit in the duct system, (b) back cancellation without persistent excitations. contribution by controller, and (c) contribution by primary source. It was proposed to use a pair of sensors to measure pres- sure signals in ducts, from which traveling waves are re- solved [14, 15]. The outbound wave could be used directly as the reference signal without cancelling feedback signals if tion of p , is equivalent to an acoustical source with strength an infinite-impulse-response (IIR) controller could be im- u and impedance Z . The downstream part, from location p p plemented accurately [14, 15]. In reality, it is very difficult to of p to the outlet, is represented by another acoustical source implement a stable ideal IIR ANC controller [16]. Most prac- with strength u and impedance Z .Characteristicimpedance s s tical ANC systems use finite-impulse-response (FIR) con- of the duct is represented by Z . trollers. The outbound wave in a duct is a linear combina- The linear system theory allows one to solve p and p 1 2 tion of primary noise and reflected version of feedback sig- in Figure 2(a) by focusing on acoustical circuits of Figures nal. Instead of using the outbound wave directly as the ref- 2(b) and 2(c) before adding two solutions together as the fi- erence, the least mean square (LMS) algorithm is applied in nal solution of Figure 2(a). For the case of u = 0, which is this study to cancel feedback signals in the outbound wave represented by Figure 2(b), one obtains before using it as the reference. Orthogonal adaptation is combined with the proposed ANC configuration to imple- Z p | = cos(kd)+ j sin(kd) p | , 2 1 u =0 u =0 p p ment a CNMIANC system. Experimental result is presented (1) to demonstrate the performance of the CNMIANC system. Z Z s s u − p | = cos(kd)+ j sin(kd) p | , s 2 1 u =0 u =0 p p Z Z p o 2. SYSTEM CONFIGURATION AND MODEL where k is the wave number. One can solve, from (1), Figure 1 illustrates the configuration of the proposed ANC system. The primary source is represented by the upstream Z Z cos(kd)+ jZ sin(kd) o p o speaker and the secondary source is the midstream speaker. p | =     u , 2 s u =0 p 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p Cross-sectional area of the duct is small enough such that (2) sound field in the duct can be modeled by a 1D sound field in the frequency range of interest. Three microphone sensors Z Z o p p | = u . are installed in the duct, measuring signals p , p ,and p , 1 s 1 2 3 u =0 p 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p respectively. Since the primary noise signal is not available (3) to the ANC system, the reference signal is recovered from p and p , while p is the error signal to be minimized by the 2 3 Similarly, for the case of u = 0, which is represented by ANC system. Figure 2(c), one obtains Let d denote the axial distance between p and p .The 1 2 acoustical two-port theory [16, 17] has been applied by many ANC researchers for the design and analysis of ANC systems. p | = cos(kd)+ j sin(kd) p | , 1 2 u =0 u =0 s s It is adopted here as an analytical tool. An equivalent acousti- (4) cal circuit is shown in Figure 2 to model the two-microphone Z Z p p u − p | = cos(kd)+ j sin(kd) p | , p 1 2 u =0 system. The upstream part, from the primary source to loca- us=0 s Z Z s o Jing Yuan 3 from which one can solve In a digital implementation of ANC system, it is recom- mended to select sampling interval δt such that its product Z Z cos(kd)+ jZ sin(kd) o s o p | = u , with sound speed c satisfies cδt = d. As a result, the delay op- 1 p u =0 s 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p o −1 erator exp(−jkd) = z becomes an exact one-sample delay (5) for discrete-time ANC systems. Z Z o s p | = u . 2 p u =0 s 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) 3. FEEDBACK CANCELLATION s p o s p (6) It is indicated by (12) that the outbound wave contains feed- Adding (2)and (6), one may write back from u that must be cancelled to recover the reference signal. Let R = (Z − Z )/(Z + Z ) denote the upstream re- 1 p o p o p = p | + p | 2 2 u =0 2 u =0 p s −jkd flection coefficient. By multiplying e R to (11), one ob- (7) tains Z Z cos(kd)+ jZ sin(kd) u + Z Z u o p o s o s p = . Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p Z − Z Z − Z Z − Z s o p o p o −jkd −jkd e R w = α   e u + u . 1 i p s The same method is applicable to (3)and (5)for 2 Z + Z 2 p o (15) p = p | + p | 1 1 u =0 1 u =0 p s A subtraction of (15)from(12)enables onetowrite (8) Z Z cos(kd)+ jZ sin(kd) u + Z Z u o s o p o p s = . −jkd Z + Z Z cos(kd)+ j Z Z + Z sin(kd) w − e R w = n, (16) s p o s p o 1 i The next step is to use complex factor α = Z /((Z + o s where Z )Z cos(kd)+ j (Z Z + Z )sin(kd)) to simplify (7)and (8). p o s p jkd −jkd α Z + Z Z + Z e − Z − Z Z − Z e The results read s o p o s o p o n= u 2 Z + Z p o p = α Z cos(kd)+ jZ sin(kd) u + Z u , 2 p o s s p (17) (9) p = α Z cos(kd)+ jZ sin(kd) u + Z u . 1 s o p p s is only contributed by the primary source u . jkd −jkd jkd jkd −jkd Since cos(kd) = 0.5(e + e )and j sin(kd) = 0.5(e − Using cos(kd) = 0.5(e + e )and j sin(kd) = −jkd jkd −jkd e ), (9)can be writtenas 0.5(e − e ), one can see that the common denomina- tor of p , p , and all transfer functions in the duct is Z + Z Z − Z 1 2 p o p o jkd −jkd p = α e + e u 2 s 2 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p Z + Z Z − Z s o s o + + u , p jkd −jkd = 0.5 Z +Z Z +Z e −0.5 Z −Z Z −Z e . 2 2 s o p o s o p o (10) (18) Z + Z Z − Z s o s o jkd −jkd p = α e + e u 1 p 2 2 Substituting (18) into the definition of α (immediately after (8)), one obtains Z + Z Z − Z p o p o + + u . 2 2 jkd −jkd 2Z =α Z +Z Z +Z e − Z −Z Z −Z e . o s o p o s o p o Let (19) Z + Z Z − Z p o s o jkd w = α u + e u (11) i p s A further substitution of (19) into (17)leads to 2 2 Z u o p Z − Z Z + Z p o s o jkd n = . (20) w = α u + e u (12) o s p Z + Z p o 2 2 represent the in- and outbound waves in the duct. By com- This is the reference signal to be recovered by the proposed paring (10)with(11)and (12), one can see that (10)are ANC system. equivalent to A question to be answered here is why not recovering the reference signal from a pressure signal such as p . The hint −jkd −jkd p = w + w e , p = w e + w . (13) 2 i o 1 i o is (8) that may be expressed as p = F (jω)u + B(jω)u .In 1 s p The in- and outbound waves can be resolved from p and p 1 2 view of (8), the acoustical feedback transfer function is via Z Z o p −1 −jkd F (jω) = . w e 1 p i 1 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p −jkd w 1 e p o 2 (21) (14) −jkd −e 1 p 1 1 Since F (jω) is a transfer function with resonant poles, it has = . −2jkd −jkd 1 − e p an infinite impulse response (IIR). In many ANC systems, a 1 −e 2 4 Advances in Acoustics and Vibration finite-impulse-response (FIR) filter F (jω) is used to approx- |R (jω)| does not have sharp peaks or dips as a function imate F (jω). This means inevitable approximation errors in of ω. In many cases, |R | is constant for a pair of fixed Z 1 p the first place. and Z .Let X (jω) denote the Fourier transform of x(t), then −1 Besides, all transfer functions in a duct are sensitive to X (jω) = L{x(t)} and x(t) = L {X (jω)} share many similar values of Z , Z , and Z .Inparticular, Z is the impedance properties. For example, if x(t) is a low-frequency function o s p s of the entire downstream segment from the location of p to of t, then the bandwidth of X (jω)isnarrowinterms of ω. the duct outlet. Objects moving near the duct outlet could Similarly, if X (jω) is a “low-frequency” function of ω, then cause changes of Z . A fracture in any downstream part may the time duration of x(t) is short (a narrow bandwidth in also cause a significant change to Z as well. If initial estimate terms of t). The fact that |R | is a “low-frequency” function s 1 of ω for each pair of fixed Z and Z implies short impulse F (jω) is remembered by an ANC system, it is a stability issue p o responses of R (z). It is, therefore, reasonable to assume that how significant will F (jω) − F (jω) turn out as a result of a −k R (z) = r z can be approximated by a FIR transfer 1 k small variation of Z . An indicative answer might be k=0 function with negligible errors (Assumption A1). If both Z and Z are constant, R is a single constant. Resonant effects −Z Z Z cos(kd)+ jZ sin(kd) ∂ o p o p o 1 F (jω) = . in the duct are hidden in wave signals w and w without af- i o ∂Z Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p fecting R . This is a major difference between recovering the (22) reference signal from traveling waves and recovering the ref- erence signal from a pressure signal. The common denominator of p , p , and all transfer func- 1 2 Even if an estimate of F (z) is obtained by initial iden- tions in the duct has an alternative form in (18), which is tification, it is less likely that online variations of environ- equivalent to mental or boundary conditions could cause significant mis- jkd −jkd match between F (z) and its initial estimate. The resultant 0.5 Z + Z Z + Z e − 0.5 Z − Z Z − Z e w s o p o s o p o ANC system is semimodel independent if its reference signal Z −Z Z −Z is recovered with (16) in combination with a MIANC adap- s o p o jkd −2jkd = 0.5 Z +Z Z +Z e 1− e s o p o tation algorithm such as orthogonal adaptation. Z +Z Z +Z s o p o jkd −2jkd = 0.5 Z + Z Z + Z e 1 − R R e , s o p o 1 2 4. COMPLETE NONINVASIVE MIANC (23) Noninvasive model-independent feedback cancellation is where R = (Z − Z )/(Z + Z ) represents the downstream 2 s o s o possible by applying LMS to (16). With assumption A1, on- reflection coefficient. line estimate of the feedback transfer function is represented Since resonant frequencies of the duct are roots of the by polynomial common denominator, it is suggested by (22)and (23) that all transfer functions in the duct, including the feedback −k transfer function F (jω), are sensitive to variance of Z at the R(z) = r (t)z , (24) s k resonant peaks. The stronger the resonance, the more sensi- k=0 tive of transfer functions with respect to Z . If an ANC system where r (t) is the kth coefficient for the tth sample. An esti- recovers the reference signal from a pressure signal like p ,a mated version of (16)would be small online variation of Z may cause a significant mismatch between F (jω) and initial estimate F (z). As a result, closed- −1 n = w − z R(z)w , (25) o i loop stability is sensitive to possible variation of Z . If the reference signal is recovered from traveling waves which has a discrete-time domain expression, with (16), the situation will be different. In a discrete-time implementation, one may rewrite (16)to n(z) = w − F (z)w , where the acoustical feedback transfer function is w i n(t) = w (t) − r (t)w (t − k − 1). (26) o k i a delayed version of upstream reflection coefficient F (z) = k=0 −1 z R (z). Here, R = (Z − Z )/(Z + Z )isonlysensitive 1 1 p o p o to Z and Z .Characteristicimpedance Z is a real constant p o o m −k Coefficients of R(z) = r (t)z are updated with the k=0 depending on sound speed and cross-sectional area between LMS algorithm as follows: p and p . It seldom changes significantly in online ANC op- 1 2 erations. As for Z , it is the impedance of the upstream por- r (t +1) = r (t)+ μn(t)w (t − k − 1), (27) k k i tion from the primary source to the location of p .Inmost applications, p and p are measured as close as possible to 1 2 where μ> 0 is a small constant representing the LMS the primary source. Impedance Z is deeply hidden in a very −k step size. Since R (z) = r z by assumption A1, the 1 k k=0 short segment of the duct. Its variation, if any, would be cer- discrete-time domain version of (16)is tainly not as significant as that of Z . No matter how significant are the possible variations of Z or Z , the passive upstream reflection always has a lim- p o n(t) = w (t) − r w (t − k − 1). (28) o k i ited magnitude |R | < 1. For each pair of fixed Z and Z , 1 p o k=0 Jing Yuan 5 Subtracting (28)from(27), one obtains Let H (z)and S(z) denote online estimates of H (z)and S(z). Path estimates H (z)and S(z) are obtained by minimizing es- timation error as follows: n(t) − n(t) = r − r (t) w (t − k − 1) k k i k=0 ε(z)=e(z) − H (z)n(z)−S(z)u (z)=ΔH (z)n(z)+ΔS(z)u (z), s s (29) (35) = Δr (t)w (t − k − 1), k i k=0 where ΔH (z) = H (z) − H (z)and ΔS(z) = S(z) − S(z)are T T T online modeling errors. Let θ = [h s ] denote online es- where Δr (t) = r − r (t) is the estimation error of r .Let k k k k T T T T Δr = [Δr , Δr ,... , Δr ] and let  (t) = [w (t − 1), w (t − timate of θ = [h s ], then h = [h h ··· h ]and 0 1 m i i i 0 1 m 2), w (t − m − 1)] . It is possible to express (29) in an inner s = [s s ··· s ] represent the coefficients of H (z)and i 0 1 m product S(z), respectively. Similar to the equivalence between (34) and e(z) = H (z)n(z)+ S(z)u (z), (35) has a discrete-time n(t) − n(t) = Δr  (t). (30) domain equivalence Estimation residues of LMS algorithms are usually expressed T T ε = e(t) − θ φ = Δθ φ , (36) t t t as inner products like (30). It has been proven that the LMS algorithm is able to drive the convergence of these inner where Δθ = θ − θ is the online coefficient error vector. The products towards zero. entire CNMIANC system performs three online tasks that are If the primary noise signal u was available, mathematical mathematically represented by the minimization of three in- model of the error signal may be expressed in the discrete- ner products. The first is inner product given in (30); the sec- time z-transform domain as e(z) = P(z)u (z)+ S(z)u (z), p s ond one is given in (36); and the third one is θ φ . where the actuation signal would be synthesized as u (z) = t Equations (30)and (36) contain estimation errors Δr and C(z)u (z). Since u is actually not available, the ANC sys- p p Δθ. Most available estimation algorithms, such as LMS and tem has to recover n(z) from the outbound wave and then the recursive least squares (RLS), are very capable of driv- synthesize u (z) = C(z)n(z) instead. After the convergence ing inner products like (30)and (36)towards zero,orat of n(z) → n (z), one may express the mathematical model of least minimizing their magnitudes [18]. A difficult problem the error signal to is how to force Δr → 0and Δθ → 0. Available solutions inject p significant levels of “persistent excitations” (invasive probing e(z) = P(z) 1+ + S(z)C(z) n(z), (31) Z signals) to the estimation system [6, 7, 9, 10, 13]. A unique feature of the proposed CNMIANC is no persistent excita- where (20)has been substituted. Let H (z) = P(z)[1 +Z /Z ], p o tions. The system works well without requiring Δr → 0and then (31)becomes Δθ → 0. For (30), minimizing the inner product in the right- e(z) = [H (z)+ S(z)C(z)]n(z). (32) hand side implies convergence of n → n in the left-hand side. It would be great if Δr → 0 as well. Otherwise, Δr may It is mathematically equivalent to another ANC system just converge to a FIR filter that filters out w from w .On i o whose primary source is available to the controller as n(z), the other hand, minimizing the inner product in (36)only with primary path transfer function H (z) and secondary implies ε → 0. The question is what does it further im- path transfer function S(z). Orthogonal adaptation is read- plies? One may consider the equivalence between (34)and ily applicable to (32) to implement a noninvasive MIANC e(z) = H (z)n(z)+ S(z)u (z), which holds if one replaces system. T T T θ = [h s ], H (z), and S(z) with respective estimates It is assumed that H (z)and S(z) can be approximated T T T θ = [h s ], H (z), and S(z). The equivalence is now be- by FIR filters with negligible errors (Assumption A2). Let T T tween forcing θ φ ≈ 0 and forcing h = [h h ··· h ]and s = [s s ··· s ]denotecoef- 0 1 m 0 1 m t ficients of H (z)and S(z), respectively, the discrete-time do- main version of e(z) = H (z)n(z)+S(z)u (z) is a discrete-time H (z)n(z)+ S(z)u (z) = H (z)+ S(z)C(z) n(z) ≈ 0. (37) s s convolution: The CNMIANC system uses online estimates of H (z)and m m S(z)tosolve C(z) that minimizes H (z)+ S(z)C(z) . This e(t) = h n(t − k) − s u (t − k), (33) k k s k=0 k=0 is equivalent to forcing θ φ ≈ 0. One can obtain where e(t), n(t), and u (t)denotesamples of e(z), T T e= ε + θ φ ≤ ε + θ φ (38) t t t t n(z), and u (z), respectively. Introducing coefficient T T T vector θ = [h s ] and regression vector φ = by adding θ φ to both sides of (36). As the CNMIANC sys- n(t) n(t−1) ··· n(t−m), u (t) u (t−1) ··· u (t−m) [ ] , s s s T T tem drives ε = Δθ φ → 0and forces |θ φ |≈ 0ultimately, onemay rewrite(33)to t t t it implies ultimate convergence of e→ 0 even though Δθ e(t) = θ φ . (34) does not necessarily converge to zero [11, 12]. t 6 Advances in Acoustics and Vibration by the dashed-black curve. For case 2, normalized PSD of e(t) is plotted with the solid-gray curve. For case 3, normal- ized PSD of e(t) is represented by the solid-black curve. Both ANC systems were able to suppress noise with good control performance as seen in Figure 3. The proposed CNMIANC has slightly worse performance since its reference was the re- covered signal n(t) instead of the true primary source u (t). This is a small price to pay in case u (t) is not available to −10 the ANC system. The proposed CNMIANC system was stable and able to recover the reference and suppress noise without −20 any persistent excitations. TheCNMIANC system wasrobustwithrespect to sud- −30 den parameter change in the duct. In the experiment, the duct outlet was changed from completely open to completely −40 200 400 600 800 1000 1200 1400 closed. Such a sudden change shifted all resonant frequencies in the duct. Path transfer functions also changed suddenly. Frequency (Hz) The CNMIANC system remained stable and converged very Figure 3: Normalized PSDs of e(t) for (a) uncontrolled case quickly. (dashed-black), (b) controlled case with u (t) available (solid-gray), and (c) controlled case with recovered n(t) (solid-black). 6. CONCLUSIONS The primary source is not necessarily available as the ref- 5. EXPERIMENTAL VERIFICATION erence signal for ANC systems in all practical applications. When the primary source is not available, the ANC system A CNMIANC system was implemented and tested in an ex- must recover the reference signal from a sound field to which periment, with a configuration shown in Figure 1. Cross- ANC is applied. Feedback cancellation is an important issue sectional area of the duct was 12× 15 cm . Two microphones in ANC systems that recover reference signals from sound were placed 30 cm downstream from the primary speaker fields. In most MIANC systems, persistent excitations are with a space of d = 10 cm between p and p . The distance 1 2 required for online modeling of feedback path model and between p and the secondary speaker is represented by L in adaptive feedback cancellation [9, 13]. In this study, a CN- Figure 1. To guarantee a causal ANC system, the value of L MIANC system is proposed that recovers reference signal must satisfy L> 2d such that the outbound wave is at least from traveling waves without persistent excitations. The cor- two samples ahead of sound propagation in duct. The sam- responding feedback path model is the upstream reflection pling interval of the controller was 0.29 millisecond with a coefficient and hence closer to an FIR filter than pressure sampling frequency of 3.448 Hz, which satisfies d = cδt with feedback transfer functions (IIR path models in resonant c = 344 m/s and exp(jkd) = z.The cutoff frequency of an- ducts). Theoretical analysis and experimental results are pre- tialias filters was chosen to be 1200 Hz. The in- and out- sented to demonstrate the stable operation of the proposed bound waves were recovered from pressure signals with (14). CNMIANC system. The reference signal was recovered with (25). Coefficients of R(z) were adapted with (27). Another online modeling pro- REFERENCES cess used (34)toobtaincoefficients of H (z)and S(z). The ANC transfer function was solved by online minimization [1] C.H.Hansenand S. D. Snyder, Active Control of Noise and of H (z)+ S(z)C(z) . The CNMIANC system was imple- Vibration, E & FN Spon, London, UK, 1997. mented in a dSPACE 1103 board. [2] P. A. Nelson and S. J. Elliott, Active Control of Sound,Academic Error signal e(t) and primary noise u (t)werecollected Press, London, UK, 1992. as vectors e and u for three cases. In case 1, there was no p [3] M. A. Vaudrey, W. T. Baumann, and W. R. Saunders, “Stabil- control action. In case 2, u (t) was available as the reference ity and operating constraints of adaptive LMS-based feedback control,” Automatica, vol. 39, no. 4, pp. 595–605, 2003. signal for an ANC system to suppress noise in the duct. In [4] E. Bjarnason, “Analysis of the filtered-X LMS algorithm,” IEEE case 3, u (t) was not available and the CNMIANC system Transactions on Speech and Audio Processing,vol. 3, no.6,pp. had to recover n(t)from p and p for controller synthe- 1 2 504–514, 1995. sis. For each respective case, power spectral densities (PSD’s) [5] S. D. Snyder and C. H. 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Ser, “Cross-updated active noise control system with online secondary path modeling,” IEEE Transactions on Speech and Audio Processing, vol. 9, no. 5, pp. 598–602, 2001. [8] R. L. Clark and D. G. Cole, “Active damping of enclosed sound fields through direct rate feedback control,” Journal of the Acoustical Society of America, vol. 97, no. 3, pp. 1710–1716, [9] X.Qiu andC.H.Hansen, “Analgorithm foractivecontrol of transformer noise with on-line cancellation path modelling basedonthe perturbation method,” Journal of Sound and Vi- bration, vol. 240, no. 4, pp. 647–665, 2001. [10] M. Zhang, H. Lan, and W. Ser, “A robust online secondary path modeling method with auxiliary noise power schedul- ing strategy and norm constraint manipulation,” IEEE Trans- actions on Speech and Audio Processing, vol. 11, no. 1, pp. 45– 53, 2003. [11] J. Yuan, “Orthogonal adaptation for active noise control,” Jour- nal of the Acoustical Society of America, vol. 120, no. 1, pp. 204– 210, 2006. [12] J. Yuan, “Orthogonal adaptation for multichannel feedfor- ward control,” Journal of the Acoustical Society of America, vol. 120, no. 6, pp. 3723–3729, 2006. [13] M. T. Akhtar, M. Abe, and M. Kawamata, “On active noise control systems with online acoustic feedback path modeling,” IEEE Transactions on Audio, Speech and Language Processing, vol. 15, no. 2, pp. 593–600, 2007. [14] J.-S. Hu, “Feedforward and feedback control strategy for ac- tive noise cancellation in ducts,” Journal of Dynamic Systems, Measurement and Control, vol. 118, no. 2, pp. 372–378, 1996. [15] J.-S. Hu, “Active sound attenuation in finite-length ducts using close-form transfer function models,” Journal of Dynamic Sys- tems, Measurement and Control, vol. 117, no. 2, pp. 143–154, [16] M. R. Bai, Y. Lin, and J.-D. Wu, “Analysis and DSP implemen- tation of a broadband duct ANC system using spatially feed- forward structure,” Journal of Vibration and Acoustics, vol. 123, no. 2, pp. 129–136, 2001. [17] M. L. Munjal and L. J. Eriksson, “An analytical, one- dimensional, standing-wave model of a linear active noise controlsysteminaduct,” Journal of the Acoustical Society of America, vol. 84, pp. 1066–1093, 1988. [18] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1984. 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Noninvasive Model Independent Noise Control with Adaptive Feedback Cancellation

Advances in Acoustics and Vibration , Volume 2008 – Mar 17, 2008

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Hindawi Publishing Corporation
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Copyright © 2008 Jing Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2008/863603
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 863603, 7 pages doi:10.1155/2008/863603 Research Article Noninvasive Model Independent Noise Control with Adaptive Feedback Cancellation Jing Yuan Department of Mechanical Engineering, Faculty of Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Correspondence should be addressed to Jing Yuan, mmjyuan@polyu.edu.hk Received 12 November 2007; Accepted 2 February 2008 Recommended by Marek Pawelczyk An active noise control (ANC) system is model dependent/independent if its controller transfer function is dependent/independent on initial estimates of path models in a sound field. Since parameters of path models in a sound field will change when boundary conditions of the sound field change, model-independent ANC systems (MIANC) are able to tolerate variations of boundary conditions in sound fields and more reliable than model-dependent counterparts. A possible way to implement MIANC systems is online path modeling. Many such systems require invasive probing signals (persistent excitations) to obtain accurate estimates of path models. In this study, a noninvasive MIANC system is proposed. It uses online path estimates to cancel feedback, recover reference signal, and optimize a stable controller in the minimum H norm sense, without any forms of persistent excitations. Theoretical analysis and experimental results are presented to demonstrate the stable control performance of the proposed system. Copyright © 2008 Jing Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION changes to path transfer functions P(z)and S(z). Since a model-dependent ANC system only remembers initial path estimates P(z)and S(z), variation of P(z)and S(z)may Most active noise control (ANC) systems are model depen- cause mismatch with initial estimates P(z)and S(z)tode- dent. Let P(z)and S(z) denote estimates of primary and sec- gradeANC performance. In casesofseveremismatchbe- ondary path transfer functions P(z)and S(z). Either S(z) tween path transfer functions and their initial estimates, a or both P(z)and S(z) must be obtained by initial system model-dependent ANC system may generate constructive in- identification for model-dependent ANC systems. Controller stead of destructive interference, or even become unstable. transfer function C(z) of a model-dependent ANC system is Model-independent ANC (MIANC) systems depend on either designed by minimizing P(z)+S(z)C(z),oradapted online path modeling or invariant properties of sound fields with the aid of S(z)[1, 2]. If estimates P(z)and S(z)contain to update or design controllers [6–8]. These systems avoid too much error, a model-dependent ANC system may gen- initial path modeling and are adaptive to variations of en- erate constructive instead of destructive interference. This is vironmental or boundary conditions of sound fields. Many mathematically equivalent to P(z)+ S(z)C(z) > P(z) adaptive MIANC systems require invasive persistent excita- even if P(z)+ S(z)C(z) is minimized. If phase error in tions to obtain accurate path estimates and ensure closed- S(z) exceeds 90 in some frequency, an ANC system adapted loop stability [6, 7, 9, 10]. Noninvasive MIANC systems are by the filtered-X least mean square (FXLMS) algorithm may able to ensure closed-loop stability without persistent ex- become unstable [3–5]. An operator of a model-dependent citations, which are possible by a recently developed algo- ANC system must have the knowledge and skill to obtain ac- rithm, known as orthogonal adaptation [11, 12], if the pri- curate estimates of path models by initial system identifica- mary noise signal is directly available as the reference signal. tion for each individual application. In many real applications, the primary noise signal is not During the operation of an ANC system, changes of en- necessarily available and the reference signal must be recov- vironmental or boundary conditions may cause significant ered from the sound field [1, 2]. When an ANC system is 2 Advances in Acoustics and Vibration 1 p p u Z Z p s cos(kd) jZ sin(kd) p p p 1 2 3 sin(kd) d cos(kd) MIANC system (a) Figure 1: Configuration of the proposed MIANC system. p p 1 2 cos(kd) jZ sin(kd) sin(kd) cos(kd) active, a measured signal is a linear combination of primary and secondary signals. Feedback of ANC signal in the mea- surement is mathematically modeled by a feedback transfer (b) function F (z) from the controller to the reference sensor. Ac- curate estimation of F (z)isasimportant as accurate estima- 1 p u Z p p tion of P(z)or S(z)[9, 13]. A complete noninvasive MIANC cos(kd) jZ sin(kd) (CNMIANC) system must be able to suppress the noise sig- sin(kd) Z cos(kd) nal without injecting probing signals for online modeling of j P(z), S(z), and F (z). Most available methods for adaptive feedback cancellation require persistent excitations [9, 13]. (c) In this study, a new method is presented for adaptive feed- Figure 2: (a) Acoustical two-port circuit in the duct system, (b) back cancellation without persistent excitations. contribution by controller, and (c) contribution by primary source. It was proposed to use a pair of sensors to measure pres- sure signals in ducts, from which traveling waves are re- solved [14, 15]. The outbound wave could be used directly as the reference signal without cancelling feedback signals if tion of p , is equivalent to an acoustical source with strength an infinite-impulse-response (IIR) controller could be im- u and impedance Z . The downstream part, from location p p plemented accurately [14, 15]. In reality, it is very difficult to of p to the outlet, is represented by another acoustical source implement a stable ideal IIR ANC controller [16]. Most prac- with strength u and impedance Z .Characteristicimpedance s s tical ANC systems use finite-impulse-response (FIR) con- of the duct is represented by Z . trollers. The outbound wave in a duct is a linear combina- The linear system theory allows one to solve p and p 1 2 tion of primary noise and reflected version of feedback sig- in Figure 2(a) by focusing on acoustical circuits of Figures nal. Instead of using the outbound wave directly as the ref- 2(b) and 2(c) before adding two solutions together as the fi- erence, the least mean square (LMS) algorithm is applied in nal solution of Figure 2(a). For the case of u = 0, which is this study to cancel feedback signals in the outbound wave represented by Figure 2(b), one obtains before using it as the reference. Orthogonal adaptation is combined with the proposed ANC configuration to imple- Z p | = cos(kd)+ j sin(kd) p | , 2 1 u =0 u =0 p p ment a CNMIANC system. Experimental result is presented (1) to demonstrate the performance of the CNMIANC system. Z Z s s u − p | = cos(kd)+ j sin(kd) p | , s 2 1 u =0 u =0 p p Z Z p o 2. SYSTEM CONFIGURATION AND MODEL where k is the wave number. One can solve, from (1), Figure 1 illustrates the configuration of the proposed ANC system. The primary source is represented by the upstream Z Z cos(kd)+ jZ sin(kd) o p o speaker and the secondary source is the midstream speaker. p | =     u , 2 s u =0 p 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p Cross-sectional area of the duct is small enough such that (2) sound field in the duct can be modeled by a 1D sound field in the frequency range of interest. Three microphone sensors Z Z o p p | = u . are installed in the duct, measuring signals p , p ,and p , 1 s 1 2 3 u =0 p 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p respectively. Since the primary noise signal is not available (3) to the ANC system, the reference signal is recovered from p and p , while p is the error signal to be minimized by the 2 3 Similarly, for the case of u = 0, which is represented by ANC system. Figure 2(c), one obtains Let d denote the axial distance between p and p .The 1 2 acoustical two-port theory [16, 17] has been applied by many ANC researchers for the design and analysis of ANC systems. p | = cos(kd)+ j sin(kd) p | , 1 2 u =0 u =0 s s It is adopted here as an analytical tool. An equivalent acousti- (4) cal circuit is shown in Figure 2 to model the two-microphone Z Z p p u − p | = cos(kd)+ j sin(kd) p | , p 1 2 u =0 system. The upstream part, from the primary source to loca- us=0 s Z Z s o Jing Yuan 3 from which one can solve In a digital implementation of ANC system, it is recom- mended to select sampling interval δt such that its product Z Z cos(kd)+ jZ sin(kd) o s o p | = u , with sound speed c satisfies cδt = d. As a result, the delay op- 1 p u =0 s 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p o −1 erator exp(−jkd) = z becomes an exact one-sample delay (5) for discrete-time ANC systems. Z Z o s p | = u . 2 p u =0 s 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) 3. FEEDBACK CANCELLATION s p o s p (6) It is indicated by (12) that the outbound wave contains feed- Adding (2)and (6), one may write back from u that must be cancelled to recover the reference signal. Let R = (Z − Z )/(Z + Z ) denote the upstream re- 1 p o p o p = p | + p | 2 2 u =0 2 u =0 p s −jkd flection coefficient. By multiplying e R to (11), one ob- (7) tains Z Z cos(kd)+ jZ sin(kd) u + Z Z u o p o s o s p = . Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p Z − Z Z − Z Z − Z s o p o p o −jkd −jkd e R w = α   e u + u . 1 i p s The same method is applicable to (3)and (5)for 2 Z + Z 2 p o (15) p = p | + p | 1 1 u =0 1 u =0 p s A subtraction of (15)from(12)enables onetowrite (8) Z Z cos(kd)+ jZ sin(kd) u + Z Z u o s o p o p s = . −jkd Z + Z Z cos(kd)+ j Z Z + Z sin(kd) w − e R w = n, (16) s p o s p o 1 i The next step is to use complex factor α = Z /((Z + o s where Z )Z cos(kd)+ j (Z Z + Z )sin(kd)) to simplify (7)and (8). p o s p jkd −jkd α Z + Z Z + Z e − Z − Z Z − Z e The results read s o p o s o p o n= u 2 Z + Z p o p = α Z cos(kd)+ jZ sin(kd) u + Z u , 2 p o s s p (17) (9) p = α Z cos(kd)+ jZ sin(kd) u + Z u . 1 s o p p s is only contributed by the primary source u . jkd −jkd jkd jkd −jkd Since cos(kd) = 0.5(e + e )and j sin(kd) = 0.5(e − Using cos(kd) = 0.5(e + e )and j sin(kd) = −jkd jkd −jkd e ), (9)can be writtenas 0.5(e − e ), one can see that the common denomina- tor of p , p , and all transfer functions in the duct is Z + Z Z − Z 1 2 p o p o jkd −jkd p = α e + e u 2 s 2 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p Z + Z Z − Z s o s o + + u , p jkd −jkd = 0.5 Z +Z Z +Z e −0.5 Z −Z Z −Z e . 2 2 s o p o s o p o (10) (18) Z + Z Z − Z s o s o jkd −jkd p = α e + e u 1 p 2 2 Substituting (18) into the definition of α (immediately after (8)), one obtains Z + Z Z − Z p o p o + + u . 2 2 jkd −jkd 2Z =α Z +Z Z +Z e − Z −Z Z −Z e . o s o p o s o p o Let (19) Z + Z Z − Z p o s o jkd w = α u + e u (11) i p s A further substitution of (19) into (17)leads to 2 2 Z u o p Z − Z Z + Z p o s o jkd n = . (20) w = α u + e u (12) o s p Z + Z p o 2 2 represent the in- and outbound waves in the duct. By com- This is the reference signal to be recovered by the proposed paring (10)with(11)and (12), one can see that (10)are ANC system. equivalent to A question to be answered here is why not recovering the reference signal from a pressure signal such as p . The hint −jkd −jkd p = w + w e , p = w e + w . (13) 2 i o 1 i o is (8) that may be expressed as p = F (jω)u + B(jω)u .In 1 s p The in- and outbound waves can be resolved from p and p 1 2 view of (8), the acoustical feedback transfer function is via Z Z o p −1 −jkd F (jω) = . w e 1 p i 1 2 Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p −jkd w 1 e p o 2 (21) (14) −jkd −e 1 p 1 1 Since F (jω) is a transfer function with resonant poles, it has = . −2jkd −jkd 1 − e p an infinite impulse response (IIR). In many ANC systems, a 1 −e 2 4 Advances in Acoustics and Vibration finite-impulse-response (FIR) filter F (jω) is used to approx- |R (jω)| does not have sharp peaks or dips as a function imate F (jω). This means inevitable approximation errors in of ω. In many cases, |R | is constant for a pair of fixed Z 1 p the first place. and Z .Let X (jω) denote the Fourier transform of x(t), then −1 Besides, all transfer functions in a duct are sensitive to X (jω) = L{x(t)} and x(t) = L {X (jω)} share many similar values of Z , Z , and Z .Inparticular, Z is the impedance properties. For example, if x(t) is a low-frequency function o s p s of the entire downstream segment from the location of p to of t, then the bandwidth of X (jω)isnarrowinterms of ω. the duct outlet. Objects moving near the duct outlet could Similarly, if X (jω) is a “low-frequency” function of ω, then cause changes of Z . A fracture in any downstream part may the time duration of x(t) is short (a narrow bandwidth in also cause a significant change to Z as well. If initial estimate terms of t). The fact that |R | is a “low-frequency” function s 1 of ω for each pair of fixed Z and Z implies short impulse F (jω) is remembered by an ANC system, it is a stability issue p o responses of R (z). It is, therefore, reasonable to assume that how significant will F (jω) − F (jω) turn out as a result of a −k R (z) = r z can be approximated by a FIR transfer 1 k small variation of Z . An indicative answer might be k=0 function with negligible errors (Assumption A1). If both Z and Z are constant, R is a single constant. Resonant effects −Z Z Z cos(kd)+ jZ sin(kd) ∂ o p o p o 1 F (jω) = . in the duct are hidden in wave signals w and w without af- i o ∂Z Z + Z Z cos(kd)+ j Z Z + Z sin(kd) s p o s p fecting R . This is a major difference between recovering the (22) reference signal from traveling waves and recovering the ref- erence signal from a pressure signal. The common denominator of p , p , and all transfer func- 1 2 Even if an estimate of F (z) is obtained by initial iden- tions in the duct has an alternative form in (18), which is tification, it is less likely that online variations of environ- equivalent to mental or boundary conditions could cause significant mis- jkd −jkd match between F (z) and its initial estimate. The resultant 0.5 Z + Z Z + Z e − 0.5 Z − Z Z − Z e w s o p o s o p o ANC system is semimodel independent if its reference signal Z −Z Z −Z is recovered with (16) in combination with a MIANC adap- s o p o jkd −2jkd = 0.5 Z +Z Z +Z e 1− e s o p o tation algorithm such as orthogonal adaptation. Z +Z Z +Z s o p o jkd −2jkd = 0.5 Z + Z Z + Z e 1 − R R e , s o p o 1 2 4. COMPLETE NONINVASIVE MIANC (23) Noninvasive model-independent feedback cancellation is where R = (Z − Z )/(Z + Z ) represents the downstream 2 s o s o possible by applying LMS to (16). With assumption A1, on- reflection coefficient. line estimate of the feedback transfer function is represented Since resonant frequencies of the duct are roots of the by polynomial common denominator, it is suggested by (22)and (23) that all transfer functions in the duct, including the feedback −k transfer function F (jω), are sensitive to variance of Z at the R(z) = r (t)z , (24) s k resonant peaks. The stronger the resonance, the more sensi- k=0 tive of transfer functions with respect to Z . If an ANC system where r (t) is the kth coefficient for the tth sample. An esti- recovers the reference signal from a pressure signal like p ,a mated version of (16)would be small online variation of Z may cause a significant mismatch between F (jω) and initial estimate F (z). As a result, closed- −1 n = w − z R(z)w , (25) o i loop stability is sensitive to possible variation of Z . If the reference signal is recovered from traveling waves which has a discrete-time domain expression, with (16), the situation will be different. In a discrete-time implementation, one may rewrite (16)to n(z) = w − F (z)w , where the acoustical feedback transfer function is w i n(t) = w (t) − r (t)w (t − k − 1). (26) o k i a delayed version of upstream reflection coefficient F (z) = k=0 −1 z R (z). Here, R = (Z − Z )/(Z + Z )isonlysensitive 1 1 p o p o to Z and Z .Characteristicimpedance Z is a real constant p o o m −k Coefficients of R(z) = r (t)z are updated with the k=0 depending on sound speed and cross-sectional area between LMS algorithm as follows: p and p . It seldom changes significantly in online ANC op- 1 2 erations. As for Z , it is the impedance of the upstream por- r (t +1) = r (t)+ μn(t)w (t − k − 1), (27) k k i tion from the primary source to the location of p .Inmost applications, p and p are measured as close as possible to 1 2 where μ> 0 is a small constant representing the LMS the primary source. Impedance Z is deeply hidden in a very −k step size. Since R (z) = r z by assumption A1, the 1 k k=0 short segment of the duct. Its variation, if any, would be cer- discrete-time domain version of (16)is tainly not as significant as that of Z . No matter how significant are the possible variations of Z or Z , the passive upstream reflection always has a lim- p o n(t) = w (t) − r w (t − k − 1). (28) o k i ited magnitude |R | < 1. For each pair of fixed Z and Z , 1 p o k=0 Jing Yuan 5 Subtracting (28)from(27), one obtains Let H (z)and S(z) denote online estimates of H (z)and S(z). Path estimates H (z)and S(z) are obtained by minimizing es- timation error as follows: n(t) − n(t) = r − r (t) w (t − k − 1) k k i k=0 ε(z)=e(z) − H (z)n(z)−S(z)u (z)=ΔH (z)n(z)+ΔS(z)u (z), s s (29) (35) = Δr (t)w (t − k − 1), k i k=0 where ΔH (z) = H (z) − H (z)and ΔS(z) = S(z) − S(z)are T T T online modeling errors. Let θ = [h s ] denote online es- where Δr (t) = r − r (t) is the estimation error of r .Let k k k k T T T T Δr = [Δr , Δr ,... , Δr ] and let  (t) = [w (t − 1), w (t − timate of θ = [h s ], then h = [h h ··· h ]and 0 1 m i i i 0 1 m 2), w (t − m − 1)] . It is possible to express (29) in an inner s = [s s ··· s ] represent the coefficients of H (z)and i 0 1 m product S(z), respectively. Similar to the equivalence between (34) and e(z) = H (z)n(z)+ S(z)u (z), (35) has a discrete-time n(t) − n(t) = Δr  (t). (30) domain equivalence Estimation residues of LMS algorithms are usually expressed T T ε = e(t) − θ φ = Δθ φ , (36) t t t as inner products like (30). It has been proven that the LMS algorithm is able to drive the convergence of these inner where Δθ = θ − θ is the online coefficient error vector. The products towards zero. entire CNMIANC system performs three online tasks that are If the primary noise signal u was available, mathematical mathematically represented by the minimization of three in- model of the error signal may be expressed in the discrete- ner products. The first is inner product given in (30); the sec- time z-transform domain as e(z) = P(z)u (z)+ S(z)u (z), p s ond one is given in (36); and the third one is θ φ . where the actuation signal would be synthesized as u (z) = t Equations (30)and (36) contain estimation errors Δr and C(z)u (z). Since u is actually not available, the ANC sys- p p Δθ. Most available estimation algorithms, such as LMS and tem has to recover n(z) from the outbound wave and then the recursive least squares (RLS), are very capable of driv- synthesize u (z) = C(z)n(z) instead. After the convergence ing inner products like (30)and (36)towards zero,orat of n(z) → n (z), one may express the mathematical model of least minimizing their magnitudes [18]. A difficult problem the error signal to is how to force Δr → 0and Δθ → 0. Available solutions inject p significant levels of “persistent excitations” (invasive probing e(z) = P(z) 1+ + S(z)C(z) n(z), (31) Z signals) to the estimation system [6, 7, 9, 10, 13]. A unique feature of the proposed CNMIANC is no persistent excita- where (20)has been substituted. Let H (z) = P(z)[1 +Z /Z ], p o tions. The system works well without requiring Δr → 0and then (31)becomes Δθ → 0. For (30), minimizing the inner product in the right- e(z) = [H (z)+ S(z)C(z)]n(z). (32) hand side implies convergence of n → n in the left-hand side. It would be great if Δr → 0 as well. Otherwise, Δr may It is mathematically equivalent to another ANC system just converge to a FIR filter that filters out w from w .On i o whose primary source is available to the controller as n(z), the other hand, minimizing the inner product in (36)only with primary path transfer function H (z) and secondary implies ε → 0. The question is what does it further im- path transfer function S(z). Orthogonal adaptation is read- plies? One may consider the equivalence between (34)and ily applicable to (32) to implement a noninvasive MIANC e(z) = H (z)n(z)+ S(z)u (z), which holds if one replaces system. T T T θ = [h s ], H (z), and S(z) with respective estimates It is assumed that H (z)and S(z) can be approximated T T T θ = [h s ], H (z), and S(z). The equivalence is now be- by FIR filters with negligible errors (Assumption A2). Let T T tween forcing θ φ ≈ 0 and forcing h = [h h ··· h ]and s = [s s ··· s ]denotecoef- 0 1 m 0 1 m t ficients of H (z)and S(z), respectively, the discrete-time do- main version of e(z) = H (z)n(z)+S(z)u (z) is a discrete-time H (z)n(z)+ S(z)u (z) = H (z)+ S(z)C(z) n(z) ≈ 0. (37) s s convolution: The CNMIANC system uses online estimates of H (z)and m m S(z)tosolve C(z) that minimizes H (z)+ S(z)C(z) . This e(t) = h n(t − k) − s u (t − k), (33) k k s k=0 k=0 is equivalent to forcing θ φ ≈ 0. One can obtain where e(t), n(t), and u (t)denotesamples of e(z), T T e= ε + θ φ ≤ ε + θ φ (38) t t t t n(z), and u (z), respectively. Introducing coefficient T T T vector θ = [h s ] and regression vector φ = by adding θ φ to both sides of (36). As the CNMIANC sys- n(t) n(t−1) ··· n(t−m), u (t) u (t−1) ··· u (t−m) [ ] , s s s T T tem drives ε = Δθ φ → 0and forces |θ φ |≈ 0ultimately, onemay rewrite(33)to t t t it implies ultimate convergence of e→ 0 even though Δθ e(t) = θ φ . (34) does not necessarily converge to zero [11, 12]. t 6 Advances in Acoustics and Vibration by the dashed-black curve. For case 2, normalized PSD of e(t) is plotted with the solid-gray curve. For case 3, normal- ized PSD of e(t) is represented by the solid-black curve. Both ANC systems were able to suppress noise with good control performance as seen in Figure 3. The proposed CNMIANC has slightly worse performance since its reference was the re- covered signal n(t) instead of the true primary source u (t). This is a small price to pay in case u (t) is not available to −10 the ANC system. The proposed CNMIANC system was stable and able to recover the reference and suppress noise without −20 any persistent excitations. TheCNMIANC system wasrobustwithrespect to sud- −30 den parameter change in the duct. In the experiment, the duct outlet was changed from completely open to completely −40 200 400 600 800 1000 1200 1400 closed. Such a sudden change shifted all resonant frequencies in the duct. Path transfer functions also changed suddenly. Frequency (Hz) The CNMIANC system remained stable and converged very Figure 3: Normalized PSDs of e(t) for (a) uncontrolled case quickly. (dashed-black), (b) controlled case with u (t) available (solid-gray), and (c) controlled case with recovered n(t) (solid-black). 6. CONCLUSIONS The primary source is not necessarily available as the ref- 5. EXPERIMENTAL VERIFICATION erence signal for ANC systems in all practical applications. When the primary source is not available, the ANC system A CNMIANC system was implemented and tested in an ex- must recover the reference signal from a sound field to which periment, with a configuration shown in Figure 1. Cross- ANC is applied. Feedback cancellation is an important issue sectional area of the duct was 12× 15 cm . Two microphones in ANC systems that recover reference signals from sound were placed 30 cm downstream from the primary speaker fields. In most MIANC systems, persistent excitations are with a space of d = 10 cm between p and p . The distance 1 2 required for online modeling of feedback path model and between p and the secondary speaker is represented by L in adaptive feedback cancellation [9, 13]. In this study, a CN- Figure 1. To guarantee a causal ANC system, the value of L MIANC system is proposed that recovers reference signal must satisfy L> 2d such that the outbound wave is at least from traveling waves without persistent excitations. The cor- two samples ahead of sound propagation in duct. The sam- responding feedback path model is the upstream reflection pling interval of the controller was 0.29 millisecond with a coefficient and hence closer to an FIR filter than pressure sampling frequency of 3.448 Hz, which satisfies d = cδt with feedback transfer functions (IIR path models in resonant c = 344 m/s and exp(jkd) = z.The cutoff frequency of an- ducts). Theoretical analysis and experimental results are pre- tialias filters was chosen to be 1200 Hz. The in- and out- sented to demonstrate the stable operation of the proposed bound waves were recovered from pressure signals with (14). CNMIANC system. The reference signal was recovered with (25). Coefficients of R(z) were adapted with (27). Another online modeling pro- REFERENCES cess used (34)toobtaincoefficients of H (z)and S(z). The ANC transfer function was solved by online minimization [1] C.H.Hansenand S. D. Snyder, Active Control of Noise and of H (z)+ S(z)C(z) . 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