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Optimal and Adaptive Virtual Unidirectional Sound Source in Active Noise Control

Optimal and Adaptive Virtual Unidirectional Sound Source in Active Noise Control Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 647318, 12 pages doi:10.1155/2008/647318 Research Article Optimal and Adaptive Virtual Unidirectional Sound Source in Active Noise Control Dariusz Bismor Institute of Automatic Control, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland Correspondence should be addressed to Dariusz Bismor, dariusz.bismor@polsl.pl Received 21 November 2007; Revised 22 February 2008; Accepted 5 May 2008 Recommended by Marek Pawelczyk One of the problems concerned with active noise control is the existence of acoustical feedback between the control value (“active” loudspeaker output) and the reference signal. Various experiments show that such feedback can seriously decrease effects of attenuation or even make the whole ANC system unstable. This paper presents a detailed analysis of one of possible approaches allowing to deal with acoustical feedback, namely, virtual unidirectional sound source. With this method, two loudspeakers are used together with control algorithm assuring that the combined behaviour of the pair makes virtual propagation of sound only in one direction. Two different designs are presented for the application of active noise control in an acoustic duct: analytical (leading to fixed controller) and adaptive. The algorithm effectiveness in simulations and real experiments for both solutions is showed, discussed, and compared. Copyright © 2008 Dariusz Bismor. 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 the method requires a model of the acoustic feedback path and therefore is computationally more expensive. Active noise control is mainly concerned with low-frequency The approach to acoustic feedback cancellation pre- sounds that cannot be suppressed in passive manner at sented in this paper is called virtual unidirectional sound reasonable cost. Other prerequisite for using ANC is that source (VUSS). The method, in authors opinion, has some it does not obstruct media flow in ducts, for example, air advantages over well-known two loudspeaker active noise conditioning ducts. Unfortunately, almost all duct applica- control system in duct applications. First, it clearly separates tions have acoustic feedback; a phenomenon that can heavily the subsystem responsible for neutralisation of undesired decrease or even destroy the results of otherwise properly set- feedback from active noise control part. Second, its adaptive up control system [1, 2]. version can also be applied in case of varying acoustic There are several methods for avoiding acoustic feedback path responses. Last, virtual unidirectional sound source can in duct applications [2–6]. The best one that avoids acoustic also be applied in more general active control of sound feedback completely is to use nonacoustic reference sensor. applications, not only in active noise control. The paper Unfortunately, this usually limits the application to cancel attempts to explain the theory behind VUSS as well as to the tonal sounds and does not allow for the reduction of give some results showing its excellent performance in case broadband noise. Another approach to acoustic feedback is of ANC. to internally process signal from the reference microphone The paper is organised as follows. Section 2 sum- in a way allowing to compensate for this effect [2, 4–6]. marises physical basics necessary to understand principles This technique, usually called “feedback neutralisation,” uses of VUSS operation and its advantage over one-loudspeaker a model of acoustic feedback path to filter the controller applications. In Section 3, the idea of VUSS is explained output and then subtract it from the reference signal. In the together with the original detailed analysis of optimal (fixed) simplest case, the model is obtained offline and does not design (Section 3.1) and adaptive approach (Section 3.2). account for time variation of the feedback path. Moreover, Finally, Section 4 gives the results of simulations and real 2 Advances in Acoustics and Vibration duct, marked as p (x)and p (x). Assume also that the a1+ a2+ sound wave produced by the primary source is propagating downstream the duct, as marked on the figure. 1.5 Assume that the complex pressure of sound wave gener- ated by first “active” loudspeaker is given by −ikx p (x) = Be for x> 0, a1+ (2) ikx 0.5 p (x) = Be for x< 0, a1− and the complex pressure of sound wave produced by the second “active” loudspeaker is given by −1 −0.5 0 0.5 1 −ik(x−l) p (x) = Ce for x> l, a2+ (3) ik(x−l) Figure 1: Acoustic pressure distribution in duct: one loudspeaker p (x) = Ce for x< l, a2− case (solid) and two loudspeaker case (dashed). where B and C are the complex amplitudes of the sound waves. In the situation described above it is possible to set another condition on the system regardless of primary sound source attenuation requirement. In the case of unidirectional p (x) p+ p (x)+ p (x) p (x)+ p (x) a1− a2− a1+ a2+ sound source, it may be the requirement of self-cancellation x = 0 x = l of the secondary sound waves propagating upstream the duct: Figure 2: Acoustic duct with two active loudspeakers. p (x)+ p (x) =0for x< 0. (4) a1− a2− experiments of feedforward active noise control system using In this case, the secondary sources have to be driven in virtual unidirectional sound source. the way providing [3] −ikl B =−Ce . (5) 2. ACOUSTICAL PHENOMENA IN DUCTS In the same time to achieve the main ANC goal, which The simplest solution to acoustic feedback cancellation is to cancel the primary sound downstream the second problem in case where omnidirectional microphone must be loudspeaker completely, it is necessary to set the complex used as the reference sensor is called feedback neutralisation amplitude: [2, 4–6]. With this approach, only one loudspeaker is −A used, but the signal obtained from reference microphone C = . (6) 2i sin(kl) is processed in the way allowing for compensation of its influence. If no compensation is performed, severe problems The acoustic pressure level distribution for two loudspeakers can result. In the simplest case of attenuation of one tone, case has been shown with dashed line on Figure 1.The the pressure distribution upstream secondary source [3] figure shows that the sound pressure has been perfectly takes shape of standing wave, presented with solid line cancelled at coordinates x> l while the absolute value of the on Figure 1. If, for particular frequency of this tone, the pressure level has not been affected by the secondary sources reference microphone is placed in standing wave nodal point, at negative coordinates x. The zone between the secondary it will give measurement equal to zero regardless of waves’ loudspeakers (0 <x <l)isatransientzone. actual magnitude. Although two-loudspeaker system offers significant Another situation occurs when two loudspeakers are improvement over one loudspeaker system, it has its draw- used, as presented in Figure 2. Assume first loudspeaker is backs too. Specifically, the above control law says that for placed in the duct at position x = 0 and the second at some frequencies for which sin(kl) is close to zero, both the position x = l.Denote p (x)as the complex pressure [3]at p+ secondary sources will have to produce sound waves with location x produced by primary source emitting tonal sound: amplitude very large, compared with the amplitude A of the −ikx primary waveform. Consider, for example, the case where p (x) = Ae,(1) p+ the loudspeakers are separated by the distance l = 0.3m. where A is the complex amplitude of the sound wave and In this case the sound waves of frequency 570 Hz and all where acoustic wave number k is equal to k = ω/c ,with ω its multiplies will result in sin(kl) equal to, or close to, being wave angular frequency and c being the sound speed zero, depending on the actual speed of sound, and thus on in air. temperature, humidity, and so on. This fact will certainly Assume the two loudspeakers on Figure 2 produce sound influence the performance of ANC system trying to attenuate waves propagating upstream the duct, marked as p (x) such sound waves (see, e.g., the drop in attenuation just a1− and p (x), and sound waves propagating downstream the below 600 Hz on Figure 10). a2− |p(x)| Dariusz Bismor 3 S S 11 12 path between the loudspeakers and the M2 microphone. The S S 21 22 value of the delay ensuring causality of the algorithm should be chosen before running the algorithm. The error signals M1 M2 e (i)and e (i) are used by adaptive algorithm to update 1 2 parameters of the transfer functions W (z)and W (z)(in L1 L2a L2b 1 2 x(i) e(i) case of optimal design they are used only for observing the u (i) u (i) 1 2 performance of the system). W (z) W (z) 1 2 ++ 3.1. Optimal and suboptimal filter designs Tuning −− 0 e (i) e (i) 1 2 In this subsection, an attempt to derive analytical optimal t (i) and suboptimal filter formulae will be made. The need for −Δ t(i) suboptimal formula will be explained and feasibility of both the solutions will be discussed. Figure 3: Block diagram of virtual unidirectional sound source. 3.1.1. Formulation using transfer functions Suppose that the whole system presented in Figure 3 is linear 3. VIRTUAL UNIDIRECTIONAL SOUND SOURCE and the signal to be cancelled is deterministic, or random wide-sense stationary [8]. Suppose that W and W are It is well known that introducing the second “active” loud- 1 2 infinite impulse response (IIR) filters. Figure 3 shows that speaker allows to deal with many one-loudspeaker systems difficulties [4]. Moreover, the analysis outlined in Section 2 the control values u (i)and u (i) can be expressed as a 1 2 filtration of set point value t(i) by the W and W filters. proves that it allows also for (hypothetically) perfect cancel- 1 2 Using the notation of difference equation, this filtration can lation of the acoustic feedback. One of possible systems using be expressed as two secondary loudspeakers is called virtual unidirectional sound source. u (i) = W (z)·t(i) = w t(i)+ w t(i − 1) +··· 1 1 1,0 1,1 The idea of virtual unidirectional sound source (VUSS) is to use digital signal processing algorithm to drive two loudspeakers in such way that the sound produced by them = w t(i − n), 1,n propagates only downstream the duct [7]. In fact, although n=0 (7) the sound generated by each loudspeaker propagates in both u (i) = W (z)·t(i) = w t(i)+ w t(i − 1) +··· 2 2 2,0 2,1 directions, the processing algorithm tries to assure that the sound waves propagating upstream the duct are actively = w t(i − n), 2,n cancelled by themselves while those propagating downstream n=0 the duct are amplified. The additional advantage of this approach is that it is sometimes possible to equalise the where amplitude spectrum of the secondary path transfer function −1 −2 (transfer function between signals t(i)and e(i)on Figure 3) W (z) = w + w z + w z +··· , 1 1,0 1,1 1,2 (8) that plays very important role in active control of sound [4]. −1 −2 W (z) = w + w z + w z +··· . 2 2,0 2,1 2,2 The block diagram of VUSS has been shown in Figure 3. The system contains two “active” loudspeakers, L2a and L2b, In the above equation, w and w are nth elements of the 1,n 2,n as well as two microphones: M1, generally called reference W and W filter impulse response functions, and are also 1 2 microphone, and M2 called error microphone (the names −1 called filter coefficients. Using this notation, the z is treated of the microphones correspond to the role they play in −n as discrete shift operator so that z t(i) = t(i − n). active noise control). The signal to be processed by VUSS From Figure 3 it also follows that the reference signal is the set point value (or antinoise signal) t(i)produced x(i) due to the secondary sources measured by the M1 by ANC algorithm. It is filtered by two filters with transfer microphone can be expressed with the following difference functions, W (z)and W (z), and sent to the loudspeakers, 1 2 equation: L2a and L2b, respectively. The error signals are obtained by comparing values acquired from the microphones M1 and x(i) = S (z)·u (i)+ S (z)·u (i), (9) 11 1 12 2 M2 with their required values (set points). As the goal of VUSS system is to cancel the influence of the set point value where S (z)and S (z) are impulse response functions of 11 12 t(i) at the microphone M1 point, the signal x(i)iscompared electroacoustical paths between the reference signal x(i)and with zero (e (i) = x(i) − 0 = x(i)). Similarly, as the desired 1 the u (i), u (i) control values, including amplifiers, filters, 1 2 signal at the microphone M2 point is the delayed set point loudspeakers, and others. value, the second error signal is calculated as e (i) = e(i) − Similarly, the signal e(i) measured by the microphone M2 t (i). The need for delaying the set point value t(i)comes o due to the secondary sources can be given by from condition on causality of the system: no filtration can compensate for the delay time introduced by the acoustic e(i) = S (z)·u (i)+ S (z)·u (i). (10) 21 1 22 2 4 Advances in Acoustics and Vibration Considering the above and the definitions of error signals Moreover, assuming the S transfer functions are of FIR nn given in the previous subsection, the error signals can be type, the suboptimal solutions are also FIR type. expressed as It is, however, important to notice that in case of suboptimal solution the secondary path transfer function e (i) = S ·u (i)+ S ·u (i), 1 11 1 12 2 (transfer function between signals t(i)and e(i)on Figure 3) (11) is given by e (i) = S ·u (i)+ S ·u (i) − t (i), 2 21 1 22 2 o S(z) =−S ·S + S ·S ,or 12 21 11 22 where the dependence of S transfer functions on z has been nn (17) S(z) = S ·S − S ·S . omitted to simplify the notation. 12 21 11 22 The goal of the tuning algorithm is to drive error signals (Again, the dependence of S transfer functions on z has nn to zero: been omitted to clarify the notation.) This shows that in case of no demand for equalisation of the secondary path transfer e (i) = 0, e (i) = 0. (12) 1 2 function amplitude spectrum it can assume arbitrary shape, potentially “worse” than in case of single secondary source. Substituting (7)and (11) into (12), recognising that −Δ This effect must be taken into account when estimating t (i) = z t(i)gives forall t(i)= 0 o / secondary path models required by active noise control S ·W (z)+ S ·W (z) = 0, algorithm (e.g., by using longer filters). 11 1 12 2 (13) −Δ S ·W (z)+ S ·W (z) = z , 21 1 22 2 3.1.2. Formulation using reference signal matrix where Δ samples delay time was introduced to assure The following subsection presents result obtained using the causality of the system. approach presented in [3, 9], which is based on impulse Solving the above equation set gives the transfer func- response of appropriate electroacoustic paths and on the tions of optimal filters: use of filtered-reference signals. Using this approach, the lth error signal (l ={1, 2} in this case) can be expressed as a −Δ sum of desired signals and contributions from m secondary W (z) =−z , 1opt S ·S − S ·S 11 22 12 21 sources (with m = 2 in this case) as: (14) −Δ 2 J−1 W (z) = z . 2opt S ·S − S ·S 11 22 12 21 e (i) =−d (i)+ s (j )u (i − j ), (18) l l lm m m=1 j=0 Unfortunately, (14) cannot be used in practise due to non- minimumphase nature of transfer functions S leading to nn where s (j ) is the j th coefficient of impulse response of lm instability of VUSS filters. Moreover, even when all the electroacoustic path between the lth sensor and the mth transfer functions S are minimumphase, the subtraction in nn actuator and d (i)isdesired valueofthe lth sensor signal. denominator of (14) can still result in instability of the whole The length of impulse response can be arbitrary number J filter. This is referred to in the literature as “unconstrained guaranteeing desired accuracy. controller” [3]. In the same manner, the signals driving actuators can be The suboptimal design of virtual unidirectional sound expressed as source can be performed after omitting the second condition N−1 from (12) set in mathematical derivations. In consequence, it u (i) = w t(i − n), (19) m m,n means we do not expect the transfer function from set point n=0 value t(i) to microphone M2 signal e(i)tobepuredelay, but we allow it to take more complex form. Furthermore, it where w is the nth parameter of the N th order w FIR m,n m means that the amplitude spectrum of this transfer function filter impulse response. will not be flat over the frequency range under consideration. Substituting (19) into (18) yields However, we still request cancellation of the secondary J−1 2 N−1 sources influence on the reference signal x(i). e (i) =−d (i)+ s (j )w t(i − j − n). (20) l l lm m,n With control goal stated above, there are two equivalent m=1 j=0 n=0 solutions that will be called suboptimal, namely, To derive a matrix formulation for the above equation it is W (z) =−S (z), 1sub 12 necessary to reorganise the order of filtration of the set point (15) value t(i). To do this, it is necessary to assume that the filters W (z) = S (z), 2sub 11 s and w are time invariant. If the filtered reference signal lm m,n or is defined as J−1 W (z) = S (z), 1sub 12 (16) r (i) = s (j )t(i − j ), (21) lm lm W (z) =−S (z). j=0 2sub 11 Dariusz Bismor 5 the error signals can be written as 3.1.3. Formulation in frequency domain 2 N−1 Applying the Fourier transform to (7)and (11)and substi- e (i) =−d (i)+ w r (i − n). (22) l l m,n lm tuting the transformed (7) into (11) yields m=1 n=0 iωT iωT iωT iωT p p p p E e = S e W e T e 1 11 1 After defining iωT iωT iωT p p p + S e W e T e , 12 2 w r (i) 1,i l1 w(i) = , r (i) = . (23) w r (i) 2,i l2 iωT iωT iωT iωT p p p p (33) E e = S e W e T e 2 21 1 Equation (22) can be expressed as iωT iωT iωT p p p + S e W e T e 22 2 N−1 iωT −T e e (i) =−d (i)+ w (n)r (i − n). (24) l l l n=0 iωT with z = e ,where T is a sampling period. Theabove equationscan be expressedinmoreconve- Finally, the vector of two error signals can be defined as nient matrix form as e (i) e (i) e(i) = 1 2 (25) iωT iωT iωT iωT iωT p p p p p e e =−d e + S e W e T e , (34) and the vector of two desired signals can be defined as iωT iωT iωT p p p where e( e ), d( e ), and W( e ) are transformed vectors defined by (25), (30), and (29), respectively, and the d(i) = d (i) d (i) , (26) 1 2 matrix of transfer functions is defined as ⎡ ⎤ leading to the conclusive formula iωT iωT p p S e S e 11 12 jωT p ⎣ ⎦ S e = . (35) iωT iωT e(i) =−d(i)+ R(i)W, (27) p p S e S e 21 22 where The unconstrained controller can be found by minimising ⎡ ⎤ T T T the cost function equal to expectation of the sum of squared r (i) r (i − 1) ··· r (i − N +1) 1 1 1 ⎣ ⎦ R(i) = (28) errors, independently at each frequency [3]: T T T r (i) r (i − 1) ··· r (i − N +1) 2 2 2 H H J = E e e = trace E ee . (36) is called matrix of filtered reference signals, and Thus, the optimal controller is equal to T T T w (0) w (1) ... w (N − 1) W = (29) iωT −1 iωT iωT −1 iωT p p p p W e = S e S e S e , (37) opt td tt is the vector containing all the coefficients of both W and where matrices of power and cross spectral densities are W filters (the filters are time independent). defined as It is interesting to notice that in case of virtual unidi- rectional sound source the vector of desired signals can be iωT iωT iωT p p ∗ p S e = E d e T e , td instantiated as (38) iωT iωT ∗ iωT p p p S e = E T e T e . tt d(i) = . (30) t(i − Δ) Assume now that the set point value t(i) used during VUSS learning phase is a sequence of white noise with expected Following the methodology presented in [9] the optimal value σ . The power spectral density matrix S is then equal tt filter solution can be expressed as to −1 T T W = E R (i)R(i) E R (i)d(i) , (31) iωT 2 opt p S e = σ , (39) tt where E{·} denotes expectation operator. independent on ω, and the cross spectral density matrix S td Thematrixtobeinvertedisofdimension N × N,where is equal to N is the filter length. Fortunately, it can be proved that the matrix is block Toeplitz matrix [3], so effective iterative iωT S e = . (40) methods can be applied for inversion. The expectation td −iωT Δ 2 e σ operator tells that statistical properties of filtered set point value t(i) will be taken into account. The expression under After evaluating inverse of transfer function block matrix iωT the right-hand side expectation operator, in the above p S( e ), the optimal controller obtained with white noise equation, takes particularly the simple form of excitation becomes −iωT Δ T T T T e −S R (i)d(i) = r (i) r (i − 1) ··· r (i − N +1) t(i − Δ). iωT 2 2 2 W e = , (41) opt S S − S S 11 11 22 12 21 (32) 6 Advances in Acoustics and Vibration iωT M1 L2a L2b M2 where the dependence of S transfer function on e has nn been omitted to clarify the notation. The above equation is consistent with the result obtained L1 in Section 3.1.1. It is, however, still the unconstrained x(i) u (i) u (i) e(i) 1 2 form of the controller which is not feasible for practical W (z) W (z) 1 2 implementation. t(i) 3.2. Adaptive filter solution F (z) This subsection presents an alternative approach to optimal + 0 Adaptation and suboptimal filter designs discussed above. Now we will try to develop an adaptive algorithm with the goal defined by Figure 4: Active noise control system using virtual unidirectional set of (12). We will assume that W (z)and W (z)are in form 1 2 sound source. of finite impulse response filters so the filter coefficients can be stored in a vector similar to this defined by (29), but now with coefficients varying slowly (compared to timescales of result has been shown by Wang and Ren [10], the sufficient plant dynamics) in time. condition for stability of adaptation. Nevertheless it is not a Although there are many algorithms that can be used necessary condition and it has been obtained with the small to tune filter coefficients [4], two-channel filtered-x LMS step size assumption. algorithm was chosen for the following derivation and During this research it was useful to introduce an experiments. Its advantages are simplicity and robustness, additional weight parameter β ∈ (0, 1) to specify which even if speed of convergence is not the best. Using two- of the goals defined in (12)shouldhavemorebearing on channel FXLMS algorithm, the update equation is given by adaptation process. For such case the definition of error vector e(i) was modified as follows: W(i +1) = W(i)+ μR (i)e(i), (42) ⎡ ⎤ (1 − β)e (i) (1 − β)0 where μ is step size and R(i)and e(i)are defined by (28)and ⎣ ⎦ e (i) = = e(i) = Be(i). (45) 0 β (25), respectively. βe (i) The matrix of reference signals R(i) is generated by filtration of the set point value by electroacoustic path When β is close to zero, the algorithm is better excited along transfer functions S (z)to S (z). In practise, the latter the modes responsible for neutralisation of the acoustic 11 22 can be only estimated yielding a matrix of estimated plant feedback effect, while when β is close to one the algorithm responses R(i). The estimation must take place before puts more stress on equalising the secondary path transfer function. running FXLMS algorithm and is usually done by separate identification procedure. The same procedure can also be Introducing the weight parameter β modifies the steady- state vector of filter coefficients giving turned on during active noise control phase after detecting substantial changes in plant dynamics (e.g., changes in air −1 temperature). W = E[R(i)BR(i)] E[R(i)Bd(i)]. (46) To study convergence properties of the above algorithm, it is necessary to substitute (27) into (42) with estimated The maximum step size parameter, μ, assuring convergence plant responses matrix, producing of the whole algorithm is given by T T 2Re λ W(i +1) = W(i)+ μ − R (i)d(i)+ R (i)R(i)W(i) . (43) max 0 <μ<   , (47) max If the adaptive algorithm is stable, it will converge to the solution setting the expectation value of the term in bracket where λ denotes maximum eigenvalue of R(i)BR(i) max to zero [3]. The steady-state vector of filter coefficients will matrix rather than R(i)R(i) as was in the original solution therefore be equal to [3]. The weight parameter β can therefore influence, to some degree, the eigenvalues of the matrix under consideration. −1 W = E R(i)R(i) E R(i)d(i) . (44) 4. ACTIVE NOISE CONTROL WITH VUSS The above result for the adaptive algorithm steady state is in coincidence with the optimal solution presented in 4.1. Active noise control algorithms Section 3.1.2 (31) if and only if the estimated matrix of reference signals R is equal to the true matrix of reference Feedforward active noise control system using virtual uni- signal. This validates, however, the methodology. directional sound source has been shown in Figure 4. This The precise consequences of nonperfect matching of true system uses a finite impulse response filter as control filter plant responses contained in the matrix of reference signals with different adaptation algorithms. As in case of all on adaptation process are still unknown. The strongest feedforward algorithms, only the reference value x(i) is used Dariusz Bismor 7 by the control filter F (z) to produce the set point value t(i). VUSS algorithm has to “pay more attention” to neutralising The set point value is then processed by VUSS filters W (z) acoustic feedback and secondary path equalisation is only its and W (z)togivetwo controlvalues u (i)and u (i) that additional task. 2 1 2 are amplified and sent to the loudspeakers (via amplifiers and reconstruction filters not shown on the figure). The RLS Algorithm adaptation algorithm on the other hand uses the reference The second algorithm of ANC filter adaptation algorithms signal x(i) as well as the error signal e(i) to tune the control filter coefficients. tested was recursive least squares (RLS) algorithm (see, e.g., The duct used in experiments described below was made [9]). RLS algorithm uses very similar update equation in the outofwood.Itwas 4 m long,with0.2 × 0.4 m rectangular form section. One of the duct ends was terminated with noise f (i +1) = f (i)+ k(i)·e(i), (49) generating loudspeaker, while the other was opened. The attenuating loudspeakers were of 0.16 m diameter and were where k(i) is the gain vector showing how much the value of situated approximately in the middle of the duct. The e(i) will modify different filter coefficients. The gain vector is distance between the middles of the loudspeakers was equal calculated as to 0.3 m. The reference microphone was located 1.23 m from the middle of L2a loudspeaker, while the error microphone k(i) = P(i)·x(i), (50) was separated from L2b loudspeaker by the distance of 0.23 m. where P(i) is a matrix updated in each step according to the It should be emphasised that the goal of active noise following equation: control algorithm is in contradiction to the goal of the virtual unidirectional sound source adaptation algorithm 1 P(i − 1)x(i − 1)x (i − 1)P(i − 1) P(i) = P(i − 1) − . [11]. The former tries to assure the compensation of sound T λ λ + x (i − 1)P(i − 1)x(i − 1) waves from both the primary and secondary sources at the (51) microphone M2 point without taking any notice of what happens at the microphone M1 point. The goal of the The RLS algorithm usually converges faster than the LMS latter is to compensate sound waves from the secondary algorithm [12]. sources only at the microphone M1 point and try to assure Because RLS algorithm needs to update P matrix of the set point value appearing without any alternation at size equal to the number of filter coefficients in each the microphone M2 point. This leads to a conclusion that adaptation step, it requires computational effort of N , both the algorithms should never operate at the same time. whereas LMS algorithm requires computational effort of N Indeed, the experiments show that when both the adaptation only [13]. There is, however, a family of RLS algorithm algorithms are in operation the whole system goes unstable. implementations called fast RLS algorithms that allows to Therefore, the VUSS adaptation algorithm is usually run omit computation of P matrix and use a selection of row on the beginning of experiments and switched off after vector instead [13, 14]. This work used one of such fast algo- adaptation is completed. Next, only ANC algorithm is in rithms, called fast transversal filter [15]. The algorithm was use. Only when significant changes in the environment are parametrised as follows. The initial value of the minimum detected (e.g., temperature change above some level), the sum of backward a posteriori prediction-error squares was ANC adaptation is temporarily frozen and the VUSS is equal to 1 and the exponential weighting factor was equal to retuned. 0.9999. The algorithm showed no stability issues, as it was expected on floating-point arithmetic platform. Both the FXLMS and RLS algorithms were implemented FXLMS algorithm in C programming language. Testing and debugging were The first of ANC filter adaptation algorithms tested was performed using developed simulation platform for Linux operating system. The platform allowed to emulate DSP Filtered-x LMS algorithm [4, 9]. Assume that the F (z)isan processor board behaviour and therefore the next step, FIR filter with coefficients (see Figure 4) in the ith sample of time stored in a form of a vector f (i). In that case, the FXLMS moving the code onto Texas Instruments TMS320C31 pro- cessor board, was purely automatic. Another PC-computer algorithm update equation is given by program was used to tune various algorithm parameters f (i +1) = f (i)+ μ·x (i)·e(i), (48) and to acquire the results. The sampling frequency of 2 kHz was chosen as frequency band up to 1 kHz was of the where μ is the step size and x (i) is the vector of reference author interest. It appeared that, due to hardware limitations, signal x(i) samples filtered by the secondary path transfer appliable filter lengths were up to 70 with this sampling function estimate, that is, an estimate of the transfer function frequency. between set point value signal t(i) and error signal e(i). If VUSS algorithm was working perfectly, the secondary 4.2. Testing sounds path transfer function would be equal to simple time delay. Unfortunately, the study from Section 3.1 leads to The set of signals chosen for experiments is presented with a conclusion that such perfect situation is impossible: the dashed line on Figures 12–17. Testing signals N1 and N2 8 Advances in Acoustics and Vibration Table 1: The attenuation in simulations. Signal Analytical solution + FXLMS Analytical solution + FXLMS Analytical solution + RLS N1 9dB 15dB 16dB N2 16 dB 30 dB 33 dB N3 10 dB 15 dB 16 dB N4 15 dB 22 dB 22 dB N5 9dB 17dB 18dB N6 6dB 9 dB 8dB 0.6 7 0.5 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz) Figure 5: Amplitude spectrum of transfer function between set 1 point value t(i)and x(i) (solid) and e(i)(dotted). 0 200 400 600 800 1000 Frequency (Hz) Figure 7: Noise N5 attenuation using analytical VUSS solution. — Spectrum with ANC, - - - spectrum without ANC. 2.5 with 250 Hz being the dominant. The N5 signal was recorded in water power plant turbine proximity. It has the harmonics 1.5 of 100 Hz distinguishable among broadband noise. The last signal N6 was recorded in small bureau with an electric propeller fan turned on. It is similar to N5, but the dominant frequency is higher. All the signals were filed using 1000 samples. This ensem- 0.5 ble was repeated many times and (after amplification) sent to loudspeaker L2 (see Figure 4). The results of attenuation were measured with microphone M2 and calculated as 0 200 400 600 800 1000 Frequency (Hz) MSV of e(i) without attenuation T = 10 log [dB]. (52) MSV of e(i)withattenuation Figure 6: Noise N2 attenuation using analytical VUSS solution. — Spectrum with ANC, - - - spectrum without ANC. 4.3. Simulation results The simulations were performed using 150th order FIR were generated offline as white noise filtered with high- filters models of the duct paths, identified offline using order bandpass filter. Bandpass was 200–300 Hz in case of N1 methodology presented in [16]. The algorithm for subopti- signal and 700–800 Hz in case of signal N2. The former was mal VUSS filter solution described in Section 3.1 was tested chosen to show low-frequency attenuation capabilities and first. As expected, it allowed for very effective acoustical the latter to show high frequency attenuation. The N3 is a feedback cancellation but at the cost of the secondary path signal acquired in close vicinity of a food processor. It has the being substantially degenerated. The amplitude spectrum of dominant frequency of about 270 Hz, but with substantial transfer function between set point value t(i) and both the amount of broadband noise. The N4 is a signal collected near reference signal x(i) and error signal e(i)ispresented on a power transformer. It has all the harmonics of 50 Hz visible Figure 5. Amplitude Amplitude Amplitude Dariusz Bismor 9 4 45 3.5 2.5 1.5 0.5 0 0 0 200 400 600 800 1000 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 8: Noise N2 attenuation using adaptive VUSS solution. — Figure 10: Attenuation (in dB) of tonal sounds for adaptive filter Spectrum with ANC, - - - spectrum without ANC. tuning. — RLS algorithm, - - - FXLMS algorithm. 4 10 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 11: Comparison of attenuation of tonal sounds for analyt- ical and adaptive VUSS filter tuning. — FXLMS algorithm with Figure 9: Noise N5 attenuation using adaptive VUSS solution. — Spectrum with ANC, - - - spectrum without ANC. analytical tuning, - - - RLS algorithm with analytical tuning, ... RLS algorithm with adaptive tuning. After tuning phase, the system was engaged to active was also infinite in case of adaptive VUSS filters tuning. noise control algorithms: FXLMS and Fast RLS. The ANC But the attenuation of testing signals was slightly better: filter having 70 coefficients proved to be long enough to 9–30 dB in case of FXLMS algorithm and 8–33 dB in case cancel out the noise in case of tonal sounds, so this was the of RLS algorithm, see Table 1. The examples of spectra of chosen value. However, in case of N1–N6 signals attenuation the signals observed at the error microphone before and was only (as for simulations of feedforward controller) 6– after attenuation have been shown on Figures 8 and 9.The 16 dB (with FXLMS algorithm, which proved to be better), difference in amplitude spectrum of the N2 and N5 signals see Table 1. The examples of spectra of the signals observed without attenuation visible on these figures and on Figure 6 at the error microphone before and after attenuation have and 7 comes from the fact that this is the spectrum measured been presented on Figures 6 and 7. with the M2 microphone, not the spectrum of the signal In the following experiments, adaptive algorithm driving the L1 loudspeaker. described in Section 3.2 was responsible for VUSS filters Table 1 summarises the results obtained during sim- tuning. The value of the β parameter (see (45)) has been ulations for broadband signals N1–N6. The simulation chosen as 0.5 to put the same effort into satisfying both experiments proved adaptive VUSS filters tuning superiority the goals defined by (12). The attenuation of tonal sounds over analytical suboptimal filter design. Amplitude Amplitude Attenuation (dB) Attenuation (dB) 10 Advances in Acoustics and Vibration 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 12: Noise N1 (broadband noise 200–300 Hz) attenuation. Figure 14: Noise N3 (food processor) attenuation. — Spectrum — Spectrum with ANC, - - - spectrum without ANC. with ANC, - - - spectrum without ANC. 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 15: Noise N4 (power transformer) attenuation. — Spec- Figure 13: Noise N2 (broadband noise 700–800 Hz) attenuation. trum with ANC, - - - spectrum without ANC. — Spectrum with ANC, - - - spectrum without ANC. figure shows that in frequency range from 150 to 950 Hz the 4.4. Real experiments attenuation was between 14 and 33 dB (average 25 dB) for To perform the real experiments using analytical filter design FXLMS ANC algorithm and between 15 and 44 dB (average it was necessary to identify the models of electroacoustic 30 dB) for RLS ANC algorithm. paths S (z) − S (z) (see Figure 3) first. The identification Thedropinattenuation aboveafrequencyofabout 11 22 of FIR models of these paths was performed using LMS 420 Hz, especially distinct in case of RLS algorithm, can algorithm prior to each experiment. Moreover, in case of be explained as the ducts first cut-on frequency appears at both the analytical and adaptive filter designs it was necessary 425 Hz. The figure shows that the performances of both to identify the model of the secondary path before the ANC. FXLMS and RLS algorithms above first cut-on frequency The identification of FIR model of this path was performed are similar. However, below a frequency of 400 Hz the after VUSS was tuned by online identification procedure. performance of RLS algorithm is up to 14 dB better than in The procedure, however, was disabled during the ANC. case of FXLMS algorithm. The first ANC laboratory experiments used tonal sounds. The next step was to compare the performance of ANC The frequency range between 100 and 950 Hz has been system in case of analytical and adaptive filter design. During checked with resolution of 20 Hz. The VUSS filter design was these experiments frequency range from 100 to 900 Hz was performed using adaptive method. The attenuation obtained tested with resolution of 100 Hz. The results are shown on during these experiments has been shown on Figure 10.The Figure 11. In all cases the efficiency of ANC algorithm with Amplitude (mV) Amplitude (mV) Amplitude (mV) Amplitude (mV) Dariusz Bismor 11 Table 2: The attenuation in real experiments. Signal Adaptive solution + FXLMS Adaptive solution + RLS N1 12 dB 14 dB N2 7 dB 7 dB N3 4 dB 8 dB N4 13 dB 16 dB N5 3 dB 8 dB N6 3 dB 5 dB 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 16: Noise N5 (power plant turbine) attenuation. — Figure 17: Noise N6 (electric propeller fan) attenuation. — Spectrum with ANC, - - - spectrum without ANC. Spectrum with ANC, - - - spectrum without ANC. one of the strengths of VUSS, the ability to adapt to time analytical VUSS tuning was worse than efficiency of the same algorithm with adaptive VUSS tuning. Therefore, only varying feedback paths. adaptive VUSS design was considered for the following tests. Finally, the noise signals described in Section 4.2 were REFERENCES used in real experiments. Although both FXLMS and RLS ANC algorithms were checked for performance, the only [1] D. Bismor, “Ro ´z ˙nice pomidzy obiektami automatyki prze- spectra presented on Figure 12 through Figure 17 are those mysłowej a obiektami akustycznymi w sw ´ ietle dosw ´ iadczen ´ identyfikacyjnych,” in Materiały XXV Zimowej Szkoły Zwal- obtained with RLS algorithm, as this algorithm performance czania Zagroz ˙en ´ Wibroakustycznych,Gliwice-Ustron, ´ Poland, was superior to FXLMS in all experiments. The attenuation factors for both FXLMS and RLS algorithms are presented in [2] M. T. Akhtar, M. Abe, and M. Kawamata, “On active noise Table 2. control systems with online acoustic feedback path modeling,” IEEE Transactions on Audio, Speech and Language Processing, 5. CONCLUSIONS vol. 15, no. 2, pp. 593–600, 2007. [3] S. Elliott, Signal Processing for Active Noise Control,Academic Press, London, UK, 2001. The idea of virtual unidirectional sound source presented in [4] S.Kuo andD.Morgan, Active Noise Control Systems,John this paper is based on theoretical study of wave propagation Wiley & Sons, New York, NY, USA, 1996. in duct. VUSS itself is a special case of two-reference, two- [5] J.Poshtan,S.Sadeghi, andM.H.Kahaei, “Aninvestigation output system with a detailed analysis presented in Section 3. on the effect of acoustic feedback in a single-channel active Its application to active noise control in an acoustic duct noise control system,” in Proceedings of the IEEE Conference on proved to be effective, resulting in 20–40 dB attenuation of Control Applications (CCA ’03), vol. 1, pp. 430–434, Istanbul, tonal sounds and 5–16 dB attenuation of complex signals Turkey, June 2003. with broadband noise. The best results were obtained with [6] T.Habib,M.Akhtar, andM.Arif, “Acousticfeedback path adaptive VUSS design together with RLS active noise control modeling and neutralization in active noise control systems,” algorithm. These results are comparable with similar results in Proceedings of the IEEE Multitopic Conference (INMIC ’06), reported by other authors. The results, however, do not show pp. 89–93, Islamabad, Pakistan, December 2006. Amplitude (mV) Amplitude (mV) 12 Advances in Acoustics and Vibration [7] V. Valim ¨ aki ¨ and S. Uosukainen, “Adaptive design of a unidirectional source in a duct,” in Proceedings of the 23th International Conference on Noise and Vibration Engineering (ISMA23 ’98), vol. 3, pp. 1253–1260, Leuven, Belgium, September 1998. [8] E. Wong, Procesy Stochastyczne w Teorii Informacji i Układow ´ Dynamicznych, WNT, Warszawa, Poland, 1976. [9] S. Haykin, Adaptive Filter Theory, Prentice-Hall, New York, NY, USA, 4th edition, 2002. [10] A. K. Wang and W. Ren, “Convergence analysis of the multi- variable filtered-X LMS algorithm with application to active noise control,” IEEE Transactions on Signal Processing, vol. 47, no. 4, pp. 1166–1169, 1999. [11] D. Bismor, “Generation of effect of virtual undirectional source of sound using adaptive techniques,” Archives of Control Sciences, vol. 13, no. 2, pp. 215–230, 2003. [12] D. Bismor, “RLS algorithm in active noise control,” in Proceedings of the 6th International Congress on Sound and Vibration (ICSV ’99), Lyngby, Denmark, July 1999. [13] A. H. Sayed, Fundamentals of Adaptive Filtering, John Wiley & Sons, New York, NY, USA, 2003. [14] L. Rutkowski, Filtry Adaptacyjne i Adaptacyjne Przetwarzanie Sygnałow ´ , WNT, Warszawa, Poland, 1994. [15] J. Cioffi and T. Kailath, “Fast, recursive-least-squares transver- sal filters for adaptive filtering,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 32, no. 2, pp. 304–337, 1984. [16] D. Bismor, Adaptive Algorithms for Active Noise Control in an Acoustic Duct, Jacek Skalmierski Computer Studio, Gliwice, Poland, 1999. 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Optimal and Adaptive Virtual Unidirectional Sound Source in Active Noise Control

Advances in Acoustics and Vibration , Volume 2008 – Jun 22, 2008

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Hindawi Publishing Corporation
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Copyright © 2008 Dariusz Bismor. 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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1687-6261
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1687-627X
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10.1155/2008/647318
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2008, Article ID 647318, 12 pages doi:10.1155/2008/647318 Research Article Optimal and Adaptive Virtual Unidirectional Sound Source in Active Noise Control Dariusz Bismor Institute of Automatic Control, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland Correspondence should be addressed to Dariusz Bismor, dariusz.bismor@polsl.pl Received 21 November 2007; Revised 22 February 2008; Accepted 5 May 2008 Recommended by Marek Pawelczyk One of the problems concerned with active noise control is the existence of acoustical feedback between the control value (“active” loudspeaker output) and the reference signal. Various experiments show that such feedback can seriously decrease effects of attenuation or even make the whole ANC system unstable. This paper presents a detailed analysis of one of possible approaches allowing to deal with acoustical feedback, namely, virtual unidirectional sound source. With this method, two loudspeakers are used together with control algorithm assuring that the combined behaviour of the pair makes virtual propagation of sound only in one direction. Two different designs are presented for the application of active noise control in an acoustic duct: analytical (leading to fixed controller) and adaptive. The algorithm effectiveness in simulations and real experiments for both solutions is showed, discussed, and compared. Copyright © 2008 Dariusz Bismor. 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 the method requires a model of the acoustic feedback path and therefore is computationally more expensive. Active noise control is mainly concerned with low-frequency The approach to acoustic feedback cancellation pre- sounds that cannot be suppressed in passive manner at sented in this paper is called virtual unidirectional sound reasonable cost. Other prerequisite for using ANC is that source (VUSS). The method, in authors opinion, has some it does not obstruct media flow in ducts, for example, air advantages over well-known two loudspeaker active noise conditioning ducts. Unfortunately, almost all duct applica- control system in duct applications. First, it clearly separates tions have acoustic feedback; a phenomenon that can heavily the subsystem responsible for neutralisation of undesired decrease or even destroy the results of otherwise properly set- feedback from active noise control part. Second, its adaptive up control system [1, 2]. version can also be applied in case of varying acoustic There are several methods for avoiding acoustic feedback path responses. Last, virtual unidirectional sound source can in duct applications [2–6]. The best one that avoids acoustic also be applied in more general active control of sound feedback completely is to use nonacoustic reference sensor. applications, not only in active noise control. The paper Unfortunately, this usually limits the application to cancel attempts to explain the theory behind VUSS as well as to the tonal sounds and does not allow for the reduction of give some results showing its excellent performance in case broadband noise. Another approach to acoustic feedback is of ANC. to internally process signal from the reference microphone The paper is organised as follows. Section 2 sum- in a way allowing to compensate for this effect [2, 4–6]. marises physical basics necessary to understand principles This technique, usually called “feedback neutralisation,” uses of VUSS operation and its advantage over one-loudspeaker a model of acoustic feedback path to filter the controller applications. In Section 3, the idea of VUSS is explained output and then subtract it from the reference signal. In the together with the original detailed analysis of optimal (fixed) simplest case, the model is obtained offline and does not design (Section 3.1) and adaptive approach (Section 3.2). account for time variation of the feedback path. Moreover, Finally, Section 4 gives the results of simulations and real 2 Advances in Acoustics and Vibration duct, marked as p (x)and p (x). Assume also that the a1+ a2+ sound wave produced by the primary source is propagating downstream the duct, as marked on the figure. 1.5 Assume that the complex pressure of sound wave gener- ated by first “active” loudspeaker is given by −ikx p (x) = Be for x> 0, a1+ (2) ikx 0.5 p (x) = Be for x< 0, a1− and the complex pressure of sound wave produced by the second “active” loudspeaker is given by −1 −0.5 0 0.5 1 −ik(x−l) p (x) = Ce for x> l, a2+ (3) ik(x−l) Figure 1: Acoustic pressure distribution in duct: one loudspeaker p (x) = Ce for x< l, a2− case (solid) and two loudspeaker case (dashed). where B and C are the complex amplitudes of the sound waves. In the situation described above it is possible to set another condition on the system regardless of primary sound source attenuation requirement. In the case of unidirectional p (x) p+ p (x)+ p (x) p (x)+ p (x) a1− a2− a1+ a2+ sound source, it may be the requirement of self-cancellation x = 0 x = l of the secondary sound waves propagating upstream the duct: Figure 2: Acoustic duct with two active loudspeakers. p (x)+ p (x) =0for x< 0. (4) a1− a2− experiments of feedforward active noise control system using In this case, the secondary sources have to be driven in virtual unidirectional sound source. the way providing [3] −ikl B =−Ce . (5) 2. ACOUSTICAL PHENOMENA IN DUCTS In the same time to achieve the main ANC goal, which The simplest solution to acoustic feedback cancellation is to cancel the primary sound downstream the second problem in case where omnidirectional microphone must be loudspeaker completely, it is necessary to set the complex used as the reference sensor is called feedback neutralisation amplitude: [2, 4–6]. With this approach, only one loudspeaker is −A used, but the signal obtained from reference microphone C = . (6) 2i sin(kl) is processed in the way allowing for compensation of its influence. If no compensation is performed, severe problems The acoustic pressure level distribution for two loudspeakers can result. In the simplest case of attenuation of one tone, case has been shown with dashed line on Figure 1.The the pressure distribution upstream secondary source [3] figure shows that the sound pressure has been perfectly takes shape of standing wave, presented with solid line cancelled at coordinates x> l while the absolute value of the on Figure 1. If, for particular frequency of this tone, the pressure level has not been affected by the secondary sources reference microphone is placed in standing wave nodal point, at negative coordinates x. The zone between the secondary it will give measurement equal to zero regardless of waves’ loudspeakers (0 <x <l)isatransientzone. actual magnitude. Although two-loudspeaker system offers significant Another situation occurs when two loudspeakers are improvement over one loudspeaker system, it has its draw- used, as presented in Figure 2. Assume first loudspeaker is backs too. Specifically, the above control law says that for placed in the duct at position x = 0 and the second at some frequencies for which sin(kl) is close to zero, both the position x = l.Denote p (x)as the complex pressure [3]at p+ secondary sources will have to produce sound waves with location x produced by primary source emitting tonal sound: amplitude very large, compared with the amplitude A of the −ikx primary waveform. Consider, for example, the case where p (x) = Ae,(1) p+ the loudspeakers are separated by the distance l = 0.3m. where A is the complex amplitude of the sound wave and In this case the sound waves of frequency 570 Hz and all where acoustic wave number k is equal to k = ω/c ,with ω its multiplies will result in sin(kl) equal to, or close to, being wave angular frequency and c being the sound speed zero, depending on the actual speed of sound, and thus on in air. temperature, humidity, and so on. This fact will certainly Assume the two loudspeakers on Figure 2 produce sound influence the performance of ANC system trying to attenuate waves propagating upstream the duct, marked as p (x) such sound waves (see, e.g., the drop in attenuation just a1− and p (x), and sound waves propagating downstream the below 600 Hz on Figure 10). a2− |p(x)| Dariusz Bismor 3 S S 11 12 path between the loudspeakers and the M2 microphone. The S S 21 22 value of the delay ensuring causality of the algorithm should be chosen before running the algorithm. The error signals M1 M2 e (i)and e (i) are used by adaptive algorithm to update 1 2 parameters of the transfer functions W (z)and W (z)(in L1 L2a L2b 1 2 x(i) e(i) case of optimal design they are used only for observing the u (i) u (i) 1 2 performance of the system). W (z) W (z) 1 2 ++ 3.1. Optimal and suboptimal filter designs Tuning −− 0 e (i) e (i) 1 2 In this subsection, an attempt to derive analytical optimal t (i) and suboptimal filter formulae will be made. The need for −Δ t(i) suboptimal formula will be explained and feasibility of both the solutions will be discussed. Figure 3: Block diagram of virtual unidirectional sound source. 3.1.1. Formulation using transfer functions Suppose that the whole system presented in Figure 3 is linear 3. VIRTUAL UNIDIRECTIONAL SOUND SOURCE and the signal to be cancelled is deterministic, or random wide-sense stationary [8]. Suppose that W and W are It is well known that introducing the second “active” loud- 1 2 infinite impulse response (IIR) filters. Figure 3 shows that speaker allows to deal with many one-loudspeaker systems difficulties [4]. Moreover, the analysis outlined in Section 2 the control values u (i)and u (i) can be expressed as a 1 2 filtration of set point value t(i) by the W and W filters. proves that it allows also for (hypothetically) perfect cancel- 1 2 Using the notation of difference equation, this filtration can lation of the acoustic feedback. One of possible systems using be expressed as two secondary loudspeakers is called virtual unidirectional sound source. u (i) = W (z)·t(i) = w t(i)+ w t(i − 1) +··· 1 1 1,0 1,1 The idea of virtual unidirectional sound source (VUSS) is to use digital signal processing algorithm to drive two loudspeakers in such way that the sound produced by them = w t(i − n), 1,n propagates only downstream the duct [7]. In fact, although n=0 (7) the sound generated by each loudspeaker propagates in both u (i) = W (z)·t(i) = w t(i)+ w t(i − 1) +··· 2 2 2,0 2,1 directions, the processing algorithm tries to assure that the sound waves propagating upstream the duct are actively = w t(i − n), 2,n cancelled by themselves while those propagating downstream n=0 the duct are amplified. The additional advantage of this approach is that it is sometimes possible to equalise the where amplitude spectrum of the secondary path transfer function −1 −2 (transfer function between signals t(i)and e(i)on Figure 3) W (z) = w + w z + w z +··· , 1 1,0 1,1 1,2 (8) that plays very important role in active control of sound [4]. −1 −2 W (z) = w + w z + w z +··· . 2 2,0 2,1 2,2 The block diagram of VUSS has been shown in Figure 3. The system contains two “active” loudspeakers, L2a and L2b, In the above equation, w and w are nth elements of the 1,n 2,n as well as two microphones: M1, generally called reference W and W filter impulse response functions, and are also 1 2 microphone, and M2 called error microphone (the names −1 called filter coefficients. Using this notation, the z is treated of the microphones correspond to the role they play in −n as discrete shift operator so that z t(i) = t(i − n). active noise control). The signal to be processed by VUSS From Figure 3 it also follows that the reference signal is the set point value (or antinoise signal) t(i)produced x(i) due to the secondary sources measured by the M1 by ANC algorithm. It is filtered by two filters with transfer microphone can be expressed with the following difference functions, W (z)and W (z), and sent to the loudspeakers, 1 2 equation: L2a and L2b, respectively. The error signals are obtained by comparing values acquired from the microphones M1 and x(i) = S (z)·u (i)+ S (z)·u (i), (9) 11 1 12 2 M2 with their required values (set points). As the goal of VUSS system is to cancel the influence of the set point value where S (z)and S (z) are impulse response functions of 11 12 t(i) at the microphone M1 point, the signal x(i)iscompared electroacoustical paths between the reference signal x(i)and with zero (e (i) = x(i) − 0 = x(i)). Similarly, as the desired 1 the u (i), u (i) control values, including amplifiers, filters, 1 2 signal at the microphone M2 point is the delayed set point loudspeakers, and others. value, the second error signal is calculated as e (i) = e(i) − Similarly, the signal e(i) measured by the microphone M2 t (i). The need for delaying the set point value t(i)comes o due to the secondary sources can be given by from condition on causality of the system: no filtration can compensate for the delay time introduced by the acoustic e(i) = S (z)·u (i)+ S (z)·u (i). (10) 21 1 22 2 4 Advances in Acoustics and Vibration Considering the above and the definitions of error signals Moreover, assuming the S transfer functions are of FIR nn given in the previous subsection, the error signals can be type, the suboptimal solutions are also FIR type. expressed as It is, however, important to notice that in case of suboptimal solution the secondary path transfer function e (i) = S ·u (i)+ S ·u (i), 1 11 1 12 2 (transfer function between signals t(i)and e(i)on Figure 3) (11) is given by e (i) = S ·u (i)+ S ·u (i) − t (i), 2 21 1 22 2 o S(z) =−S ·S + S ·S ,or 12 21 11 22 where the dependence of S transfer functions on z has been nn (17) S(z) = S ·S − S ·S . omitted to simplify the notation. 12 21 11 22 The goal of the tuning algorithm is to drive error signals (Again, the dependence of S transfer functions on z has nn to zero: been omitted to clarify the notation.) This shows that in case of no demand for equalisation of the secondary path transfer e (i) = 0, e (i) = 0. (12) 1 2 function amplitude spectrum it can assume arbitrary shape, potentially “worse” than in case of single secondary source. Substituting (7)and (11) into (12), recognising that −Δ This effect must be taken into account when estimating t (i) = z t(i)gives forall t(i)= 0 o / secondary path models required by active noise control S ·W (z)+ S ·W (z) = 0, algorithm (e.g., by using longer filters). 11 1 12 2 (13) −Δ S ·W (z)+ S ·W (z) = z , 21 1 22 2 3.1.2. Formulation using reference signal matrix where Δ samples delay time was introduced to assure The following subsection presents result obtained using the causality of the system. approach presented in [3, 9], which is based on impulse Solving the above equation set gives the transfer func- response of appropriate electroacoustic paths and on the tions of optimal filters: use of filtered-reference signals. Using this approach, the lth error signal (l ={1, 2} in this case) can be expressed as a −Δ sum of desired signals and contributions from m secondary W (z) =−z , 1opt S ·S − S ·S 11 22 12 21 sources (with m = 2 in this case) as: (14) −Δ 2 J−1 W (z) = z . 2opt S ·S − S ·S 11 22 12 21 e (i) =−d (i)+ s (j )u (i − j ), (18) l l lm m m=1 j=0 Unfortunately, (14) cannot be used in practise due to non- minimumphase nature of transfer functions S leading to nn where s (j ) is the j th coefficient of impulse response of lm instability of VUSS filters. Moreover, even when all the electroacoustic path between the lth sensor and the mth transfer functions S are minimumphase, the subtraction in nn actuator and d (i)isdesired valueofthe lth sensor signal. denominator of (14) can still result in instability of the whole The length of impulse response can be arbitrary number J filter. This is referred to in the literature as “unconstrained guaranteeing desired accuracy. controller” [3]. In the same manner, the signals driving actuators can be The suboptimal design of virtual unidirectional sound expressed as source can be performed after omitting the second condition N−1 from (12) set in mathematical derivations. In consequence, it u (i) = w t(i − n), (19) m m,n means we do not expect the transfer function from set point n=0 value t(i) to microphone M2 signal e(i)tobepuredelay, but we allow it to take more complex form. Furthermore, it where w is the nth parameter of the N th order w FIR m,n m means that the amplitude spectrum of this transfer function filter impulse response. will not be flat over the frequency range under consideration. Substituting (19) into (18) yields However, we still request cancellation of the secondary J−1 2 N−1 sources influence on the reference signal x(i). e (i) =−d (i)+ s (j )w t(i − j − n). (20) l l lm m,n With control goal stated above, there are two equivalent m=1 j=0 n=0 solutions that will be called suboptimal, namely, To derive a matrix formulation for the above equation it is W (z) =−S (z), 1sub 12 necessary to reorganise the order of filtration of the set point (15) value t(i). To do this, it is necessary to assume that the filters W (z) = S (z), 2sub 11 s and w are time invariant. If the filtered reference signal lm m,n or is defined as J−1 W (z) = S (z), 1sub 12 (16) r (i) = s (j )t(i − j ), (21) lm lm W (z) =−S (z). j=0 2sub 11 Dariusz Bismor 5 the error signals can be written as 3.1.3. Formulation in frequency domain 2 N−1 Applying the Fourier transform to (7)and (11)and substi- e (i) =−d (i)+ w r (i − n). (22) l l m,n lm tuting the transformed (7) into (11) yields m=1 n=0 iωT iωT iωT iωT p p p p E e = S e W e T e 1 11 1 After defining iωT iωT iωT p p p + S e W e T e , 12 2 w r (i) 1,i l1 w(i) = , r (i) = . (23) w r (i) 2,i l2 iωT iωT iωT iωT p p p p (33) E e = S e W e T e 2 21 1 Equation (22) can be expressed as iωT iωT iωT p p p + S e W e T e 22 2 N−1 iωT −T e e (i) =−d (i)+ w (n)r (i − n). (24) l l l n=0 iωT with z = e ,where T is a sampling period. Theabove equationscan be expressedinmoreconve- Finally, the vector of two error signals can be defined as nient matrix form as e (i) e (i) e(i) = 1 2 (25) iωT iωT iωT iωT iωT p p p p p e e =−d e + S e W e T e , (34) and the vector of two desired signals can be defined as iωT iωT iωT p p p where e( e ), d( e ), and W( e ) are transformed vectors defined by (25), (30), and (29), respectively, and the d(i) = d (i) d (i) , (26) 1 2 matrix of transfer functions is defined as ⎡ ⎤ leading to the conclusive formula iωT iωT p p S e S e 11 12 jωT p ⎣ ⎦ S e = . (35) iωT iωT e(i) =−d(i)+ R(i)W, (27) p p S e S e 21 22 where The unconstrained controller can be found by minimising ⎡ ⎤ T T T the cost function equal to expectation of the sum of squared r (i) r (i − 1) ··· r (i − N +1) 1 1 1 ⎣ ⎦ R(i) = (28) errors, independently at each frequency [3]: T T T r (i) r (i − 1) ··· r (i − N +1) 2 2 2 H H J = E e e = trace E ee . (36) is called matrix of filtered reference signals, and Thus, the optimal controller is equal to T T T w (0) w (1) ... w (N − 1) W = (29) iωT −1 iωT iωT −1 iωT p p p p W e = S e S e S e , (37) opt td tt is the vector containing all the coefficients of both W and where matrices of power and cross spectral densities are W filters (the filters are time independent). defined as It is interesting to notice that in case of virtual unidi- rectional sound source the vector of desired signals can be iωT iωT iωT p p ∗ p S e = E d e T e , td instantiated as (38) iωT iωT ∗ iωT p p p S e = E T e T e . tt d(i) = . (30) t(i − Δ) Assume now that the set point value t(i) used during VUSS learning phase is a sequence of white noise with expected Following the methodology presented in [9] the optimal value σ . The power spectral density matrix S is then equal tt filter solution can be expressed as to −1 T T W = E R (i)R(i) E R (i)d(i) , (31) iωT 2 opt p S e = σ , (39) tt where E{·} denotes expectation operator. independent on ω, and the cross spectral density matrix S td Thematrixtobeinvertedisofdimension N × N,where is equal to N is the filter length. Fortunately, it can be proved that the matrix is block Toeplitz matrix [3], so effective iterative iωT S e = . (40) methods can be applied for inversion. The expectation td −iωT Δ 2 e σ operator tells that statistical properties of filtered set point value t(i) will be taken into account. The expression under After evaluating inverse of transfer function block matrix iωT the right-hand side expectation operator, in the above p S( e ), the optimal controller obtained with white noise equation, takes particularly the simple form of excitation becomes −iωT Δ T T T T e −S R (i)d(i) = r (i) r (i − 1) ··· r (i − N +1) t(i − Δ). iωT 2 2 2 W e = , (41) opt S S − S S 11 11 22 12 21 (32) 6 Advances in Acoustics and Vibration iωT M1 L2a L2b M2 where the dependence of S transfer function on e has nn been omitted to clarify the notation. The above equation is consistent with the result obtained L1 in Section 3.1.1. It is, however, still the unconstrained x(i) u (i) u (i) e(i) 1 2 form of the controller which is not feasible for practical W (z) W (z) 1 2 implementation. t(i) 3.2. Adaptive filter solution F (z) This subsection presents an alternative approach to optimal + 0 Adaptation and suboptimal filter designs discussed above. Now we will try to develop an adaptive algorithm with the goal defined by Figure 4: Active noise control system using virtual unidirectional set of (12). We will assume that W (z)and W (z)are in form 1 2 sound source. of finite impulse response filters so the filter coefficients can be stored in a vector similar to this defined by (29), but now with coefficients varying slowly (compared to timescales of result has been shown by Wang and Ren [10], the sufficient plant dynamics) in time. condition for stability of adaptation. Nevertheless it is not a Although there are many algorithms that can be used necessary condition and it has been obtained with the small to tune filter coefficients [4], two-channel filtered-x LMS step size assumption. algorithm was chosen for the following derivation and During this research it was useful to introduce an experiments. Its advantages are simplicity and robustness, additional weight parameter β ∈ (0, 1) to specify which even if speed of convergence is not the best. Using two- of the goals defined in (12)shouldhavemorebearing on channel FXLMS algorithm, the update equation is given by adaptation process. For such case the definition of error vector e(i) was modified as follows: W(i +1) = W(i)+ μR (i)e(i), (42) ⎡ ⎤ (1 − β)e (i) (1 − β)0 where μ is step size and R(i)and e(i)are defined by (28)and ⎣ ⎦ e (i) = = e(i) = Be(i). (45) 0 β (25), respectively. βe (i) The matrix of reference signals R(i) is generated by filtration of the set point value by electroacoustic path When β is close to zero, the algorithm is better excited along transfer functions S (z)to S (z). In practise, the latter the modes responsible for neutralisation of the acoustic 11 22 can be only estimated yielding a matrix of estimated plant feedback effect, while when β is close to one the algorithm responses R(i). The estimation must take place before puts more stress on equalising the secondary path transfer function. running FXLMS algorithm and is usually done by separate identification procedure. The same procedure can also be Introducing the weight parameter β modifies the steady- state vector of filter coefficients giving turned on during active noise control phase after detecting substantial changes in plant dynamics (e.g., changes in air −1 temperature). W = E[R(i)BR(i)] E[R(i)Bd(i)]. (46) To study convergence properties of the above algorithm, it is necessary to substitute (27) into (42) with estimated The maximum step size parameter, μ, assuring convergence plant responses matrix, producing of the whole algorithm is given by T T 2Re λ W(i +1) = W(i)+ μ − R (i)d(i)+ R (i)R(i)W(i) . (43) max 0 <μ<   , (47) max If the adaptive algorithm is stable, it will converge to the solution setting the expectation value of the term in bracket where λ denotes maximum eigenvalue of R(i)BR(i) max to zero [3]. The steady-state vector of filter coefficients will matrix rather than R(i)R(i) as was in the original solution therefore be equal to [3]. The weight parameter β can therefore influence, to some degree, the eigenvalues of the matrix under consideration. −1 W = E R(i)R(i) E R(i)d(i) . (44) 4. ACTIVE NOISE CONTROL WITH VUSS The above result for the adaptive algorithm steady state is in coincidence with the optimal solution presented in 4.1. Active noise control algorithms Section 3.1.2 (31) if and only if the estimated matrix of reference signals R is equal to the true matrix of reference Feedforward active noise control system using virtual uni- signal. This validates, however, the methodology. directional sound source has been shown in Figure 4. This The precise consequences of nonperfect matching of true system uses a finite impulse response filter as control filter plant responses contained in the matrix of reference signals with different adaptation algorithms. As in case of all on adaptation process are still unknown. The strongest feedforward algorithms, only the reference value x(i) is used Dariusz Bismor 7 by the control filter F (z) to produce the set point value t(i). VUSS algorithm has to “pay more attention” to neutralising The set point value is then processed by VUSS filters W (z) acoustic feedback and secondary path equalisation is only its and W (z)togivetwo controlvalues u (i)and u (i) that additional task. 2 1 2 are amplified and sent to the loudspeakers (via amplifiers and reconstruction filters not shown on the figure). The RLS Algorithm adaptation algorithm on the other hand uses the reference The second algorithm of ANC filter adaptation algorithms signal x(i) as well as the error signal e(i) to tune the control filter coefficients. tested was recursive least squares (RLS) algorithm (see, e.g., The duct used in experiments described below was made [9]). RLS algorithm uses very similar update equation in the outofwood.Itwas 4 m long,with0.2 × 0.4 m rectangular form section. One of the duct ends was terminated with noise f (i +1) = f (i)+ k(i)·e(i), (49) generating loudspeaker, while the other was opened. The attenuating loudspeakers were of 0.16 m diameter and were where k(i) is the gain vector showing how much the value of situated approximately in the middle of the duct. The e(i) will modify different filter coefficients. The gain vector is distance between the middles of the loudspeakers was equal calculated as to 0.3 m. The reference microphone was located 1.23 m from the middle of L2a loudspeaker, while the error microphone k(i) = P(i)·x(i), (50) was separated from L2b loudspeaker by the distance of 0.23 m. where P(i) is a matrix updated in each step according to the It should be emphasised that the goal of active noise following equation: control algorithm is in contradiction to the goal of the virtual unidirectional sound source adaptation algorithm 1 P(i − 1)x(i − 1)x (i − 1)P(i − 1) P(i) = P(i − 1) − . [11]. The former tries to assure the compensation of sound T λ λ + x (i − 1)P(i − 1)x(i − 1) waves from both the primary and secondary sources at the (51) microphone M2 point without taking any notice of what happens at the microphone M1 point. The goal of the The RLS algorithm usually converges faster than the LMS latter is to compensate sound waves from the secondary algorithm [12]. sources only at the microphone M1 point and try to assure Because RLS algorithm needs to update P matrix of the set point value appearing without any alternation at size equal to the number of filter coefficients in each the microphone M2 point. This leads to a conclusion that adaptation step, it requires computational effort of N , both the algorithms should never operate at the same time. whereas LMS algorithm requires computational effort of N Indeed, the experiments show that when both the adaptation only [13]. There is, however, a family of RLS algorithm algorithms are in operation the whole system goes unstable. implementations called fast RLS algorithms that allows to Therefore, the VUSS adaptation algorithm is usually run omit computation of P matrix and use a selection of row on the beginning of experiments and switched off after vector instead [13, 14]. This work used one of such fast algo- adaptation is completed. Next, only ANC algorithm is in rithms, called fast transversal filter [15]. The algorithm was use. Only when significant changes in the environment are parametrised as follows. The initial value of the minimum detected (e.g., temperature change above some level), the sum of backward a posteriori prediction-error squares was ANC adaptation is temporarily frozen and the VUSS is equal to 1 and the exponential weighting factor was equal to retuned. 0.9999. The algorithm showed no stability issues, as it was expected on floating-point arithmetic platform. Both the FXLMS and RLS algorithms were implemented FXLMS algorithm in C programming language. Testing and debugging were The first of ANC filter adaptation algorithms tested was performed using developed simulation platform for Linux operating system. The platform allowed to emulate DSP Filtered-x LMS algorithm [4, 9]. Assume that the F (z)isan processor board behaviour and therefore the next step, FIR filter with coefficients (see Figure 4) in the ith sample of time stored in a form of a vector f (i). In that case, the FXLMS moving the code onto Texas Instruments TMS320C31 pro- cessor board, was purely automatic. Another PC-computer algorithm update equation is given by program was used to tune various algorithm parameters f (i +1) = f (i)+ μ·x (i)·e(i), (48) and to acquire the results. The sampling frequency of 2 kHz was chosen as frequency band up to 1 kHz was of the where μ is the step size and x (i) is the vector of reference author interest. It appeared that, due to hardware limitations, signal x(i) samples filtered by the secondary path transfer appliable filter lengths were up to 70 with this sampling function estimate, that is, an estimate of the transfer function frequency. between set point value signal t(i) and error signal e(i). If VUSS algorithm was working perfectly, the secondary 4.2. Testing sounds path transfer function would be equal to simple time delay. Unfortunately, the study from Section 3.1 leads to The set of signals chosen for experiments is presented with a conclusion that such perfect situation is impossible: the dashed line on Figures 12–17. Testing signals N1 and N2 8 Advances in Acoustics and Vibration Table 1: The attenuation in simulations. Signal Analytical solution + FXLMS Analytical solution + FXLMS Analytical solution + RLS N1 9dB 15dB 16dB N2 16 dB 30 dB 33 dB N3 10 dB 15 dB 16 dB N4 15 dB 22 dB 22 dB N5 9dB 17dB 18dB N6 6dB 9 dB 8dB 0.6 7 0.5 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz) Figure 5: Amplitude spectrum of transfer function between set 1 point value t(i)and x(i) (solid) and e(i)(dotted). 0 200 400 600 800 1000 Frequency (Hz) Figure 7: Noise N5 attenuation using analytical VUSS solution. — Spectrum with ANC, - - - spectrum without ANC. 2.5 with 250 Hz being the dominant. The N5 signal was recorded in water power plant turbine proximity. It has the harmonics 1.5 of 100 Hz distinguishable among broadband noise. The last signal N6 was recorded in small bureau with an electric propeller fan turned on. It is similar to N5, but the dominant frequency is higher. All the signals were filed using 1000 samples. This ensem- 0.5 ble was repeated many times and (after amplification) sent to loudspeaker L2 (see Figure 4). The results of attenuation were measured with microphone M2 and calculated as 0 200 400 600 800 1000 Frequency (Hz) MSV of e(i) without attenuation T = 10 log [dB]. (52) MSV of e(i)withattenuation Figure 6: Noise N2 attenuation using analytical VUSS solution. — Spectrum with ANC, - - - spectrum without ANC. 4.3. Simulation results The simulations were performed using 150th order FIR were generated offline as white noise filtered with high- filters models of the duct paths, identified offline using order bandpass filter. Bandpass was 200–300 Hz in case of N1 methodology presented in [16]. The algorithm for subopti- signal and 700–800 Hz in case of signal N2. The former was mal VUSS filter solution described in Section 3.1 was tested chosen to show low-frequency attenuation capabilities and first. As expected, it allowed for very effective acoustical the latter to show high frequency attenuation. The N3 is a feedback cancellation but at the cost of the secondary path signal acquired in close vicinity of a food processor. It has the being substantially degenerated. The amplitude spectrum of dominant frequency of about 270 Hz, but with substantial transfer function between set point value t(i) and both the amount of broadband noise. The N4 is a signal collected near reference signal x(i) and error signal e(i)ispresented on a power transformer. It has all the harmonics of 50 Hz visible Figure 5. Amplitude Amplitude Amplitude Dariusz Bismor 9 4 45 3.5 2.5 1.5 0.5 0 0 0 200 400 600 800 1000 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 8: Noise N2 attenuation using adaptive VUSS solution. — Figure 10: Attenuation (in dB) of tonal sounds for adaptive filter Spectrum with ANC, - - - spectrum without ANC. tuning. — RLS algorithm, - - - FXLMS algorithm. 4 10 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 11: Comparison of attenuation of tonal sounds for analyt- ical and adaptive VUSS filter tuning. — FXLMS algorithm with Figure 9: Noise N5 attenuation using adaptive VUSS solution. — Spectrum with ANC, - - - spectrum without ANC. analytical tuning, - - - RLS algorithm with analytical tuning, ... RLS algorithm with adaptive tuning. After tuning phase, the system was engaged to active was also infinite in case of adaptive VUSS filters tuning. noise control algorithms: FXLMS and Fast RLS. The ANC But the attenuation of testing signals was slightly better: filter having 70 coefficients proved to be long enough to 9–30 dB in case of FXLMS algorithm and 8–33 dB in case cancel out the noise in case of tonal sounds, so this was the of RLS algorithm, see Table 1. The examples of spectra of chosen value. However, in case of N1–N6 signals attenuation the signals observed at the error microphone before and was only (as for simulations of feedforward controller) 6– after attenuation have been shown on Figures 8 and 9.The 16 dB (with FXLMS algorithm, which proved to be better), difference in amplitude spectrum of the N2 and N5 signals see Table 1. The examples of spectra of the signals observed without attenuation visible on these figures and on Figure 6 at the error microphone before and after attenuation have and 7 comes from the fact that this is the spectrum measured been presented on Figures 6 and 7. with the M2 microphone, not the spectrum of the signal In the following experiments, adaptive algorithm driving the L1 loudspeaker. described in Section 3.2 was responsible for VUSS filters Table 1 summarises the results obtained during sim- tuning. The value of the β parameter (see (45)) has been ulations for broadband signals N1–N6. The simulation chosen as 0.5 to put the same effort into satisfying both experiments proved adaptive VUSS filters tuning superiority the goals defined by (12). The attenuation of tonal sounds over analytical suboptimal filter design. Amplitude Amplitude Attenuation (dB) Attenuation (dB) 10 Advances in Acoustics and Vibration 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 12: Noise N1 (broadband noise 200–300 Hz) attenuation. Figure 14: Noise N3 (food processor) attenuation. — Spectrum — Spectrum with ANC, - - - spectrum without ANC. with ANC, - - - spectrum without ANC. 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 15: Noise N4 (power transformer) attenuation. — Spec- Figure 13: Noise N2 (broadband noise 700–800 Hz) attenuation. trum with ANC, - - - spectrum without ANC. — Spectrum with ANC, - - - spectrum without ANC. figure shows that in frequency range from 150 to 950 Hz the 4.4. Real experiments attenuation was between 14 and 33 dB (average 25 dB) for To perform the real experiments using analytical filter design FXLMS ANC algorithm and between 15 and 44 dB (average it was necessary to identify the models of electroacoustic 30 dB) for RLS ANC algorithm. paths S (z) − S (z) (see Figure 3) first. The identification Thedropinattenuation aboveafrequencyofabout 11 22 of FIR models of these paths was performed using LMS 420 Hz, especially distinct in case of RLS algorithm, can algorithm prior to each experiment. Moreover, in case of be explained as the ducts first cut-on frequency appears at both the analytical and adaptive filter designs it was necessary 425 Hz. The figure shows that the performances of both to identify the model of the secondary path before the ANC. FXLMS and RLS algorithms above first cut-on frequency The identification of FIR model of this path was performed are similar. However, below a frequency of 400 Hz the after VUSS was tuned by online identification procedure. performance of RLS algorithm is up to 14 dB better than in The procedure, however, was disabled during the ANC. case of FXLMS algorithm. The first ANC laboratory experiments used tonal sounds. The next step was to compare the performance of ANC The frequency range between 100 and 950 Hz has been system in case of analytical and adaptive filter design. During checked with resolution of 20 Hz. The VUSS filter design was these experiments frequency range from 100 to 900 Hz was performed using adaptive method. The attenuation obtained tested with resolution of 100 Hz. The results are shown on during these experiments has been shown on Figure 10.The Figure 11. In all cases the efficiency of ANC algorithm with Amplitude (mV) Amplitude (mV) Amplitude (mV) Amplitude (mV) Dariusz Bismor 11 Table 2: The attenuation in real experiments. Signal Adaptive solution + FXLMS Adaptive solution + RLS N1 12 dB 14 dB N2 7 dB 7 dB N3 4 dB 8 dB N4 13 dB 16 dB N5 3 dB 8 dB N6 3 dB 5 dB 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) Figure 16: Noise N5 (power plant turbine) attenuation. — Figure 17: Noise N6 (electric propeller fan) attenuation. — Spectrum with ANC, - - - spectrum without ANC. Spectrum with ANC, - - - spectrum without ANC. one of the strengths of VUSS, the ability to adapt to time analytical VUSS tuning was worse than efficiency of the same algorithm with adaptive VUSS tuning. Therefore, only varying feedback paths. adaptive VUSS design was considered for the following tests. Finally, the noise signals described in Section 4.2 were REFERENCES used in real experiments. Although both FXLMS and RLS ANC algorithms were checked for performance, the only [1] D. Bismor, “Ro ´z ˙nice pomidzy obiektami automatyki prze- spectra presented on Figure 12 through Figure 17 are those mysłowej a obiektami akustycznymi w sw ´ ietle dosw ´ iadczen ´ identyfikacyjnych,” in Materiały XXV Zimowej Szkoły Zwal- obtained with RLS algorithm, as this algorithm performance czania Zagroz ˙en ´ Wibroakustycznych,Gliwice-Ustron, ´ Poland, was superior to FXLMS in all experiments. The attenuation factors for both FXLMS and RLS algorithms are presented in [2] M. T. Akhtar, M. Abe, and M. Kawamata, “On active noise Table 2. control systems with online acoustic feedback path modeling,” IEEE Transactions on Audio, Speech and Language Processing, 5. CONCLUSIONS vol. 15, no. 2, pp. 593–600, 2007. [3] S. Elliott, Signal Processing for Active Noise Control,Academic Press, London, UK, 2001. The idea of virtual unidirectional sound source presented in [4] S.Kuo andD.Morgan, Active Noise Control Systems,John this paper is based on theoretical study of wave propagation Wiley & Sons, New York, NY, USA, 1996. in duct. VUSS itself is a special case of two-reference, two- [5] J.Poshtan,S.Sadeghi, andM.H.Kahaei, “Aninvestigation output system with a detailed analysis presented in Section 3. on the effect of acoustic feedback in a single-channel active Its application to active noise control in an acoustic duct noise control system,” in Proceedings of the IEEE Conference on proved to be effective, resulting in 20–40 dB attenuation of Control Applications (CCA ’03), vol. 1, pp. 430–434, Istanbul, tonal sounds and 5–16 dB attenuation of complex signals Turkey, June 2003. with broadband noise. 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Ren, “Convergence analysis of the multi- variable filtered-X LMS algorithm with application to active noise control,” IEEE Transactions on Signal Processing, vol. 47, no. 4, pp. 1166–1169, 1999. [11] D. Bismor, “Generation of effect of virtual undirectional source of sound using adaptive techniques,” Archives of Control Sciences, vol. 13, no. 2, pp. 215–230, 2003. [12] D. Bismor, “RLS algorithm in active noise control,” in Proceedings of the 6th International Congress on Sound and Vibration (ICSV ’99), Lyngby, Denmark, July 1999. [13] A. H. Sayed, Fundamentals of Adaptive Filtering, John Wiley & Sons, New York, NY, USA, 2003. [14] L. Rutkowski, Filtry Adaptacyjne i Adaptacyjne Przetwarzanie Sygnałow ´ , WNT, Warszawa, Poland, 1994. [15] J. Cioffi and T. Kailath, “Fast, recursive-least-squares transver- sal filters for adaptive filtering,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 32, no. 2, pp. 304–337, 1984. [16] D. Bismor, Adaptive Algorithms for Active Noise Control in an Acoustic Duct, Jacek Skalmierski Computer Studio, Gliwice, Poland, 1999. 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