Master Thesis. August 25, Examination committee: Prof. dr. ir. A. de Boer Dr. ir. A.P. Berkhoff Ir. J.J. de Jong PDEng

Transcription

1 Master Thesis Active control of time-varying broadband noise and vibrations with parallel fast-array recursive least squares filters using on-line system identification H.J. Meijer (s89273) August 25, 217 Examination committee: Prof. dr. ir. A. de Boer Dr. ir. A.P. Berkhoff Ir. J.J. de Jong PDEng Document number: ET-MS3/TM-583 University of Twente. Faculty of Engineering Technology (ET) Department of Applied Mechanics

2 Summary Least mean square (LMS) based algorithms are very popular in the context of feedforward broadband active noise control (ANC). Their implementation is straightforward and their computational burden is light. However they are poor at tracking moving noise sources, such as cars and planes. The availability of affordable, high performance digital signal processors makes it possible to implement more advanced signal processing algorithms with better tracking performance. One such an algorithm is the modified filtered reference mixed windowed recursive least squares (mod fx mixed windowed RLS) algorithm, which uses convex mixing of two parallel fast array RLS filters to establish a numerically stable ANC algorithm. This report presents the results of this algorithm combined with online identification of the transfer path between the noise canceling speaker and the error microphone.

3 Contents 1 Introduction Background Motivation Hypothesis Approach Methods Modified filtered reference algorithms Algorithm A: LMS ANC with VSS LMS identification Algorithm B: mixed windowed RLS ANC with VSS LMS identification Algorithm C: mixed windowed RLS ANC with mixed windowed RLS identification Simulation setup Experiment setup Hardware en parameters Calibration Simulation results Simulation results for ANC with offline system identification Simulation results for ANC with online system identification Algorithm A Algorithm B Algorithm C Experimental results Experimental results for ANC with offline system identification Experimental results for ANC with online system identification and a change in the primary path Experimental results for ANC with online system identification and a change in the secondary path Error reduction Experimental results for ANC with offline system identification Experimental results for ANC with online system identification Conclusions Hypothesis

4 5.2 Discussion Outlook Bibliography 25 A Appendices 28 A.1 Least mean squares algorithm A.2 Modified filtered reference least mean squares algorithm A.3 Mixed windowed RLS algorithm

5 Chapter 1 Introduction This chapter introduces the subject of this thesis by presenting some background in Section 1.1, the motivation in Section 1.2, the hypothesis in Section 1.3 and the approach in Section Background Active Noise Control (ANC) uses feedforward to produce a sound cancelling pressure wave with secondary/control speaker from a reference signal measurement coming from the primary/disturbance speaker/source to minimize the sound pressure at the error microphone (Fig. 1.1). Filtered reference Least Means Square (fxlms) (Fig. 1.2) and filtered error LMS algorithms are commonly used for feed forward broadband ANC. They are easy to implement and have a low computational cost. However they have a low rate of convergence and have a poor tracking performance (Elliott, 21; Bjarnason, 1992). The main reason for this low convergence rate is the assumption that both the adaptive filter and secondary path estimate are linear time-invariant and therefore can be interchanged (Elliott, 21). This only holds if the adaptive filter changes slowly in comparison to the secondary path dynamics. In the case of the primary noise source moving relatively to the ANC system or when the reference microphone is moving, the primary Figure 1.1. Diagram of a duct with a disturbance speaker, a reference microphone, a noise cancellation actuator and an error microphone in which P denotes the primary path and G the secondary path Figure 1.2. Block diagram for an adaptive feedforward ANC system with filtered reference (fx) structure.

6 Master Thesis HJ Meijer 6 path can change rapidly, which violates the assumption of a system with slowly variating dynamics. Some examples of moving noise sources are airplanes and cars and examples of ANC with such sources are given by Omoto et al. (22) and Van Ophem and Berkhoff (213). To improve the convergence rate changes to the fxlms algorithm have been proposed in literature, for example the modified fxlms (Bjarnason, 1992). Alternatively, due to the increasing availability of affordable, high performance digital signal processors it also becomes possible to implement more advanced algorithms, such as the modified filtered reference Recursive Least Squares (RLS) algorithm. It has a high rate of convergence, but is computationally demanding. Fraanje et al. (25) showed that the modified filtered RLS algorithm is a special case of a Kalman filter, when it is assumed that there is no uncertainty in the secondary path model. This allows an efficient implementation, namely a fast-array implementation as described by Sayed (23). However this implementation was demonstrated to be numerically unstable. (Van Ophem and Berkhoff, 213) Ophem and Berkhoff (216) present a numerically stable algorithm: the mixed windowed RLS filter. It has a fast convergence rate, fast tracking performance and the linear computational complexity of the fast-array Kalman filter. But it does not suffer from the previously mentioned numerical problems. This is accomplished by two finite length growing memory recursive least squares filters running in parallel. When the control signal is calculated, a convex combination of the two filters is used. The filter parameters are periodically reset to prevent the numerical error growth. The mixing parameters are chosen in such a way that the total available information used to calculate the control signal will be approximately equal at every time instance. The performance of the filter has been demonstrated in numerical simulations and real-time lab experiments. Experiments show that the algorithm performs better numerically, while achieving a comparable convergence rate and tracking performance compared to the fastarray sliding window recursive least squares filter proposed by Sayed (23) and Park et al. (1993). Alternatively, Houacine (1991) uses a regularization approach to solve the numerical instability. In the current state of the mixed windowed RLS algorithm it depends on a secondary path model which has to be identified beforehand. In practical implementations over time the secondary path dynamics will change, for example due to temperature changes. So it is desirable to implement online system identification, which adapts the secondary path model to these changes. There are two different approaches in ANC systems for online secondary path modeling. The first approach involves injecting an identification noise into the ANC system and uses a system identification method to model the secondary path. The second approach models the secondary path from the output of the ANC system, avoiding the injection of additional noise into the ANC system. From a detailed comparison Bao et al. (1993a) concludes that the first approach is better than the second approach at among others convergence rate, speed of response to changes of primary noise (i.e. tracking performance) and computational complexity. Basic technique using identification noise for online secondary path modeling in ANC systems is proposed by Eriksson and Allie (1989). It consists of two adaptive filters: the control filter and the secondary path modeling filter. In this system the injected identification noise acts as a disturbance for the control filter and will degrade the overall performance of the ANC system. So the main challenge is developing an algorithm that can converge

7 Master Thesis HJ Meijer 7 with a low level of identification noise. Improvements have been proposed by Bao et al. (1993b), Kuo and Vijayan (1997) and Zhang et al. (21). These methods introduce an extra adaptive filter, which increases the design complexity. Akhtar et al. (26) presents an LMS algorithm using a variable step size (VSS LMS), which is varied in accordance with the power of the disturbance signal, for the secondary path modeling filter. This algorithm avoids adding a third adaptive filter, yet achieves an improved performance within simulations. Concurrently with the research in this thesis Liebich et al. (217) developed an Kalman algorithm with online model parameter estimation of the uncertainty in the secondary path. The focus of this paper was on the cancellation filter adaptation and assumed a perfect secondary path estimation. 1.2 Motivation Thus in the literature a good method, mixed windowed RLS, is found to deal with the noise canceling of moving noise sources and one, VSS LMS, to deal with changes in the real physical situation of the secondary path. The motivation for this thesis is to further develop the mixed windowed RLS ANC algorithm by combining it with online identification of the secondary path using VSS LMS. This leads to the following hypothesis this thesis investigates. 1. The combined algorithm converges. 2. The combined algorithm can track a change in the primary transfer path. 3. The combined algorithm can track a change in the secondary transfer path. 4. The combined algorithm can achieve a similar error reduction as the algorithm using offline identification. 1.4 Approach This thesis approaches this hypothesis by examining different combinations of (VSS) LMS and mixed windowed RLS with numerical simulations and real-time lab experiments. The scope for this thesis is limited to a SISO implementation, one dimensional acoustic broadband noise and the assumption that the feedback from the secondary/control speaker to the reference microphone is neglected. Some of the numerical simulation results were earlier presented at conferences (Berkhoff et al., 216a,b). The structure of this thesis is as follows. First, Chapter 2 describes the algorithms that are investigated and the methods used for the numerical simulations and real-time lab experiments. Next, Chapter 3 presents the results of the numerical simulations and Chapter 4 those of the real-time lab experiments. Lastly, Chapter 5 uses the results to answer the hypothesis. 1.3 Hypothesis The mixed windowed RLS algorithm can be combined with online identification of the secondary transfer path. This will be investigated by checking the following sub-hypotheses:

8 Chapter 2 Methods This chapter covers the algorithm, software and hardware implementations used in the course of this research. First, Section 2.1 describes the algorithms examined in this thesis, which are based on the modified fx mixed windowed RLS and the modified fxlms ANC algorithms. Next, the software, approach and parameters used for the numerical simulations can be found in Section 2.2. And lastly, Section 2.3 discusses the hardware, approach and parameters used in the real-time lab experiments. 2.1 Modified filtered reference algorithms This section describes three algorithms based on the modified filtered reference least mean squares (mod fxlms) scheme (Elliott, 21; Bjarnason, 1992) and using an identification noise signal, v, for online system identification of the secondary path. This is the transfer path between the control signal and the error signal. Furthermore, all three use adaptive finite impulse response (FIR) filters for the controller and the secondary path model, but differ in the method used to update them. Appendix A.2 gives the shared structure of the three algorithms. Of note are the adaptive control filter, C and the adaptive modeling filter, Ĝ. The control filter reduces the primary/disturbance noise, d, by minimizing the modified error, ε and the modeling filter identifies the secondary path model by minimizing the estimated error, ê. Three algorithms using different combinations of update methods are explained in the following subsections: Algorithm A in Section is the same as Akhtar et al. s algorithm using a LMS update for the control filter and variable stepsize LMS (VSS LMS) update for the secondary path model filter. This is a well performing algorithm and is thus used as comparison for the two following novel algorithms. Algorithm B in Section uses mixed windowed RLS for the control filter update and VSS LMS for the model filter update. Algorithm C in Section uses mixed windowed RLS for both the control as the model filter update Algorithm A: LMS ANC with VSS LMS identification This subsection covers the update method for algorithm A. The control filter is updated with LMS using a constant step size and the secondary path model filter is updated with LMS using a variable step size. A block diagram of the algorithm can be seen in Fig The n c impulse response coefficients of the control filter c i are updated every iteration i.

9 Master Thesis HJ Meijer 9 Figure 2.1. Algorithm A: LMS based ANC system with VSS LMS based online secondary path modeling. The update equation based on LMS is: c i = c i 1 µ cˆr nc,iε i (2.1) where µ c is the constant step size parameter for the control filter, ˆr nc,i is the vector of the past n c samples of the filtered reference signal ˆr i and ε i is the modified error. A more detailed derivation can be found in Appendix A.1 The n g impulse response coefficients of the secondary path model filter ĝ i are updated in a similar way: ĝ i = ĝ i 1 + µ g,i v ng,iê i (2.2) in which µ g,i is a variable step size which is a function of the signal powers of the residual error e i and the estimated error ê i, v ng,i is the vector of the past n g samples of the identification noise signal v i Algorithm B: mixed windowed RLS ANC with VSS LMS identification This subsection presents the update method for algorithm B, which updates the control filter with the mixed windowed RLS algorithm and the secondary path modeling filter with the VSS LMS algorithm. This combination was presented at a conference (Berkhoff et al., 216a). A block diagram is shown in Fig The mixed windowed RLS algorithm uses two parallel growing memory RLS algorithms which are reset every time a window length L has passed. These filters are mixed using mixing parameters α i and β i to create a constant length finite memory RLS algorithm.

10 Master Thesis HJ Meijer 1 Figure 2.2. Algorithm B: mixed windowed RLS based ANC system with VSS LMS based online secondary path modeling. Ophem and Berkhoff (216) gives a complete description of the control algorithm and the update procedure of which a version with the sign convention and symbols used in this thesis can be found in Appendix A.3. According to this procedure the n c impulse response coefficients of the control filter c mix,i are updated every iteration i using the following update equation based on mixed windowed RLS: ε 1,i, ε 2,i. The n g impulse response coefficients of the secondary path model filter ĝ i are updated in the same way as algorithm A in Section 2.1.1: ĝ i = ĝ i 1 + µ g,i v ng,iê i (2.4) c mix,i = α i (c 1,i 1 K 1,i R 1 1,i ε 1,i) + β i (c 2,i 1 K 2,i R 1 2,i ε 2,i) (2.3) where K 1,i, K 2,i are Kalman gains, R 1,i, R 2,i are the expected values of the modified errors in which µ g,i is a variable step size which is a function of the signal powers of the residual error e i and the estimated error ê i, v ng,i is the vector of the past n g samples of the identification noise signal v i.

11 Master Thesis HJ Meijer Algorithm C: mixed windowed RLS ANC with mixed windowed RLS identification This subsection describes the update method for algorithm C, which implements a mixed windowed RLS based update for both the control filter and the secondary path model filter. This combination was presented at a conference (Berkhoff et al., 216b). A block diagram is shown in Fig In the same way as algorithm B in Section the n c impulse response coefficients of the control filter c mix,i are updated every iteration i using the following update equation based on mixed windowed RLS: c mix,i = α i (c 1,i 1 K 1,i R 1 1,i ε 1,i) + β i (c 2,i 1 K 2,i R 1 2,i ε 2,i) (2.5) where K 1,i, K 2,i are Kalman gains, R 1,i, R 2,i are the expected values of the modified errors ε 1,i, ε 2,i. The n g impulse response coefficients of the secondary path model filter ĝ mix,i are updated in a similar way as those of the control filter: ĝ mix,i = α i (ĝ 1,i 1 K m,1,i R 1 m,1,iê1,i) + β i (ĝ 2,i 1 K m,2,i R 1 m,2,iê2,i) (2.6) where K m,1,i, K m,2,i are Kalman gains, R m,1,i, R m,2,i are the expected values of the estimated errors ê 1,i, ê 2,i.

12 Master Thesis HJ Meijer 12 Figure 2.3. Algorithm C: mixed windowed RLS based ANC system with mixed window RLS based online secondary path modeling. 2.2 Simulation setup This section describes the model and parameters used in the simulations. The primary and secondary transfer path models were generated with a model of a duct (Fig. 2.4 with an cross sectional area of 343m 2 and length of 1.372m. It has a reflection coefficient of R =.9 at one end and of R L =.95 at the other. The primary/disturbance, the secondary/control speaker and error mi- Figure 2.4. Diagram of a duct with a disturbance speaker, a noise cancellation actuator and an error microphone

13 Master Thesis HJ Meijer 13 crophone are located.1715m,.5145m and 1.25m respectively from R. The disturbance signal at the primary source consists of random values from a normal distribution with a mean and a standard deviation of 1. This signal is used directly as the reference signal and filtered by the primary transfer path model for the disturbance signal at the error microphone. The simulations use a regularization coefficient of δ =.8, Bartlett windows with window length of L = 6 samples, FIR filters with n w = 25 filter coefficients and a sample frequency of f s = 2Hz. It initializes the expected value of the innovations at R = 1 and the Kalman gain as a zero vector with length n w = 25. in which the goal is to minimize the acoustic pressure at the open ended side of a duct, caused by a white noise source on the closed side of a duct (Fig The duct has a length of 3.1m. At 45cm from the end of the duct a secondary noise source was placed. At the open ended side of the pipe an error microphone was placed and a digital reference signal was used both as the algorithm reference signal x and as the signal driving the white noise source at the closed end of the duct, so that no feedback from the secondary loudspeaker to the reference signal occurs. A sample rate of f s = 2Hz was chosen and a FIR filter with n c = 25 filter coefficients was used for the control filter and a FIR filter with n g = 25 was chosen for the secondary plant model filter. 2.3 Experiment setup This section describes the hardware and parameters used for the experiments in Section And Section describes the procedure and materials used to calibrate the error microphone Hardware en parameters This experiment setup was used earlier in Van Ophem and Berkhoff (213). SISO versions of the algorithms are implemented on an embedded PC running an RT-Linux operating system. They are designed in SIMULINK/- MATLAB using custom blocks that are written in the C programming language. Using the REALTIME WORKSHOP, the code used for the simulation in SIMULINK can be identically used to generate the real-time code. Analog I/O for the system was implemented on a dedicated module and provides 16 analog inputs and 16 analog outputs. The sample rate of the controller was set to 2kHz. The algorithms are tested in an experiment Calibration Calibration of the error microphone (Brüel & Kjær 4193 Fig. 2.5) was done as follows: A calibrator (Fig. 2.6) generates a 1kHz sine wave at 94dB (= 1P a) for an additional microphone connected to a Brüel & Kjær Nexus conditioning amplifier set to 1V/P a (Fig. 2.7) and a Fluke 115 multimeter. The resulting measurement was 126mV (F ig. 2.8) Using the previous setup without the calibrator, but with a tone generator and a Tang Band speaker a 3Hz sine wave at 126mV (= 94dB = 1P a) is generated (Fig. 2.9). This is measured with the error microphone connected to the experimental setup. The experimental setup measured a root mean square (rms) value of.662. So 1P a on the error microphone gives a rms value of.662. This value is used as a conversion factor to obtain the sound wave pressure from the measurement values.

14 Master Thesis HJ Meijer 14 Figure 2.5. The microphone used to measure the error signal (B&K 4193) Figure 2.6. calibrator generating a 1kHz sine wave at 94dB Figure 2.7. Brüel & Kjær Nexus conditioning amplifier set to 1V /P a Figure 2.8. Fluke 115 multimeter measuring the calibration signal Figure 2.9. Setup to calibrate the error microphone using an additional calibrated microphone

15 Chapter 3 Simulation results This chapter presents the simulation results. For all the simulations in this chapter the data is filtered with a first order Butterworth filter and the ANC algorithms are activated at time 1s. They also all use a reference signal x with a rms value of 1, which results in the disturbance noise signal d shown in Fig The change in the primary path can be seen at time 11s. This change simulates moving the primary/disturbance speaker 1 sample (= 1/2Hz 343m/s =.1715m) closer to the error microphone. Section 3.1 shows how the ANC algorithms behave with an offline identified secondary path model and Section 3.2 how they behave with online identification of the secondary path model. 3.1 Simulation results for ANC with offline system identification In this section two algorithms using a offline identified secondary path model are presented to serve as comparison for the algorithms with online identification in Section 3.2. Figure 3.2 shows Algorithm A without online identification, which then becomes identical to the modified fxlms algorithm (Appendix A.2. And in Fig. 3.3 the behavior of Algorithm B and C without online identification, i.e. the RMS [-] time [s] Figure 3.1. Disturbance noise d modified fx mixed windowed RLS algorithm can be seen. All initial filter coefficients of the control filter and the secondary path modeling filter are zero. The reduction of the residual error signal obtained between 1s and 11s starting from zero (i.e. convergence) and the reduction of the residual error signal obtained between 11s and 16s, after the change of the primary path at 11s (i.e. tracking), were observed to be considerably higher in Fig. 3.2 compared to Fig. 3.3

16 Master Thesis HJ Meijer RMS [-] RMS [-] time [s] Figure 3.2. Algorithm A without online identification: Residual error, e (LMS using offline identified model) time [s] Figure 3.4. Part of the residual error due to identification noise w RMS [-] The secondary path identification signal v is generated at the secondary/control speaker when the ANC is activated at time 1s with a rms value of.8 and is reduced at time 3.5s to.1. Fig. 3.4 shows the part of the residual error due to this signal, w time [s] Figure 3.3. Algorithm B,C without online identification: Residual error, e (mixed windowed RLS using offline identified model) 3.2 Simulation results for ANC with online system identification In this section the performance is presented of the ANC algorithms using online identification. They use an identification signal v, of which a high level is needed to quickly identify the secondary path model, but a low level is desired for better error reduction. So for the following simulations the identification noise is designed for these cases. For practical use this will need to be automated. The simulation results for algorithms A, B and C are presented in Section 3.2.1, Section and Section respectively Algorithm A Fig. 3.5 shows the residual error e for ANC with VSS LMS identification, of which the variable step size µ s varies between µ min =.2 and µ max =.5. And Fig. 3.6 shows the residual error e for ANC with LMS, which uses a fixed step size of µ =.2. There are no notable differences between these two figures. The first peak in Fig. 3.5 can be explained by the high level of identification noise (Fig. 3.4). After 3.5 seconds this noise becomes low level and Algorithm A gets very close to the version without online identification (Fig. 3.2).

17 Master Thesis HJ Meijer 17 RMS [-] time [s] Figure 3.5. Algorithm A: Residual error, e (LMS with VSS LMS online modeling) RMS [-] time [s] Figure 3.6. Algorithm A with fixed step size LMS: Residual error, e (LMS with LMS online modeling) Algorithm B Fig. 3.7 shows the residual error e for ANC with VSS LMS identification, of which the variable step size µ s varies between µ min =.2 and µ max =.5. And Fig. 3.8 shows the residual error e for ANC with LMS, which uses a fixed step size of µ =.2. Contrary to Algorithm A some small differences can be observed: Algorithm B with a fixed step size obtains more error reduction between 4.5s and 11s and from time 13.5s converges more smoothly than with a variable step size. And as with the ANC using offline identified models (Fig. 3.3 compared to Fig. 3.2), the convergence and tracking for Algorithm B (Fig. 3.7 and Fig. 3.8) are considerably higher compared to Algorithm A (Fig. 3.7 and Fig. 3.8) Algorithm C Fig. 3.9 shows the residual error e for ANC with mixed windowed RLS identification. The convergence and tracking performance do not significantly differ compared to Algorithm B with a fixed step size (Fig. 3.8). So for the purposes of tracking changes in the primary path Algorithm B with fixed step size performs as well as Algorithm C and is computationally less complex. From Section and Section can be observed that Algorithm A and B, using a variable step size perform similar or even worse compared to their versions with a fixed step size. From the results presented in Akhtar et al. s paper, it was expected to observe an improved performance of using VSS LMS compared to LMS. The simulations in this chapter were not able to reproduce Akhtar s results. This could be due to the combination with mixed windowed RLS, the parameters not being optimized or the initialization of the secondary path model coefficients to zero (Akhtar used secondary path models that were converged to a certain extent). Considering this and the better performance of Algorithm B and C compared to A, only Algorithm B with fixed step size and Algorithm C will be further investigated with real-time lab experiments in Chapter 4.

18 Master Thesis HJ Meijer RMS [-] time [s] Figure 3.7. Algorithm B: Residual error, e (mix windowed RLS with VSS LMS online modeling) RMS [-] time [s] Figure 3.8. Algorithm B with fixed step size LMS: Residual error, e (mix windowed RLS with LMS online modeling) RMS [-] time [s] Figure 3.9. Algorithm C: Residual error, e (mix windowed RLS with mix windowed RLS online modeling)

19 Chapter 4 Experimental results In this chapter the results of the real-time lab experiments are discussed. For all the experiments in this chapter the time domain data is smoothed with a first order Butterworth filter. And the initial control and modeling filter coefficients are zero, unless otherwise stated. They all use a reference signal x, which is sent to the primary/disturbance speaker. It consists of a mean white Gaussian noise signal with a standard deviation of.5. First, Section 4.1 shows how the ANC algorithms with an offline identified secondary path model behave. Next, Section 4.2 presents the behavior in the case of a change in the primary path and Section 4.3 in the case of a change in the secondary path. Lastly, Section 4.4 looks into the frequency domain data and average error reduction. pressure [Pa] time [s] Figure 4.1. fxlms with offline identified model: Convergence of residual error, e Experimental results for ANC with offline system identification pressure [Pa] Fig. 4.1 shows the convergence of the residual error for the fxlms algorithm with offline identified model and Fig. 4.2 for the Algorithm B,C without online identification (mixed windowed RLS algorithm). At time 18s the ANC algorithm is activated, which sends a control signal to the secondary/control speaker. Fig. 4.1 converges to time [s] Figure 4.2. Algorithm B,C without online identification: Convergence of residual error, e (mixed windowed RLS using offline identified model).

20 Master Thesis HJ Meijer 2 a smaller error than Fig. 4.2, but not significantly. Considering the performance of LMS based ANC in the simulations, this was unexpected and needs further investigation. 4.2 Experimental results for ANC with online system identification and a change in the primary path pressure [Pa] time [s] Figs. 4.3 and 4.4 show the convergence and tracking (between 29s and 4s) behavior of the residual error for the mixed windowed RLS algorithm. Fig. 4.3 shows Algorithm B with fixed step size and Fig. 4.4 Algorithm C (mixed windowed RLS). At time 9s the ANC algorithm is activated, which produces a control signal and a system identification noise sent to the secondary/control speaker. This system identification noise consists of a mean white Gaussian noise signal with a standard deviation of.1. At time 19s the system identification noise is reduced to a signal with a standard deviation of.25. At time 29s a sample delay is introduced to the signal sent through the primary/disturbance speaker. This has the same effect as suddenly increasing the distance between the primary/disturbance speaker and the error microphone (i.e. the primary path) with 17cm Algorithm C converges faster and to a smaller error between 9s and 19s than Algorithm B with fixed step size. Between 19s and 29s there only a small difference between the algorithms. Finally, between 29s and 4s Algorithm C again outperforms Algorithm B with fixed step size. Figure 4.3. Algorithm B with fixed step size: Tracking behavior for a sample delay in the primary path of residual error, e (mixed windowed RLS with LMS online modeling). pressure [Pa] time [s] Figure 4.4. Algorithm C: Tracking behavior for a sample delay in the primary path of residual error, e (mixed windowed RLS with mixed windowed RLS online modeling). 4.3 Experimental results for ANC with online system identification and a change in the secondary path Figs. 4.5 and 4.6 show the convergence of the residual error for the mixed windowed RLS algorithm. Fig. 4.5 uses an LMS algorithm and

21 Master Thesis HJ Meijer 21 Fig. 4.6 the mixed windowed RLS algorithm for the online identification of the secondary path model to filter the reference signal. The figures first show the disturbance signal without control. The signal is used as reference signal and is sent to the primary/disturbance speaker. It consists of a mean white Gaussian noise signal with a standard deviation of.5. After approximately 9 seconds the ANC algorithm is activated, which produces a control signal and a system identification noise sent to the secondary/control speaker. This system identification noise consists of a mean white Gaussian noise signal with a standard deviation of.1. The control and secondary path modeling filters are initialized with their coefficients at zero. After about 19 seconds the system identification noise is reduced to a signal with a standard deviation of.25. Around 29 seconds the system identification noise is increased to the original standard deviation of.1 and a sample delay is introduced to the signal sent through the secondary/control speaker. Lastly, after 39 seconds the standard deviation of system identification noise signal is again reduced to.25. It can be seen in Fig. 4.5 that the algorithm using LMS online identification becomes unstable, when the sample delay is introduced in the secondary path. At the 31 second mark the ANC algorithm is disabled. pressure [Pa] time [s] Figure 4.5. Algorithm B with fixed step size: Behavior of residual error, e for a sample delay in the secondary path at time 28s. Due to unstable behavior ANC is deactivated at time 31s (mixed windowed RLS with LMS online modeling). pressure [Pa] time [s] Figure 4.6. Algorithm C: Behavior of residual error, e for a sample delay in the secondary path at time 28s (mixed windowed RLS with mixed windowed RLS online modeling). 4.4 Error reduction This section shows the error reduction with frequency domain data. Section gives the experimental results for two ANC algorithms using an offline identified secondary path model. Next Section shows the noise reduction obtained with algorithms B with fixed step size and C.

22 Master Thesis HJ Meijer Experimental results for ANC with offline system identification Control off Control on: -16. db Fig. 4.7 shows the power spectral density spectra of the residual error with and without control for an fxlms algorithm and Fig. 4.8 for the mixed windowed RLS algorithm. db re 1 Pa 2 /Hz These algorithms use an offline identified secondary path model to filter the reference signal and the spectra are determined from the time domain data shown in Fig. 4.1 and Fig The error reduction is determined from the data in Fig. 4.7: the mean power spectral density with control in relation to the mean spectral density without control results in an average error reduction of 16.dB. Likewise, the data shown in Fig. 4.8 results in an average error reduction of 18.3dB Experimental results for ANC with online system identification Figs. 4.9 and 4.1 show the power spectral density spectra of the residual error with and without control for Algorithm B with fixed step size and Algorithm C respectively. The spectra are determined from 2 samples without control and 2 samples with control using a converged secondary path model and a reduced identification noise signal with a standard deviation of.25. An average error reduction of 14.7dB can be obtained with Algorithm B with fixed step size and an average reduction of 14.8dB with Algorithm C. This is significantly less compared to the 18.3dB reduction of Algorithm B,C (Fig. 4.8) using offline identification frequency [Hz] Figure 4.7. fxlms with offline identified model: Comparison of the power spectrum density with control and without control. (See Fig. 4.1 for the time domain data) db re 1 Pa 2 /Hz Control off Control on: db frequency [Hz] Figure 4.8. Algorithm B,C without online identification: Comparison of the power spectrum density with control and without control (mixed windowed RLS using offline identified model). (See Fig. 4.2 for the time domain data) which the identification noise is further reduced to a standard deviation of.5 resulting in an average error reduction of 2.7dB and 2.5dB respectively. By reducing the identification noise level further, similar noise reduction to the algorithm with offline identification can be achieved. This is shown in Fig and Fig. 4.12, in

23 Master Thesis HJ Meijer Control off Control on: -2.7 db -2-3 Control off Control on: db -4-5 db re 1 Pa 2 /Hz db re 1 Pa 2 /Hz frequency [Hz] Figure 4.9. Algorithm B with fixed step size: Comparison of the power spectrum density with and without control (mixed windowed RLS with LMS online modeling) Control off Control on: db frequency [Hz] Figure Algorithm B with fixed step size and further reduced identification noise level: Comparison of the power spectrum density with and without control (mixed windowed RLS with LMS online modeling and further reduction of the identification noise signal to a standard deviation of.5) Control off Control on: -2.5 db db re 1 Pa 2 /Hz db re 1 Pa 2 /Hz frequency [Hz] Figure 4.1. Algorithm C: Comparison of the power spectrum density with and without control (mixed windowed RLS with mixed windowed RLS online modeling) frequency [Hz] Figure Algorithm C with further reduced identification noise level: Comparison of the power spectrum density with and without control (mixed windowed RLS with mixed windowed RLS online modeling and a further reduction of the identification noise signal to a standard deviation of.5).

24 Chapter 5 Conclusions This chapter answers the main hypothesis by checking the sub-hypotheses with the results from the previous chapter in Section 5.1, offers some disscussion in Section 5.2 and gives an outlook in Section Hypothesis The combined algorithm converges. Both the simulation results in Section 3.2 as the experimental results in Section 4.2 and in Section 4.3 respectively show that the combined algorithm converges. The results are shown for online identification using the LMS and the mixed windowed RLS algorithms. The combined algorithm can track a change in the primary transfer path. The experiment results in Section 4.2 confirm that the combined algorithm can track a change of a sample delay in the primary transfer path. The combined algorithm with online identification using the mixed windowed RLS algorithm shows a faster recovery and lower MSE than using the LMS algorithm. The combined algorithm can track a change in the secondary transfer path. Section 4.3 shows that if the identification noise level is increased to the level used to obtain the initial convergence of the algorithm a sample delay in the secondary path can be tracked by the combined algorithm using mixed windowed RLS for the online identification. The combined algorithm can achieve a similar noise reduction as the algorithm using offline identification. In Section and Section it can be seen that the combined algorithm obtains 14.7dB reduction using online identification with LMS and 14.8dB with mixed windowed RLS. This is significantly less compared to the 18.3dB reduction of the mod fx windowed RLS algorithm with with offline identification. By reducing the identification noise level further, similar noise reduction can be achieved. The stable version of the efficient fast array RLS algorithm can be combined with online identification of the secondary transfer path. From the results obtained in simulations and experiments it can be concluded that in this thesis the stable version of the efficient fast array RLS algorithm is successfully combined with online identification of the secondary transfer path. In the form of LMS that has a convergence rate that satisfies implementations focusing on tracking fast changes in the primary path and in the form of mixed windowed RLS, which can deal with challenges involving fast changes in the secondary path.

25 Master Thesis HJ Meijer Discussion This section offers a critical view of the results obtained in this thesis. To start with, all the simulations and measurements in this thesis consist of single simulations/measurements and can give an indication of behavior, but to obtain a more reliable insight into the behavior of these algorithms sets of simulations/experiments should be done. Flowing from this is the lack of indepth analysis into among others long term stability, robustness and optimal parameter settings. Furthermore, the results were helped by manually manipulating the identification noise level and it is still a question if the same results will obtained with an automated system. expanding the mixed windowed algorithm with online system identification into a full kalman filter by taking the uncertainty in the secondary path model into account (e.g. by applying Liebich et al. (217) online model parameter estimation algorithm). expanding the algorithm to a MIMO ANC system In the simulations in Chapter 3 the variable step size had minimal effect compared to fixed step size. This could have been investigated more thoroughly. Lastly the simulation model was not representative of the experimental setup and simulates higher level of sound pressure, but still satisfies 1-D acoustic wave characteristic. It also simulates the change in the primary in a similar way as the real-time lab experiments. 5.3 Outlook The algorithm was further developed in this thesis, but it is still far from practical implementation. Future research can look into: applying different kinds of noise sources to expand the insight in the algorithm. finding a way to vary the identification noise level and apply it to these algorithms. the possibilities of tracking large and sudden changes in the secondary path.

26 Bibliography M. T. Akhtar, M. Abe, and M. Kawamata. A new variable step size lms algorithmbased for improved online secondary path modeling in active noise control systems. IEEE Transactions on Audio, Speech, and Language Processing, 14(2):72 726, 26. ISSN doi: 1.119/TSA C. Bao, P. Sas, and H. V. Brussel. Comparison of two on-line identification algorithms for active noise control. Proc.Recent Advances in Active Control of Sound Vibration, pages 38 51, 1993a. C. Bao, P. Sas, and H. Van Brussel. Adaptive active control of noise in 3-d reverberant enclosures. Journal of Sound and Vibration, 161(3):51 514, 1993b. A. P. Berkhoff, H. J. Meijer, and S. Van Ophem. Parallel fast-array recursive least squares filters for active noise control with on-line system identification. In W. Krop, O. von Estorff, and B. Schulte-Fortkamp, editors, Proceedings of the INTER-NOISE th International Congress and Exposition on Noise Control Engineering: Towards a Quieter Future, pages , Berlin, 216a. German Acoustical Society (DEGA). A. P. Berkhoff, H. J. Meijer, and S. Van Ophem. Active control of timevarying broadband noise using online system identification with parallel fastarray recursive least squares filters. In P. Sas, D. Moens, and A. Van de Walle, editors, Proceedings of ISMA International Conference on Noise and Vibration Engineering and USD216 - International Conference on Uncertainty in Structural Dynamics, pages , Leuven, 216b. KU Leuven, Departement Werktuigkunde. E. Bjarnason. Active noise cancellation using a modified form of the filtered-x LMS algorithm. Proceedings of Eusipco 92, 6th European Signal Processing Conference, pages , S. J. Elliott. Signal Processing for Active Control. Academic Press, London, 21. L. J. Eriksson and M. C. Allie. Use of random noise for on-line transducer modeling in an adaptive active attenuation system. Journal of the Acoustical Society of America, 85 (2):797 82, R. Fraanje, A. H. Sayed, M. Verhaegen, and N. J. Doelman. A fast-array kalman filter solution to active noise control. International Journal of Adaptive Control and Signal Processing, 19(2 3): , 25. ISSN doi: 1.12/acs.841. A. Houacine. Regularized fast recursive least squares algorithms for adaptive filtering. IEEE Transactions on Signal Processing, 39(4):86 871, S. M. Kuo and D. Vijayan. A secondary path modeling technique for active noise control systems. IEEE Transactions on Speech and Audio Processing, 5(4): , S. Liebich, J. Fabry, P. Jax, and P. Vary. Kalman-like time-domain filter for noise cancellation headphones. In EUSIPCO 217 Proceedings, 217.

27 Master Thesis HJ Meijer 27 A. Omoto, D. Morie, and K. Fujiwara. Behavior of adaptive algorithms in active noise control systems with moving noise sources. Acoustical Science and Technology, 23(2): 84 89, 22. S. V. Ophem and A. P. Berkhoff. A numerically stable, finite memory, fast array recursive least squares filter for broadband active noise control. International Journal of Adaptive Control and Signal Processing, 3 (1):31 45, 216. P. Park, Y. M. Cho, and T. Kailath. Chandrasekhar recursion for structured timevarying systems and its application to recursive least squares problems. In Proceedings of the IEEE Conference on Control Applications, volume 2, pages , A. H. Sayed. Fundamentals of Adaptive Filtering. John Wiley & Sons, Inc., NY, 23. S. Van Ophem and A. P. Berkhoff. Multichannel kalman filters for active noise control. The Journal of the Acoustical Society of America, 133(4): , 213. doi: / M. Zhang, H. Lan, and W. Ser. Cross-updated active noise control system with online secondary path modeling. IEEE Transactions on Speech and Audio Processing, 9(5):598 62, 21.

28 Appendix A Appendices A.1 Least mean squares algorithm From Elliott (21): For an adaptive FIR filter the filter coefficients are adjusted in response to each new set of data to evolve in a direction which minimizes the mean-square error. The filter converges to the optimal filter for stationary signals. And it will readjust its coefficients to track the statistics of nonstationary signals, provided the changes in the statistics occur slowly compared with the convergence time of the adaptive filter. Fig. A.1 shows a block diagram of the LMS algorithm. The most widely used algorithm for adapting FIR filters is the steepest-descent algorithm: c i (n + 1) = c i (n) µ J c i (n) (A.1) where c i are the i coefficients of the adaptive filter, µ is the step size and J is the cost function. The cost function is equal to the average mean square value of the error signal: [ ] J = E e 2 (n) (A.2) e(n) = x nc,i(n)c T i (n) + d(n) = x T n c,i(n)c i (n) + d(n) (A.3) [ ] J = c T i (n)e x nc,i(n)x T n c,i(n) c i (n) [ ] + 2c T i (n)e [x nc,i(n)d(n)] + E d 2 (n) (A.4) J (n) = c i [ ] 2E x nc,i(n)x T n c,i(n)c i (n) + x nc,i(n)d(n) = 2E [x nc,i(n)e(n)] (A.5) Instead of infrequently updating the filter coefficients with an averaged estimate of the gradient, the coefficients are updated at every sample time using an instantaneous estimate of the gradient: e 2 (n) c i (n) = 2x n c,i(n)e(n) This gives the LMS algorithm: (A.6) For the model shown in Fig. A.1 the error, the cost function and the derivative of the cost function with respect to the filter coefficients then become: c i (n + 1) = c i (n) αx nc,i(n)e(n) with α = 2µ the convergence coefficient. (A.7)

29 Master Thesis HJ Meijer 29 Figure A.1. LMS algorithm using the symbol convention used in this thesis A.2 Modified filtered reference least mean squares algorithm This subsection describes the structure of the modified fxlms algorithm. Fig. A.2 shows a block diagram of the modified fxlms algorithm. The adaptive control filter tries to find a set of FIR filter coefficients c i R nc, which minimize the modified error ɛ i. This error is calculated by adding the output ỹ i of the adaptive filter to the estimated disturbance ˆd i : ε i = ˆd i + ỹ i. (A.8) The output of the adaptive filter is calculated by multiplying the filtered reference signal ˆr i with the filter coefficients c i ỹ i = ˆr T n c,ic i. (A.9) ˆr nc,i is a vector with the last n c values of the filtered reference signal: [ ] T ˆr nc,i = ˆr i ˆr i 1 ˆr i nc+1. (A.1) The filtered reference signal ˆr i is calculated by filtering the measured reference signal x i with the estimated secondary path FIR model coefficients ĝ i : ˆr i = x T n c,iĝ i. (A.11) x nc,i is a vector with the last n c values of the measured reference signal: [ x nc,i = x i x i 1 x i nc+1 ] T. (A.12) The estimated value of the error signal without identification noise ê i is obtained by subtracting the estimated identification noise ŵ i from the measured error e i : ê i = e i ŵ i. (A.13) The estimated value of the disturbance ˆd i is calculated by subtracting the estimated output of the secondary path ŷ i from the estimated value of the error signal without identification noise ê i : ˆd i = ê i ŷ i. (A.14) The estimated output of the secondary path ŷ i is determined by filtering the control signal u i with the estimated secondary path FIR model coefficients ĝ i :

30 Master Thesis HJ Meijer 3 Figure A.2. Modified FxLMS algorithm using the symbol convention used in this thesis. ŷ i = u T n c,iĝ i. (A.15) u nc,i is a vector with the last n c values of the control signal: used in this update procedure was used to create Fig. 3.3, but for the other algorithms it was changed to a FIR representation. [ u nc,i = u i u i 1 u i nc+1 ] T. (A.16) The control signal u i is calculated by filtering the measured reference signal x i with the adaptive control filter coefficients c i : u i = x T n c,ic i. (A.17) This filter structure is well known in the context of ANC and has been applied both to filtered reference LMS and RLS algorithms (Elliott, 21; Fraanje et al., 25). A.3 Mixed windowed RLS algorithm In this appendix Table A.1 shows the update procedure for the mixed windowed RLS algorithm using the symbol and sign convention of this thesis. The state space representation