Active Control of Impulsive Noise

Transcription

1 Active Control of Impulsive Noise A project Submitted to the Faculty of the Department of Mathematics and Statistics University of Minnesota Duluth by Ronghua Zhu In partial fulfillment of the requirements for the degree of Master of Science in Applied and Computational Mathematics August, 9

2 Abstract This report presents about the comparison of the performance among four different adaptive algorithms for suppressing impulsive noise. They are filtered-x least mean square (FXLMS) algorithm, least mean M-estimate (LMM) algorithm, filtered-x least mean M-estimate (FXLMM) algorithm, and a modified FXLMM algorithm. A designed noise, which is composed of impulsive noise and background noise, is used as the input signal in the simulation. Extensive simulations are executed to demonstrate the effectiveness of these four algorithms. From the convergence speed and stability point of view, FXLMM achieves the best performance among them. Besides, some analysis on the stability and modeling error are presented. i

3 Acknowledgement I would like to thank my advisor Dr. Zhuangyi Liu for the encouragement and directions he gave to my project. My other advisor Dr. Xun Yu is the mentor of the project who offered me valuable knowledge and inspirations about how to do a good research. I also appreciate the help from Dr. Steven Trogdon at the beginning stage of my research. Last but not least, I would like to thank my parents who continuously support my study. Finally, I thank all my friends, the faculty and the staff in Mathematics and Statistics Department for their help and support! ii

4 Contents Abstract... i Acknowledgement... ii Chapter 1 Introduction to Active Noise Control system... 1 Chapter Broadband Feedforward Adaptive Filter Algorithms FXLMS Algorithm Derivation Analysis of the FXLMS Algorithm on the step size μ LMM ALGORITHM Derivation Parameter Estimation FXLMM Algorithm A Modified FXLMM Algorithm Effects of Modeling Errors of the FXLMM algorithm Chapter 3 Simulation Results FXLMS LMM Algorithm FXLMM A Modified FXLMM... 8 Reference... 3 APPENDIX iii

5 Chapter 1 Introduction to Active Noise Control system With the development of industrial equipments such as fans, transformers, and engines, acoustic problems become more and more significant. Passive devices, such as, enclosures, barriers, and silencers are applied to attenuate the undesired noise but they did not work well at low frequencies. Active noise control (ANC) is the technique which brings revolutionary changes to the field of noise cancellation. ANC works on the principle of acoustic superposition between the primary input noise and the noise generated by the secondary source. An electroacoustic or electromechanical system is usually employed as the secondary source to cancel the primary noise. In a 1936 s patent, Lueg first proposed the basic idea of acoustic ANC, who utilized a microphone and a loudspeaker to cancel the sound [1]. An important reason why ANC is more efficient than passive devices at low frequencies is that ANC uses a digital signal processing (DSP) system as the noise canceller, which can execute complicated mathematical operations with enough speed and precision in real time. Therefore, with the use of DSP, the amplitude and phase of both the primary and the secondary noises match closely. As a consequence, maxim- um noise suppression is achieved with temporal stability, and reliability. In 198 s, development of powerful digital signal processors tremendously propelled the research on ANC. These cheap, high-performance devices promoted the low-cost implementation of adaptive algorithms and widespread application of ANC systems. ANC is an effective way to reduce large amounts of noise and there are several areas where ANC is applied such as in the automotive, industrial, and transportation sectors. Nowadays, ANC techniques are also applied in the healthcare industry because more and more healthcare providers need to control low-frequency noise, especially in sensitive medical applications like infant incubators. Essentially, there are two control methods: feedforward control and feedback 1

6 control in ANC. The difference between them is whether a separate reference signal is used to drive the secondary source via the noise controller [3]. In this report, we will focus on feedforward control, where a reference signal is highly correlated with the primary noise and detected before it propagates past the secondary source. Feedforward approach of ANC for broadband noise in ducts is illustrated in Figure 1 [1]. Figure 1 Single-channel active noise system using broadband feedforward control In this block diagram, x(n) is the reference signal produced by the reference microphone after sensing the primary noise. Then, using the information on x(n) and e(n), ANC system generates a signal y(n) to drive the loudspeaker. Thus, a secondary source noise is generated, which is 18 o out of phase with the reference signal, so that undesired primary noise is cancelled. At last, the error microphone picks up the residual error e(n) after acoustic superposition is implemented and sends e(n) to ANC system. In this process, e(n) is gradually minimized by adjusting the coefficients of the adaptive filter based on a certain adaptive algorithm until the system reaches its optimum. The most well-known adaptive algorithm used for ANC system is the Least Mean Square (LMS) algorithm, which was first published by Widrow and Hoff in 196 [8]. However, ANC system using LMS is unstable because of secondary path effects, that is, the signal from the adaptive filter suffers a phase shift by passing through the

7 secondary path. Morgan first proposed the solution to this problem in 198 and one year later Widrow et al., and Burgess independently brought forward a similar proposal [3]. The concept of this solution is to make up the phase shift by introducing an identical one into a reference signal path to the adaptive filter which still applies LMS algorithm. Based on the way of realization, the solution is called the filtered-x LMS (FXLMS) algorithm. Nevertheless, these algorithms could not effectively suppress impulsive noise, which is caused by noise disturbance with low probability occurrence and large amplitude. Over the past twenty years, various adaptive algorithms have been explored by researchers to control impulsive noise and great progress has been made. In the rest of paper, I will present four of them which are all gradient-based adaptive algorithms. This report is organized as follows. In Chapter, Section.1 describes the FXLMS algorithm in detail including the derivation and the analysis of the FXLMS algorithm; Section. presents the Least Mean M-Estimate (LMM); Section.3 describes the Filtered-x LMM (F- XLMM) algorithm which is developed based on the LMM algorithm; Section.4 mentions a new modified FXLMM algorithm; Section.5 describes the effect of modeling error of the FXLMS algorithm. Simulation results are discussed in Chapter 3. At last, the definition of the Z-transform and off-line modeling technique for S(z) are introduced in Appendix. 3

8 Chapter Broadband Feedforward Adaptive Filter Algorithms This project studies active control of impulsive noise by using various adaptive algorithms. In practical applications, impulsive noises do exist in ANC system and the study of them has been a source of increasing interest. As a result, there are a number of adaptive algorithms developed on the basis of gradient-descent technique. The FXLMS algorithm is one of the most well known algorithms for ANC. The optimization criterion of FXLMS algorithm is minimization of the mean-squareerror. However, the FXLMS algorithm becomes unstable in ANC with the presence of impulsive noise. Therefore, besides the FXLMS algorithm, the Least Mean M-estimate (LMM) algorithm [4], the filtered-x LMM (FXLMM) algorithm [6], and a modified FXLMM algorithm are applied to ANC system with additive impulsive noise to the input. The objective of this report is, by comparing these algorithms in suppressing impulsive noises, to indicate the superiority of the LMM algorithm. Since all of them are gradient-based algorithms and LMS algorithm is the most basic gradient-based algorithm, the LMS algorithm is outlined at the beginning of this chapter. The idea behind LMS adaptive filter algorithm is to use the method of gradient descent to find a coefficient vector which minimizes the cost function-mean Square Error (MSE). As shown in Figure, x(n) is the reference input, d(n) is the primary Figure Electrical configuration of ANC system using LMS algorithm 4

9 input, y(n) is the output from the digital filter W(z), and error e(n) is the difference between d(n) and y(n). n is the current sample number. The digital filters are usually specified in the Z-domain, where the characteristics of the filters are more obvious. The Z transform is explained further in Appendix A. In general, for adaptive filtering there are two types of digital filter structures can be used: finite impulse response (FIR) and infinite impulse response (IIR). However, we restrict to the class of FIR filter in this report. The transversal FIR filter is depicted in Figure 3. Given a set of L filter weights, w i (n), i=,1,,l-1, and the input data sequence x(n),, x(n-l+1), the output is linearly expressed as L 1 y n = w i n x(n i) i= Here the filter weights w i (n) are time-varying and updated by the adaptive algorithm. Figure 3 Block diagram of digital FIR filter Suppose we define the input signal vector at time n as x(n) x n, x n 1,, x n L + 1 T and the weight vector at time n as w(n) [w n, w 1 n,, w L 1 n ] T where L is the order of the filter, which is defined as the maximum delay, in samples, used in creating each output sample. Thus, the output y(n) and the error e(n) in Fig. can be shown by the vector operation as follows y(n)= w T (n) x(n)= x T (n) w(n), e(n)=d(n) - y(n)=d(n) - w T (n) x(n). 5

10 We assume that d(n) and x(n) are statistically stationary. Although the cost function is ε n E[e n ], where E[.] is the mean operator, the squared error e (n) is used to estimate the mean square error. Recalling the gradient descent approach, the updated equation of the coefficients of the digital filter is [1] w n + 1 = w n μ ε(n). (1) where μ is a convergence factor usually called step size. If μ is very small, then at each iteration the weights change a little amount resulting in the slow convergence of the filter. Conversely, the larger the value of μ, the faster the filter converges. However, when μ is too large, the weights might change too fast that the filter will diverge and unstable. The factor is included to simplify the final form of the algorithm. Due to ε n = e (n), the gradient of ε n with respect to w(n) is ε n = e n e(n) Then, from e(n)=d(n) - w T (n) x(n), we can get e n = x(n). Therefore, the gradient estimate is ε n = x(n)e(n) Substituting ε n = x(n)e(n) into Eq. (1) instead of ε(n), it yields w n + 1 = w n + μx(n)e(n) () This is the well-known LMS algorithm. Eq. () indicates that if w n converges, in other words, if w n does not change with time any more, then the error e(n) should be zero. The analysis of LMS algorithm can be found in [1]..1 FXLMS Algorithm.1.1 Derivation If LMS algorithm is used without modification, the performance of ANC system is 6

11 unstable. The reason is we have ignored the transfer functions on the secondary path, which are generated by the use of two microphones to pick up the reference signal and the residual acoustic noise, along with a loudspeaker to produce the canceling sound. Furthermore, the acoustic superposition in the space from the loudspeaker to the error microphone is replaced by a summing junction. Therefore, it is necessary to compensate for the secondary-path transfer function S(z) from y(n) to e(n), which also includes the D/A converter, A/D converter, power amplifier, antialiasing filter etc [1]. The corresponding block diagram is Figure 4. Figure 4 Simplified block diagram of ANC system After the secondary-path transfer function S(z) is introduced into the controller, to guarantee convergence of the system, the conventional LMS algorithm has to be modified. This is because y (n) suffers a phase shift after passing through S(z). The instantaneous measurement of the gradient, x(n)e(n), is thus no longer an unbiased estimate of the true gradient [3]. To solve this problem, Morgan proposed to place an identical filter S(z) in the reference signal path to the weight update of the LMS algorithm in 198 [1]. That is why this method is so called the filtered-x LMS (FXLMS) algorithm. The block diagram of the FXLMS algorithm based on single-channel feedforward ANC system is illustrated in Figure 5. FXLMS is considered as the most effective algorithm for broadband feedforward active noise control. As illustrated in Fig. 4, the residual signal e(n) is [1] 7

12 Figure 5 Block diagram of ANC system using the FXLMS algorithm e n = d n y (n) = d n s n y n = d n s n [w T n x n ] (3) where * is the linear convolution operator. Thus, s n y n = 8 i= s i y(n i). s(n) is the impulse response of secondary path S(z) at time n. The objective of the adaptive filter is the same as LMS algorithm s, which is to minimize the instantaneous squared error, ε n = e (n). Therefore, apply the gradient descent method here except that change ε(n) for ε n as below w n + 1 = w n μ ε(n) (4) where ε(n) is an instantaneous estimate of the MSE gradient at time n, and the expression is ε n = e n = e n e(n). Now, take the derivative of Eq. (3) with respect to w(n), we have e n = s n x n x n where x n = [x n, x n 1,, x n L + 1 ] T and x n = s n x n. Hence, the estimate of the gradient is ε n = x n e(n). (5) Plugging Eq. (5) into Eq. (4), the FXLMS algorithm s full expression is obtained as w n + 1 = w n + μx n e(n). (6)

13 This equation indicates that the weight update is related to S(z). However, S(z) is unknown in practice and has to be estimated by another filter S(z). Thus, as shown in Figure 5, the reference signal x(n) is passing through S(z) to realize the filtered reference signal x n. This leads to x n = s n x(n). For S(z), we can use off-line modeling method to estimate S(z) at the beginning of the training stage, then apply it to ANC system at the end of the training period. For the off-line modeling method, please see Appendix B..1. Analysis of the FXLMS Algorithm on the step size μ For the FXLMS algorithm, the weight update equation is w n + 1 = w n + μx (n)e(n). This equation implies the step size μ is a constant and it has nothing to do with the error e(n). On the other hand, in a frequency-domain, let us consider the traditional LMS algorithm with a transfer function S(z) in the secondary path, which is the same as Figure 5 except S z = 1. On the basis of the frequency-domain model, convergence in this case requires that [1] 1 μ P x ω S(ω) < 1, (7) where P x ω = E[ X ω n ] is the power of X ω n, which is the Fourier transform of x(n) at frequency ω. It is obvious that μ is not barely a constant because P x ω is a function related to n. As a matter of fact, μ should be a function with n as an independent variable. Thus, as the simulation showed later in chapter 3, the FXLMS algorithm is invalid for the random number series, which is generated by a random number block with mean= and variance=1 in MATLAB Simulink, since P x ω S(ω) can be very large for some n. Therefore, the aim of many developed algorithms has been to adaptively change the step size of the adaptive filter to obtain better performance especially when the impulsive noise is added to the input noise. The algorithms I will introduce later such 9

14 as LMM, FXLMM and a modified FXLMM algorithm all use this idea to achieve better performance.. LMM ALGORITHM The performance of the adaptive filters using conventional linear adaptive algorithms could be affected in case the desired or the input signal is interrupted by impulsive noise. As a result, many nonlinear techniques are applied to reduce the impact of the impulsive noise. In this section, one of the nonlinear algorithms - the Least Mean M- Estimate (LMM) algorithm is presented. This algorithm is proposed by Y. X. Zou in [4] in Instead of using the mean square objective function E[e n ] as LMS and FXLMS algorithms, the LMM utilizes the mean M-estimate error objective function J Mρ E[ρ e n ] on the basis of robust statistical estimation, where ρ is a robust M-estimate function. As the classical LMS algorithm, LMM is a stochastic gradient based algorithm and it only has a computational complexity of order O(L) [4]...1 Derivation The basic broadband ANC system shown in Figure 6 can be described in a system identification framework. The x(n) and y(n) are the input and output of the adaptive filter W(z). The error e(n) at time n is e n = d n y n = d n w T n x(n), where x(n) x n, x n 1,, x n L + 1 T,w n = [w n, w 1 n,, w L 1 n ] T are the weight vector and the input signal vector, respectively. d(n) is the desired or reference signal. x(n) and d(n) might be interfered by noises η s n and η o n respectively in practical application. Thus, they can be considered as contaminated Gaussian noise. As mentioned before, the cost function 1

15 Figure 6 System identification structure for the adaptation is proposed as: J Mρ = E[ρ e n ]. (8) ρ is defined as the more general Hampel s three-part redescending M-estimate function, which is given as follows [4] ρ e = e, e < ξ ξ e ξ ξ e < 1 ξ 1 + ξ ( e ) 1, 1 e < ξ 1 + ξ, e where ξ, 1, and are the threshold parameters. Suppose the optimal filter (9) coefficients w o minimizes the objective function J Mρ, vector differentiation of Eq. (8) with respect to w(n) gives w o as the solution to where ψ(e) ρ e e is illustrated in Figure 7. J Mρ w = E ψ e n x n = (1) is called the score function and its expression is as follows and For the convenience of notation, we define the weight function q(e) as q(e) ψ(e)/e. Substituting e n = d n w T (n)x(n) into (1), it yields 11

16 ψ e = e, e < ξ ξ sgn e, ξ e < 1 ξ e 1 sgn e, 1 e <, e (11) Figure 7 Score function of Hampel s three-part redescending M-estimate function E[q e n d n w T n x(n) x n ] =, which further leads to E q e n x n x T n w = E q e n d n x n. Hence, M-estimate normal equation in matrix notation is R xρ w = P ρ (1) where R xρ E q e n x n x T n and P ρ = E q e n d n x n are respectively, the M-estimate autocorrelation matrix of the input x(n) and the M-estimate cross-correlation vector between d(n) and x(n). They are used for the similar purpose as the conventional correlation matrix R and the cross-correlation vector p in the LMS algorithm. In the LMM algorithm, the objective of the adaptive filter is still to minimize the mean M-estimate objective function J Mρ. This is achieved by the most widely used method-the gradient descent, which updates the coefficient vector in the negative gradient direction with step size μ: 1

17 w n + 1 = w n μ wρ (13) where wρ is the gradient of J Mρ, it can be approximated by w J Mρ = J M ρ w wρ = w ρ e n = ρ e n e (n) e (n) w(n) = ψ e n x n = q e n e n x(n). (14) Therefore, the LMM algorithm is w n + 1 = w n μ wρ w n μ wρ = w n + μq e n e n x(n). (15) When e < ξ, the weight function q(e(n))=1. Then Eq. (15) is identical to the LMS algorithm. When e is greater than ξ, the value of the function q(e) begins to decrease, until e. Thus, the LMM algorithm can effectively suppress the effect of large noise or impulsive noise during the adaptive filtering. Furthermore, Eq. (15) implies that the step size of the algorithm μ is not a constant any more. As a matter fact, μ is changing with the error e(n) in the form of q function and e(n) is a function of time n. Therefore, μ varies from time to time to fit with the error e(n). In chapter 3, the advantage of this type of method will be explained by the simulation results... Parameter Estimation In this section, we are going to talk about how to choose the values for the threshold parameters ξ, 1, and for the function ρ. Although the distribution of error 13

18 e(n) is unknown, for simplicity, we assume that it belongs to Gaussian distribution but interfered by additive impulsive noise. For the first step, we will estimate the variance of e(n) without impulses so that the possible impulses in e(n) might be detected or rejected. Since we know that the probability of e(n) greater than a given threshold T h is defined as, θ T n = P r e n > T = erfc( T ), where ς e n erfc r = 1 erf r = ( π ) r e x dx is the complementary error function and ς e n is the estimated standard deviation of the error e(n) without impulses. Thus, applying different threshold parameters T h, impulses can be detected with different degrees of confidence. Suppose θ ξ P r e n > ξ, θ 1 n P r e n > 1, and θ n P r e n > are the probabilities that e(n) is greater than ξ, 1, and, respectively. By choosing appropriate θ ξ, θ 1, and θ, the values of ξ, 1, and can be fixed if the estimation ς e n is known. After the examination of a number of methods for estimating ς e n, Y. X. Zou proposed that the following robust recursive estimator for ς e n is both effective and computational inexpensive [4] ς e n = λ ς ς e n 1 + C 1 1 λ ς med(a e n ) (16) where C 1 = 1.483(1 + 5 N ω 1 ) is a finite sample correction factor, A e n = {e n,, e n N ω + 1 }, and λ ς is the forgetting factor..3 FXLMM Algorithm The FXLMM Algorithm is developed by P. Thanigai in [6]. The difference between LMM algorithm and FXLMM algorithm is the same as the one between LMS algorithm and FXLMS algorithm. Thus, the transfer function S(z) is added in the secondary path and also the identical one S(z) in the reference signal path to the weight update of the LMM algorithm. The FXLMM algorithm s full expression is 14

19 obtained as w(n + 1) = w n + μq e n e n x (n) (17) where x n = s n x n is the filtered reference signal and q(e) ψ(e)/e is the same as the function defined in the LMM algorithm. As demonstrated in LMM algorithm, μ is a function of time n since μ is related to e(n)..4 A Modified FXLMM Algorithm Since the substance of the LMM or the FXLMM algorithm is adapting the step size μ with respect to the error e(n), in fact, there are various ways that μ can follow. Meanwhile, FXLMS is one of the most effective algorithms for small amplitude noise attenuation. Thus, in the interval e < ξ, the FXLMS algorithm is applied as FXLMM. In the interval e > ξ, let the step size μ decreases following a line between (ξ, ξ) and (, ), whose slope is greater thanψ(r) s in this interval, so that it drops off faster than the FXLMM algorithm. Therefore, we can observe how the smaller step size μ in the interval e > ξ would affect the performance of ANC system, whose input signal contains impulsive noise. In this part, based on the FXLMM algorithm, a triangular function Φ(r) is tested, which is illustrated in Figure 8 instead of the score functionψ(r) in the LMM algorithm. This idea comes from the FXLMM algorithm. Therefore, I choose a triangular function and the parameter estimation is identical to the LMM algorithm. We define the weight function p(e) as p(e) Φ(e)/e, thus, the weight update equation is w(n) = w n 1 + μp e n e n x (n) From this equation, it implies that when e < ξ, the weight function p(e(n))=1, then the weight update is identical to the FXLMS algorithm. When e is greater than ξ, the value of the function p(e) begins to decrease, until e. The goal 15

20 e, Φ e = e ξ e < ξ ξ sgn e, ξ e <, e Figure 8 two-part redescending estimate function Φ e (18) of this algorithm is still to change μ adaptively in order to check the performance of this algorithm with additive impulsive noise in comparison with LMM algorithm or FXLMM algorithm..5 Effects of Modeling Errors of the FXLMM algorithm Some researchers discussed the effect of secondary path modeling error on the optimal step size μ and convergence time t. The analysis is done under the condition that a narrowband reference signal is disturbed by broadband noise. Numeric results suggest that phase error of 4 o hardly affect the convergence speed of the algorithm [1]. It is found in [9] that if the reference signal is a synchronously sampled sinusoid, then the single-channel adaptive canceller behaves exactly like a linear, time invariant system between the desired and error signals. The input signal of the adaptive filter is x(n) as follows: x n = cos(ω n) 16

21 Thus, x n i = co s ω n ω i = 1 [exp jω n exp( jω i) + exp jω n exp(jω i)] Using the FXLMS algorithm for the update of the weight vector w(n) is given by w i n + 1 = w i n + μe n x n i i =,, L 1. The filtered reference signal x (n) is generated by passing x(n) through an estimate filter S(z) on the secondary path. But since S(z) is only excited by x(n) at the freq- uency ω, we assume S e jω = Ae jφ is the modulus and phase of S(z) at this frequency. Therefore, the filtered reference signal is x n = Acos(ω n + φ) x n i = Acos ω n i + φ = A [exp jω n exp j φ ω i + exp jω n exp j φ ω i Take the Z transform of the product of e(n) and x n i and suppose e(n) E(z). Then Z e n x n i = A [exp j φ ω i E ze jω + exp j φ ω i E ze jω ]. Again taking the Z transform of the update equation zw i z = W i z + Aμ [exp j φ ω i E ze jω + exp j φ ω i E ze jω ] W i z = Aμ U(z)[exp j φ ω i E ze jω + exp j φ ω i E ze jω ] where U z = 1 z 1. The filter s output y(n) is formed by y n = L 1 i= w i n x(n i). Then take the Z transform of each term on the right hand side for y(n) Y z = 1 = Aμ 4 L 1 i= L 1 i= W i ze jω e jω i + W i ze jω e jω i U(ze jω )e jω i [exp j φ ω i E ze jω + exp j φ ω i E z ] +U ze jω e jω i + exp j φ ω i E z + exp j φ ω i E ze jω 17

22 = Aμ 4 L 1 i= U(ze jω )[exp j φ ω i E ze jω + exp( jφ) E z ] +U(ze jω )[exp(jφ) E z + exp j φ ω i E ze jω ] = Aμ 4 L 1 i= E z [U(ze jω ) exp( jφ) + U(ze jω ) exp jφ ] + E ze jω +U(ze jω ) exp j φ ω i + E ze jω ]U(ze jω ) exp j φ ω i = AμL 4 {E z U ze jω exp jφ + U ze jω exp jφ L 1 L 1 i= i= }. + E ze jω U ze jω e jφ e jω i + E ze jω U ze jω e jφ e jω i But the sum of the first L terms of a geometric series is (19) e ±jω i = 1 e ±j ω L L 1 i= 1 e ±j ω = e jω L e jω e jω L e jω L j e jω e jω j = e jω (L 1) sin (ω L) sin (ω ). If the reference signal x(n) is synchronously sampled and L = kπ ω where k is an integer, then ω L = kπ, which result in sin (ω L) sin (ω ) yields Y z = AμL exp jφ exp jφ E z 4 1 ze j ω + 1 ze j ω = AμL G(z) E(z) sin (kπ ) = =. Therefore, Eq. (19) sin (kπ I) E z cos φ z cos (ω φ) 1 zcos ω +z () with G z = Y(z) = β z cos w φ cos φ E(z) z z cos w +1 where β = AμL, A = S(ω ). Since e(n) = d n s n y n, taking the Z transform of both sides yields E z = D z S z Y(z) 1 = D(z) Y(z) S(z) E(z) E(z) = D(z) S z G(z). E(z) Thus, E(z) = 1 D(z) 1+S z G(z) = z z cos w +1 z z cos w +1+βS z [z cos w φ cos φ ]. (1) 18

23 For very slow convergence, μ. Hence β. Therefore, the dynamic properties of the secondary path are unimportant. Setting S(z)=1 and use only β and φ to express S z, Eq.(1) yields H z = z z cos w +1 z [ cos w β cos w φ ]z+1 β cos φ. () As a second-order system, the stability of H(z) can be investigated by the position of its poles. If the poles are within the unit circle, it implies that the system is stable. For small β, there are two conjugate poles at a distance of 1 β cos φ from the origin in H(z). Since μ >, L >, S(w ) >, we know that β = μl S(w ), β >. If cos φ <, then the distance of the pole from the origin can be greater than 1. So the stability condition must be: cos φ >. Therefore, 9 o > φ > 9 o. This result shows that the difference of phase response between the estimate of the plant response S z and the true phase response of the plant S z for the system should be within ±9 o for the ANC system to converge. 19

24 amplitude amplitude Chapter 3 Simulation Results In this chapter, the simulation results are provided to verify the effectiveness of FXLMS, LMM, FXLMM, and a modified FXLMM algorithm to impulsive noise. Since these algorithms are invalid for the random number series as illustrated in Figure 9 and Figure 1, a designed input noise is used instead. The primary noise x(n) is composed of two parts. One part is background noise, produced by mixing sinusoids with the same amplitude.5 at 3 different frequencies, π 7, π 3, π 5 (rad/sec), respectively. The other part is impulsive noise, which is generated by the product of a sinusoid and a designed impulse. The sine wave s amplitude is and frequency is π 5(rad/sec). The impulse s period is 1 seconds, amplitude is 1 and the pulse width is 1 % of the period. The adaptive filter W(z) uses the FIR filter structure with tap-weight length L=3. In order to obtain stable convergence, the constant step size μ for these four algorithms is selected as 1-5 experimentally. The initial values of the weights are set at zero level w()=. The sampling rate is 1-4 and the simulation time is 1 seconds for all algorithms. 5 x(n) e(n) Figure 9 Comparison diagram of x(n) and e(n) with the random number series as input

25 w(n) Figure 1 divergence of w(n) with the random number series as input 3.1 FXLMS Using the designed input signal x(n), FXLMS algorithm gradually reduces x(n) as shown in Figure 11 and 1. The corresponding output y(n) is plotted in Figure13. From Figure14 we can see, the disturbance of w(n) is very small with impulsive noise present. 1

26 amplitude amplitude amplitude amplitude 4 x(n) e(n) Figure 11 input signal x(n) versus error e(n) at the beginning of the simulation 4 x(n) e(n) Figure 1 input signal x(n) versus error e(n) in a period time at the end of the simulation

27 amplitude 4 y(n) Figure 13 the corresponding output y(n) of Figure 1.6 w(n) Figure 14 Performance of w(n) 3. LMM Algorithm In this simulation part, θ ξ, θ 1, and θ are chosen to be.5,.5, and.1, respectively. Thus, we have 95%, 97.5%, and 99% confidence to reduce the weight of 3

28 amplitude amplitude impulsive noise in the interval [ξ, 1 ] and 1,, and [, ]. According to θ T = erfc( T ς e n ), the threshold parameters are as follows: ξ = k ξς e n = 1.96ς e n, 1 = k 1 ς e n =.4ς e n, and = k ς e n =.57ς e n, where ς e =. ς e n is estimated based on ς e n = λ ς ς e n 1 + C 1 1 λ ς med(a e n ). Therefore, let the forgetting factor λ ς be.99, and the window length N ω be 14 [4]. Then, these parameters are calculated off-line with non- impulsive noises as the input signal. However, during the simulation of the FXLMM algorithm, this approach did not work well even with the designed signal x(n). Hence, a block which can compute ς e n in every iteration is added to the model so that the parameters are adaptively updated as well. And the result is much better than off-line approach. Moreover, this method is also applied to the FXLMM algorithm s and the modified FXLMM algorithm s simulation models. 3 x(n) x 118 e(n) Figure 15 input signal x(n) versus error e(n) at the beginning of the simulation 4

29 amplitude 5 x 118 y(n) Figure 16 the corresponding output y(n) of Figure 15 5 x 116 w(n) Figure 17 Performance of w(n) Figure 15~17 illustrate the simulation results without using the estimate of the transfer function S(z) in the weight update equation, LMM algorithm does not work on impulsive noise at all. Furthermore, it diverges very fast with μ= 1-5. Therefore, 5

30 amplitude amplitude S(z) in the reference signal path to the weight update of the LMM algorithm plays a key role in either impulsive or broadband noise attenuation. 3.3 FXLMM As mentioned before, the estimate of the secondary path transfer function S(z) is added to the model which is used in the LMM algorithm. The following figures are the performance of the ANC system at the first seconds by applying the FXLMM algorithm. Comparing to the FXLMS algorithms, FXLMM is converging much faster. As illustrated in Fig. 1, w(n) is only affected by the impulsive noise and more smooth than FXLMS algorithms in other part. 4 x(n) e(n) Figure 18 input signal x(n) versus error e(n) at the beginning of the simulation 6

31 amplitude amplitude amplitude 4 x(n) e(n) Figure 19 input signal x(n) versus error e(n) in a period time at the end of the simulation 4 y(n) Figure the corresponding output y(n) of Figure 19 7

32 amplitude amplitude.6 w(n) Figure 1 Performance of w(n) 3.4 A Modified FXLMM As shown in Fig. and 3, this algorithm cannot converge as well as FXLMM algorithm in the presence of the impulsive noise. Moreover, the disturbance of w(n) at impulsive noise is greater than other algorithms. 4 x(n) e(n) Figure input signal x(n) versus error e(n) at the beginning of the simulation 8

33 amplitude amplitude amplitude 4 x(n) e(n) Figure 3 input signal x(n) versus error e(n) in a period time at the end of the simulation 4 y(n) Figure 4 the corresponding output y(n) of Figure 3.8 w(n) Figure 5 Performance of w(n) 9

34 After observing four algorithms simulation results, the conclusion is summarized as follows: (a) FXLMS algorithm: The performance criterion is e (n) and the step size μ is a constant. Thus, FXLMS does not converge very fast at the beginning. On the other hand, this algorithm shows the excellent attenuation function to broadband noise. Numerically speaking, the maximum amplitude of the impulsive noise is ±.97 and it can be at least reduced to ±1.87. (b) LMM algorithm: This algorithm uses a new performance criterion-hampel s three part redescending M-estimate function ρ(e n ) instead of the traditional e (n). The other advantage of LMM is the step size μ is not a constant anymore and it alters following the function q( ) with respect to e(n). However, LMM ignores the estimate of secondary path transfer function S(z) so that the error signal e(n) is not synchronous with the reference signal, due to the presence of S(z). Therefore, the biased estimate of the true gradient causes the instability. (c) FXLMM algorithm: The model of ANC system using FXLMM contains both the secondary path transfer function S(z) and its estimate S(z) like FXLMS algorithm which solves the phase shift problem. For this reason, the error signal is correctly aligned in time with the reference signal so that the true gradient is unbiasly estimated. Also, the performance criterion and the step size μ are set identically to the LMM algorithm. The later one implies that μ is adaptively changing with respect to e(n) following the function q( ), which is more suitable for an algorithm to reduce impulsive noise than a constant. Therefore, FXLMM algorithm converges faster from the beginning of the simulation and the weight of the adaptive filter w(n) is more stable than the FXLMS algorithm. The impulse of the input signal x(n) decreases to at least after using FXLMM algorithm. (d) A Modified FXLMM algorithm: The only thing modified for this algorithm is the score function ψ(e). Therefore, the step size μ is changing following the function p( ). In the interval ξ,, the step size μ is decreasing faster than FXLMM algorithm. Thus, for impulsive noise present part this algorithm 3

35 converges slowly, but fast for background noise. From the numerical point of view, by use of this algorithm, the impulsive noise in x(n) decreases to less than In summary, FXLMS, FXLMM, and the modified FXLMM algorithms reduce the amplitude of the impulsive noise to the same level. Furthermore, FXLMM algorithm converges fastest and is most stable among these four algorithms. Thus, FXLMM algorithm is the most effective algorithm in suppressing the impulsive noise in this simulation. It would be interesting to investigate the performance of ANC system with different types of algorithms and more randomized input noise is not tested. 31

36 Reference [1] Sen M. Kuo, Dennis R. Morgan, Active Noise Control Systems Algorithms and DSP Implementations, New York: Wiley, [] Sen M. Kuo, Active Noise Control: A Tutorial Review, Proceedings of the IEEE, vol. 87, no. 6, pp , June [3] S. J. Elliott and P. A. Nelson, Active Noise Control, IEEE Signal Processing Magazine, vol. 1, pp. 1-35, October [4] Yuexian Zou, S.C. Chan, and T.S. Ng, Least Mean M-Estimate Algorithms for Robust Adaptive Filtering in Impulse Noise, IEEE Trans. On Circuits and Systems II, vol. 47, pp , 1999 [5] Y. Zou, S. C. Chan, and T. S. Ng, A Recursive Least M-Estimate (RLM) Adaptive Filter for Robust Filtering in Impulse Noise, IEEE Signal Processing Letters, vol. 7, no. 11, November [6] Priya Thanigai, Sen M. Kuo and Ravi Yenduri, Nonlinear Active Noise Control for Infant Incubators in Neo-natal Intensive Care Units, Acoustics, Speech and Signal Processing, 7. ICASSP 7. IEEE International Conference on vol. 1, pp. I-19-I-11, April 7 [7] [8] [9] S.J. Elliott, I. M. Stothers, and P. A. Nelson, A multiple error LMS algorithm and its application to the active control of sound and vibration, IEEE Trans. Acoustic, Speech, Signal Processing, ASSP-35, , October

37 APPENDIX The content of this appendix is from[7] and [1] and it is added to help the readers to understand the algorithms introduced in this report more clearly. A. The Z-transform [7] The Z-transform is a mathematical transform tool which can convert a time-domain discrete sequence into a complex frequency-domain signal. It is a discrete equivalent of the Laplace transform. The Z-transform can be defined as either a one-sided or two-sided transform. 1. Two-sided Z-transform Suppose x(n) is a discrete-time signal and the function X(z) is defined as X z = Z x n = x(n)z n n= where n is an integer and z is generally a complex number as follows z = Ae jφ = A(cosφ + jsinφ) where A is the magnitude of z, and φ is the phase in radians.. One-sided Z-transform This case is for the discrete sequence x(n) which is defined only for n. X z = Z x n = x(n)z n n= In signal processing, this definition is used when the signal is causal. For more information, please see [7]. B. Off-line modeling technique for S(z) [1] Figure a1 illustrates the block diagram of the direct off-line modeling. The white noise is generated inside of the DSP, which is used as the input to a loudspeaker and to an adaptive filter S(z). 33

38 Figure a1 Experiment setup for off-line secondary-path modeling The procedure of off-line modeling is summed up as below: (1) Generate sampled white noise signal x(n) () Receive the signal d(n) from error microphone (3) The followings are the adaptive filter algorithm a. Compute the output of S(z) L 1 y n = s i n x(n i) i= where s i n is the ith coefficient of the estimation filter S(z) at time n. b. Compute error signal e(n)=d(n)-y(n). c. Update the coefficients based on the LMS algorithm s i n + 1 = s i n + μx n i e n, i =,1,, L 1. (4) Go back to Step 1 for next iteration until adaptive filter S(z) converges to optimal solution. 34

39 The iteration is stopped after the LMS algorithm converges and coefficients s i n, i =,1,, L 1 are applied to the ANC system. However, S(z) could not adaptively respond to changes in the secondary path on an on-line basis. Therefore, it is sometimes desirable to perform adaptive on-line modeling of the secondary path and the most recent estimate S(z) is used in the controller adaptation algorithm. 35