Modified filtered-x hierarchical lms algorithm with sequential partial updates for active noise control

applied sciences Article Modified Filtered-X Hierarchical LMS Algorithm with Sequential Partial Updates for Active Noise Control Pedro Ramos Lorente 1,* , Raúl Martín Ferrer 2, Fernando Arranz Martínez 1and Guillermo Palacios-Navarro 1   Citation: Ramos Lorente, P.; Martín Ferrer, R.; Arranz Martínez, F.; Palacios-Navarro, G. Modified Filtered-X Hierarchical LMS Algorithm with Sequential Partial Updates for Active Noise Control. Appl. Sci. 2021,11, 344. https:// doi.org/10.3390/app11010344 Received: 26 November 2020 Accepted: 28 December 2020 Published: 31 December 2020 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Electronic Engineering and Communications, University of Zaragoza, 44003 Teruel, Spain; [email protected] (F.A.M.); [email protected] (G.P.-N.) 2Department of Computer Science and Systems Engineering, University of Zaragoza, 44003 Teruel, Spain; [email protected] *Correspondence: [email protected] Abstract: In the field of active noise control (ANC), a popular method is the modified filtered-x LMS algorithm. However, it has two drawbacks: its computational complexity higher than that of the conventional FxLMS, and its convergence rate that could still be improved. Therefore, we propose an adaptive strategy which aims at speeding up the convergence rate of an ANC system dealing with periodic disturbances. This algorithm consists in combining the organization of the filter weights in a hierarchy of subfilters of shorter length and their sequential partial updates (PU). Our contribution is threefold: (1) we provide the theoretical basis of the existence of a frequency-dependent parameter, called gain in step-size. (2) The theoretical upper bound of the step-size is compared with the limit obtained from simulations. (3) Additional experiments show that this strategy results in a fast algorithm with a computational complexity close to that of the conventional FxLMS. Keywords: adaptive signal processing; active control of periodic noise; modified filtered-x LMS; hierarchical filter; sequential partial updates 1. Introduction Attenuation of acoustic disturbances has received widespread attention in recent decades since noise seriously affects human health [ 1 – 3 ]. Thus, noise control strategies have been applied in different scenarios, such as aircraft, road vehicles, or the proximity of air conditioning ducts, where the noise level has to be reduced to improve intelligibility. Apart from passive techniques based on the absorption and reflection properties of materials [ 4 , 5 ], acoustic noise reduction can be done by using active noise control (ANC) techniques based on the principle of destructive wave interference. Thus, to cancel the annoying noise at a given location, an anti-noise is generated with the same amplitude as the undesired disturbance, but with an appropriate phase shift. This is carried out by means of secondary sources, generating a zone of silence around an acoustical sensor. As the properties—power, frequency, etc.—of the undesired acoustic disturbance may be time-variant, adaptive control systems have to be implemented to attenuate the noise [6]. One may find in [ 7 ] a review of ANC techniques for noise cancellation inside automobiles—that is our field of interest—during the past 15 years, including commercial developments available in mass production vehicles. The most popular adaptive algorithm used in DSP-based implementations of ANC systems is the filtered-x LMS (FxLMS) algorithm, originally proposed by Morgan [ 8 ]. Figure 1shows the way the electro-acoustic elements are arranged and the block diagram of this solution. Appl. Sci. 2021,11, 344. https://doi.org/10.3390/app11010344 https://www.mdpi.com/journal/applsci Appl. Sci. 2021,11, 344 2 of 24 Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 25 (a) (b) Figure 1. Single-channel active noise control system using the FxLMS algorithm. (a) Physical arrangement of the electroacoustic elements. (b) Equivalent block diagram. The primary path 𝑃(𝑧) comprises the elements from the reference microphone to the error microphone, whereas the secondary path 𝑆(𝑧) includes the elements from the secondary source to the error microphone, namely the D/A converter, the power amplifier, the loudspeaker, the acoustic path, the error microphone, and the A/D converter. The adaptive control filter is denoted as 𝑊(𝑧). Due to the presence of a secondary path that is fed by the output of the ANC system, deriving the LMS-based ANC solution leads to a specific recursive equation. Indeed, the adaptive filter taps are updated by adding a weighted term defined as the product of the reference signal filtered by the secondary path and the so-called error (The error e(n) is defined as the difference between the antinoise and the undesired disturbance). Therefore, the FxLMS-based solution requires an accurate estimate 𝑆 󰆻(𝑧) of the secondary path [8]. Moreover, the convergence of the adaptive filter depends on the step size. In [9] (The version of the FxLMS with leakage addressed in [9] is often used in practical implementations to constrain the power of the output y(n) of the canceller. Then, the leaky FxLMS algorithm reduces undesirable effects due to numerical errors in finite-precision machines, overload of the secondary source, etc.), a stochastic analysis of the FxLMS based on the first and second order moments of the weight-error vector makes it possible to derive the upper step-size bound, whereas a convergence condition for the FxLMS with deterministic reference can be found in [10]. A complete statistical convergence analysis of the FxLMS algorithm without assuming a specific model for the reference signal can be found in [11]. Even if the computational complexity of the FxLMS is quite low, it needs to be reduced as much as possible to be implemented in DSP-based real time applications. In addition, the FxLMS algorithm suffers from slow convergence mainly due to the output delay caused by 𝑆(𝑧). Moreover, errors in the estimate of the secondary path result in instability of the FxLMS algorithm [12–14]. Therefore, various methods have been proposed to avoid the above drawbacks. Thus, to reduce the computational complexity of the control algorithm, the delayed-x LMS [15,16] can be considered. This control strategy is based on the hypothesis that the secondary path model for the FxLMS method does not have to be accurate and can be represented by a delay. To effectively remove the delay of the secondary path within the coefficient updates, the modified FxLMS (Mod FxLMS) algorithm [17,18] has been proposed. It is based on the estimation of the undesired noise by filtering the output of the ANC by the estimate of the secondary path 𝑆 󰆻(𝑧) and by adding the resulting output 𝑦(𝑛) to the error measured by the error microphone. Having estimated the undesired noise 𝑑 󰆻(𝑛), the secondary path and the adaptive filter are swapped in the updates path. Then, the error signal of the adaptive algorithm is calculated as the difference between the estimated noise and the output of the adaptive filter. Hence, the behavior of the system is similar to that of the Figure 1. Single-channel active noise control system using the FxLMS algorithm. ( a ) Physical arrangement of the electroacoustic elements. (b) Equivalent block diagram. The primary path P(z) comprises the elements from the reference microphone to the error microphone, whereas the secondary path S(z) includes the elements from the secondary source to the error microphone, namely the D/A converter, the power amplifier, the loudspeaker, the acoustic path, the error microphone, and the A/D converter. The adaptive control filter is denoted as W(z) . Due to the presence of a secondary path that is fed by the output of the ANC system, deriving the LMS-based ANC solution leads to a specific recursive equation. Indeed, the adaptive filter taps are updated by adding a weighted term defined as the product of the reference signal filtered by the secondary path and the so-called error (The error e(n) is defined as the difference between the antinoise and the undesired disturbance). Therefore, the FxLMS-based solution requires an accurate estimate e S(z) of the secondary path [ 8 ]. Moreover, the convergence of the adaptive filter depends on the step size. In [ 9 ] (The version of the FxLMS with leakage addressed in [ 9 ] is often used in practical implementations to constrain the power of the output y(n) of the canceller. Then, the leaky FxLMS algorithm reduces undesirable effects due to numerical errors in finite-precision machines, overload of the secondary source, etc.), a stochastic analysis of the FxLMS based on the first and second order moments of the weight-error vector makes it possible to derive the upper step-size bound, whereas a convergence condition for the FxLMS with deterministic reference can be found in [ 10 ]. A complete statistical convergence analysis of the FxLMS algorithm without assuming a specific model for the reference signal can be found in [11]. Even if the computational complexity of the FxLMS is quite low, it needs to be reduced as much as possible to be implemented in DSP-based real time applications. In addition, the FxLMS algorithm suffers from slow convergence mainly due to the output delay caused by S(z) . Moreover, errors in the estimate of the secondary path result in instability of the FxLMS algorithm [ 12 – 14 ]. Therefore, various methods have been proposed to avoid the above drawbacks. Thus, to reduce the computational complexity of the control algorithm, the delayed-x LMS [ 15 , 16 ] can be considered. This control strategy is based on the hypothesis that the secondary path model for the FxLMS method does not have to be accurate and can be represented by a delay. To effectively remove the delay of the secondary path within the coefficient updates, the modified FxLMS (Mod FxLMS) algorithm [ 17 , 18 ] has been proposed. It is based on the estimation of the undesired noise by filtering the output of the ANC by the estimate of the secondary path e S(z) and by adding the resulting output e y(n) to the error measured by the error microphone. Having estimated the undesired noise e d(n) , the secondary path and the adaptive filter are swapped in the updates path. Then, the error signal of the adaptive algorithm is calculated as the difference between the estimated noise and the output of the adaptive filter. Hence, the behavior of the system is similar Appl. Sci. 2021,11, 344 3 of 24 to that of the conventional LMS algorithm. Nevertheless, the secondary path impulse response is assumed to be accurately estimated (A frequency domain analysis about the behaviour of the Mod FxLMS algorithm in the presence of secondary path modelling errors is proposed in [ 19 ]). The convergence rate of the Mod FxLMS algorithm is increased at the cost of an additional computational complexity, which turns out to be the main drawback of the approach. A trade-off has hence to be found between convergence rate and computational complexity [ 20 ]. Reduced-complexity implementations of the Mod FxLMS have been proposed in [ 21 , 22 ], but the convergence speed can still be improved. In [ 23 ] a new delay-less frequency-domain ANC algorithm is proposed. The proposal not only removes the delay in the weight adaptation (as the modified filter-x scheme implies) but also removes the delay in the signal path. The proposed strategy exhibits lower computational complexity than other state-of-the-art frequency-domain FxLMS algorithms [24]. In this paper, we propose to combine the Mod FxLMS structure and the hierarchical LMS (HLMS) algorithm, initially developed in the field of channel equalization by Woo [ 25 ]. One may find a performance analysis of the HLMS algorithm in [ 26 ]. In [ 27 ], the meansquared error in a two-level HLMS algorithm is analyzed; in this example, the HLMS is used as a predictive strategy that can significantly speed up the convergence rate during the initial stage of the algorithm. In the HLMS adaptive algorithm, the filter coefficients are organized into a hierarchy of subfilters of shorter length distributed in α levels (details on the hierarchical arrangement of subfilters are provided in Section 2). The output signals of the subfilters at level (l−1) are the input signals of the subfilters placed at the next level l, with lvarying from 1 to α . Then, the number of subfilters per level is divided from level (l − 1) to level lby a factor given by the length of the subfilters at level (l − 1). At the last level of the hierarchy, namely level α , there is only one subfilter. Since the subfilters have shorter length than a conventional FIR filter, they can converge faster, as recalled in the Appendix A.3 of the Appendix A. However, the computational complexity associated to this multi-level structure is higher than that of the conventional LMS algorithm. To address the above problem, we suggest using partial updates (PU) of the adaptive filter coefficients. A widely used PU algorithm is the sequential PU LMS algorithm with decimation factor N[ 28 ]. This algorithm updates a subset of size L/N, out of Lcoefficients— wj(n), 1 ≤j≤L—per iteration of a conventional L-length FIR filter according to wj(n+1) = wj(n) + µx(n−j+1)e(n)if (n−j+1)mod N=0 wj(n)otherwise (1) where µ is the step-size of the algorithm, x(n) the input signal, and e(n) the error. Nevertheless, the higher the decimation factor Nis, the lower the convergence rate will be. In [ 29 ], we have shown that, in the context of a conventional adaptive FIR filter, this lower convergence rate can be compensated, under the assumption of a periodic input signal, by an affordable increase in the step-size µ . As the maximum step-size that ensures convergence with a sequential PU algorithm is Ntimes larger than the maximum step-size for a full updates adaptive algorithm, one can introduce a parameter called gain in step-size, that determines the factor by which the step-size µ can be multiplied to improve the convergence rate of the sequential PU adaptive algorithm. Note that the theoretical analysis of the strategy developed in [ 29 ] excludes the use of certain frequencies corresponding to notches appearing in the gain in step-size whose width and exact location depend on the system parameters, namely the decimation factor, the sampling frequency and the length of the adaptive filter. In this paper, our purpose is hence to study the relevance of the combination of the Mod FxLMS, the HLMS and the sequential PU LMS with gain in step size. The resulting ANC approach is called the modified filtered-x hierarchical sequential PU LMS algorithm with gain in step-size (G µ —Mod Fx H Seq LMS). The other contributions of this paper consist in: Appl. Sci. 2021,11, 344 4 of 24 (1) deriving the theoretical gain in step-size of the control strategy. It is defined as the ratio between the upper bounds on the step-sizes evaluated in the two following cases: when only a subset of the weights of the hierarchical filter proposed by Woo [ 25 ] is updated at each iteration and when every tap—regardless the position of the weight in the hierarchy of subfilters—is updated at every cycle. We will see that the frequency response of this gain in the step-size exhibits notches. Their width and location depend on the length of the slowest subfilter of the hierarchy, the decimation factor, and the sampling frequency. Therefore, this phenomenon has to be taken into account when the input signal contains harmonics at frequencies corresponding to the location of the notches; (2) carrying out computer-based experiments to confirm that the predicted theoretical gain in step of the G µ —Mod Fx H Seq LMS algorithm is well suited to the maximum affordable increase in step-size obtained by simulations; (3) completing additional computer-based simulations to test the performance of the Gµ—Mod Fx H Seq LMS algorithm for active attenuation of periodic disturbances. The paper is organized as follows. In Section 2, we propose the modified filtered-x hierarchical sequential PU LMS algorithm with gain in step-size. Section 3deals with the convergence analysis of the proposed algorithm. The approach consists in applying to the hierarchical filter used in our proposal the results provided in the Appendix Afor a conventional adaptive FIR filter. Results of computer-based simulations are provided in Section 4. We carry out a comparison between the theoretical prediction and the experimental behavior of the proposed algorithm. The experiments also include a comparative study of various ANC strategies in terms of convergence rate and computational complexity. Section 5is devoted to discussion. 2. Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size In this section, we propose the G µ -Mod Fx H Seq LMS algorithm by combining the Mod FxLMS, the HLMS, and the sequential PU LMS with gain in step-size. Our goal is to derive an ANC adaptive algorithm with a faster convergence rate than the conventional FxLMS with a similar computational complexity. Figure 2shows the block diagram of the proposed algorithm. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 25 (1) deriving the theoretical gain in step-size of the control strategy. It is defined as the ratio between the upper bounds on the step-sizes evaluated in the two following cases: when only a subset of the weights of the hierarchical filter proposed by Woo [25] is updated at each iteration and when every tap—regardless the position of the weight in the hierarchy of subfilters—is updated at every cycle. We will see that the frequency response of this gain in the step-size exhibits notches. Their width and location depend on the length of the slowest subfilter of the hierarchy, the decimation factor, and the sampling frequency. Therefore, this phenomenon has to be taken into account when the input signal contains harmonics at frequencies corresponding to the location of the notches; (2) carrying out computer-based experiments to confirm that the predicted theoretical gain in step of the Gμ—Mod Fx H Seq LMS algorithm is well suited to the maximum affordable increase in step-size obtained by simulations; (3) completing additional computer-based simulations to test the performance of the Gμ—Mod Fx H Seq LMS algorithm for active attenuation of periodic disturbances. The paper is organized as follows. In Section 2, we propose the modified filtered-x hierarchical sequential PU LMS algorithm with gain in step-size. Section 3 deals with the convergence analysis of the proposed algorithm. The approach consists in applying to the hierarchical filter used in our proposal the results provided in the Appendix for a conventional adaptive FIR filter. Results of computer-based simulations are provided in Section 4. We carry out a comparison between the theoretical prediction and the experimental behavior of the proposed algorithm. The experiments also include a comparative study of various ANC strategies in terms of convergence rate and computational complexity. Section 5 is devoted to discussion. 2. Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size In this section, we propose the Gμ-Mod Fx H Seq LMS algorithm by combining the Mod FxLMS, the HLMS, and the sequential PU LMS with gain in step-size. Our goal is to derive an ANC adaptive algorithm with a faster convergence rate than the conventional FxLMS with a similar computational complexity. Figure 2 shows the block diagram of the proposed algorithm. Figure 2. Block diagram of the modified filtered-x hierarchical LMS algorithm with sequential partial updates. According to Figure 2, the filtered reference 𝑥′(𝑛) is the input of the control adaptive filter whereas the reference signal 𝑥(𝑛) is filtered by a slave filter, which is a copy of the Figure 2. Block diagram of the modified filtered-x hierarchical LMS algorithm with sequential partial updates. According to Figure 2, the filtered reference x0(n) is the input of the control adaptive filter whereas the reference signal x(n) is filtered by a slave filter, which is a copy of the control adaptive filter. By cascading the slave filter and the estimate of the secondary path Appl. Sci. 2021,11, 344 5 of 24 e S(z) , and then by passing the reference signal x(n) through the resulting filter, one can obtain an estimation e y(n) of the antinoise and hence an estimation e d(n) of the undesired noise. Inaccuracy of the secondary path estimate and its effects are discussed in [ 12 – 14 ] in the context of filtered x LMS algorithms. At that stage, the output z(n) of the adaptive control filter is directly subtracted from the estimated noise e d(n) to provide the numerical error en(n). One may find in [ 30 ] a complete review of techniques of estimation of the error signal using signal processing algorithms. As this error is used to update the adaptive control filter, the limitations imposed on the step-size µ for the standard version of the FxLMS algorithm are now overcome. Let us now focus our attention on the hierarchical filter. Given the number Lof taps at the first level of the hierarchy, the number of subfilters at the lth level is given by Nl=L l ∏ r=1 βr = α ∏ r=1 βr l ∏ r=1 βr = α ∏ r=1+1 βr(2) where βl denotes (with this notation, we implicitly assume that the subfilters at the same level have the same number of taps and this number may vary from one level to another) the number of weights of a subfilter at level l, varying from 1 to α . As the subfilter length may vary from one level to another, the step-size bound of every subfilter can be different. In the sequel, the coefficients of the ith hierarchically arranged subfilter impulse response at the lth level are denoted as wl i(n) = hwl i1(n)wl i2(n)··· wl iβl(n)i, 1 ≤l≤α, 1 ≤i≤Nl(3) where wl ij denotes the weight for the jth tap of the ith subfilter at the lth level. In addition, zl ij and yl ij , respectively, denote the input signals of the the jth tap of the ith subfilter at the lth level of the adaptive and the slave hierarchical filters. The outputs of the adaptive and the slave hierarchical filters, respectively denoted as z(n) and y(n) , are given by the last loop of the multilevel filtering, that is, z(n)=zα+1 11 (n) and y(n)=yα+1 11 (n) . The error signal of the ith subfilter at the lth level is denoted as el i . These error signals are obtained by subtracting the output of every subfilter from the estimated noise e d(n) . It should be noted that the necessity of using the estimated noise e d(n) to update the subfilters placed at the intermediate levels of the hierarchy is already solved as we use the Mod FxLMS version of the ANC algorithm. Figure 3shows the architecture of a 2-level hierarchical filter. In this example, the number of subfilters at levels 1 and 2 are N1=L/β and N2= 1, respectively. The number of coefficients of every subfilter at levels 1 and 2 are βand L/β, respectively. The main drawback of the HLMS is the high computational complexity inherently associated to its multi-level structure. Sequential PU of the coefficients of the hierarchical filter are used to reduce the computational complexity. PU are applied to every coefficient at every level, of the hierarchical organization of taps, from the first tap of the first subfilter to the last tap of the last subfilter. For instance, in Figure 3, the shadowed coefficients of the hierarchical filter are the N-equally-spaced taps that have to be updated at a given time n. At the following iterations of the updating process, namely n+ 1, n+ 2, . . . ,n+N − 1, the next subsets of equally spaced coefficients of the hierarchical filter are updated. Appl. Sci. 2021,11, 344 6 of 24 Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 25 Figure 3. Two-level hierarchical filter. The main drawback of the HLMS is the high computational complexity inherently associated to its multi-level structure. Sequential PU of the coefficients of the hierarchical filter are used to reduce the computational complexity. PU are applied to every coefficient at every level, of the hierarchical organization of taps, from the first tap of the first subfilter to the last tap of the last subfilter. For instance, in Figure 3, the shadowed coefficients of the hierarchical filter are the N-equally-spaced taps that have to be updated at a given time n. At the following iterations of the updating process, namely n + 1, n + 2, …, n + N − 1, the next subsets of equally spaced coefficients of the hierarchical filter are updated. Due to PU, the algorithm suffers from a reduction in convergence rate as N increases. Then, by using the gain in step-size, the slower convergence rate of the sequential PU adaptive algorithm can be compensated. The strategy hence gives the same performance as that of the full updates algorithm in terms of convergence rate, but with lower computational complexity. In previous works [29], this strategy is analyzed in the context of a conventional adaptive FIR filter. The Gμ—Mod Fx H Seq LMS deals with periodic disturbances. These periodic noises, such as engine noise, are very often the subject of cancellation in ANC applications. This is due to two reasons. First, these disturbances are the most annoying and, second, it is usually easy to find a good reference signal to cancel them. In the Algorithm 1, the Gμ— Mod Fx H Seq LMS algorithm is given: Figure 3. Two-level hierarchical filter. Due to PU, the algorithm suffers from a reduction in convergence rate as Nincreases. Then, by using the gain in step-size, the slower convergence rate of the sequential PU adaptive algorithm can be compensated. The strategy hence gives the same performance as that of the full updates algorithm in terms of convergence rate, but with lower computational complexity. In previous works [ 29 ], this strategy is analyzed in the context of a conventional adaptive FIR filter. The G µ —Mod Fx H Seq LMS deals with periodic disturbances. These periodic noises, such as engine noise, are very often the subject of cancellation in ANC applications. This is due to two reasons. First, these disturbances are the most annoying and, second, it is usually easy to find a good reference signal to cancel them. In the Algorithm 1, the Gµ—Mod Fx H Seq LMS algorithm is given: Appl. Sci. 2021,11, 344 7 of 24 Algorithm 1 Gµ—Mod Fx H Seq LMS algorithm for i = 1to # iterations /* MAIN LOOP*/ y1 ∀i∀j(n) = x(n)/*First level of slave hierarchical filter is filled with x(n)*/ /* SLAVE HIERARCHICAL FILTER */ for l = 1to α/* From first to top (α)level of the hierarchy fo*/ for i = 1to α ∏ r=l+1 βr/* From first to last subfilter at each level */ /* Computing the output of every subfilter */ yl+1 p q (n) = wl iT(n)yl ip=di βle,q=i−bi−1 βlcβl end of for (i) end of for (l)→y(n) = yα+1 1 1 (n)/* END OF SLAVE HIERARCHICAL FILTER */ e y(n) = ~ sT(n)yα+1 11 (n) /* Computing the antinoise signal where */ /* es(n) = [es1es2. . . esL]Tand */ /* yα+1 1 1 (n) = hyα+1 11 (n). . . yα+1 1 1 (n−Ls+1))iT*/ /* Measured error em(n) = d(n)−y0(n)*/ e d(n) = e y(n) + em(n)/* Computing the estimated noise */ x0(n) = ~ sT(n)x(n)/* Filtering the referencex(n) = [x(n). . . x(n−Ls+1))]T*/ z1 ∀i∀j(n) = x0(n)/* First level of adaptive hierarchical filter is filled with x0(n) *//* ADAPTIVE HIERARCHICAL FILTER */ for l = 1to α/* From first to top level of the hierarchy */ for i = 1to α ∏ r=l+1 βr/* From first to last subfilter at each level */ /* Computing the output of every subfilter */ zl+1 p q (n) = wl iT(n)zl ip=di βle,q=i−bi−1 βlcβl el i(n) = e d(n)−wl iT(n)zl i(n)/* Computing the error of every subfilter */ for j = 1to βl/* For every tap, Sequential partial updates */ if (k−((i−1)βl+j) + 1)mod N== 0 wl i j(k+1) = wl i j(k) + Gµµlel i(n)zl i j(n) else wl i j(k+1) = wl i j(k) end of if end of for (j) end of for (i) end of for (l)→z(n) = zα+1 1 1 (n)/* END OF ADAPTIVE HIERARCHICAL FILTER */ end of for (n)/* END OF MAIN LOOP */ 3. Convergence Analysis In the first part of this section, we establish the assumptions taken into account in the convergence analysis. In the second sub-section, we derive the gain in step-size of the Gµ—Mod Fx H Seq LMS algorithm. 3.1. Assumptions in the Convergence Analysis In [ 29 ], we have derived an upper bound on the step-size for the Fx sequential PU LMS algorithm updating a conventional FIR filter. This analysis is based on two assumptions, Appl. Sci. 2021,11, 344 8 of 24 namely the independence theory between the reference signal and the filter weights, and the slow convergence condition (The reader is referred to [ 29 ] for more information on the assumption of independence theory and the slow convergence condition. Despite the fact that such assumptions might be initially questionable when dealing with periodic inputs, we confirm in [ 29 ] the feasibility of assuming both conditions in the analysis of a FIR-based Fx sequential PU LMS strategy to attenuate periodic disturbances). When using a more complex filtering structure [ 25 ] based on hierarchically arranged subfilters wl i(n) , 1 ≤l≤α , 1 ≤i≤N , the overall convergence of the hierarchical structure is assumed to be constrained by the hierarchical arranged subfilter wslow(n) that converges with the slowest convergence rate. By applying to wslow(n) the convergence analysis for a conventional FIR filter recalled in the Appendix A, one can obtain the analytical expression of the gain in step size of the Gµ—Mod Fx H Seq LMS algorithm. Let us now give the criteria to recognize the slowest subfilter wslow(n) of the arrangement. The convergence conditions of wl i(n) , 1 ≤l≤α , 1 ≤i≤N , depends on its length and on its input signal. The larger the subfilter is, the smaller the maximum step-size is. Therefore, the larger the subfilter is, the slower the convergence will be (See Appendix A.3 of the Appendix A, where the dependence of the step-size bound on the length of a filter is derived). As far as the input signal is concerned, Woo [ 25 ] states that the influence of the input on the convergence is related to the level of the subfilter in the hierarchy. The eigenvalue spread of the input-signal autocorrelation matrix becomes smaller from level lto level l+ 1 because the hierarchical structure tends to average the eigenvalues of the related input-signal autocorrelation matrix. Then, assuming that the number of taps β of every subfilter is the same, regardless the position in the hierarchy, the bottle-neck in the convergence process is located at the subfilters of the first level. Nevertheless, in our approach, every hierarchically arranged subfilter, and more particularly wslow(n) , is updated by the sequential PU algorithm. Therefore, we have to consider that the logical subfilter (we consider a logical subfilter as the set of N-equallyspaced taps of a filter updated at every iteration of the updating process according to a sequential LMS algorithm with decimation factor N(see Appendix A)) is formed by the subset of β /N coefficients of the β -length subfilter wslow(n) . These β /N coefficients are updated in one iteration of the sequential LMS algorithm with decimation factor N. Therefore, the convergence condition of the hierarchical arrangement is established on the basis of the joint convergence of the Nlogical subfilters into which the slowest subfilter wslow(n) is decomposed. Having determined the element that limits the convergence rate, we derive the gain in step-size for the hierarchical structure in the next section. 3.2. Gain in Step-Size of the Gµ—Mod Fx H Seq LMS Algorithm Results derived in the Appendix Afor a conventional FIR filter (theoretical derivation of the gain in step-size for a conventional adaptive FIR filter whose coefficients are partially updated according to the sequential LMS algorithm can be found in the Appendix A.4 of the Appendix), are extended to the β -length slowest subfilter of the hierarchy wslow(n) , when sequential PU with decimation factor Nare applied to the hierarchical filter. The role of this slowest subfilter can be played by any of the subfilters located at the first level of the hierarchy. To obtain more easily the factor by which the step-size parameter µ of the proposed algorithm can be increased with regard to the step-size of the full updates approach, we impose the use of the same value for µ for every subfilter of the hierarchy. Then, we consider the bound on the step-size of the hierarchical filter as the maximum value of µ that ensures convergence in all the subfilters. Since β is shorter than the total number Lof taps at the first level of the hierarchy ( β=α √L typically), and, provided that the decimation factor N> 1, the number β /N of subfilter coefficients that are effectively updated per iteration is small. When β /N is not integer, this ratio must be rounded to the nearest integer either towards zero, β/N , or towards infinity, β/N . In that case, the β -length hierarchical subfilter is decomposed Appl. Sci. 2021,11, 344 9 of 24 into β−Nβ/N logical subfilters of length β/N , and N−β+Nβ/N logical subfilters of length β/N . The β/N -length logical subfilters have a slower convergence rate than that of the β/N -length logical subfilters (in Appendix A.3 of the Appendix A, it is proved that the larger an adaptive filter, the smaller the bound on its step-size). As there is at least one β/N -length logical subfilter per iteration at the first level, the convergence rate of a β/N -length logical subfilter determines the convergence of the hierarchy of the subfilters. For the sake of simplicity to obtain the dependence of the gain in step-size on the length β and on the decimation factor N, let a single tone of normalized frequency f 0 be the input signal of the hierarchical structure. Thus, the gain in step-size is given by Gµ(1, f0,β,N)=maxn1 4hβ±sin(β2πf0) sin(2πf0)io max(1 4"lβ Nm±sinlβ Nm2πN f0 sin(2πN f0)#). (4) We have carried out several computer-based simulations to compare the theoretical prediction given by Equation (4), with the experimental results. This study confirms that this worst-case hypothesis makes it possible to accurately predict the behavior of the experimental convergence process. In Section 4.1, we provide a comparison between the theoretical gain in step-size, given by Equation (4), with the affordable increase in step-size obtained by MATLAB simulation. 4. Simulation Results The purpose of this section is twofold. First, we compare the theoretical prediction of the gain in step-size—Equation (4)—with computer-based results. Then, we analyze the relevance of the G µ —Mod Fx H Seq LMS algorithm in an ANC system when dealing with harmonic disturbances. 4.1. Gain in Step-Size: Simulation vs. Theory Let us consider the following simulation protocol: it corresponds to the 1 × 1 × 1 arrangement, that is, 1 reference sensor, 1 error microphone and 1 secondary source (see Figure 1). The first level of the hierarchical filter consists of 384 coefficients organized in 16 subfilters of 24 taps. Therefore, in the second level of the hierarchy, one has a 16-length subfilter. The reference is a single sinusoidal signal whose frequency varies in 41.6 Hz steps from 41.6 to 4000 Hz. The sampling frequency is 8000 samples/s. Primary and secondary paths are set to filters modelling real-world active noise control systems. We use filters 25th order IIR filters provided by Kuo and Morgan in [ 6 ] (in the book-attached floppy disc featuring C and assembly programs for implementing ANC systems). Plant models from this well-known reference are considered among researchers in the topic as a valid benchmark. Figures 4and 5show the magnitude and phase of the primary and secondary paths, respectively. Appl. Sci. 2021,11, 344 16 of 24 when the sequential PU LMS algorithm is used with decimation factor (N<L; in addition, and for the sake of simplicity, we assume throughout the paper that the ratio L/Nis integer) N, the weights of the L-length filter are updated by means of the following recursion w(n+1) = w(n) + µe(n)I(N) 1+nmodNx(n)(A3) where the L×Lmatrix I(N) p, with p=1+n mod N ≤Nis defined by I(N) p=diag 0··· 0 |{z } p−1 10··· 0 | {z } N−1 10··· 0 | {z } N−1 1 0 ···0 1 0··· 0 | {z } N−p!(A4) As one has I(N) px(n) = 0··· 0 | {z } p−1 x(n−p+1)0··· 0 | {z } N−1 x(n−p−N+1)0··· 0x(n−L+N−p+1)0··· 0 | {z } N−p  T(A5) Equation (A3) leads to wp+αN(n+1) = wp+αN(n) + µe(n)x(n−p−αN+1)f or α=0, . . . , L N−1 wp+αN(n)otherwise.(A6) At that stage, let us introduce the pth L/N-length “logical subfilter”, with pcyclically varying from 1 to N. It is defined by the set of L/N equally-spaced taps wpwp+Nwp+2N. . . wp+L−N of the L-length filter response vector w(n) . At time n+p−1 , where p= 1 +n mod N , the pth logical subfilter is updated by means of Equation (A6). Therefore, the Nsubfilters require the same signal input samples to be updated. They are stored in the vector x(N)(n) that corresponds to the N-decimated version of x(n). x(N)(n) = [x(n)x(n−N)x(n−2N)··· x(n−L+N)]T. (A7) At time n+N , x(N)(n) is shifted, inserting a new sample at the position formerly given by the index n, while the older data is lost. See Table A1, which shows the subsets of coefficients of the filter response vector w(n)=[w1w2. . . wL]T to be updated according to Equation (A3) during N+ 1 consecutive iterations and their corresponding samples of the input vector. Table A1. Coefficients to be updated—defining a logical subfilter—during N + 1 consecutive iterations and their corresponding samples of the input vector. # Iteration Coefficients That Form the Logical Subfilter Updated at the Current Iteration Samples of the Input Vector Used to Update the Logical Subfilter 1[w1w1+Nw1+2N. . . wL−N+1]T[x(n)x(n−N)x(n−2N). . . x(n−L+1))]T . . .. . .. . . N[wNw2Nw3N. . . wL]T[x(n)x(n−N)x(n−2N). . . x(n−L+1))]T N+ 1 [w1w1+Nw1+2N. . . wL−N+1]T[x(n+N)x(n)x(n−N). . . x(n−L+N+1))]T In light of Equation (A7), the (L/N) × (L/N) Toeplitz autocorrelation matrix R(N) of the decimated input vector x(N)(n)is given by R(N)=Ehx(N)(n)x(N)T(n)i. (A8) In the sequel, the convergence properties of the (L/N)-length logical subfilters is analyzed based on the eigenvalues of the autocorrelation matrix R(N). Appl. Sci. 2021,11, 344 17 of 24 Appendix A.2. Eigenvalues of the Autocorrelation Matrix of a Periodic Signal Consisting of K Harmonics Let us assume that the input signal x(n)of an adaptive filter is defined as follows x(n) = K ∑ k=1 Ckcos(2πk f0n+ϕk)(A9) where f0 is the normalized fundamental frequency, {φk}k=1,...,K the initial random phases mutually independent and uniformly distributed from 0 to 2 π and {Ck}k=1,...,K the amplitudes of the harmonics. Given Equation (A9), the autocorrelation function of input signal x(n)can be expressed as rxx(τ) = K ∑ k=1 C2 k 2cos(2πk f0τ). (A10) Therefore, the autocorrelation matrix of the input vector x(n) can be expressed as the sum of Kmatrices Rkof size L×Las follows R= K ∑ k=1 C2 kRk(A11) where Rk=1 2                  1 cos(2πk f0)··· ··· cos[2πk(L−1)f] cos(2πk f0)1.... . . . . ........... . . . . .......cos(2πk f0) cos[2πk(L−1)f]··· ··· cos(2πk f0)1                  . (A12) The largest eigenvalue λk,max(k f0)of each matrix Rkis given by [31] λk,max(k f0) = max1 4L±sin(L2πk f0) sin(2πk f0) (A13) where the subscript krefers to the index of the submatrix Rk. According to the triangle inequality [ 32 ], appendix E, the largest eigenvalue of a sum of matrices is bounded by the sum of the largest eigenvalues of each of its components. Therefore, the largest eigenvalue λtot,max of Ris bounded by λtot,max ≤ K ∑ k=1 C2 kλk,max(k f0) = K ∑ k=1 C2 kmax1 4L±sin(L2πk f0) sin(2πk f0) (A14) where the subscript tot refers to the whole autocorrelation matrix R. As far as the sequential PU LMS algorithm with decimation factor Nis concerned, the convergence condition of the whole filter might be translated to the parallel convergence of Nlogical subfilters of length L/N updated by a N-decimated input signal x(N)(n) [ 29 ]. Adjusting the above approach to the case of sequential PU LMS, where the size of the autocorrelation matrix is L/N and the sampling frequency is divided by N, we deal with K matrices R(N) kof size (L/N)×(L/N) defined by Appl. Sci. 2021,11, 344 18 of 24 R(N) k=1 2                      1 cos(2πNk f0)··· ··· cosh2πNkL N−1fi cos(2πNk f0)1.... . . . . ........... . . . . .......cos(2πNk f0) cosh2πNkL N−1fi··· ··· cos(2πNk f0)1                      . (A15) Therefore, the largest eigenvalue λ(N) k,max (k f0) of each matrix R(N) k can be expressed as follows λ(N) k,max(k f0) = max   1 4 L N±sinL N2πkN f0 sin(2πkN f0)   . (A16) Considering the triangle inequality, the largest eigenvalue λ(N) tot,max of the (L/N) × (L/N) matrix R(N)=K ∑ k=1 C2 kR(N) kis bounded by λ(N) tot,max ≤ K ∑ k=1 C2 kλ(N) k,max(k f0) = K ∑ k=1 C2 kmax   1 4 L N±sinL N2πkN f0 sin(2πkN f0)   . (A17) It should be noticed that for N= 1 the sequential PU LMS algorithm reduces to the conventional full updates LMS algorithm and Equations (A16) and (A17) reduce to Equations (A13) and (A14), respectively. Appendix A.3. Effect of the Length of the Filter on the Step-Size Bound In this section, we analyze the effect of the length of an adaptive filter on the maximum value of the step-size that ensures convergence of the adaptive algorithm. The analysis deals with the case of the reference signal defined in Equation (A.9). The dependence of the step-size bound on the number of coefficients of the filter is studied not only for the full updates LMS algorithm, but also for the sequential PU LMS algorithm with decimation factor N. (a) Full updates LMS algorithm Let the input signal of the LMS algorithm be the periodic signal given by Equation (A9) . The convergence in mean of the weights of the filter is guaranteed if the step-size satisfies the inequality [33] 0<µLMS <2 λtot,max . (A18) Thus, combining Equations (A14) and (A18), we obtain a more restrictive bound on the step-size that ensures convergence in mean. 0<µLMS <2 K ∑ k=1 C2 kmaxn1 4hL±sin(L2πk f0) sin(2πk f0)io. (A19) Hence, the bound on µLMS depends on the frequency f0 , the length of the filter L, and the weights Ck of the input signal. To simplify the graphical representation of the bound, the input signal of a L-length adaptive filter updated by the conventional LMS algorithm is defined by the following single tone of normalized frequency f0. x(n) = cos(2πf0n+ϕ). (A20) Appl. Sci. 2021,11, 344 19 of 24 Due to Equations (A19) and (A20), the bound on the step-size is then given by 0<µLMS <2 maxn1 4hL±sin(L2πf0) sin(2πf0)io=2 λtot,max . (A21) Figure A1 shows the size bound of the conventional LMS algorithm for L= 16, 32, 64, and 128. The normalized frequency of the input vector varies from 0 to 0.5. Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 of 25 Hence, the bound on 𝜇 depends on the frequency 𝑓, the length of the filter L, and the weights 𝐶 of the input signal. To simplify the graphical representation of the bound, the input signal of a L-length adaptive filter updated by the conventional LMS algorithm is defined by the following single tone of normalized frequency 𝑓. . )2cos()( 0 φ π += nfnx (A20) Due to Equations (A19) and (A20), the bound on the step-size is then given by () () . 2 2sin 2sin 4 1 max 2 0 max,tot 0 0 λ π π μ =            ± << f fL L LMS (A21) Figure A1 shows the size bound of the conventional LMS algorithm for L = 16, 32, 64, and 128. The normalized frequency of the input vector varies from 0 to 0.5. . Figure A1. Step-size bound of the conventional full updates LMS algorithm for different filter lengths, L = 16, 32, 64, and 128. The input vector is a single tone whose normalized frequency varies from 0 to 0.5. According to Figure A1, and considering that a large step-size guarantees fast convergence rate, we conclude that a shorter filter can converge faster than a large one. (b) Sequential PU LMS algorithm. Let the L × 1 input vector of the sequential PU LMS algorithm be given by the N-decimated version 𝐱()(𝑛) of the periodic signal 𝐱(𝑛) expressed by Equation (A9). A similar analysis as the one carried out in the previous section for the conventional LMS algorithm yields a more restrictive bound on the step-size that ensures convergence in mean for the case of the sequential PU LMS algorithm. Figure A1. Step-size bound of the conventional full updates LMS algorithm for different filter lengths, L= 16, 32, 64, and 128. The input vector is a single tone whose normalized frequency varies from 0 to 0.5. According to Figure A1, and considering that a large step-size guarantees fast convergence rate, we conclude that a shorter filter can converge faster than a large one. (b) Sequential PU LMS algorithm. Let the L × 1 input vector of the sequential PU LMS algorithm be given by the Ndecimated version x(N)(n) of the periodic signal x(n) expressed by Equation (A9). A similar analysis as the one carried out in the previous section for the conventional LMS algorithm yields a more restrictive bound on the step-size that ensures convergence in mean for the case of the sequential PU LMS algorithm. 0<µSeqLMS <2 K ∑ k=1 C2 kmax1 4L N±sin(L N2πkN f0) sin(2πkN f0)<2 λ(N) tot,max . (A22) This bound on µSeqLMS depends on the frequency f0 , the length of the filter L, the weights of the input signal Ck , and the decimation factor N. As we did in the previous section to simplify the graphical representation of the bound, we reduce the number of harmonics of the input signal to K= 1. In so doing, the L × 1 input vector of the sequential Appl. Sci. 2021,11, 344 20 of 24 PU LMS algorithm is given by the N-decimated version x(N)(n) of a sinusoidal signal defined in Equation (A20). The bound on the step-size is then reduced to 0<µSeqLMS <2 max1 4L N±sin(L N2πN f0) sin(2πN f0)=2 λ(N) tot,max . (A23) Figures A2 and A3 show the size bound of the sequential PU LMS algorithm given by Equation (A23) for decimation factors N= 2 and N= 4, respectively. Results are given for L= 16, 32, 64, and 128. The normalized frequency of the input vector varies from 0 to 0.5. Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 of 25 () . 2 2sin 2sin 4 1 max 2 0)( max,tot 10 0 2 N K kk SeqLMS kNf kNf N L N L C λ π π μ <                                ± <<  = (A22) This bound on 𝝁𝑺𝒆𝒒𝑳𝑴𝑺 depends on the frequency 𝒇𝟎, the length of the filter L, the weights of the input signal 𝑪𝒌, and the decimation factor N. As we did in the previous section to simplify the graphical representation of the bound, we reduce the number of harmonics of the input signal to K = 1. In so doing, the L × 1 input vector of the sequential PU LMS algorithm is given by the N-decimated version 𝐱(𝑵)(𝒏) of a sinusoidal signal defined in Equation (A20). The bound on the step-size is then reduced to () . 2 2sin 2sin 4 1 max 2 0)( max,tot 0 0 N SeqLMS Nf Nf N L N L λ π π μ =                                ± << (A23) Figures A2 and A3 show the size bound of the sequential PU LMS algorithm given by Equation (A23) for decimation factors N = 2 and N = 4, respectively. Results are given for L = 16, 32, 64, and 128. The normalized frequency of the input vector varies from 0 to 0.5. Figure A2. Step-size bound of the sequential PU LMS algorithm with decimation factor N = 2 for different filter lengths, L = 16, 32, 64, and 128. The input vector is an N-decimated single tone whose normalized frequency varies from 0 to 0.5. Figure A2. Step-size bound of the sequential PU LMS algorithm with decimation factor N= 2 for different filter lengths, L= 16, 32, 64, and 128. The input vector is an N-decimated single tone whose normalized frequency varies from 0 to 0.5. According to Figures A2 and A3, and considering that a large step-size guarantees fast convergence rate, we conclude that a shorter filter can converge faster than a large one when the sequential PU LMS is used. Appl. Sci. 2021,11, 344 21 of 24 Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 25 Figure A3. Step-size bound of the sequential PU LMS algorithm with decimation factor N = 4 for different filter lengths, L = 16, 32, 64, and 128. The input vector is an N-decimated single tone whose normalized frequency varies from 0 to 0.5. According to Figures A2 and A3, and considering that a large step-size guarantees fast convergence rate, we conclude that a shorter filter can converge faster than a large one when the sequential PU LMS is used Appendix A.4. The Gain in Step-Size By defining the gain in step-size Gμ as the ratio between the bounds on the step-sizes in two different cases—N > 1 (sequential PU LMS) and N = 1 (conventional LMS)—we obtain the factor by which the step-size parameter can be multiplied when the adaptive algorithm uses sequential PU () {} {} {} {} () () () . 2sin 2sin 4 1 max 2sin 2sin 4 1 max )( )( max 2 max 2 ,,, 10 0 2 10 0 2 1 0 )( max, 2 1 0max, 2 max,tot )( max,tot 0     = = = =                                ±            ± = === K kk K kk K k N kk K kkk N LMS SeqLMS kNf kNf N L N L C kf kfL LC kfC kfC bound bound NLfKG π π π π λ λ λ λ μ μ μ (A24) To more easily visualize the dependence of the gain in step-size on the length of the filter L and on the decimation factor N, the number of harmonics of the input signal is set Figure A3. Step-size bound of the sequential PU LMS algorithm with decimation factor N= 4 for different filter lengths, L= 16, 32, 64, and 128. The input vector is an N-decimated single tone whose normalized frequency varies from 0 to 0.5. Appendix A.4. The Gain in Step-Size By defining the gain in step-size G µ as the ratio between the bounds on the step-sizes in two different cases—N> 1 (sequential PU LMS) and N= 1 (conventional LMS)—we obtain the factor by which the step-size parameter can be multiplied when the adaptive algorithm uses sequential PU Gµ(K,f0,L,N)=bound{µSeqLMS} bound{µLMS}= 2 maxλ(N) tot,max 2 max{λtot,max} = K ∑ k=1 C2 kλk,max(k f0) K ∑ k=1 C2 kλ(N) k,max(k f0) = K ∑ k=1 C2 kmax1 4L±sin(L2πk f0) sin(2πk f0) K ∑ k=1 C2 kmax(1 4"L N±sin(L N2πkN f0) sin(2πkN f0)#). (A24) To more easily visualize the dependence of the gain in step-size on the length of the filter Land on the decimation factor N, the number of harmonics of the input signal is set to K= 1. Now, the gain in step-size, that is, the ratio between the bounds on the step-size when N> 1 and N= 1, is given by Gµ(1, f0,L,N)=boundµSeqLMS bound{µLMS}=maxn1 4hL±sin(L2πf0) sin(2πf0)io max1 4L N±sin(L N2πN f0) sin(2πN f0). (A25) Figures A4 and A5 show, respectively, the gain in step-size for a single tone when different decimation factors and different filter lengths are considered. According to Figures A4 and A5 show that the step-size can be multiplied by Nas long as certain frequencies, at which a notch in the gain in step-size appears, are avoided. The location Appl. Sci. 2021,11, 344 22 of 24 of these critical frequencies, as well as the number and width of the notches, will be analyzed as a function of the sampling frequency Fs , the length of the adaptive filter L, and the decimation factor N. According to Equations (A24) and (A25), with increasing decimation factor N, the step-size can be multiplied by Nand, as a result of that affordable compensation, the sequential PU LMS algorithm convergence is as fast as the full updates LMS algorithm as long as the undesired disturbance is free of components located at the notches of the gain in step-size. Appl. Sci. 2021, 11, x FOR PEER REVIEW 23 of 25 to K = 1. Now, the gain in step-size, that is, the ratio between the bounds on the step-size when N > 1 and N = 1, is given by () {} {} () () () . 2sin 2sin 4 1 max 2sin 2sin 4 1 max ,,,1 0 0 0 0 0                                ±            ± == Nf Nf N L N L f fL L bound bound NLfG LMS SeqLMS π π π π μ μ μ (A25) Figures A4 and A5 show, respectively, the gain in step-size for a single tone when different decimation factors and different filter lengths are considered. According to Figures A4 and A5 show that the step-size can be multiplied by N as long as certain frequencies, at which a notch in the gain in step-size appears, are avoided. The location of these critical frequencies, as well as the number and width of the notches, will be analyzed as a function of the sampling frequency s F, the length of the adaptive filter L, and the decimation factor N. According to Equations (A24) and (A25), with increasing decimation factor N, the step-size can be multiplied by N and, as a result of that affordable compensation, the sequential PU LMS algorithm convergence is as fast as the full updates LMS algorithm as long as the undesired disturbance is free of components located at the notches of the gain in step-size. Figure A4. Gain in step-size for a single tone and different decimation factors, N = 1, 2, 4, and 8. The length of the filter is set to L = 256 taps. Figure A4. 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