用於主動消噪系統之新可變長度與步階大小的FXLMS 演算法

國

立

交

通

大

學

電機學院

電信學程

碩

士

論

文

用於主動消噪系統之新可變長度與步

階大小的 FXLMS 演算法

Active Noise Cancellation with a New

Variable Tap Length and Step Size

FXLMS Algorithm

研 究 生：楚 斐 韜

指導教授：吳 文 榕

博士

張 大 中

博士

用於主動消噪系統之新可變長度與步階大小的

FXLMS 演算法

Active Noise Cancellation with a New Variable

Tap Length and Step Size FXLMS Algorithm

研 究 生 ： 楚斐韜 Student: Fei-Tao Chu

指導教授 ： 吳文榕 博士 Advisor: Wen-Rong Wu, Ph.D.

張大中 博士 Dah-Chung Chang, Ph.D.

國立交通大學

電機學院

電信學程

碩士論文

A Thesis

Submitted to College of Electrical and Computer

Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Communication Engineering

Jan 2013

用於主動消噪系統之新可變長度與

步階大小的 FXLMS 演算法

研究生：楚斐韜

指導教授：吳文榕

博士

張大中

博士

國立交通大學

電機學院

電信學程

碩士班

摘 要

Active Noise Cancellation with a New Variable

Tap Length and Step Size FXLMS Algorithm

Student:Fei-Tao Chu Adivisor:Wen-Rong Wu, Ph.D.

Dah-Chung Chang, Ph.D.

Degree Program of Electrical and Computer Engineering

National Chiao Tung University

ABSTRACT

The filtered-X least mean square (FxLMS) algorithm is widely used for active noise cancellation (ANC). Some variants of FxLMS algorithms have been studied to reduce computational complexity or to improve con-vergence rate. In general applications, a long tap length is usually re-quired for the conventional FxLMS method which convergence rate is very slow though its structure is possibly very easy to implement. In this paper, a new ANC system is proposed with a variable tap length and step size FxLMS algorithm. Taking into account the effect of the lowpass filter in the secondary path of an ANC system, the impulse response of

ing envelope to develop our algorithm. Simulation results show that the proposed algorithm does provide a significant performance improvement on convergence rate and noise reduction ratio compared to the fixed tap FxLMS and previously proposed variable step size FxLMS algorithms.

Acknowledgement

It is really a long journey to complete my thesis. First of all, I sin-cerely express my gratitude to my thesis advisors, Prof. Wen-Rong Wu and Prof. Dah-Chung Chang, for their guidance and direction. In ad-dition, the reviewer, Prof. Ta-Sung Lee, offered his precious advice on my thesis, which inspired me very much. In this year, my family's and friends' support and encouragement are highly appreciated.

Contents

Symbol Description . . . vii

Chapter1 Introduction . . . 1

1.1 Review and Background . . . 1

1.2 Organization . . . 3

Chapter2 The FxLMS ANC System . . . 4

Chapter3 The New Variable Tap Length and Step Size FxLMS Algorithm for ANC . . . 8

3.1 Proposed Algorithm . . . 8

3.2 Recursive Form and Convergence . . . 16

Chapter4 Practice Methods and Online Secondary Path Sys-tems . . . 19

4.1 Proposed Algorithm with Online Secondary Path Es-timation . . . 19

4.2 Proposed Algorithm Practice in Real System . . . 21

Chapter5 Simulation Results . . . 24

5.1 Proposed Algorithm on FxLMS . . . 24

5.2 Proposed Algorithm on Online Secondary Path Mod-eling FxLMS . . . 29

5.3 Proposed Algorithm Practice in Real System on On-line Secondary Path Modeling FxLMS . . . 33

Chapter6 Conclusion . . . 39

References . . . 40

List of Figures

Figure 2.1 Block diagram of ANC system using the FXLMS

algo-rithm. . . 5

Figure 4.1 ANC system with online secondary-path estimation (Zhang's method). . . 21

Figure 5.1 Frequency response of S(z). . . 25

Figure 5.2 Impulse response of P(z). . . 26

Figure 5.3 Comparison of the MSD convergence performance for different algorithms. . . 27

Figure 5.4 Convergence comparison of tap-length and step size for proposed algorithm and other FxLMS algorithms: (a) Tap length M (n); (b) Step size µ(n). . . 28

Figure 5.5 The transient effect of noise reduction: (a) Input refer-ence noise x(n); (b) Residual error e(n). . . 29

Figure 5.6 Comparison of NRR (dB) performance for different ANC algorithms. . . 30

Figure 5.7 The MSD of proposed algorithm with online secondary path modeling. . . 31

Figure 5.8 The average of MSD andΛofW (z)with different shifted maximum power center. . . 32

Figure 5.9 The steady-state MSD of different variances of back-ground noise v(n). . . 33

Figure 5.10 The steady-state MSD of different variances of auxiliary noisesε(n). . . 34

Figure 5.11 The impulse response of real P(z). . . 35

Figure 5.12 The frequency response of real P(z). . . 36

Figure 5.13 The impulse response of real S(z). . . 37

Figure 5.14 Comparison of NRR (dB) performance for different on-line secondary path modeling FxLMS algorithm. . . 37

Figure 5.15 The result of the comparison of NRR (dB) performance for different online secondary path modeling FxLMS algorithm be-fore 2000 iterations. . . 38

Figure 5.16 Convergence comparison of tap-length and step size for proposed algorithm and other online secondary path modeling FxLMS algorithms: (a) Tap lengthM (n); (b) Step sizeµ(n). . . 38

Symbol Description

P (z) : unknown plant

S(z) : secondary path

W (z) : adaptive filter

v(n) : unwanted background noise

x(n) : undesired noise

d(n) : output of P (z)

e(n) : residual error signal

s(n) : the impulse response ofS(z)

K : sufficient tap length forW (z)

w(n) : coefficient vector

x(n) : input noise vector

bξ(n) : instantaneous estimate of the MSE gradient at timen

b

S(z) : estimated of secondary path

bs(n) : estimated impulse response ofS(z)b x′(n) : the output ofS(z)b

Px : the power of the input signalx(n)

∆ : accounts for the secondary path delay

M : the left tap length of the maximum output impulse response

N : right tap length including the maximum output impulse

response

wo

K : the optimal coefficients for W (z)

τ1 : left decaying factor

τ2 : right decaying factor

rw(i) : zero-mean i.i.d. Gaussian random sequence

σ2

rw : variance of zero-mean i.i.d. Gaussian random sequence

L(n) : left hand-sides tap length at timen R(n) : right hand-side tap length at timen

µ(n) : step size at timen

g_K(n) : total coefficients error

Λ(n) : the MSD of g_K(n) ∥.∥2 2 : ℓ2 norm E[·] : taking expectation σ2_x : variances ofx(n) σ2 v : variances ofv(n) σ2 x′ : variances ofx′(n) σ2 e : variances ofe(n) ε(n) : auxiliary noise

ˆ

u(n) : the output ofS(z)b byε(n)

ξ(n) : convolution output of the auxiliary noiseε(n) and the deviation of S(z)b

σ2

ξ : variances ofξ(n)

Chapter 1

Introduction

1.1

Review and Background

The acoustic noise reduction problem [1] [2] has been explored for many years, which is widely used in headphones, mobile phones, au-tomobiles, and some industries which need to remedy the circumstance of noise disturbance. Instead of using passive methods, the active noise cancellation (ANC) system improves the efficiency in noise control with lower volume and cost. One of most widely used algorithms for the broadband ANC system is the filtered-X LMS (FxLMS) algorithm [3]. The FxLMS algorithm uses secondary path modeling and a control filter to compensate for uncertain effects in the system, e.g., ADC, DAC, error microphone, pre-amplifier, etc., because the FxLMS algorithm can lead to smaller residual noise in the broadband ANC system.

The FxLMS algorithm used for ANC has some variants such as the lattice ANC [4], frequency domain ANC, delayless subband ANC [5] [6], etc. The lattice structure filter fails to provide a satisfying convergence rate when the primary noise is broadband. Although the LMS processing in frequency domain or subband turns to obtain a faster convergence rate

complexity to implement the discrete-time Fourier transform (DFT) or filter banks. Compared with the above variants, the transversal filter structure is relatively simple but a long tap length is required such that the maximum step size is limited in the FxLMS algorithm and consequently, the convergence rate is significantly slow [7] [8]. To remain the advan-tage of the simple LMS algorithm while increase the convergence rate, we develop a new variable tap length and step size FxLMS algorithm for ANC. In the literature, there are some existing variable tap length LMS algorithm [9] [10] [11], which considered a constant exponential decay envelope for the unknown impulse response plant in a system identifi-cation model. Based on minimizing the mean square deviation (MSD) of filter coefficients, the principle of the variable tap length algorithm is to first approach the modeled part of the plant's impulse response with a smaller tap length for using a larger step size which value is inverse to the tap length. Then, the tap length is progressively increased and finally converged to satisfy the minimum MSD criterion with a continuously de-creasing step size. Hence, a fast convergence rate can be obtained.

For the application of ANC, the secondary path contains a lowpass fil-ter model that results in a double-sided decaying envelope for the impulse response. However, the maximum output of the unknown primary plant is not necessarily at the middle of the impulse response. Hence, in

or-der to develop a new variable tap length FxLMS algorithm for ANC, we consider the unsymmetric and double-sided exponential decay impulse response in our case. The adaptation method for the variable step size is also developed in this thesis. Moreover, we propose a recursive form for optimal tap length estimation in order to simplify the computational complexity. Numerical results show that the new variable tap length and step size FxLMS algorithm has a fast convergence rate compared to the conventional FxLMS and variable step size algorithms. From the evalua-tion of noise reducevalua-tion for ANC, the proposed algorithm achieves a better convergence performance than other compared methods as well.

1.2

Organization

The rest of this thesis is organized as follows: Chapter 2 describes the basic ANC system model. The proposed variable tap length and step size algorithm is addressed in Chapter 3, where a recursive form for variable tap length estimation and the convergence problem are also mentioned. In Chapter 4, we discuss how to achieve in the real system and the appli-cation for online secondary path estimation. The simulation results are collected in Chapter 5 and the conclusion is drawn in Chapter 6.

Chapter 2

The FxLMS ANC System

for the loudspeaker systemS(z)to cancel an undesired noisex(n)through P (z). The unwanted background noisev(n)is usually uncorrelated tox(n)

and added to the cancellation error signal. In this model, the objective of W (z) is to minimize the residual error signal e(n). Denote by d(n) the output of P (z) and s(n) the impulse response of S(z). Consider that K

is a sufficient tap length for W (z), the coefficient vector of W (z) at time

index n is w(n) = [w0(n) w1(n) · · · wK−1(n)]T, and the input noise vector

x(n) = [x(n) x(n− 1) · · · x(n − K + 1)]T. The residual error signal can be

expressed as

e(n) = d(n)− [xT(n)w(n)]∗ s(n) + v(n), (2.1)

where∗ denotes linear convolution.

The LMS algorithm can be used to find the recursive solution tow(n)

based on minimizing the mean square error (MSE). Let ξ(n) = e2_{(n), the}

) (n x ) (n x ) (n y ) (n d 



Figure 2.1: Block diagram of ANC system using the FXLMS algorithm. step sizeµas

w(n + 1) = w(n)− µ

2∇bξ(n) (2.2)

wherebξ(n)is an instantaneous estimate of the MSE gradient at timenwith ∇bξ(n) = 2[∇e(n)]e(n). From (2.1), we have

∇e(n) = −s(n) ∗ x(n) = −x′_{(n) (2.3)}

wherex′(n) = [x′(n) x′(n− 1) · · · x′(n− K + 1)]T. Therefore, we have

Substituting (2.4) into (2.2), we have the updated equation of w(n),

w(n + 1) = w(n) + µx′(n)e(n). (2.5) From (2.5), the transfer function S(z) of the secondary path exists in

the updated equation of adaptive filter coefficients and is conventionally called FxLMS algorithm. It is worth to note thatS(z)is usually unknown, so it should be modeled by filterS(z)b , as shown in Fig.2.1. That is,

x′(n) =bs(n) ∗ x(n), (2.6)

where bs(n) is the estimated impulse response of S(z)b . Without loss of

generality, we simply treatS(z) = S(z)b in this work. The online secondary path modeling and estimation methods can be referred to [8] [12].

In a typical ANC system, the length of the impulse responses of un-known plant and secondary path may be very long, which directly affects the tap length of the adaptive filter. As mentioned in [2], the range of the step size is

0 < µ < 2

Px(K + 2 + 2∆)

, (2.7)

wherePx is the power of the input signalx(n)forP (z) and∆accounts for the secondary path delay. When the tap length is long, the convergence

speed becomes small because of a very small step size. Consequently, a variable length and step size LMS algorithm is studied in order to improve the ANC system.

Chapter 3

The New Variable Tap Length

and Step Size FxLMS Algorithm for ANC

3.1

Proposed Algorithm

From (2.1), the z-transform of the residual error signal is

E(z) = [P (z)− W (z)S(z)]X(z) + V (z). (3.1)

As ignoring V (z), a simple insight into (3.1) is that the residual error is close to zero, i.e., E(z) = 0, after the adaptive filter converges. Hence, we can see that the control filter W (z) is to realize the optimal transfer

function with

Wo(z) = P (z)

S(z). (3.2)

In some circumstances, the power profiles of the impulse responses of

P (z) and S(z) may have exponentially decaying envelopes on both sides of the maximum output response. For example, the loudspeaker system model S(z) includes the D/A converter, reconstruction filter, power

am-plifier, etc., in which the lowpass reconstruction filter usually consists of symmetric filter coefficients for a linear-phase concern. However, the

P (z)may have an unsymmetric decaying envelope. Here, we assume that the impulse response ofW (z)also has an unsymmetric decaying envelope, where the left tap length of the maximum output impulse response is M

while the right one isN including the maximum output impulse response.

We express the optimal coefficients forW (z)as follows:

wo K = [w o −M · · · w−1o w0o w o 1· · · w o N−1] T _(3.3)

and the following exponential function is used to model the envelope of the impulse response coefficients

wo_i =          eiτ1_r w(i), i =−M, · · · , −1 e−iτ2_r w(i), i = 0, 1,· · · , N − 1, (3.4)

wherei =−M, · · · , −1, 0, 1, · · · , N −1, the decaying factorτ1 andτ2 are

pos-itive constants, andrw(i)is a zero-mean i.i.d. Gaussian random sequence with varianceσ2

rw.

The proposed FxLMS algorithm adaptively adjusts its tap length and step size as time progresses. Denote by L(n),R(n)andµ(n)the left

hand-sides tap length, right hand-side tap length and step size at time n,

ter vector and input vector, respectively, and wL(n)+R(n)(n) = [w−L(n)(n) · · · w₋₁(n) w0(n) w1(n) · · · wR(n)−1(n)]T and x′_L(n)+R(n)(n) = [x′(n + L(n)) · · · x′(n + 1) x′(n) x′(n− 1) · · · x′(n− R(n) + 1)]T_{, we can rewrite (2.5) as}      wL(n+1)(n + 1) wR(n+1)(n + 1)     =                0L(n+1)−L(n) wL(n)(n) wR(n)(n) 0R(n+1)−R(n)                + µ(n + 1)e(n)x′_{L(n+1)+R(n+1)}(n). (3.5)

Here, we assume that the modeled part ofW (z)is expanded from the

max-imum impulse response of the filter, that is, wK(n) = [0 T

L wTL(n)+R(n)(n) 0 T R]T where the left 0L denotes the 1× (M − L(n))zero vector and the right 0R denotes the 1× (N − R(n)) zero vector. Now, split wo

M +N into four parts as wo K =                wo M−L(n)′ wo L(n) wo R(n) wo N−R(n)′′                , (3.6)

and define the total coefficients error as g_K(n) =                0M−L(n) wL(n)(n) wR(n)(n) 0N−R(n)                −wo K. (3.7)

From (3.2), we can express the output ofP (z) as

d(n) = [xT

K(n) w o

K]∗ s(n). (3.8)

Substituting (3.8) into (2.1) and using (3.7), the residual error signal be-comes e(n) = [xT_K(n)wo_K − xT_L(n)+R(n)(n)wL(n)+R(n)(n)] ∗s(n) + v(n) = −[xT K(n)gK(n)]∗ s(n) + v(n) = −x′T_K(n)g_K(n) + v(n). (3.9) Substituting e(n) in (3.5) with (3.9) and subtracting wo

(3.5), we obtain g_K(n + 1) =A(n)g_K(n) + µ(n + 1)v(n)                0_M−L(n+1) x′L(n+1)(n) x′R(n+1)(n) 0N−R(n+1)                , (3.10) where A(n) =IK− µ(n + 1)                0M−L(n+1) x′L(n+1)(n) x′R(n+1)(n) 0N−R(n+1)                x′TK(n), (3.11)

and IK is theK× K identity matrix.

To develop the recursive algorithm forL(n + 1), R(n + 1) andµ(n + 1),

the MSD of g_K(n) is explored. Define

Λ(n)≡ E[∥g_K(n)∥2₂], (3.12) where∥.∥2

2denotesℓ2norm andE[·]represents taking expectation. Assume

thatx(n)andv(n)are two i.i.d. Gaussian sequences with variancesσ2

x and

σ2

we have Λ(n + 1) = η(n + 1)Λ(n) + (β(n + 1)− η(n + 1))Γ(n + 1) + γ(n + 1), (3.13) where Γ(n + 1) = E[wwwo M−L(n+1)′ w w2 2 ] + E[wwwo N−R(n+1)′′ w w2 2 ] (3.14) η(n + 1) = 1− 2µ(n + 1)σ2_x′+ (L(n + 1) + R(n + 1) + 2) ×µ2_{(n + 1)σ}4 x′, (3.15) β(n + 1) = 1 + (L(n + 1) + R(n + 1))µ2(n + 1)σ_x4′, (3.16) γ(n + 1) = (L(n + 1) + R(n + 1))µ2(n + 1)σ2_x′σ2v, (3.17) and using (3.4) for wo

K, we have E[∥wo M−L(n+1)′∥ 2 2] = 1− e2(M−L(n+1))τ1 1− e2M τ1 E[∥w o M∥ 2 2], (3.18) E[∥wo N−R(n+1)′′∥ 2 2] = e−2R(n+1)τ2 − e−2Nτw 1− e−2Nτ2 E[∥w o N∥ 2 2], (3.19)

where E[∥wo M∥ 2 2] = e−2Mτ1(₁− e2M τ1) 1− e2τ1 σ 2 rw, (3.20) E[∥wo N∥ 2 2] = 1− e−2Nτ2 1− e−2τ2 σ 2 rw. (3.21)

Substituting (3.14), (3.18), (3.19), (3.20) and (3.21) into (3.13), the MSD can be rewritten as Λ(n + 1) = η(n + 1)Λ(n) + (β(n + 1)− η(n + 1)) × ( e−2Mτ1 − e−2L(n+1)τ1 1− e2τ1 + e−2R(n+1)τ2 − e−2Nτ2 1− e−2τ2 ) σ2_rw +γ(n + 1). (3.22)

The optimal tap length and step size can be found by minimizing the MSD with respect to L(n + 1), R(n + 1) and µ(n + 1). Therefore, taking

the first-order derivative of Λ(n + 1)with respect to L(n + 1), R(n + 1)and

µ(n + 1), respectively, and setting ∂Λ(n+1) ∂L(n+1),

∂Λ(n+1) ∂R(n+1) and

∂Λ(n+1)

∂µ(n+1) to zero, after some mathematical manipulation we obtain

L(n + 1) =− 1 2τ1 lnµ(n + 1)(σ 2 x′Λ(n) + σv2)(1− e2τ1) −4τ1(1− µ(n + 1)σx2′)σr2w , (3.23)

R(n + 1) =− 1 2τ2 lnµ(n + 1)(σ 2 x′Λ(n) + σv2)(1− e−2τ2) 4τ2(1− µ(n + 1)σ_x2′)σ2rw , (3.24) and µ(n + 1) = 1− Γ(n+1)_Λ(n) (L(n + 1) + R(n + 1) + 2)σ2 x′ + (L(n+1)+R(n+1))σ2 v−2σ2_x′Γ(n+1) Λ(n) . (3.25)

We have to mention that (3.23), (3.24) and (3.25) are required to be solved simultaneously forL(n + 1),R(n + 1) andµ(n + 1). It is a tough work to get the closed-form solution of the joint equations. Taking into consideration a quasi-static assumption for L(n) ≈ L(n + 1) and R(n)≈ R(n + 1) [11], a

suboptimal solution can be efficiently found by replacing L(n + 1) and R(n + 1) byL(n)andR(n)in (3.25). That is,

µ(n + 1) = 1− Γ(n) Λ(n) (L(n) + R(n) + 2)σ2 x′ + (L(n)+R(n))σ2 v−2σ_x′2Γ(n) Λ(n) . (3.26)

From (3.26) we can observe that when v(n) is ignored and the adaptive

filter approaches the perfect tap length, σ2

v = Γ(n) = 0 and thus µ(n +

1) ≈ _(K+2)σ1 2 x′

, which is consistent with the convergence condition (2.7) excluding the effect of S(z) according to (3.2). Therefore, an alternating calculation by (3.23), (3.24) and (3.26) can be used forL(n + 1), R(n + 1)

3.2

Recursive Form and Convergence

The more useful results in using (3.23) and (3.24) are to develop their alternatives of recursive forms. First, as mentioned in [10], it can be shown thatσ2

e(n) = σx2′Λ(n) + σ2v. After re-manipulating (3.26), we have

µ(n + 1) = Λ(n)− Γ(n) 2σ2 x′(Λ(n)− Γ(n)) + (L(n) + R(n))σe2(n) = 1 2σ2 x′+ (L(n) + R(n))Φ(n) (3.27) where Φ(n) = σ 2 e(n) Λ(n)− Γ(n). (3.28)

Moreover, from (3.6), (3.7), and (3.12), we can prove that

Λ(n)− Γ(n) = E[∥g_L(n)+R(n)(n)∥2₂]. (3.29)

Now, let us move on writing the result of L(n + 1)− L(n) and R(n +

1)− R(n)based on (3.23) and (3.24). Note that althoughσ2

e(n) varies and actually vanishes as time n progresses, the statistics of two successive samples can be viewed very close, i.e., σ2

e(n) = σ2e(n + 1) = σe2, when the algorithm runs under convergence. Based on the above statement and

some mathematical manipulation, we have L(n + 1)− L(n) = − 1 2τ1 lnµ(n + 1) (1− µ(n)σ 2 x′) µ(n) (1− µ(n + 1)σ2 x′) , (3.30) and R(n + 1)− R(n) = − 1 2τ2 lnµ(n + 1) (1− µ(n)σ 2 x′) µ(n) (1− µ(n + 1)σ2 x′) . (3.31)

Substituting (3.27) into (3.30) and (3.31), we obtain the recursive form for L(n + 1) andR(n + 1)as follows:

L(n + 1) = L(n)− 1 2τ1 ln(L(n− 1) + R(n − 1))Φ(n − 1) + σ 2 x′ (L(n) + R(n))Φ(n) + σ2 x′ , (3.32) R(n + 1) = R(n)− 1 2τ2 ln(L(n− 1) + R(n − 1))Φ(n − 1) + σ 2 x′ (L(n) + R(n))Φ(n) + σ2 x′ . (3.33)

In practical use, the tap length is actually an integer number. Hence, we also need to round down L(n + 1) and R(n + 1) obtained from (3.32) and (3.33) to the nearest integers as the resultant.

The next question is whether the proposed recursion (3.32) and (3.33) can converge. Return to (3.13), it is known that for the two parameters

L(n + 1),R(n + 1)andµ(n + 1), the MSD is a convex function of them and the recursions will find minimum MSD if we can prove that ∂Λ2(n+1)

∂µ2_(n+1) > 0, ∂Λ2_{(n+1) ∂Λ}2_(n+1)

with respect toµ(n + 1), L(n + 1) andR(n + 1), respectively, we have ∂Λ2_{(n + 1)} ∂µ2_{(n + 1)} = 2σ 2 x′(L(n + 1) + R(n + 1))σ 2 e(n) + 4σ 4 x′(Λ(n)− Γ(n)), (3.34) ∂Λ2_{(n + 1)} ∂L2_{(n + 1)} = 8τ 2 1µ(n + 1)σ 2 x′(1− µ(n + 1)σ 2 x′) σ2 rw ( −e−2L(n+1)τ1) 1− e2τ1 , (3.35) and ∂Λ2(n + 1) ∂R2_{(n + 1)} = 8τ 2 2µ(n + 1)σ 2 x′(1− µ(n + 1)σ 2 x′) σ2_rwe−2R(n+1)τ2 1− e−2τ2 . (3.36)

In (3.34), Λ(n) − Γ(n) ≥ 0 based on (3.29) and in (3.35) and (3.36), 1−

µ(n + 1) > 0 in general situations. Therefore, the results of (3.34), (3.34) and (3.35) are all positive and then, we have shown that the MSD is a convex function of L(n + 1), R(n + 1)and µ(n + 1), and consequently, the

new variable tap length and step size FxLMS algorithm can converge to the minimum MSD.

Chapter 4

Practice Methods and

On-line Secondary Path Systems

4.1

Proposed Algorithm with Online Secondary

Path Estimation

In the previous section, the discussion is under the condition ofS(z) =b S(z). However in practice,S(z)b is different from S(z). In recent research the deviation ofS(z)b fromS(z)can be reduced with online secondary esti-mation. Zhang's method [13] has better performance in online secondary estimation structure, which is considered for the application of the pro-posed algorithm.

The block diagram of the Zhang's method is depicted in Fig.4.1. The coefficients ofW (z)is updated bye′(n), which differs from the traditional FxLMS and is expressed in the following:

e′(n) = d(n)− y(n) + u(n) − ˆu(n) + v(n), (4.1)

ondary path estimation FxLMS system, we set the background noisev(n)

to zero. The auxiliary noise ε(n) is generated by white noise generator, which is used for S(z)b convergence. Using the same method to obtain (3.9), we have

e′(n) = −x′T_K(n)g_K(n)− ε(n) ∗ (^s(n) − s(n)) = −x′TK(n)gK(n)− (ˆu(n) − u(n))

= −x′T_K(n)g_K(n)− ξ(n). (4.2) The residual errorξ(n)is obtained by the convolution output of the

auxil-iary noiseε(n)and the deviation ofS(z)b which equalsu(n)ˆ −u(n). Then, in

addition to σ2

v in (3.17), (3.23), (3.24), (3.25) and (3.26), σ2ξ is added. Ex-isting the deviation ofS(z)b can make the convergence rate ofW (z)slower. When S(z)b converges to S(z), the convergence of W (z)will be not influ-enced.

In practice,S(z)b needs a initial value to speed up the convergence rate

and reduces the influence for the convergence ofW (z). When the system structure and components, such as DAC, ADC, microphone, loudspeaker, etc., are fixed, an impulse signal into the DAC can be used to measure the impulse response of S(z) as the output is received behind the ADC.

) (n x ) (n x ) ( ) (n u n y  ) (n d   ) (n e ) (n y ) (z P ) (z W LMS ) ( ˆ z S ) (z S   White Noise Generator Sˆ z( ) ) (z H LMS LMS         ) ( ˆ n u ) (n e ) (n e ) (n  Copy ) (n v  ) (n  ) (n 

Figure 4.1: ANC system with online secondary-path estimation (Zhang's method).

methods.

However, when the location of the maximum impulse response output of S(z)b has the offset from that of S(z), the convergence performance is seriously affected. We can evaluate the performance of Λ to correct the maximum output location of S(z)b . The related results will be shown in

the next section.

4.2

Proposed Algorithm Practice in Real

Sys-tem

In practice, the optimal coefficients wo

size in proposed algorithm. Next, we will introduce a method to get those parameters.

First, use the fixed tap length with large step size LMS algorithm, whose tap length is large enough to cover to the maximum power of im-pulse response of the unknown plant. After hundreds of times iteration, generally optimal coefficients of the W (z) can be obtained. We express

the generally optimal coefficients forW (z)as follows:

w'o_K′ = [w′o_−M′ · · · w′o₋₁ w′o₀ w′o₁ · · · w′o_N′₋₁]T (4.3) The K′ is total tap length of the generally optimal coefficients w'o_K′, and theM′is left side tap length, andN′is right side tap length with maximum output of generally optimal coefficients.

Second, we inverse all of the w'oK′ to positive. Then compare the two hand side adjacent coefficients in order to get the each peak value like

w′o_n−1 < w′o_n > w′o_n+1, and others are set to zero. The peak coefficients can be

express as follows :

po K′ = [p

o

−M′ · · · po₋₁ po₀ po₁· · · po_N′₋₁]T (4.4) Then according to the 3.4, ignore the Gaussian random sequence and

re-verse the equation. We can rewrite the generally decaying factor if po_i > 0, τ_i′ =          1 i ln p o i, i =−M, · · · , −1 1 −iln po−i, i = 0, 1,· · · , N − 1. (4.5)

Then take the average of τ_i′ with two hand side in order to obtain the generally decaying factorτ₁′ and τ₂′. Substituteτ₁′, τ₂′,M′ andN′ into 3.20, 3.21, 3.22, 3.23 and 3.24.

Third, we can use the the two hand sideℓ2 norm of the last 10 grown

taps are less than threshold L′ and R′ for the two hand side to terminate

the growth of the tap length.

Chapter 5

Simulation Results

5.1

Proposed Algorithm on FxLMS

In this section, some computer simulations are employed to compare ANC performance of the proposed algorithm with different FxLMS

al-gorithms. The secondary path model S(z) includes the loudspeaker and

microphone system, and according to [8], a filter is used to model the sec-ondary path model S(z) and the frequency response is shown in Fig.5.1.

We set the impulse response length of plantP (z)as 1088 and the length of the loudspeaker model S(z) asL = 65. Hence, a proper length for the op-timal impulse response Wo_{(z)of W (z)is K = 1024. For simplicity, we let}

b

S(z) equalS(z) and generate P (z)from P (z) = Wo(z)S(z) and the impulse

response is shown in Fig.5.2. Once the estimateW (z)c ofW (z)is obtained,

the MSD can be easily evaluated through 100 Monte Carlo simulations by calculating ∥ˆw(n) − wo

K(n)∥22 for each iteration. Other simulation

se-tups include thatrw(i)was generated by a zero-mean white Gaussian ran-dom process with variance σ_r2_w = 0.01; the exponential decaying factorτ1

and τ2 were 0.01 and 0.005; and the reference noise and the background

noise were zero-mean i.i.d. and uncorrelated Gaussian processes with variances σ2

0 0.2 0.4 0.6 0.8 1 −6000

−4000 −2000 0

Normalized Frequency (×π rad/sample)

Phase (degrees) 0 0.2 0.4 0.6 0.8 1 −40 −30 −20 −10 0 10

Normalized Frequency (×π rad/sample)

Magnitude (dB)

Figure 5.1: Frequency response of S(z).

In Fig. 5.3, the comparison of the MSD performance in decibel (dB) is shown for 50000 iterations. Since the tap length of Wo_{(z)is 1024, we} choose the same tap length for the typical FxLMS algorithm with large and small step sizes, where the large step size, according to (2.7), is set asµmax= _(1024+2)1 _×σ2

x′

and the small step size µmin= 0.2µmaxbecauseµmin

pro-vides the 1024-tap FxLMS algorithm a steady-state performance close to the proposed algorithm in 50000 iterations. A 1024-tap FxLMS algo-rithm with variable step size which is similar to [8] is also compared by using

0 200 400 600 800 1000 1200 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Samples number Impulse Response Impulse response of P(z)

Figure 5.2: Impulse response of P(z). Pe(n)−Pe,min

Pe,max−Pe,min with Pe,max and Pe,min representing the maximum and minimum average powers of e(n), respectively and Pe(n) = _T1

∑n

i=n−T +1e2(n) where

T is an averaging constant with T = 200 herein. The maximum average

powerPe,maxuses the average of the first100 iterations ofPe(n)multiplied by1.3 and minimum average power Pe,min uses the average of the last100

iterations of Pe(n) multiplied by 0.7. From Fig. 5.3, we can find that the proposed algorithm has a much faster convergence rate than other algorithms. However, if we set a fixed step size µmax for the proposed

algorithm, the converged MSD performance becomes worse in spite of the same convergence rate. Although the variable step size FxLMS has a better convergence rate than the large step size FxLMS and a better converged performance than the small step size FxLMS, its step size is

0 1 2 3 4 5 x 104 −25 −20 −15 −10 −5 0 Number of iterations MSD(dB)

Proposed, variable step size 1024−tap FxLMS, small step size 1024−tap FxLMS, large step size 1024−tap FxLMS, variable step size Proposed, fixed step size

Figure 5.3: Comparison of the MSD convergence performance for dif-ferent algorithms.

somewhat heuristic such that the overall performance is worse than the proposed algorithm. By terminating the growth of the left tap length when theℓ2norm of the last 10 grown taps is less than10−2and10−5for the

right-side, the total tap length approaches the number about 1104 as shown in Fig. 5.4(a), however, the MSD performance almost saturates as the variable step size can still decrease slowly as shown in Fig. 5.4(b).

An experiment to see the transient effect of noise reduction using the proposed algorithm is shown in Fig. 5.5. Suppose the input noise x(n)

is exacerbated with variance σ2

x = 16 from the 25000th to the 40000th iteration as shown in Fig. 5.5(a). From Fig. 5.5(b), the ANC output

0 1 2 3 4 5 x 104 0 100 200 300 400 500 600 700 800 900 1000 Number of iterations

Tap Length L(n) and R(n)

(a)

L(n) Proposed, variable step size R(n) Proposed, variable step size L(n) Proposed, fixed step size R(n) Proposed, fixed step size

0 1 2 3 4 5 x 104 10−5 10−4 10−3 10−2 10−1 Number of iterations Step Size µ (n) (b)

Proposed, variable step size 1024−tap FxLMS, variable step size

L(n) Proposed, fixed step size

L(n) Proposed, variable step size

Figure 5.4: Convergence comparison of tap-length and step size for pro-posed algorithm and other FxLMS algorithms: (a) Tap length M (n); (b) Step size µ(n).

evaluate the noise reduction performance for ANC, we define an index, noise reduction ratio NRR (dB), which is given by

NRR(dB) = 10log ( E[d2_(n)] E[e2_(n)] ) , (5.2)

whereE[·]can be evaluated by ensemble average for simplicity. By NRR (dB), Fig. 5.6 plots the comparison of the proposed algorithm with other algorithms. In addition to a better MSD performance, the proposed algo-rithm has the superior NRR (dB) performance for the ANC application because of its fast convergence property.

0 1 2 3 4 5 x 104 −20 −10 0 10 20 Reference noise x(n) Number of iterations (a) 0 1 2 3 4 5 x 104 −5 0 5

Residual error e(n)

Number of iterations (b)

Figure 5.5: The transient effect of noise reduction: (a) Input reference noise x(n); (b) Residual error e(n).

5.2

Proposed Algorithm on Online Secondary

Path Modeling FxLMS

We have two simulations about effect of S(z)b . The online secondary path modeling is based on Zhang's method like Fig.4.1. The auxiliary noiseε(n)was zero-mean i.i.d. and uncorrelated Gaussian processes with variances σ2

ξ = 1. The initial value of S(z)b uses the s(n) with weighted standard deviation(0.5∗s(n))2)of Gaussian random noise. The step size of

b

S(z)was set toµs= _(65+2)σ0.0452

v, and the step sizeH(z)was set toµh =

0.01 (1024+2)σ2

x,

0 1 2 3 4 5 x 104 0 5 10 15 20 25 Number of iterations NRR(dB)

Proposed, variable step size 1024−tap FxLMS, small step size 1024−tap FxLMS, large step size 1024−tap FxLMS, variable step size Proposed, fixed step size

Proposed, fixed step size Proposed, variable step size 1024−tap FxLMS, large step size

1024−tap FxLMS, small step size

1024−tap FxLMS, variable step size

Figure 5.6: Comparison of NRR (dB) performance for different ANC algorithms.

is s(n), and the other one iss(n) with weighted random noise. As Fig.5.7,

the variance of auxiliary noise σ2_ξ affects the step size(3.26) during the

convergence. Although the S(z)b is almost the same as the S(z) at about

10000 iterations. That effect causes slightly different convergence speed, which make the MSD have a little offset at the end of simulation.

In the second simulation, we shift the maximum power center of ini-tial value of S(z)b with one tap on two-hand sides,each for five times. As

Fig.5.8, no matter which side the maximum power center is shifted to or

no matter how many taps, Λand MSD are very poor. However, when the

maximum power center is aligned, theΛ and MSD are the best.

sim-0 0.5 1 1.5 2 x 104 −25 −20 −15 −10 −5 0 Number of iterations MSD(dB)

Proposed with perfect S_(n) Proposed with random S(n)

Figure 5.7: The MSD of proposed algorithm with online secondary path modeling.

ulations: one is to change different variances of background noise on the traditional FxLMS, and the other one is to change different variances of auxiliary noise on the Zhang's method of online secondary path model-ing FxLMS. In the first one, the variance of background noiseσ2

v takes ten numbers that make the SNR(10log(σd2

σ2

v)) from40dB to 2dB. Then we com-pare the steady-state MSD of proposed algorithm in Fig.5.9. In the sec-ond, the variance of auxiliary noiseσ_ε2takes eleven numbers that make the

SNR(10log(σ2d σ2

ε)) from 10dB to −10dB. Then we compare the steady-state MSD of proposed algorithm on online secondary path modeling FxLMS in Fig.5.10.

−5 −4 −3 −2 −1 0 1 2 3 4 5 −60 −50 −40 −30 −20 −10 0 10 Shifted taps dB MSD after convergence Λ after convergence

Figure 5.8: The average of MSD and Λ of W (z) with different shifted

maximum power center.

line secondary path modeling FxLMS affect the covergence. In Fig.5.9,

when the SNR is smaller than 25dB, the steady-state MSD of proposed

algorithm is better than the others. According the Fig.5.7 and Fig.5.10, the smaller auxiliary noise makes the convergence ofS(z)b slower and in-directly makes the convergence ofW (z)slower. However, in the Fig.4.1,

because the residual errore(n) includes theε(n)∗ s(n), the larger auxiliary

noise makes the residual error e(n)larger, too. Therefore, the variance of auxiliary noise σ2

ε can not be set too large within the acceptable conver-gence speed, so selecting SNR between0dB to −6dB is the better choice in this case.

40 35 30 25 20 15 10 6 4 2 −30 −25 −20 −15 −10 −5 0 SNR (dB) Steady−state MSD (dB)

Proposed, variable step size 1024−tap FxLMS, small step size 1024−tap FxLMS, large step size 1024−tap FxLMS, variable step size [8]

Figure 5.9: The steady-state MSD of different variances of background noise v(n).

5.3

Proposed Algorithm Practice in Real

Sys-tem on Online Secondary Path Modeling

FxLMS

In accordance with the method described in the previous section, we use a real system of P (z) and S(z) to verify the proposed algorithm. The impulse response and frequency response of P (z) are shown as Fig.5.11

and Fig.5.12:

And the impulse response of S(z) is shown as Fig.5.13. The impulse response sample length of P (z) and S(z) are 145 and 51 taps. Based on

10 7 4 2.2 1 0 −3 −6 −7.8 −9 −10 −24.5 −24 −23.5 −23 −22.5 −22 −21.5 −21 SNR 10log(σ_d2_/σ2_ε) Steady−state MSD (dB)

Proposed, variable step size

Figure 5.10: The steady-state MSD of different variances of auxiliary noisesε(n).

The simulation is to compare noise reduction ratio NRR (dB) of pro-posed algorithm with different on online secondary path modeling FxLMS algorithms shown in Fig.5.14, Fig.5.15 and Fig.5.16. The NRR defini-tion of the on online secondary path modeling FxLMS is different from traditional FxLMS, which is given by

NRR(dB) = 10log ( E[d2_(n)] E[e′2(n)] ) . (5.3)

About the proposed algorithm setting, the generality optimal coefficients parameters M′ andN′ are set to 64 and 316, theτ₁′ andτ₂′ are set to 0.0925

and0.0626. The terminating thresholdL′ andR′are set to10−1 and5×10−4. About the normalized LMS(NLMS) algorithm [2] setting, the smoothing

0 50 100 150 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Samples number Impulse response Impulse response of P(z)

Figure 5.11: The impulse response of real P(z).

parameter is set to 0.99, the normalized step size α is set 0.4 and the tap

length L is set to 95. We use two 95-taps FxLMS algorithms one with a

large step size and the other with the small step size, where the large step size is set as µmax = ₍₉₅₊₂₎1.2_×σ2

x′

and the small step size µmin = ₍₉₅₊₂₎0.4_×σ2 x′

. A 95-taps FxLMS with variable step size is similar to [8], whose the param-etersρ,Pe,Pe,min andPe,max are the same as the previous traditional FxLMS

simulation, and the maximum and minimum step sizes are between the other two 95-taps FxLMS with large and small step size.

In Fig.5.14, we adjust the final performance of proposed algorithm, NLMS [2], variable step size [8] and 95-taps FxLMS with small step size are close, and then compare the convergence speeds. The step size of the 95-taps FxLMS with large step size is adjusted so that its convergence

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −8000 −6000 −4000 −2000 0 2000

Normalized Frequency (×π rad/sample)

Phase (degrees) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −40 −30 −20 −10 0 10

Normalized Frequency (×π rad/sample)

Magnitude (dB)

Figure 5.12: The frequency response of real P(z). final performance.

In Fig.5.15, the convergence speed of proposed algorithm is better than the NLMS [2], variable step size [8] and 95-taps FxLMS with small step size. The final performance of proposed algorithm is better than the 95-taps FxLMS with large step size.

In Fig.5.16(a), the tap length of proposed algorithm is terminated at 92 taps, which is close the tap length of optimal coefficients 95 taps. Shown in Fig.5.16(b), the step size of proposed algorithm gradually decreases slower than the others.

0 10 20 30 40 50 60 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Samples number Impulse response Impulse response of S(z)

Figure 5.13: The impulse response of real S(z).

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 2 4 6 8 10 12 14 16 18 20 Number of iterations NRR(dB)

Proposed, variable step size with limit NLMS[2]

Variable step size LMS[8] 95−tap FxLMS, small step size 95−tap FxLMS, large step size

Figure 5.14: Comparison of NRR (dB) performance for different online secondary path modeling FxLMS algorithm.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 2 4 6 8 10 12 14 16 18 Number of iterations NRR(dB)

Proposed, variable step size with limit NLMS[2]

Variable step size LMS[8] 95−tap FxLMS, small step size 95−tap FxLMS, large step size

Figure 5.15: The result of the comparison of NRR (dB) performance for different online secondary path modeling FxLMS algorithm before 2000 iterations. 0 2000 4000 6000 8000 0 10 20 30 40 50 60 70 80 Number of iterations

Tap Length L(n) and R(n)

(a)

L(n) Proposed, variable step size R(n) Proposed, variable step size

0 2000 4000 6000 8000 10−4 10−3 10−2 10−1 Number of iterations Step Size µ (n) (b)

Proposed, variable step size with limit NLMS[2]

Variable step size LMS[8]

Figure 5.16: Convergence comparison of tap-length and step size for pro-posed algorithm and other online secondary path modeling FxLMS algo-rithms: (a) Tap lengthM (n); (b) Step sizeµ(n).

Chapter 6

Conclusion

We propose a new ANC system using a variable tap length and step size FxLMS algorithm where a simple recursive form is obtained as well to estimate the tap length. Here, the new FxLMS algorithm is developed based on the assumption that the impulse response of the control filter in the ANC secondary path has an unsymmetric and exponential decay-ing enve- lope in order to deal with the lowpass filter in the loudspeaker system. The proposed FxLMS algorithm has a much faster convergence rate than the conventional and variable step size FxLMS algorithms with-out intensively computational cost of implementing complicated DFT or subband filters.

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Vita

Fei-Tao Chu was born in Taiwan R.O.C., in 1978. He received the B.S. degree in Computer Science and Information Engineering from Chung Hua University.

From June 2000 to January 2012, serving in AVID Electronics Corp, ELAN Microelectronics Corp. and WYS SoC Corp., he was engaged in the researches of the audio signal processing.

In 2013, he got the M.S. degree in the Institute of Communications Engineering in National Chiao Tung University, Hsin-Chu, Taiwan. His research interests include active noise cancellation and adaptive filtering.