Optimal Leak Factor Selection for the Output-constrained Leaky - - PowerPoint PPT Presentation

Introduction Proposed method Simulation result Conslusion

School of Electrical and Electronic Engineering

Nanyang Technological University, Singapore. Site web : eeewebc.ntu.edu.sg E-mail : DSHI003@e.ntu.edu.sg Paper ID : 4528

Optimal Leak Factor Selection for the Output-constrained Leaky Filtered-Input Least Mean Square Algorithm

Dongyuan Shi, Bhan Lam, Woon-Seng Gan, and Shulin Wen Digital Signal Processing Laboratory

Dongyuan Shi Optimal leak factor 15 avril 2020 1 / 19

Introduction Proposed method Simulation result Conslusion

Agenda

1 Introduction

Active noise control Current solution

2 Proposed method

The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

3 Simulation result

Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

4 Conslusion

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Introduction Proposed method Simulation result Conslusion Active noise control Current solution

Section Plan

1 Introduction

Active noise control Current solution

2 Proposed method 3 Simulation result 4 Conslusion

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Introduction Proposed method Simulation result Conslusion Active noise control Current solution

Active noise control : generating an anti-noise to suppress the primary disturbance at a specific location, is usually regarded as an efficient solution to deal with low-frequency noise Current issue : However, its performance is also hampered by some hardware imperfections, such as output saturation.

Output saturation :

In particular, when the hardware has insufficient output power to attenuate the disturbance, saturation distortion from the overdriven amplifier and actuator will deteriorate both the attenuation performance and residual sound

• quality. It may result in the divergence of the adaptive algorithm in severe

cases.

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Introduction Proposed method Simulation result Conslusion Active noise control Current solution

Current methods : (1. Nonlinear FxLMS algorithm, which requires an accurate estimate of the saturation amplifier model. (2. Clipping, re-scaling, 2GD-FxLMS algorithms, which truncate the

• utput signal to restrain the output power. However, these methods will

bring some amplitude distortion into the output signal. (3. Leaky FxLMS algorithm, in which the leak factor is critical to its performance and usually is determined by trial and error. (4. ···

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Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

Section Plan

1 Introduction 2 Proposed method

The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

3 Simulation result 4 Conslusion

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Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

Figure – 1. Block diagram of ANC with the saturation effect.

For alleviating saturation distortion, the LFxLMS algorithm is applied to update the control filter w(n), w(n + 1) = (1 − µγ)w(n) + µe(n)x′(n). (1) The error signal in Fig. 1 is expressed as e(n) = d(n) − f       

L−1

• l=0

slwT(n − l)x(n − l)       . (2)

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Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

To obtain the optimal solution under the output saturation, we define an active noise control cost function with a specified output constraint given by min

w J(w) = E

      

• d(n) −

L−1

• l=0

slwT(n − l)x(n − l)

• 2

      s.t. g(w) = E

• wT(n)x(n)
• 2

≤ ρ2 (3) where ρ2 denotes the maximum average output power or the rated power

• f the actuator.

To modify (3) to an equality constraint, a slack variable θ2 is introduced as θ2 = ρ2 − g (w). (4) The Lagrangian function, based on (3) and the slack variable in (4), is defined as L(w,λ,θ) = J(w) + λ

• g (w) + θ2 − ρ2

(5) where λ (λ ∈ R) is the Lagrangian factor.

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Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

The optimal control filter of (3) is derived to wo = (λoRx + Rx′)−1 Pdx′ (6) where Rx and Rx′ represent the autocorrelation matrices of the reference signal and the filtered reference signal, respectively. Pdx′ denotes the cross-correlation vector of the filtered reference signal and the disturbance. The optimal Lagrangian factor λo is obtained as λo = wT

• Pdx′ − wT
• Rx′wo

ρ2 = E{wT

• x′(n)d(n)} − E{wT
• x′(n)[x′(n)]T wo}

ρ2 = E{y′

• (n)d(n)} − E{y′
• (n)2}

ρ2 . (7)

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Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

Since anti-noise y′

• (n) is approximately linearly correlated to the

disturbance d(n), and their correlation coefficient is given by E

• [y′
• (n) − E[y′
• (n)]][d(n) − E[d(n)]]
• σ2

y′

• σ2

d

≈ 1 (8) If the output signal y(n) is assumed to be zero-mean white noise, and σ2

yo

presents the variance of the final output signal yo(n). We can get λo = σ2

yo

L−1

l=0 s2 l

ρ2        

• σ2

d

L−1

l=0 s2 l σ2 yo

− 1        . (9)

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Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor

To simplify this analysis, we assume that the reference signal x(n) is zero-mean white noise. So, the optimal control filter with the output constraint can be rewritten as wo =

• λoσ2

x I + Rx′

−1 Pdx′ (10) The optimal solution of the LFxLMS algorithm (1) is given by wo = (γI + Rx′)−1 Pdx′ (11)

Optimal leak factor

γo = σ2

x L−1

• l=0

s2

l (η − 1),

(12) Where, we define the degree of nonlinearity η2 of the system as η2 = max       σ2

d

L−1

l=0 s2 l ρ2 ,1

     

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Introduction Proposed method Simulation result Conslusion Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

Section Plan

1 Introduction 2 Proposed method 3 Simulation result

Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

4 Conslusion

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Introduction Proposed method Simulation result Conslusion Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

In the first simulation, the reference signal is a sinusoid x(n) = 2.15sin(0.16πn), and the power constraint of the system is ρ2 = 0.752/2. The secondary path is set to s = [1,0]T, and the step size of algorithms is µ = 0.005.

Figure – 2. The convergence paths of the leaky FxLMS algorithm with the optimal leak factor, for different primary paths with a sinusoid as the reference signal.

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Introduction Proposed method Simulation result Conslusion Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

The secondary path and the step size are set to s = [0,0.8]T and µ = 0.001, respectively, and the primary noise is White Gaussian noise N (0,2.5).

Figure – 3. The convergence paths of the leaky FxLMS algorithm with the optimal leak factor, for different primary paths with white noise as the reference signal.

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Introduction Proposed method Simulation result Conslusion Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

In the final simulation, we used the real measured primary and secondary paths of an air duct. To simulate the saturation effect of the ANC system, we also introduced a clipping function cascaded with the control filter that truncates parts of the output signal (above 0.75 or below −0.75). The disturbance signal (a sinusoidal reference x(n) = 0.32sin(0.1πn) is modified by the primary path with the measurement noise v(n) ∼ N(0,0.001)).

Dongyuan Shi Optimal leak factor 15 avril 2020 15 / 19

Introduction Proposed method Simulation result Conslusion Tonal noise cancelation Broadband noise cancelation Simulation based on measured pathes

Figure – 4. The spectrum of the error signal and the control signal of different algorithms.

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Introduction Proposed method Simulation result Conslusion

Section Plan

1 Introduction 2 Proposed method 3 Simulation result 4 Conslusion

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Introduction Proposed method Simulation result Conslusion

Conclusion

In the paper, an optimal leak factor is proposed for the LFxLMS algorithm, which assists ANC system to achieve the best noise reduction while restraining the average output power within a specified range to prevent saturation

• distortion. Furthermore, compared to the previous related works, this optimal

leak factor does not require any system saturation model and impulse response

• f the primary path and can be readily calculated in practice.

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Introduction Proposed method Simulation result Conslusion

Thank you for your listening !!

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