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 Dongyuan Shi Optimal leak factor 15 avril 2020 2 / 19
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 Dongyuan Shi Optimal leak factor 15 avril 2020 3 / 19
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. Dongyuan Shi Optimal leak factor 15 avril 2020 4 / 19
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 output 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. ··· Dongyuan Shi Optimal leak factor 15 avril 2020 5 / 19
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 Dongyuan Shi Optimal leak factor 15 avril 2020 6 / 19
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 L − 1 � s l w T ( n − l ) x ( n − l ) e ( n ) = d ( n ) − f (2) . l=0 Dongyuan Shi Optimal leak factor 15 avril 2020 7 / 19
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 L − 1 2 � � � � � s l w T ( n − l ) x ( n − l ) min w J ( w ) = E � d ( n ) − � � � � � (3) l=0 �� � 2 � � w T ( n ) x ( n ) � ≤ ρ 2 s.t. g ( w ) = E � � where ρ 2 denotes the maximum average output power or the rated power of the actuator. To modify (3) to an equality constraint, a slack variable θ 2 is introduced as θ 2 = ρ 2 − g ( w ) . (4) The Lagrangian function, based on (3) and the slack variable in (4), is defined as g ( w ) + θ 2 − ρ 2 � � L ( w ,λ,θ ) = J ( w ) + λ (5) where λ ( λ ∈ R ) is the Lagrangian factor. Dongyuan Shi Optimal leak factor 15 avril 2020 8 / 19
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 w o = ( λ o R x + R x ′ ) − 1 P dx ′ (6) where R x and R x ′ represent the autocorrelation matrices of the reference signal and the filtered reference signal, respectively. P dx ′ denotes the cross-correlation vector of the filtered reference signal and the disturbance. The optimal Lagrangian factor λ o is obtained as λ o = w T o P dx ′ − w T o R x ′ w o ρ 2 o x ′ ( n )[ x ′ ( n )] T w o } = E { w T o x ′ ( n ) d ( n ) } − E { w T (7) ρ 2 = E { y ′ o ( n ) d ( n ) } − E { y ′ o ( n ) 2 } . ρ 2 Dongyuan Shi Optimal leak factor 15 avril 2020 9 / 19
Introduction Proposed method Simulation result Conslusion The output-constrained LFxLMS algorithm Derivation of the optimal leak factor Since anti-noise y ′ o ( n ) is approximately linearly correlated to the disturbance d ( n ) , and their correlation coefficient is given by � � [ y ′ o ( n ) − E [ y ′ o ( n )]][ d ( n ) − E [ d ( n )]] E ≈ 1 (8) � � σ 2 σ 2 y ′ d o If the output signal y ( n ) is assumed to be zero-mean white noise, and σ 2 y o presents the variance of the final output signal y o ( n ) . We can get � � L − 1 l=0 s 2 σ 2 σ 2 y o l d λ o = − 1 . (9) ρ 2 � L − 1 l=0 s 2 l σ 2 y o Dongyuan Shi Optimal leak factor 15 avril 2020 10 / 19
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 � − 1 P dx ′ � λ o σ 2 w o = x I + R x ′ (10) The optimal solution of the LFxLMS algorithm (1) is given by w o = ( γ I + R x ′ ) − 1 P dx ′ (11) Optimal leak factor L − 1 � γ o = σ 2 s 2 l ( η − 1) , (12) x l=0 Where, we define the degree of nonlinearity η 2 of the system as σ 2 η 2 = max d l ρ 2 , 1 � L − 1 l=0 s 2 Dongyuan Shi Optimal leak factor 15 avril 2020 11 / 19
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 Dongyuan Shi Optimal leak factor 15 avril 2020 12 / 19
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 . 16 πn ) , and the power constraint of the system is ρ 2 = 0 . 75 2 / 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. Dongyuan Shi Optimal leak factor 15 avril 2020 13 / 19
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. Dongyuan Shi Optimal leak factor 15 avril 2020 14 / 19
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 . 1 πn ) 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. Dongyuan Shi Optimal leak factor 15 avril 2020 16 / 19
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