Adaptive Filters – Adaptation Control Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Electrical Engineering and Information Technology Digital Signal Processing and System Theory Slide 1
Contents of the Lecture Today: Adaptation Control: Introduction and Motivation Prediction of the System Distance Optimum Control Parameters Estimation Schemes Examples Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 2 Slide 2
Application Example – Echo Cancellation Basics Application example: Objective: Remove those components in the microphone x ( n ) x ( n ) signal that originate from the remote communication partner! Model: Echo b h ( n ) d ( n ) cancel- x ( n ) s ( n ) lation filter b d ( n ) b ( n ) b h ( n ) h ( n ) y ( n ) e ( n ) + s ( n ) b d ( n ) d ( n ) + + e ( n ) y ( n ) + b ( n ) Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 3 Slide 3
Application Example – Echo Cancellation Basic Approach Model: The loudspeaker-enclosure-microphone (LEM) system is modelled as a linear (only slowly changing) system with finite memory. Approach: Cancelling acoustic echoes by means of an adaptive filter with coefficients, operating at a sample rate kHz. For the adaptation of the filter the NLMS algorithm should be used. Advantages and disadvantages: + In contrast to former approaches (loss controls) simultaneous speech activity in both communication directions is possible now. + The NLMS algorithm is a robust and computationally efficient approach. _ Compared to former solutions more memory and a larger computational load are required. _ Stability can not be guaranteed. Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 4 Slide 4
Application Example – Echo Cancellation NLMS-Algorithm Computation of the error signal (output signal of the echo cancellation filter): Recursive computation of the norm of the excitation signal vector Adaptation of the filter vector: Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 5 Slide 5
Application Example – Echo Cancellation Convergence Examples – Part 1 Convergence without Excitation signal background noise and without local speech signals Local signal Microphone signal Error signal Microphone and error power Time in seconds Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 6 Slide 6
Application Example – Echo Cancellation Convergence Examples – Part 2 Convergence with Excitation signal background noise but without local speech signals Local signal Microphone signal Error signal Microphone and error power Time in seconds Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 7 Slide 7
Application Example – Echo Cancellation Convergence Examples – Part 3 Convergence without Excitation signal background noise but with local speech signals Local signal (step size = 1) Microphone signal Error signal Microphone and error power Time in seconds Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 8 Slide 8
Application Example – Echo Cancellation Convergence Examples – Part 4 Convergence without Excitation signal background noise but with local speech signals Local signal (step size = 0.1) Microphone signal Error signal Microphone and error power Time in seconds Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 9 Slide 9
Adaptation Control Literature Basic texts: E. Hänsler / G. Schmidt: Acoustic Echo and Noise Control – Chapter 7 (Algorithms for Adaptive Filters), Wiley, 2004 E. Hänsler / G. Schmidt: Acoustic Echo and Noise Control – Chapter 13 (Control of Echo Cancellation Systems), Wiley, 2004 Further details: S. Haykin: Adaptive Filter Theory – Chapter 6 (Normalized Least-Mean-Square Adaptive Filters), Prentice Hall, 2002 C. Breining, A. Mader: Intelligent Control Strategies for Hands-Free Telephones , in E. Hänsler, G. Schmidt, Topics on Acoustic Echo and Noise Control – Chapter 8, Springer, 2006 Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 10 Slide 10
Adaptation Control Control Approaches – Part 1 Scalar control approach: Step size Regularization Vector control approach: Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 11 Slide 11
Adaptation Control Control Approaches – Part 2 Example for a Impulse response of the system to be identified sparse impulse response For such systems a vector based control scheme can be advantageous. Example for a vector step size Coefficient index i Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 12 Slide 12
Adaptation Control How do we go on … Problem (echo cancellation performance during „double talk“) Analysis of the average system distance (taking local signals into account) Derivation of an optimal step size (using non-measurable signals) Estimation of the non-measurable signal components (leads to an implementable control scheme) Solution (robust echo cancellation due to step-size control) Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 13 Slide 13
Adaptation Control Average System Distance – Part 1 Assumptions: Adaptation using the NLMS algorithm (only step-size controlled) : White noise as excitation and (stationary) distortion: x ( n ) Statistical independence between filter vector and excitation vector. b h ( n ) h Time-invariant system: Definition of the average system distance: s ( n ) b d ( n ) d ( n ) n ( n ) + + e ( n ) y ( n ) + b ( n ) Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 14 Slide 14
Adaptation Control Average System Distance – Part 2 … Derivation during the lecture … Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 15 Slide 15
Adaptation Control Average System Distance – Part 3 Generic approach (control scheme with step size and regularization): Result: Contraction parameter Expansion parameter Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 16 Slide 16
Adaptation Control Contraction and Expansion Parameters Contraction parameter : Range: Desired: as small as possible Determines the speed of convergence without distortions Opposite to each other – a common Expansion parameter : solution (optimization) Has to be found! Range: Desired: as small as possible Determines the robustness against distortions Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 17 Slide 17
Adaptation Control Influence of the Control Parameters Values for the Expansion parameter Contraction parameter contraction and expansion parameters for the conditions: Step size Step size Regularization Regularization Step size Step size Regularization Regularization Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 18 Slide 18
Adaptation Control True and Prediction System Distance Excitation Boundary conditions of the simulation: Excitation: white noise Distortion: white noise Distortion SNR: 30 dB System distance (Simulation) (Simulation) (Theory) (Theory) Iterations Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 19 Slide 19
Adaptation Control Maximum Convergence Speed – Part 1 For the special case without any distortions and with optimal control parameters for that case we get Meaning that the average system distance can be reduced per adaptation step by a factor of . As a result adaptive filters with a lower amount of coefficients converge faster than long adaptive filters. Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 20 Slide 20
Adaptation Control Maximum Convergence Speed – Part 2 If we want to know how long it takes to improve the filter convergence by 10 dB, we can make the following ansatz: As on the previous slide we assumed an undisturbed adaptation process. By applying the natural logarithm we obtain By using the following approximations for and we get This means: At maximum speed of convergence it takes about 2N iterations until the average system distance is reduced by 10 dB. Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 21 Slide 21
Adaptation Control The „10 dB per 2N“ Rule Boundary conditions of Average system distance the simulation: Excitation: white noise Distortion: white noise SNR: 30 dB Step size: 1 Different filter lengths (500 and 1000) Average system distance Iterations Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 22 Slide 22
Adaptation Control Prediction of the Steady-State Convergence – Part 1 Recursion of the average system distance: For and appropriately chosen control parameters we obtain: Digital Signal Processing and System Theory| Adaptive Filters | Adaptation Control Slide 23 Slide 23
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