yuan chen advisor: professor paul cuff. introduction goal: remove reverberation of far-end input...
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Cascade Adaptive Filters and Applications to Acoustic Echo
CancellationYuan Chen
Advisor: Professor Paul Cuff
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Introduction
Goal: Remove reverberation of far-end input from near –end input by forming an estimation of the echo path
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Review of Previous WorkConsidered cascaded filter architecture of
memoryless nonlinearity and linear, FIR filter
Applied method of generalized nonlinear NLMS algorithm to perform adaptation
Choice of nonlinear functions: cubic B-spline, piecewise linear function
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Spline (Nonlinear) FunctionInterpolation between
evenly spaced control points:
Piecewise Linear Function:M. Solazzi et al. “An adaptive spline nonlinear
function for blind signal processing.”
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Nonlinear, Cascaded AdaptationLinear Filter Taps:
Nonlinear Filter Parameters:
Step Size Normalization:
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Optimal Filter ConfigurationFor stationary
environment, LMS filters converge to least squares (LS) filter
Choose filter taps to minimize MSE:
Solution to normal equations:
Input data matrix:
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Nonlinear Extension – Least Squares Spline (Piecewise Linear) FunctionChoose control points to minimize MSE:
Spline formulation provides mapping from input to control point “weights”:
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Optimality Conditions – Optimize with respect to control points
First Partial Derivative:
Expressing all constraints:
In matrix form:
Solve normal equations:
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Least Squares Hammerstein FilterDifficult to directly solve for both filter taps
and control points simultaneously
Consider Iterative Approach:1. Solve for best linear, FIR LS filter given
current control points2. Solve for optimal configuration of nonlinear
function control points given updated filter taps
3. Iterate until convergence
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Hammerstein OptimizationGiven filter taps,
choose control points for min. MSE:
Define, rearrange, and substitute:
Similarity in problem structure:
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ResultsEcho Reduction Loss Enhancement (ERLE):
Simulate AEC using: a.) input samples drawn i.i.d. from Gsn(0, 1) b.) voice audio inputUse sigmoid distortion and linear acoustic
impulse response
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ConclusionsUnder ergodicity and stationarity constraints,
iterative least squares method converges to optimal filter configuration for Hammerstein cascaded systems
Generalized nonlinear NLMS algorithm does not always converge to the optimum provided by least squares approach
In general, Hammerstein cascaded systems cheaply introduce nonlinear compensation