Transcript Slides

Yuan Chen
Advisor: Professor Paul Cuff
Introduction
Goal: Remove reverberation of far-end input from near
–end input by forming an estimation of the echo path
Review of Previous Work
 Considered 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
Spline (Nonlinear) Function
 Interpolation between evenly
spaced control points:
 Piecewise Linear Function:
M. Solazzi et al. “An adaptive spline nonlinear function
for blind signal processing.”
Nonlinear, Cascaded Adaptation
 Linear Filter Taps:
 Nonlinear Filter Parameters:
 Step Size Normalization:
Optimal Filter Configuration
 For stationary
environment, LMS filters
converge to least squares
(LS) filter
 Choose filter taps to
minimize MSE:
 Solution to normal
equations:
 Input data matrix:
Nonlinear Extension – Least Squares
Spline (Piecewise Linear) Function
 Choose control points to minimize MSE:
 Spline formulation provides mapping from input to
control point “weights”:
Optimality Conditions – Optimize with
respect to control points
 First Partial Derivative:
 In matrix form:
 Expressing all constraints:
 Solve normal equations:
Least Squares Hammerstein Filter
 Difficult 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
points
2. Solve for optimal configuration of nonlinear function
control points given updated filter taps
3. Iterate until convergence
Hammerstein Optimization
 Given filter taps, choose
control points for min.
MSE:
 Define, rearrange, and
substitute:
 Similarity in problem
structure:
Results
 Echo Reduction Loss Enhancement (ERLE):
 Simulate AEC using: a.) input samples drawn i.i.d.
from Gsn(0, 1) b.) voice audio input
 Use sigmoid distortion and linear acoustic impulse
response
Conclusions
 Under 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