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