Cascade Adaptive Filters and Applications to Acoustic Echo

CancellationYuan 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 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

Spline (Nonlinear) FunctionInterpolation between

evenly spaced control points:

Piecewise Linear Function:M. Solazzi et al. “An adaptive spline nonlinear

function for blind signal processing.”

Nonlinear, Cascaded AdaptationLinear Filter Taps:

Nonlinear Filter Parameters:

Step Size Normalization:

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:

Nonlinear Extension – Least Squares Spline (Piecewise Linear) FunctionChoose 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:

Expressing all constraints:

In matrix form:

Solve normal equations:

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

Hammerstein OptimizationGiven filter taps,

choose control points for min. MSE:

Define, rearrange, and substitute:

Similarity in problem structure:

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

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