1. Yuan Chen Advisor: Professor Paul Cuff

2. Introduction Goal: Remove reverberation of far-end input from near –end input by forming an estimation of the echo path

3. 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

4. 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.”

5. Nonlinear, Cascaded Adaptation Linear Filter Taps: Nonlinear Filter Parameters: Step Size Normalization:

6. 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:

7. Nonlinear Extension – Least Squares Spline (Piecewise Linear) Function Choose control points to minimize MSE: Spline formulation provides mapping from input to control point “weights”:

8. Optimality Conditions – Optimize with respect to control points First Partial Derivative: Expressing all constraints: In matrix form: Solve normal equations:

9. 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

10. Hammerstein Optimization Given filter taps, choose control points for min. MSE: Define, rearrange, and substitute: Similarity in problem structure:

11. 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

14. 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