Goals of Adaptive Signal Processing Design algorithms that learn from training data Algorithms must have good properties: attain good solutions, simple to implement, converge quickly

Learning from Examples Systems or filters (tap delay line) consist of parameters or weights that are updated Learning algorithm receives training examples (observation data) and updates parameters of system Updates depend on specified criterion Learning can be batch or on-line

Tap Delay Line D w0w0 x(n) X D w1w1 x(n-1) X w2w2 x(n-2) X  y(n)

Tap Delay Line Parameters Inputs: x(n) Delay units: (two) Weights: w  R 3 Output: y(n) = w 0 x(n) + w 1 x(n-1) + w 2 x(n-2)

Types of Learning Algorithms Supervised learning (learning with a teacher): learning algorithm receives inputs and desired outputs Unsupervised learning (no teacher) learning algorithm receives inputs Reinforcement learning (learning with a critic) learning algorithm receives inputs and an evaluation cost or penalty, possibly delayed

Supervised Learning D w0w0 x(n) X D w1w1 x(n-1) X w2w2 x(n-2) X  y(n)  - d(n) + e(n)

Supervised Learning Parameters Inputs: x(n) Outputs: y(n) Weights: w Desired Output: d(n) Error signal: e(n) = d(n) – y(n)

Adaptive Learning System (two phases) Training or learning phase (equivalent to write phase in conventional computer memory): weights are adjusted to meet certain desired criterion. Recall or test phase (equivalent to read phase in conventional computer memory): weights are fixed as system realizes some task.

How are weights updated? Iterative on-line algorithm: weights of system are adjusted on-line as training data is received. w(k+1) = L(w(k),x(k),d(k)) for supervised learning where d(k) is desired output. Cost criterion: common cost criterion Mean Squared Error: for one output J(w) =  (y(k) – d(k)) 2 Goal is to find minimum J(w) over all possible w. We will consider stochastic gradient based and least squares methods.

Comments Most filters are linear time invariant filters, but when modifying weights they become nonlinear time varying systems Focus on supervised on-line iterative learning algorithms with squared error cost functions

Adaptive Signal Processing Problems System identification: approximate unknown system Inverse modeling: find a model for inverse system for unknown noisy plant, equalization Prediction: predict value of a noisy random signal, time series (financial, biological) Interference cancellation: cancel unknown interference, noise cancellation and adaptive beamforming

Applications Adaptive equalization: remove intersymbol interference, (data transmission) Speech coding: (linear predictive coding), speech analysis and synthesis Spectrum analysis: estimate spectrum of signal in noise based on time series data Adaptive noise cancellation: adaptive echo cancellers Adaptive beamforming: signals arranged spatially, adaptive antenna arrays, cancel sidelobes