Adaptive Signal Processing Problem: Equalise through a FIR filter the distorting effect of a communication channel that may be changing with time. If the channel were fixed then a possible solution could be based on the Wiener filter approach We need to know in such case the correlation matrix of the transmitted signal and the cross correlation vector between the input and desired response. When the the filter is operating in an unknown environment these required quantities need to be found from the accumulated data. Professor A G Constantinides©

Adaptive Signal Processing The problem is particularly acute when not only the environment is changing but also the data involved are non-stationary In such cases we need temporally to follow the behaviour of the signals, and adapt the correlation parameters as the environment is changing. This would essentially produce a temporally adaptive filter. Professor A G Constantinides©

Adaptive Signal Processing A possible framework is: Algorithm Professor A G Constantinides©

Adaptive Signal Processing Applications are many Digital Communications Channel Equalisation Adaptive noise cancellation Adaptive echo cancellation System identification Smart antenna systems Blind system equalisation And many, many others Professor A G Constantinides©

Professor A G Constantinides© Applications Professor A G Constantinides©

Adaptive Signal Processing Echo Cancellers in Local Loops - + Rx1 Rx2 Tx1 Echo canceller Adaptive Algorithm Hybrid Local Loop Professor A G Constantinides©

Adaptive Signal Processing Adaptive Noise Canceller Noise Signal +Noise - + FIR filter Adaptive Algorithm PRIMARY SIGNAL REFERENCE SIGNAL Professor A G Constantinides©

Adaptive Signal Processing System Identification Unknown System Signal - + FIR filter Adaptive Algorithm Professor A G Constantinides©

Adaptive Signal Processing System Equalisation Unknown System Signal - + FIR filter Adaptive Algorithm Delay Professor A G Constantinides©

Adaptive Signal Processing Adaptive Predictors Signal - + FIR filter Adaptive Algorithm Delay Professor A G Constantinides©

Adaptive Signal Processing Adaptive Arrays Linear Combiner Interference Professor A G Constantinides©

Adaptive Signal Processing Basic principles: 1) Form an objective function (performance criterion) 2) Find gradient of objective function with respect to FIR filter weights 3) There are several different approaches that can be used at this point 3) Form a differential/difference equation from the gradient. Professor A G Constantinides©

Adaptive Signal Processing Let the desired signal be The input signal The output Now form the vectors So that Professor A G Constantinides©

Adaptive Signal Processing The form the objective function where Professor A G Constantinides©

Adaptive Signal Processing We wish to minimise this function at the instant n Using Steepest Descent we write But Professor A G Constantinides©

Adaptive Signal Processing So that the “weights update equation” Since the objective function is quadratic this expression will converge in m steps The equation is not practical If we knew and a priori we could find the required solution (Wiener) as Professor A G Constantinides©

Adaptive Signal Processing However these matrices are not known Approximate expressions are obtained by ignoring the expectations in the earlier complete forms This is very crude. However, because the update equation accumulates such quantities, progressive we expect the crude form to improve Professor A G Constantinides©

Professor A G Constantinides© The LMS Algorithm Thus we have Where the error is And hence can write This is sometimes called the stochastic gradient descent Professor A G Constantinides©

Professor A G Constantinides© Convergence The parameter is the step size, and it should be selected carefully If too small it takes too long to converge, if too large it can lead to instability Write the autocorrelation matrix in the eigen factorisation form Professor A G Constantinides©

Professor A G Constantinides© Convergence Where is orthogonal and is diagonal containing the eigenvalues The error in the weights with respect to their optimal values is given by (using the Wiener solution for We obtain Professor A G Constantinides©

Professor A G Constantinides© Convergence Or equivalently I.e. Thus we have Form a new variable Professor A G Constantinides©

Professor A G Constantinides© Convergence So that Thus each element of this new variable is dependent on the previous value of it via a scaling constant The equation will therefore have an exponential form in the time domain, and the largest coefficient in the right hand side will dominate Professor A G Constantinides©

Professor A G Constantinides© Convergence We require that Or In practice we take a much smaller value than this Professor A G Constantinides©

Professor A G Constantinides© Estimates Then it can be seen that as the weight update equation yields And on taking expectations of both sides of it we have Or Professor A G Constantinides©

Professor A G Constantinides© Limiting forms This indicates that the solution ultimately tends to the Wiener form I.e. the estimate is unbiased Professor A G Constantinides©

Professor A G Constantinides© Misadjustment The excess mean square error in the objective function due to gradient noise Assume uncorrelatedness set Where is the variance of desired response and is zero when uncorrelated. Then misadjustment is defined as Professor A G Constantinides©

Professor A G Constantinides© Misadjustment It can be shown that the misadjustment is given by Professor A G Constantinides©

Professor A G Constantinides© Normalised LMS To make the step size respond to the signal needs In this case And misadjustment is proportional to the step size. Professor A G Constantinides©

Professor A G Constantinides© Transform based LMS Algorithm Transform Inverse Transform Professor A G Constantinides©

Least Squares Adaptive with We have the Least Squares solution However, this is computationally very intensive to implement. Alternative forms make use of recursive estimates of the matrices involved. Professor A G Constantinides©

Recursive Least Squares Firstly we note that We now use the Inversion Lemma (or the Sherman-Morrison formula) Let Professor A G Constantinides©

Recursive Least Squares (RLS) Let Then The quantity is known as the Kalman gain Professor A G Constantinides©

Recursive Least Squares Now use in the computation of the filter weights From the earlier expression for updates we have And hence Professor A G Constantinides©

Professor A G Constantinides© Kalman Filters Kalman filter is a sequential estimation problem normally derived from The Bayes approach The Innovations approach Essentially they lead to the same equations as RLS, but underlying assumptions are different Professor A G Constantinides©

Professor A G Constantinides© Kalman Filters The problem is normally stated as: Given a sequence of noisy observations to estimate the sequence of state vectors of a linear system driven by noise. Standard formulation Professor A G Constantinides©

Professor A G Constantinides© Kalman Filters Kalman filters may be seen as RLS with the following correspondence Sate space RLS Sate-Update matrix Sate-noise variance Observation matrix Observations State estimate Professor A G Constantinides©

Cholesky Factorisation In situations where storage and to some extend computational demand is at a premium one can use the Cholesky factorisation tecchnique for a positive definite matrix Express , where is lower triangular There are many techniques for determining the factorisation Professor A G Constantinides©