AGC Adaptive Signal Processing DSP n n Problem

AGC Adaptive Signal Processing DSP n n 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 filter is operating in an unknown environment these required quantities need to be found from the accumulated data. Professor A G Constantinides© 1

AGC Adaptive Signal Processing DSP n n n 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© 2

AGC Adaptive Signal Processing DSP n A possible framework is: Algorithm Professor A G Constantinides© 3

AGC Adaptive Signal Processing DSP n Applications are many n n n n 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© 4

AGC DSP Applications Professor A G Constantinides© 5

AGC Adaptive Signal Processing DSP n Echo Cancellers in Local Loops Tx 1 Rx 2 Hybrid Echo canceller Adaptive Algorithm Rx 1 Local Loop - Adaptive Algorithm Rx 2 + Professor A G Constantinides© + 6

AGC Adaptive Signal Processing DSP n Adaptive Noise Canceller REFERENCE SIGNAL FIR filter Noise Adaptive Algorithm + Signal +Noise PRIMARY SIGNAL Professor A G Constantinides© 7

AGC Adaptive Signal Processing DSP n System Identification FIR filter Adaptive Algorithm Signal + Unknown System Professor A G Constantinides© 8

AGC Adaptive Signal Processing DSP n System Equalisation Signal FIR filter Unknown System Adaptive Algorithm + Delay Professor A G Constantinides© 9

AGC Adaptive Signal Processing DSP n Adaptive Predictors Signal FIR filter Delay Adaptive Algorithm Professor A G Constantinides© + 10

AGC Adaptive Signal Processing DSP n Adaptive Arrays Linear Combiner Interference Professor A G Constantinides© 11

AGC Adaptive Signal Processing DSP n n n 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© 12

AGC Adaptive Signal Processing DSP n Let the desired signal be The input signal The output Now form the vectors n So that n n Professor A G Constantinides© 13

AGC Adaptive Signal Processing DSP n The form the objective function n where Professor A G Constantinides© 14

AGC Adaptive Signal Processing DSP n We wish to minimise this function at the instant n Using Steepest Descent we write n But n Professor A G Constantinides© 15

AGC Adaptive Signal Processing DSP n n 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© 16

AGC Adaptive Signal Processing DSP n n n 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© 17

AGC The LMS Algorithm DSP n Thus we have n Where the error is n And hence can write n This is sometimes called the stochastic gradient descent Professor A G Constantinides© 18

AGC DSP Convergence The parameter is the step size, and it should be selected carefully n If too small it takes too long to converge, if too large it can lead to instability n Write the autocorrelation matrix in the eigen factorisation form Professor A G Constantinides© 19

AGC Convergence DSP n 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 n We obtain n Professor A G Constantinides© 20

AGC Convergence DSP n Or equivalently n I. e. n Thus we have n Form a new variable Professor A G Constantinides© 21

AGC Convergence DSP n n n 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© 22

AGC Convergence DSP n We require that n Or n In practice we take a much smaller value than this Professor A G Constantinides© 23

AGC Estimates DSP n Then it can be seen that as weight update equation yields the n n n And on taking expectations of both sides of it we have Or Professor A G Constantinides© 24

AGC Limiting forms DSP n n This indicates that the solution ultimately tends to the Wiener form I. e. the estimate is unbiased Professor A G Constantinides© 25

AGC Misadjustment DSP n n 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© 26

AGC Misadjustment DSP n It can be shown that the misadjustment is given by

AGC Normalised LMS DSP n n n 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© 28

AGC DSP Transform based LMS Algorithm Transform Inverse Transform Professor A G Constantinides© 29

AGC Least Squares Adaptive DSP n n 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© 30

AGC Recursive Least Squares DSP n n n Firstly we note that We now use the Inversion Lemma (or the Sherman-Morrison formula) Let Professor A G Constantinides© 31

AGC Recursive Least Squares (RLS) DSP n Let n Then n The quantity gain is known as the Kalman Professor A G Constantinides© 32

AGC Recursive Least Squares DSP n n n Now use the filter weights in the computation of From the earlier expression for have updates we And hence n Professor A G Constantinides© 33

AGC Kalman Filters DSP n n Kalman filter is a sequential estimation problem normally derived from n The Bayes approach n The Innovations approach Essentially they lead to the same equations as RLS, but underlying assumptions are different Professor A G Constantinides© 34

AGC Kalman Filters DSP n The problem is normally stated as: n n 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© 35

AGC Kalman Filters DSP n Kalman filters may be seen as RLS with the following correspondence Sate space n n n RLS Sate-Update matrix Sate-noise variance Observation matrix Observations State estimate Professor A G Constantinides© 36

AGC Cholesky Factorisation DSP n n n 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© 37