1、Professor A G Constantinides,1,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 ca
2、se 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,2,Adaptive Signal
3、 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 p
4、roduce a temporally adaptive filter.,Professor A G Constantinides,3,Adaptive Signal Processing,A possible framework is:,Professor A G Constantinides,4,Adaptive Signal Processing,Applications are many Digital Communications Channel Equalisation Adaptive noise cancellation Adaptive echo cancellation S
5、ystem identification Smart antenna systems Blind system equalisation And many, many others,Professor A G Constantinides,5,Applications,Professor A G Constantinides,6,Adaptive Signal Processing,Echo Cancellers in Local Loops,Professor A G Constantinides,7,Adaptive Signal Processing,Adaptive Noise Can
6、celler,Professor A G Constantinides,8,Adaptive Signal Processing,System Identification,Professor A G Constantinides,9,Adaptive Signal Processing,System Equalisation,Professor A G Constantinides,10,Adaptive Signal Processing,Adaptive Predictors,Professor A G Constantinides,11,Adaptive Signal Processi
7、ng,Adaptive Arrays,Professor A G Constantinides,12,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)
8、Form a differential/difference equation from the gradient.,Professor A G Constantinides,13,Adaptive Signal Processing,Let the desired signal be The input signal The output Now form the vectorsSo that,Professor A G Constantinides,14,Adaptive Signal Processing,The form the objective functionwhere,Prof
9、essor A G Constantinides,15,Adaptive Signal Processing,We wish to minimise this function at the instant n Using Steepest Descent we writeBut,Professor A G Constantinides,16,Adaptive Signal Processing,So that the “weights update equation”Since the objective function is quadratic this expression will
10、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,17,Adaptive Signal Processing,However these matrices are not known Approximate expressions are obtained by ignoring the expectations in the earlier c
11、omplete formsThis is very crude. However, because the update equation accumulates such quantities, progressive we expect the crude form to improve,Professor A G Constantinides,18,The LMS Algorithm,Thus we haveWhere the error isAnd hence can writeThis is sometimes called the stochastic gradient desce
12、nt,Professor A G Constantinides,19,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,20,Conv
13、ergence,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 forWe obtain,Professor A G Constantinides,21,Convergence,Or equivalentlyI.e.Thus we haveForm a new variable,Professor A G Consta
14、ntinides,22,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 Constantinide
15、s,23,Convergence,We require thatOrIn practice we take a much smaller value than this,Professor A G Constantinides,24,Estimates,Then it can be seen that as the weight update equation yieldsAnd on taking expectations of both sides of it we haveOr,Professor A G Constantinides,25,Limiting forms,This ind
16、icates that the solution ultimately tends to the Wiener form I.e. the estimate is unbiased,Professor A G Constantinides,26,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
17、 uncorrelated. Then misadjustment is defined as,Professor A G Constantinides,27,Misadjustment,It can be shown that the misadjustment is given by,Professor A G Constantinides,28,Normalised LMS,To make the step size respond to the signal needsIn this caseAnd misadjustment is proportional to the step s
18、ize.,Professor A G Constantinides,29,Transform based LMS,Transform,Inverse Transform,Professor A G Constantinides,30,Least Squares Adaptive,withWe have the Least Squares solutionHowever, this is computationally very intensive to implement. Alternative forms make use of recursive estimates of the mat
19、rices involved.,Professor A G Constantinides,31,Recursive Least Squares,Firstly we note thatWe now use the Inversion Lemma (or the Sherman-Morrison formula) Let,Professor A G Constantinides,32,Recursive Least Squares (RLS),LetThenThe quantity is known as the Kalman gain,Professor A G Constantinides,
20、33,Recursive Least Squares,Now use in the computation of the filter weightsFrom the earlier expression for updates we haveAnd hence,Professor A G Constantinides,34,Kalman Filters,Kalman filter is a sequential estimation problem normally derived from The Bayes approach The Innovations approach Essent
21、ially they lead to the same equations as RLS, but underlying assumptions are different,Professor A G Constantinides,35,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 formu
22、lation,Professor A G Constantinides,36,Kalman Filters,Kalman filters may be seen as RLS with the following correspondenceSate space RLS Sate-Update matrix Sate-noise variance Observation matrix Observations State estimate,Professor A G Constantinides,37,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,
