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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

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adaptive signal processing
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 processing1
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 processing2

Algorithm

Adaptive Signal Processing
  • A possible framework is:

Professor A G Constantinides©

adaptive signal processing3
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©

applications
Applications

Professor A G Constantinides©

adaptive signal processing4

Tx1

Rx2

Hybrid

Hybrid

Echo canceller

Echo canceller

Adaptive Algorithm

Adaptive Algorithm

Local Loop

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+

-

+

Rx1

Rx2

Adaptive Signal Processing
  • Echo Cancellers in Local Loops

Professor A G Constantinides©

adaptive signal processing5

REFERENCE SIGNAL

FIR filter

Noise

-

+

Adaptive Algorithm

Signal +Noise

PRIMARY SIGNAL

Adaptive Signal Processing
  • Adaptive Noise Canceller

Professor A G Constantinides©

adaptive signal processing6

FIR filter

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+

Adaptive Algorithm

Signal

Unknown System

Adaptive Signal Processing
  • System Identification

Professor A G Constantinides©

adaptive signal processing7

Signal

FIR filter

-

+

Adaptive Algorithm

Unknown System

Delay

Adaptive Signal Processing
  • System Equalisation

Professor A G Constantinides©

adaptive signal processing8

Signal

FIR filter

-

+

Adaptive Algorithm

Delay

Adaptive Signal Processing
  • Adaptive Predictors

Professor A G Constantinides©

adaptive signal processing9

Linear Combiner

Interference

Adaptive Signal Processing
  • Adaptive Arrays

Professor A G Constantinides©

adaptive signal processing10
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 processing11
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 processing12
Adaptive Signal Processing
  • The form the objective function
  • where

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adaptive signal processing13
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 processing14
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 processing15
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©

the lms algorithm
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©

convergence
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©

convergence1
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©

convergence2
Convergence
  • Or equivalently
  • I.e.
  • Thus we have
  • Form a new variable

Professor A G Constantinides©

convergence3
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©

convergence4
Convergence
  • We require that
  • Or
  • In practice we take a much smaller value than this

Professor A G Constantinides©

estimates
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©

limiting forms
Limiting forms
  • This indicates that the solution ultimately tends to the Wiener form
  • I.e. the estimate is unbiased

Professor A G Constantinides©

misadjustment
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©

misadjustment1
Misadjustment
  • It can be shown that the misadjustment is given by

Professor A G Constantinides©

normalised lms
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©

transform based lms

Algorithm

Transform based LMS

Transform

Inverse Transform

Professor A G Constantinides©

least squares adaptive
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
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
Recursive Least Squares (RLS)
  • Let
  • Then
  • The quantity is known as the Kalman gain

Professor A G Constantinides©

recursive least squares1
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©

kalman filters
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©

kalman filters1
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©

kalman filters2
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
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©

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