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Adaptive Signal Processing - PowerPoint PPT Presentation

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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Presentation Transcript

• 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.

• 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.

• A possible framework is:

• Applications are many

• Digital Communications

• Channel Equalisation

• System identification

• Smart antenna systems

• Blind system equalisation

• And many, many others

Rx2

Hybrid

Hybrid

Echo canceller

Echo canceller

Local Loop

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+

-

+

Rx1

Rx2

• Echo Cancellers in Local Loops

FIR filter

Noise

-

+

Signal +Noise

PRIMARY SIGNAL

-

+

Signal

Unknown System

• System Identification

FIR filter

-

+

Unknown System

Delay

• System Equalisation

FIR filter

-

+

Delay

Interference

• 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.

• Let the desired signal be

• The input signal

• The output

• Now form the vectors

• So that

• The form the objective function

• where

• We wish to minimise this function at the instant n

• Using Steepest Descent we write

• But

• 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

• 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

• Thus we have

• Where the error is

• And hence can write

• This is sometimes called the stochastic gradient descent

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

• 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

• Or equivalently

• I.e.

• Thus we have

• Form a new variable

• 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

• We require that

• Or

• In practice we take a much smaller value than this

• Then it can be seen that as the weight update equation yields

• And on taking expectations of both sides of it we have

• Or

• This indicates that the solution ultimately tends to the Wiener form

• I.e. the estimate is unbiased

• 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

• It can be shown that the misadjustment is given by

• To make the step size respond to the signal needs

• In this case

• And misadjustment is proportional to the step size.

Transform based LMS

Transform

Inverse Transform

• 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.

• Firstly we note that

• We now use the Inversion Lemma (or the Sherman-Morrison formula)

• Let

• Let

• Then

• The quantity is known as the Kalman gain

• Now use in the computation of the filter weights

• From the earlier expression for updates we have

• And hence

• 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

• 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

• 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