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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.

Algorithm

• A possible framework is:

• Applications are many
• Digital Communications
• Channel Equalisation
• System identification
• Smart antenna systems
• Blind system equalisation
• And many, many others

Applications

Tx1

Rx2

Hybrid

Hybrid

Echo canceller

Echo canceller

Local Loop

-

+

-

+

Rx1

Rx2

• Echo Cancellers in Local Loops

REFERENCE SIGNAL

FIR filter

Noise

-

+

Signal +Noise

PRIMARY SIGNAL

FIR filter

-

+

Signal

Unknown System

• System Identification

Signal

FIR filter

-

+

Unknown System

Delay

• System Equalisation

Signal

FIR filter

-

+

Delay

Linear Combiner

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

The LMS Algorithm
• Thus we have
• Where the error is
• And hence can write
• This is sometimes called the stochastic gradient descent

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

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

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

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

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

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

Limiting forms
• 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

Normalised LMS
• To make the step size respond to the signal needs
• In this case
• And misadjustment is proportional to the step size.

Algorithm

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.

Recursive Least Squares
• Firstly we note that
• We now use the Inversion Lemma (or the Sherman-Morrison formula)
• Let

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

Recursive Least Squares
• Now use in the computation of the filter weights
• From the earlier expression for updates we have
• And hence

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

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

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