12 adaptive filtering
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Real time DSP
Professors:Eng. Julian Bruno
Eng. Mariano Llamedo Soria

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Adaptive Filters

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Recommended bibliography
Saeed V. Vaseghi,Advanced Digital Signal Processing and
Noise Reduction, Second Edition. John Wiley & Sons Ltd. Ch 3: Probabilistic Models
Ch 6: Wiener Filters
Ch 7: Adaptive Filters
Ch 8: Linear Prediction Models
Stergios Stergiopoulos,Advanced Signal Processing
Handbook. CRC Press LLC, 2001
Ch 2: Adaptive Systems for Signal Process  Simon Haykin
B FarhangBoroujeny.Adaptive Filters. Theory andApplications. John Wiley & Sons.
NOTE:Many images used in this presentation were extracted from the recommended bibliography.

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Stochastic Processes
The term stochastic process is broadly used to
describe a random process that generates
sequential signals such as speech or noise.
In signal processing terminology, a stochasticprocess is a probability model of a class of
random signals, e.g. Gaussian process, Markov
process, Poisson process,etc.

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StrictSense Stationary Processes
A random processX(m) is stat ionaryin a
str ict sense if all its distributions and
statistical parameters such as the mean, the
variance, the power spectral composition andthe higherorder moments of the process, are
timeinvariant.
E[x(m)] = x ; mean
E[x(m)x(m + k)] = rxx(k) ; variance
E[X(f,m)2] = E[X(f)2] = Pxx(f) ; power spectrum

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WideSense Stationary Processes
A process is said to be w ide sense
stat ionaryif the mean and the
autocorrelation functions of the process are
time invariant:
E[x(m)] =x
E[x(m)x(m + k)]= rxx(k)

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NonStationary Processes
A random process is nonstat ionary if its distributions or
statistics vary with time.
Most stochastic processes such as video signals, audio
signals, financial data, meteorological data, biomedical
signals, etc., are nonstationary, because they are generatedby systems whose environments and parameters vary over
time.

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Adaptive Filters
An adaptive filter is in reality a nonl inear
device, in the sense that it does not obey the
principle of superposition.
Adaptive filters are commonly classified as:
Linear
An adaptive filter is said to be linear if the estimate of
quantity of interest is computed adaptively (at the outputof the filter) as a linear combination of the available set
of observations applied to the filter input.
Nonlinear
Neural Networks

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Linear Filter Structures
The operation of a linear adaptive filtering algorithm involves
two basic processes:
a f i l ter ing process designed to produce an output in response to a
sequence of input data
an adaptive process, the purpose of which is to provide mechanismfor the adaptive control of an adjustable set of parameters used in the
filtering process.
These two processes work interactively with each other.
There are three types of filter structures with finite memory : transversal filter,
lattice predictor,
and systolic array.

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Linear Filter Structures
For stationary inputs, the resulting solution is commonlyknown as the Wiener fi l ter, which is said to be optimum inthe meansquaresense.
A plot of the meansquare value of the error signal vs. the
adjustable parameters of a linear filter is referred to as theerrorperformance surface.
The minimum point of this surface represents the Wienersolution.
The Wiener filter is inadequate for dealing with situations inwhich nonstationarity of the signal and/or noise is intrinsicto the problem.
A highly successful solution to this more difficult problem isfound in the Kalman fi l ter, a powerful device with a wide
variety of engineering applications.

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Lattice Predictor
It has the advantage of simplifying the computation

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Systolic Array
The use of systolic arrays has madeit possible to achieve a high throughput,
which is required for many advanced signal
processing algorithms to operate
in real time

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Linear Adaptive Filtering Algorithms
Stochastic Gradient Approach
LeastMeanSquare (LMS) algorithm
Gradient Adaptive Lattice (GAL) algorithm
LeastSquares Estimation
Recursive leastsquares (RLS) estimation
Standard RLS algorithm
Squareroot RLS algorithms
Fast RLS algorithms

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Wiener Filters
The design of a Wiener filter requires a prioriinformation about the statistics of the data to be
processed.
The filter is optimum only when the statisticalcharacteristics of the input data match the a priori
information on which the design of the filter is
based.
When this information is not known completely,
however, it may not be possible to design the
Wiener filter or else the design may no longer be
optimum.

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The filter inputoutput relation is given by:
The Wiener filter error signal, e(m) is defined as thedifference between the desired signalx(m) and the filter
output signalx (m) :
error signal e(m) for N samples of the signals x(m) and y(m):
Wiener Filters: Least Square
Error Estimation
Wi Fil S

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Wiener Filters: Least Square
Error Estimation
The Wiener filter coefficients are obtained by minimising
an average squared error function E[e2(m)] with respect
to the filter coefficient vector w
Ryy=E [y(m)yT
(m)] is the autocorrelation matrix of theinput signal
rxy=E [x(m)y(m)] is the crosscorrelation vector of the
input and the desired signals
Wi Filt L t S

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Wiener Filters: Least Square
Error Estimation
For example, for a filter with only two coefficients (w0,
w1), the mean square error function is a bowlshaped
surface, with a single minimum point
Wi Filt L t S

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Wiener Filters: Least Square
Error Estimation
The gradient vector is defined as
Where the gradient of the mean square error function withrespect to the filter coefficient vector is given by
The minimum mean square error Wiener filter is obtained by
setting equation to zero
Wi Filt L t S

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Wiener Filters: Least Square
Error Estimation
The calculation of the Wiener filter coefficients requires the
autocorrelat ionmatrix of the input s ignaland the
crosscorrelat ionvector of the input and the desired
signals.
The optimum wvalue is wo= Ryy1ryx

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The LMS Filter
The LMS adaptation method is defined as
The instantaneous gradient of the squared error can beexpressed as
Substituting this equation into the recursion update equation
of the filter parameters, yields the LMS adaptation equation

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The LMS Filter
The main advantage of the LMS algorithm is its simplicity
both in terms of the memory requirement and the
computational complexity which is O(P), where P is the filter
length
Leaky LMS Algorithm The stability and the adaptability of the recursive LMS adaptation can
improved by introducing a socalledleakage factor as
w(m +1) =.w(m) + .[y(m).e(m)]
When the parameter

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LMS Algorithm
Wk=zeros(1,L+1); % Vector Inicial de Pesos
yk=zeros(size(xk)); % Seal de salida inicial del FIR
ek=zeros(size(xk)); % Seal inicial de error
for i=L+1:N1
for n=1:L+1
xk_i(1,n)=xk(i+1n); % Vector x isimo
end
yk(i)=xk_i*Wk'; % seal a la salida del FIR
ek(i)=dk(i)yk(i); % Seal de error
Wk=Wk+2*mu*ek(i)*xk_i; % Vector de pesos isimoend

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Inverse modeling
Predictive deconvolution
Adaptive equalization
Blind equalization

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Prediction
Linear predictive codingAdaptive differential pulsecode modulation
Autoregressive spectrum analysis
Signal detection

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