ETE 315Adaptive Filter

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A.H.M. Asadul Huq, Ph.D.http://asadul.drivehq.com/students.htm

asadul.huq@ulab.edu.bd

Vol-7

Adaptive Digital Filter

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References1. Digital Signal Processing Principles … 4/e – John G.

Proakis et al.2. Adaptive filter theory – Simon Haykin3. Adaptive Filters Theory and Applications – B.

Farhang-Boroujeny4. Digital Signal Processing A practical Approach –

Emmanuel C. Ifeachor (P 645 – 680, 2/e)

Fixed versus Adaptive Filter DesignFixed

Determine the values of the coefficients of the digital filter that meet the desired specifications and the values are not changed once they are implemented.

Adaptive

The coefficient values are not fixed. They are adjusted to optimize some measure of the filter performance using incoming input data and error.

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w0, w1, w2, …, wN-1

W0(n), w1(n), w2(n), …, wN-1(n)

Introduction to Adaptive Filter [Efea 541]

• An adaptive filter is a digital filter with self-adjusting characteristics.

• It adapts, automatically, to changes in its input signals.

• A variety of recursive algorithms have been developed for the operation of adaptive filters, e.g., LMS, RLS, etc.

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Continued … Introduction to the Adaptive Filter [Far 2]

• The figure shows a filter emphasizing the way it is used in typical problems.

• The filter is used to reshape certain input signals in such a way that its output is a good estimate of the given desired signal.

• The process of selecting or adapting in this case the filter parameters (coefficients) so as to achieve the best match between the desired signal and the filter output is often done by optimizing an appropriately defined performance function.

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Continued … Introduction to the Adaptive Filter• The performance function can be defined in a statistical or

deterministic framework.• In the statistical approach, the most commonly used performance

function is the mean-square value of the error signal, i.e. the difference between the desired signal and the filter output. For stationary input and desired signals, minimizing the mean-square error results in the well-known Wiener filter, which is said to be optimum in the mean-square sense.

• Most algorithms are practical solutions to Wiener filters.• In the deterministic approach, the usual choice of performance

function is a weighted sum of the squared error signal. Minimizing this function results in a filter which is optimum for the given set of data.

• Depending on the certain statistical properties of the data, the deterministic solution will approach the statistical solution, i.e. the Wiener filter, for large data lengths.

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AF Classes

• Identification – The AF provides a model that represents an unknown system.

• Inverse Modeling – The AF provides an inverse model that represents an unknown system.

• Prediction – The AF provides the best prediction of the present value of a random signal.

• Interference Canceling – The AF is used in such a way that it can cancel unknown interference contained in an information signal.

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Applications of AFAF Classes Applications

Identification System IdentificationLayered Earth Modeling

Inverse Modeling Adaptive equalization

Prediction ADPCMSignal detection

Interference canceling Adaptive noise CancelingEcho Cancellation

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Adaptive Filter Structure [Far 3]x(n)

Z-1Z-1Z-1

x(n-1) x(n-2) x(n-M+1)

w0(n) w1(n) w2(n) wM-1(n)

Adaptive Algorithm

y(n)d(n)-

+

e(n)

1

0

)()()(M

ii inxnwny

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Continued .. Adaptive Filter Structure• Adaptive filter is implemented using a transversal (FIR),

lattice or even IIR structure.• FIR structure is the most widely used because of its simplicity

and guaranteed stability• Where, the filter has input x(n) and an output, y(n). The

sequence d(n) is called the desired signal. wi(n)s are the filter tap weights (coefficients) and M is the filter length. The tap weights vary in time and are controlled by a suitable adaptive algorithm.

• The output, y(n), is generated as a linear combination of the delayed samples of the input sequence, x(n), according to the equation -

1

0

)()()(M

ii inxnwny

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Adaptive Filter Algorithms [Ifea 648]• LMS – Least Mean Square• RLS – Recursive Least Squares• Kalman Filter Algorithms

LMS• The most efficient in terms of computation and storage

requirements• Does not suffer from the numerical instability problem.• Popular

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The LMS Algorithm[Far 138, Ifea 654, Hay 299, Proa 902-905]

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Continued .. The LMS Algorithm• The algorithm was derived by Widrow and Hoff in 1959• The algorithm adapts each of the coefficient values of the tap vector

in the direction of the squared amplitude of the error signal• Algorithm adaptation step size may be fixed to a suitable value.• Input vector = {x(n)}=x(n), x(n-1) … x(n-M+1)• Tap vector = {w(n)}=w0(n), w1(n), … wM-1(n) where M-1 is the number of delay elements.• Expected {w(n)} is computed using LMS algorithm.• During the process of filtering, d(n) is supplied as the desired

response.• Given the input vector, {x(n)}, and tap vector {w(n)}, the filter

produces an output y(n) which is an estimate of d(n).• Now, calculate e(n)=d(n)-y(n)

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Continued .. The LMS Algorithm

)()()( nnny T xw

)()()( nyndne

)()()()1( nennn xww

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1. Filter output

2. Estimation error

3. Tap-weight update

We write here 3 basic relations of the LMS algorithm [Hay 303, Far 141]

Summary of the LMS Algorithm [Far 141]

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AF Application: Noise Cancellation[Hay 48, Far 21, Proa 896]

• Adaptive Noise Cancelling (ANC) is performed by subtracting noise from a signal (where noise has been mixed) for the purpose of improved signal-to-noise ratio.

• The filtering and subtraction are controlled by the adaptive process.

• Basically an adaptive noise canceller is a dual input, closed adaptive control system.

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Adaptive Noise Canceller (ANC)

+

Adaptive Filter

eRecovered

signal

d = s + nPrimary input(Noisy Signal)

xRef. input

(Noise samples) Adaptive noise canceller

y

Noiseless Signal

Noise/interference

-

s

n

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Fig. (Proa 13.1.14) AF Canceller

ANC System Operation• Primary input=noisy signal = d=s+n• Ref input =noise samples = x• Adaptive filter output =

• Now, error signal = noise canceller output = e = d-y =s+n-y• The adaptive algorithm adapts the filter coefficients

so that y becomes equal to n.• So, AF output e = s = clean signal !!

1

0

)()()(N

ii inxnwny

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HOME WORKS

• Application of Adaptive filtering to system identification(System Modeling) problem [Proa P882].

• Adaptive Channel Equalization [Proa 883].• Adaptive Echo Canellation [Proa 887].

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DSP LectureADAPTIVE FILTER

THE ENDTHE END

THANK YOUTHANK YOU

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