10 adaptive filter

8/13/2019 10 Adaptive Filter

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Adaptive Filters and

Applications

Agfianto Eko Putra

http://agfi.staff.ugm.ac.id
http://agfi.staff.ugm.ac.id/http://agfi.staff.ugm.ac.id/


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Objectives

Principles of...

Adaptive filters

Adaptive least mean square (LMS) algorithm

Illustrates how to apply the adaptive filters tosolve realworld application problems

Adaptive noise cancellation,

System modeling, Adaptive line enhancement, and

Telephone echo cancellation

2Adaptive Filters and Applications


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Introduction to LMS Adaptive FIR

Filters An adaptive filter is a digital

filter that has self-adjustingcharacteristics: It is capable of adjusting its

filter coefficients

automatically to adapt theinput signal via an adaptivealgorithm.

Adaptive filters workgenerally for Adaptation of signal-changing

environments, Spectral overlap between

noise and signal, andunknown, or time-varying,noise.

Adaptive Filters and Applications 3


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Filters


When interference noise is strong and its spectrum overlaps that of the

desired signal, the conventional approach will fail to preserve the desired

signal spectrum while removing the interference using a traditional filter...


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Filters

We only discuss about... Adaptive finite impulse response (FIR) filters with a

simple and popular least mean square (LMS)algorithm.

Further exploration into... Adaptive infinite impulse response (IIR) filters,

Adaptive lattice filters, their associated algorithms andapplications, and so on...

Can be found in comprehensive texts by Haykin(1991), Stearns (2003), and Widrow and Stearns(1985).



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The adaptive filter contains a digital filter with adjustable coefficient(s) and the LMS

algorithm to modify the value(s) of the coefficient(s) for filtering each sample.

When the noise estimate y(n) equals or approximates the noise n(n) in the corrupted

signal, that is, y(n) = n(n), the error signal e(n) = s(n) + n(n) - y(n) = s(n) willapproximate the clean speech signal s(n) the noise is canceled.


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In our example...

The adaptive filter is set to be a one-tap FIR

filter to simplify numerical algebra.

The filter adjustable coefficient wn is adjusted

based on the LMS algorithm.



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In our example...



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In our example...



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In our example...

In general, the FIR filter with multiple-taps is

used and has the following format:



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Basic Wiener Filter Theory and Least

Mean Square Algorithm


Many adaptive algorithms can be viewed as

approximations of the discrete Wiener filter

The Wiener filter adjusts its weight(s) to produce filter output y(n), which

would be as close as the noise n(n) contained in the corrupted signal d(n).


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Example 10.1.

Given a quadratic MSE function for the

Wiener filter:

Find the optimal solution for wto achieve the

minimum MSEJmin

and determineJmin

.



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Example 10.1: Solution



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Notice for Equation (10.7)

1. Optimal coefficient(s) can be different for every block ofdata, since the corrupted signal and reference signal areunknown. The autocorrelation and cross-correlation mayvary.

2. If a larger number of coefficients (weights) are used, theinverse matrix of R-1may require a larger number ofcomputations and may come to be illconditioned. This willmake real-time implementation impossible.

3. The optimal solution is based on the statistics, assumingthat the size of the data block, N, is sufficiently long. Thiswill cause a long processing delay that will make real-timeimplementation impossible.



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The steepest descent algorithm

Solving the Wiener solution Equation (10.7) requires

a lot of computations, including matrix inversion for

a general multiple-taps FIR filter.

The well-known textbook by Widrow and Stearns(1985) describes a powerful LMS algorithm by using

the steepest descent algorithmto minimize the MSE

sample by sample and locate the filter coefficient(s).



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Ilustration of the steepest descent

algorithm



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Example 10.2.

Given a quadratic MSE function for the

Wiener filter:

Use the steepest descent method with an

initial guess w0= 0as and = 0.04to find the

optimal solution for w*and determine Jminby

iterating three times.



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Develop the LMS Algorithm...

To develop the LMS algorithm in terms of

sample-based processing...



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Develop the LMS Algorithm...



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The LMS Algorithm...



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Noise Cancellation



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Example 10.3.



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Example 10.3.



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Examine the MSE function



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Examine the MSE function



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Simulation Example

Sample rate = 8,000 Hz

Original speech data: wen.dat

Speech corrupted by Gaussian noise with a power of 1delayed by 5 samples from the noise reference

Noise reference containing Gaussian noise with apower of 1

Adaptive FIR filter used to remove the noise

Number of FIR filter taps = 21

Convergence factor for the LMS algorithm chosen to be0.01 (


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Simulation Example



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Simulation Example: Result



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System Modeling

The adaptive filter can keep tracking the

behavior of an unknown system by using the

unknown systems input and output...



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Example 10.4



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Contoh lain...

Diasumsikan sebuah sistem yang tak-diketahui

(unknown system) merupakan tapis IIR Bandpass

orde ke-4 dengan frekuensi cutoff bawah 1400Hz

dan atas 1600Hz bekerja pada 8kHz. Digunakan masukan nada 500, 1500 dan 2500Hz.

Tanggap frekuensi dari sistem yang tak-diketahui

tersebut ditunjukkan pada Gambar 10.12!



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Gambar 10.12


0 500 1000 1500 2000 2500 3000 3500 4000-200

-100

0

100

200

Frequency (Hz)

Phase(degree

s)

0 500 1000 1500 2000 2500 3000 3500 4000-80

-60

-40

-20

0

Frequency (Hz)

Magnitude(dB)


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Contoh lain...

Bentuk gelombang masukan x(n) dengan tiga nadaditunjukkan pada plot pertama di Gambar 10.13.

Luaran dari sistem yang tak-diketahui mengandung nada1500Hz saja, dua nada lainnya ditolak oleh sistem yang tak-diketahui.

Hasil dari tapis adaptif: Gunakan tapis FIR adaptif dengan 21 tap dan faktor

konvergensinya 0.01;

Pada ranah waktu, luaran gelombang dari sistem tak-diketahuid(n) dan luaran tapis adaptif y(n) hampir identik setelah cuplikan

ke-70 saat algoritma LMS konvergen; Sinyal ralat e(n) juga diplot untuk menunjukkan bahwa tapis

adaptif tetap melalukan penjajakan terhadap luaran sistem tak-diketahui tanpa ada perbedaan setelah 50 cuplikan pertama.



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Figure 10.13


0 100 200 300 400 500 600 700 800

-2

0

2

Systemi

nput

0 100 200 300 400 500 600 700 800

-1

0

1

System

output

0 100 200 300 400 500 600 700 800

-1

0

1

ADFoutput

0 100 200 300 400 500 600 700 800

-1

0

1

Error

Number of samples


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Figure 10.13


0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

Syst.inputspect.

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

Syst.outputspect.

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

ADFoutputspect.

Frequency (Hz)


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Li E h U i Li


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Line Enhancement Using Linear

Prediction

If a signal frequency content is very narrow compared with the bandwidth

and changes with time, then the signal can efficiently be enhanced by the

adaptive filter, which is line enhancement.

The adaptive filter is actually a linear predictor of the desired narrow band

signal. A two-tap adaptive FIR filter can predict one sinusoid.

The value of is usually determined by experiments or experience in

practice to achieve the best enhanced signal.



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Simulation Example

Our simulation example has the followingspecifications:

Sampling rate = 8,000 Hz

Corrupted signal = 500 Hz tone with unitamplitude added with white Gaussian noise

Adaptive filter = FIR type, 21 taps

Convergence factor = 0.001

Delay value = 7

LMS algorithm applied



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Figure 10.16


0 100 200 300 400 500 600 700 800

-2

-1

0

1

2

Noisysignal

0 100 200 300 400 500 600 700 800

-2

-1

0

1

2

ADFoutput(enhancedsignal)

Number of samples


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Figure 10.17


0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

N

oisysignalspectrum

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

ADFoutputspe

ctrum

Frequency (Hz)

C li P i di I t f


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Canceling Periodic Interferences

Using Linear Prediction

An audio signal may be corrupted by periodic

interference and no noise reference is

available. Such examples include the playback

of speech or music with the interference oftape hum, turntable rumble, or vehicle engine

or power line interference.

We can use the modified line enhancementstructure as shown in Figure 10.18.



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Figure 10.18



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ECG Interference Cancellation

As we discussed in Chapters 1 and 8, inrecording of electrocardiograms, there oftenexists unwanted 60-Hz interference, along

with its harmonics, in the recorded data. This interference comes from the power line,

including effects from magnetic induction,displacement currents in leads or in the body

of the patient, and equipmentinterconnections and imperfections.



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Figure 10.19


Echo Cancellat ion in Long Distance


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Echo Cancellat ion in Long-Distance

Telephone Circuits

Long-distance telephone transmission oftensuffers from impedance mismatches.

This occurs primarily at the hybrid circuit

interface. Balancing electric networks within the hybrid can

never perfectly match the hybrid to thesubscriber loop due to temperature variations,

degradation of the transmission line, and so on. As a result, a small portion of the received signal

is leaked for transmission.



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Figure 10.20a



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Figure 10.20b



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Summary

1. Adaptive filters can be applied to signal-changingenvironments, spectral overlap between noise and signal,and unknown, or time-varying, noises.

2. Wiener filter theory provides optimal weight solutionsbased on statistics. It involves collection of a large block of

data, calculation of an autocorrelation matrix and a cross-correlation matrix, and inversion of a large size of theautocorrelation matrix.

3. The steepest descent algorithm can find the optimalweight solution using an iterative method, so a large

matrix inversion is not needed. But it still requirescalculating an autocorrelation and cross-correlationmatrix.



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Summary

4. The LMS is a sample-based algorithm, which does notneed collection of data or computation of statistics anddoes not involve matrix inversion.

5. The convergence factor for the LMS algorithm is boundedby the reciprocal of the product of the number of filter

coefficients and input signal power.6. The LMS adaptive FIR filter can be effectively applied for

noise cancellation, system modeling, and lineenhancement.

7. Further exploration includes other applications such ascancellation of periodic interference, biomedical ECGsignal enhancement, and adaptive telephone echocancellation.



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References...

Haykin, S. (1991).Adaptive Filter Theory, 2nd ed.Englewood Cliffs, NJ: Prentice Hall.

Ifeachor, E. C., and Jervis, B. W. (2002). DigitalSignal Processing: A Practical Approach, 2nd ed.

Upper Saddle River, NJ: Prentice Hall. Stearns, S. D. (2003). Digital Signal Processing

with Examples in MATLAB.Boca Raton, FL: CRCPress LLC.

Widrow, B., and Stearns, S. (1985).AdaptiveSignal Processing.Upper Saddle River, NJ:Prentice Hall.

10 adaptive filter

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