Adaptive Filters.ppt

© M.S Ramaiah School of Advanced Studies - Bangalore 1
PEPs- ESD 521
ADAPTIVE FILTERS
 The characteristics of FIR and IIR filters are time invariant
because they have fixed coefficients.
 These filters cannot be applied for time-varying signals
and noises.
 The characteristics of adaptive filters are time varying and
can adapt to an unknown and/or changing environment.
Therefore, coefficients of adaptive filters cannot be
determined by filter design software such as FDATool.

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Structures and Algorithms
of Adaptive Filters
 An adaptive filter shown in figure below consists of two functional
blocks
 A digital filter to perform the desired filtering and
 An adaptive algorithm to automatically adjust the coefficients (or
weights) of that filter.
 In the figure, d(n) is a desired signal, y(n) is the output of a digital
filter driven by an input signal x(n), and error signal e(n) is the
difference between d(n) and y(n).
 The adaptive algorithm adjusts the filter coefficients to minimize a
predetermined cost function that is related to e(n).
Figure : Block diagram
of adaptive filter

PEPs- ESD 521
of Adaptive Filters
 Assuming the adaptive FIR filter used in Figure with L
coefficients, wl(n), l = 0, 1, . . . , L - 1, the filter output
signal is computed as
 where the filter coefficients wl(n) are time varying and
updated by an adaptive algorithm. It is important to note
that the length of filter is L and the order of the filter is L -
1. We define the input vector at time n as
and the weight vector at time n as

PEPs- ESD 521
of Adaptive Filters
 The output signal y(n) can be expressed with the vector form as
follows:
 where T denotes the transpose operation of the vector. The filter output
y(n) is compared with the desired response d(n), which results in the
error signal
 The objective of the adaptive algorithm is to update the filter
coefficients to minimize some predetermined performance criterion (or
cost function).
 The most commonly used cost function is based on the mean square
error (MSE) defined as
where E denotes the expectation operation.
 The MSE defined above is a function of filter coefficient vector w(n).

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Design and Applications of Adaptive Filters
 The MATLAB Filter Design Toolbox provides the function
adaptfilt for implementing adaptive filters.
 The function below returns an adaptive filter h of type
algorithm. h = adaptfilt.algorithm(. . .);
 MATLAB supports many adaptive algorithms.
 For example, the LMS-type adaptive algorithms based on FIR filters
are summarized in table below
 The Filter Design Toolbox also provides many advanced
algorithms such as recursive least squares, affine
projection, and frequency-domain algorithms.
 For example, we can construct an adaptive FIR filter with the LMS
algorithm object as follows:
 h = adaptfilt.lms(l,step,leakage,coeffs,states);

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Table A: Summary of LMS-
Type Adaptive Algorithms for
FIR Filters
Table B: Input
Arguments for
adaptfilt.lms

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 Adaptive system identification is illustrated in Figure below, where
P(z) is an unknown system to be identified and W(z) is an adaptive
filter used to model P(z).
 The adaptive filter adjusts itself to cause its output to match that of
the unknown system.
 If the input signal x(n) provides sufficient spectral excitation, the
adaptive filter output y(n) will approximate d(n) in an optimum sense
after convergence.
 When the difference between the physical system response d(n) and
the adaptive model response y(n) has been minimized, the adaptive
model W(z) approximates P(z) from the input/output viewpoint.
 When the plant is time varying, the adaptive algorithm will keep the
modeling error small by continually tracking time variations of the
plant dynamics.

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Figure : Block diagram of adaptive system identification

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Case Study Example 4.9
 In the figure, an unknown system P(z) is an FIR filter designed by
the following function:
p = fir1(15,0.5);
 The excitation signal x(n) used for system identification is generated
as follows:
x = randn(1,400);
 Use an adapting filter W(z) with the LMS algorithm to identify P(z).
 The length of the adaptive filter is 16, and the step size is 0.01. The
adaptive filtering is conducted as follows:
ha = adaptfilt.lms(16,mu);
[y,e] = filter(ha,x,d);

PEPs- ESD 521
% Example 4.9
% Description: Adaptive system identification of unknown system P(z), which is an FIR filter
x = randn(1,400); % excitation signal x(n)
p = fir1(15,0.5); % FIR P(z) to be identified
d = filter(p,1,x); % FIR filtering x(n) by P(z)
% to obtain desired signal d(n)
mu = 0.01; % step size mu
ha = adaptfilt.lms(16,mu); % construct adaptive filter object
[y,e] = filter(ha,x,d); % adaptive filtering to get y(n) and e(n)
plot(1:400,[d;y;e]);
title('Adaptive System Identification');
legend('d(n)','y(n)','e(n)');
xlabel('Time index, n'); ylabel('Amplitude');
pause;
title('Adaptive System Identification');
stem([p.',ha.coefficients.']);
legend('Actual','Estimated');
xlabel('Coefficient #'); ylabel('Coefficient Values'); grid on;

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Figure 4.23 Adaptive system identification results. Top: signals d(n),
y(n), and e(n). Bottom: coefficients of P(z) and W(z)

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EXERCISE 4.9
 Modify example4_9.m for the following simulations:
1. Design P(z) with different orders. Set W(z) with the same, higher, and
lower orders and compare the identification results. Note that the step
size value should be changed according to the filter length.
2. Using a filter length of 16, change the step size value from 0.01 to 0.05,
0.1, 0.005, and 0.001. Evaluate the performance of the adaptive filter.
3. Change the adaptive algorithm from lms to other functions defined in
Table, and evaluate the performance of the system identification results.
4. Table B shows that leakage is the leakage factor. It must be a scalar
between 0 and 1. If it is less than 1, the leaky LMS algorithm is
implemented. It defaults to 1 (no leakage). Try different leaky factors and
compare the performance with Figure 4.23.
5. Plot e2(n) instead of e(n). In addition, use the following equation to
smooth the curve:
6. Try different values of α and initial values of Pˆe(0).

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Adaptive prediction
 Linear prediction has been successfully applied to a wide range of
applications such as speech coding and separating signals from
noise.
 As illustrated in Figure, the adaptive predictor consists of a digital
filter in which the coefficients wl(n) are updated by the LMS
algorithm.
 For example, consider the adaptive predictor for enhancing multiple
sinusoids embedded in white noise.
 In this application, the structure shown in figure is called the adaptive
line enhancer (ALE), which provides an efficient means for the
adaptive tracking of the sinusoidal components of a received signal
d(n) and separates these narrowband signals from broadband noise.
 In this figure, delay  = 1 is adequate for decorrelating the white noise
component between d(n) and x(n), and longer delay may be required for
other broadband noises.
 This technique has been shown effective in practical applications
when there is insufficient a priori knowledge of the signal and noise
parameters.

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Adaptive prediction
Figure 4.24: Block diagram of adaptive prediction

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 As shown in Figure 4.24, assume that signal d(n)
consists of a desired sine wave that is corrupted by
white noise. An adaptive filter will form a bandpass filter
to pass the sine wave in y(n).
 As shown in Figure 4.25, adaptive filter output y(n)
gradually approaches a clean sine wave, while the error
signal e(n) is reduced to the noise level.

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 Example 4.10
 Description: (1) Mixing sinewave with white noise
 (2) Use adaptive filter for enhancing sinewave
f = 400; fs = 4000; % signal parameters
N = 400; n = 0:1:N-1; % length and time index
sn = sin(2*pi*f*n/fs); % generate sinewave
noise=randn(size(sn)); % generate random noise
dn = sn+0.2*noise; % mixing sinewave with white noise
xn = dn; xn(1)=0; % generate x(n) using delay = 1
L = 64; % filter length
mu = 0.0005; % step size mu
ha = adaptfilt.lms(L,mu); % construct adaptive filter object
[y,e] = filter(ha,xn,dn); % adaptive filtering to get y(n) and e(n)
plot(n,y,':',n,e,'-');
title('Adaptive Line Enhancement');

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Figure 4.25 Adaptive line enhancement of narrowband signal

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EXERCISE 4.10
 Modify example4_10.m for the following simulations:
1. Add more sine waves into d(n) and redo the simulation with
different values of μ and L. We may need a higher order of
filter when the sinusoidal frequencies are close. Accordingly, it
requires a smaller value of the step size and results in slow
convergence.
2. Plot the magnitude response of converged filter W(z) and
confirm that it will approximate a bandpass filter. What is the
center frequency of the passband?
3. Redo Exercise 1 with different delay values Δ.

PEPs- ESD 521
Adaptive noise cancellation
 Adaptive noise cancellation employs an adaptive filter to cancel the noise
components in the primary signal picked up by the primary sensor.
 The primary sensor is placed close to the signal source to pick up the
desired signal. The reference sensor is placed close to the noise source to
sense the noise only.
 The primary input d(n) consists of signal plus noise, which is highly
correlated with x(n) because they are derived from the same noise source.
The reference input consists of noise x(n) alone.
 The adaptive filter uses the reference input x(n) to estimate the noise picked
up by the primary sensor.
 The filter output y(n) is then subtracted from the primary signal d(n),
producing e(n) as the desired signal plus reduced noise.
 To apply the adaptive noise cancellation effectively, it is critical to avoid the
signal components from the signal source being picked up by the reference
sensor. This “cross talk” effect will degrade the performance because the
presence of the signal components in the reference signal will cause the
adaptive filter to cancel the desired signal.

PEPs- ESD 521
Figure : Block diagram of adaptive
noise cancellation
Figure : Block diagram of adaptive
channel equalizer

PEPs- ESD 521
 The transmission of high-speed data through a channel is
limited by intersymbol interference caused by distortion in the
transmission channel. This problem can be solved by using an
adaptive equalizer in the receiver that counteracts the channel
distortion.
 The received signal s(n) is different from the original signal
x(n) because it was distorted by the overall channel transfer
function C(z), which includes the transmit filter, the
transmission medium, and the receive filter.
 To recover the original signal x(n), we need to process s(n)
with the equalizer W(z), which is the inverse of the channel’s
transfer function C(z), to compensate for the channel
distortion. That is, C(z)W(z) = 1 such that xˆ(n) = x(n). Note
that d(n) may not be available during data transmission.

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Selection of step size μ
 The optimum step size μ is difficult to determine.
 Improper selection of μ might make the convergence
speed unnecessarily slow or introduce excess MSE.
 If the signal is changing and real-time tracking capability
is crucial, we can use a larger μ. If the signal is
stationary and convergence speed is not important, we
can use a smaller μ to achieve better performance in a
steady state.
 In some practical applications, we can use a larger μ at
the beginning for faster convergence and use a smaller
μ after convergence to achieve better steady-state
performance.

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Case study Example 4.11
 To examine how step size affects the performance of the
algorithm.
 Similar to Example 4.10, we fix the filter length to L = 64;
however, we set μ values to 0.001, 0.005, and 0.01.
 The simulation results confirm that faster convergence
can be achieved by using a larger step size.

PEPs- ESD 521
 Example 4.11
 Description: Examine how step size affects performance of
adaptive FIR filter with LMS algorithm
f = 400; fs = 4000; % signal parameters
N = 300; n = 0:1:N-1; % length and time index
sn = sin(2*pi*f*n/fs); % generate sinewave
noise=randn(size(sn)); % generate random noise
dn = sn+0.2*noise; % mixing sinewave with white noise
xn = dn; xn(1)=0; % generate x(n) using delay = 1
L = 64; % filter length
mu1 = 0.001; % step size mu1
ha1 = adaptfilt.lms(L,mu1); % construct adaptive filter object
[y1,e1] = filter(ha1,xn,dn);% adaptive filtering to get y(n) and e(n)
% learning curves
c1 = e1.*e1;
c2 = e2.*e2;
c3 = e3.*e3;
alpha = 0.1; b=alpha; a=[1 -(1-alpha)];
d1 = filter(b, a, c1);
plot(1:N,[d1;d2;d3]);
title('Learning curves for different mu values');

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Case study Example 4.11
Figure 4.28 Learning curves for different step size values

PEPs- ESD 521
Session Summary
 Digital IIR filters can be designed by beginning with the
design of an analog filter in the s-domain and using
mapping technique to transform it into the z-domain
 The adaptive predictor consists of a digital filter in which
the coefficients wl(n) are updated by the LMS algorithm.
 Adaptive noise cancellation employs an adaptive filter to
cancel the noise components in the primary signal
picked up by the primary sensor.

Adaptive Filters.ppt

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