Applications of Adaptive Filters

1
Applications of Adaptive Filters
2
Applications of Adaptive Filters

• Overview
• Adaptive Noise Canceling Applied to a Sinusoidal
  Interference
• Adaptive Line Enhancement
• Adaptive Arrays

3
Applications of Adaptive Filters

• The ability of an adaptive filter to operate
  satisfactorily in an unknown environment and
  track time variations of input statistics makes
  the adaptive filter a powerful device for
  signal-processing and control applications.

4
Applications of Adaptive Filters

• AFs have been successfully applied in such
  diverse fields as communications, radar, sonar,
  seismology, and biomedical engineering.

5
Four Basic Classes of Adaptive Filtering
Applications

• Identification
• Inverse modeling
• Prediction
• Interference cancellation

6
Identification

• An AF is used to provide a linear model that
  represents the best fit (in the some sense) to an
  unknown plant .

7
Inverse Modeling

• AF is to provide an inverse model that represents
  the best fit (in some sense) to an unknown noisy
  plant.

8
Prediction

• AF is to provide the best prediction (in some
  sense) of the present value of a random signal .
• Depending on the application of interest, the
  adaptive output or the estimation (prediction)
  error may serve as the system output.

9
Interference Cancellation

• AF is used to cancel unknown interference
  contained (alongside an information bearing
  signal component) in a primary signal, with the
  cancellation being optimized in some sense.

10
Adaptive Noise Canceling

• Applied to a Sinusoidal Interference

11
Adaptive Noise Cancelling

• The traditional method of suppressing a
  sinusoidal interference corrupting an
  information-bearing signal is to use a fixed
  notch filter tuned to the frequency of the
  interference.

12
Adaptive Noise Canceling

• If the notch is required to be very sharp and the
  interfering sinusoid is known to drift slowly,
  what can we do?

13
Adaptive Noise Canceling

• One such solution is provided by the use of
  adaptive noise canceling, an application that is
  different from the previous three in that it is
  not based on a stochastic excitation.

14
Adaptive Noise Canceller
Block diagram of adaptive noise canceller
15
Adaptive Noise Canceller

• Two important characteristics
• The canceller is tunable, and the tuning
  frequency moves with .
• The notch in the frequency response of the
  sinusoidal interference by choosing a small
  enough value for the step-size parameter .

16
Adaptive Noise Canceling

• Primary input
• Reference input
• where the amplitude A and the phase are
  different from those in the primary input, but
  the angular frequency is the same.

17
Adaptive Noise Canceling

• Using the real form of the LMS algorithm,
• where M is the length of the transversal filter
  and the constant is the step-size parameter.
  

(1)
18
Adaptive Noise Cancelling
lump the sinusoidal input of , the
transversal filter, and the weight-update
equation of the LMS algorithm into a single
(open-loop) system.
19
Adaptive Noise Cancelling

The adaptive system with input and
output varies with time and cannot be
represented by a transfer function.
20
Adaptive Noise Canceling

• Let the adaptive system be excited with
  .
• The output consists of three components
• one proportional to .
• The second proportional to .
• The third proportional to .

21
Adaptive Noise Canceling

• The corresponding value of the tap input is
• where
• Taking the z-transform of product

(2)
22
Adaptive Noise Canceling

• Taking the z-transform of Eq.1
• Using the z-transform given in Eq. (2)

23
Compute the output
24
Adaptive Noise Canceling
25
Adaptive Noise Canceling
26
Expression for

• A time-invariant component
• is independent of the phase .

27

• Accordingly, we may justifiably ignore the
  time-varying component of the z-transform and so
  approximate by retaining the
  time-invariant component only

28

• The open-loop transfer function

29

• The model is recognized as a closed-loop feedback
  system whose transfer function is related to the
  open-loop transfer function via the
  equation

30

• It is the transfer function of a second-order
  digital notch filter with a notch at the
  normalized angular frequency.

31
(No Transcript)
32

• For a small value of the step-size parameter
  (i.e., a slow adaptation rata),

33

• The poles of are located at
• The two poles of lie inside the unit
  circle, a radial distance approximately equal to
• behind the zeros.

34

• The fact that the poles of lie inside
  the unit circle means that the adaptive noise
  canceller is stable, as it should be for
  practical use in real time.

35
the Half-Power Points
36

• Since the zeros of lie on the unit
  circle, the adaptive noise canceller has (in
  theory) a notch of infinite depth (in dB) at
  .
• The sharpness of the notch is determined by the
  closeness of the poles of to its zeros.

37

• The 3-dB bandwidth B is determined by locating
  the two half-power points on the unit circle that
  are times as far from the poles as they
  are from the zeros.

38

• The 3-dB bandwidth of the adaptive noise
  canceller
• The smaller we make , the smaller the
  bandwidth
• is, and the sharper the notch is.

39
Adaptive Line Enhancement
40
Adaptive Line Enhancement

• The adaptive line enhancer (ALE) is a system that
  may be used to detect a sinusoidal signal buried
  in a wideband noise background.

41
Adaptive Line Enhancement

• The ALE is a degenerate form of the adaptive
  noise canceller in that its reference signal,
  instead of being derived separately, consists of
  a delayed version of the primary (input) signal.
• The reference signal
• the prediction depth, or
  decorrelation delay

42
Adaptive Line Enhancement

• input signal
• where is an arbitrary phase shift and the
  noise is assumed to have zero mean and variance
  .

43
Adaptive Line Enhancement

• The ALE acts as a signal detector by virtue of
  two actions
• is large enough to remove the
  correlation between the noise in the
  original input signal and the noise
  in the reference signal, while a sample phase
  shift equal to is introduced between
  sinusoidal components in these two inputs.

44
Adaptive Line Enhancement

• The tap weights of the transversal filter are
  adjusted by the LMS algorithm so as to minimize
  the mean-square value of the error signal and
  thereby compensate for the unknown phase shift
  .

45
Adaptive Line Enhancement

• The output signal
• where denotes a phase shift and is
  the length of the transversal filter.
• The scaling factor

46
Adaptive Line Enhancement

• The ALE acts as a self-turning filter whose
  frequency response exhibits a peak at the angular
  frequency of the incoming sinusoid.
• Hence it is called spectral line enhancer or
  simply line enhancer.

47
Adaptive Line Enhancement

• The signal-to-noise ratio at the input
• It had been shown that the power spectral density
  of the ALE output

48
Adaptive Line Enhancement

• The steady-state model of the converged weight
  vector consists of the Wiener solution ,
  acting in parallel with a slowly fluctuating,
  zero-mean random component due to
  gradient noise.

49
Adaptive Line Enhancement
50
Adaptive Line Enhancement

• Recognizing that the ALE input itself consists of
  two componentssinusoid of angular frequency
• and wideband noise of zero
  mean and variance .
• There are four components in the power spectrum.

51
Adaptive Line Enhancement

• A sinusoidal component of angular frequency
• and average power
• A sinusoidal component of angular frequency
• and average power
• A wideband noise components of variance
• A narrowband filtered noise component centered on

52
Adaptive Line Enhancement

• The power spectrum of the ALE output consists of
  a sinusoid at the center of a pedestal of
  narrowband filtered noise.
• When an adequate SNR exists at the ALE input, the
  ALE output is, on the average, approximately to
  the sinusoidal component present.

53
Adaptive Line Enhancement

• component due to Wiener filter
• Component due to stochastic filter
• Wideband noise due to the action of the
  stochastic filter
• Narrowband filtered noise due to Wiener filter

54
Adaptive Arrays
55
Adaptive Arrays

• Adaptive filtering has also been widely applied
  to multiple data sequences that result from
  antenna, hydrophone, and seismometer arrays,
  where the sensors (antennas, hydrophones, or
  seismometers) are arranged in some spatial
  configuration.
• Each element of the array of sensors provides a
  signal sequence.

56
Adaptive Arrays

• By properly combining the signals from the
  various sensors, it is possible to change the
  directivity pattern of the array.

57
Adaptive Arrays

• If the signals are simply linearly summed, the
  sequence
• which results in the antenna directivity pattern
  shown in Fig. (a).

58
Adaptive Arrays

• Suppose that an interference signal is received
  from a direction corresponding to one of the side
  lobes in the array.
• By properly weighting the sequence prior
  to combining, it is possible to alter the side
  lobe pattern such that the array contains a null
  in the direction of interference, as shown in
  Fig. (b).

59
Adaptive Arrays

• If the signals are simply linearly summed,
• where the are the weights.

60
Adaptive Arrays

• linear antenna array with antenna pattern
• (b) linear antenna array with a null placed in
  the direction of the interference

61
Adaptive Arrays

• We may also change or steer the direction of the
  main antenna lobe by simply introducing delays in
  the output of the sensor signals prior to
  combining.

62
Adaptive Arrays

• From K sensors we have a combined signal of the
  form
• The choice of weights and may be
  used to place nulls in specific directions.

63
Adaptive Arrays

• We may simply filter each sequence prior to
  combining.
• The output sequence
• where is the impulse response of the
  filter for processing the kth sensor output and
  the are the delays that steer the beam
  pattern.