ADAPTIVE FILTERS FOR REMOVAL OF INTERFERENCE

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ADAPTIVE FILTERS FOR REMOVAL OF INTERFERENCE

• 2004235124 ???

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• Adaptive Filter Overview
• Adaptive Noise Cancellor
• The Least Mean Squares Adaptive Filter
• The Recursive Least Squares Adaptive Filter
• Selecting An Appropriate Filter
• Application Removal of Artifacts in the ECG
• Application Adaptive Cancellation of the
  Maternal ECG to obtain the Fetal ECG
• Application Adaptive Cancellation of
  Muscle-Contraction Interference in Knee-Joint
  Vibration Signals

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Adaptive Filter Overview
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Adaptive Filter Overview (1)

• Signal and noise are stationary ? Filter with
  fixed tap weights or coefficients
• Frequency filter not suitable when signal/noise
  vary with time or signal and interference
  overlap.
• Ex ECG signals of a fetus and the mother

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Adaptive Filter Overview (2)

• Fixed filtering cannot separate them.
• Such a situation calls for the use of a filter
  that can learn and adapt. This requires the
  filter to automatically adjust its impulse
  response as the characteristics of the signal
  and/of noise vary.

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The adaptive noise canceller
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The adaptive noise canceller (1)

• x(n) v(n) m(n)
• x(n) primary input to the filter, observed
  signal
• v(n) signal of interest
• m(n) primary noise
• Adaptive filtering requires a second input r(n),
  reference input

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The adaptive noise canceller (2)

• r(n) is uncorrelated with v(n), closely
  correlated with the noise m(n)
• ANC? noise m(n)? ?? ??? y(n)? ??? ?? r(n)?
  filtering ??? ??? ???.
• Assume v(n), m(n), r(n), y(n) are stationary and
  have zero means.
• e(n) x(n) y(n)
• v(n) m(n) y(n)
• y(n) m(n) is the estimate of the primary noise
  obtained at the output of the adaptive filter.

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The adaptive noise canceller (3)

• Take the square and expectation(statistical
  average)
• Ee2 (n) Ev2 (n) Em(n) y(n) 2
  2Ev(n)m(n) y(n))
• Since m(n) and y(n) are uncorrelated with v(n)
• Ev(n)m(n) y(n) Ev(n)Em(n) y(n)
  0
• rewritten
• Ee2(n) Ev2 (n) Em(n) y(n) 2
• ??? Adaptive FIR ??? ??? ?? ??? ????? ?? ???? ???
  ?? ???? least-squared e(n)? ???.

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The adaptive noise canceller (4)

• min Ee2 (n) Ev2 (n) min Em(n) y(n) 2
  
• Ee2 (n) is minimized, min Em(n) y(n) 2 is
  also minimized
• and since e(n) v(n) m(n) y(n). when Em(n)
  y(n) 2 minimized, Ee(n) v(n) 2
  minimized
• Adapting the filter to minimize the total output
  power means causing the output e(n) to be the
  MMSE(minimum mean square error) estimate of the
  signal of interest v(n)
• Minimizing the total output power minimizes the
  output noise power and maximizes the output SNR.

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The adaptive noise canceller (5)

• The output y(n) of the adaptive filter in
  response to its input r(n) is given by
• wk are the tap weights, M is the order of the
  filter
• Define the tap-weight vector at time n
• w(n) w0(n), w1(n), ..wM-1 (n) T and
• r(n) r(N), r(n-1), .., r(n-M-1) T
• so e(n) x(n) - w T(n)r(n)

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The adaptive noise canceller (6)

• 2 methods to maximize the output SNR
• LMS(least-mean-squares)
• RLS(Recursive least-squares)

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The least mean squares adaptive filter
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The least mean squares adaptive filter (1)

• Square the estimation error e(n) To adjust the
  tap-weight vector to minimize the MSE
• Squared error ? 2? ?? ??? ???? ??? ??
  ????(hyper-paroboloidal, bowl-like)? ??. ? ????
  ??? ???? ???? (???? ?? ????) gradient-based
  method of steepst descent? ????.
• In LMS algorithm w(n1) w(n) µ?(n)
• The parameter µ controls the stability and rate
  of convergence of the algorithm. The larger the
  value of µ, the larger is the gradient of the
  noise and the faster is the convergence.

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The least mean squares adaptive filter (2)

• The LMS algorithm approximates ?(n) by the
  derivative of the squared error with respect to
  the tap-weight vector
• w(n1) w(n) 2 µe(n)r(n) widrow-Hoff LMS
  algorithm.

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The least mean squares adaptive filter (3)

• Application
• VAG signals recorded from the mid-patella(???)
  and the tibial(??) tuberosity(??)
• Reference distal(??) rectus(??) femoris(???)
  muscle-contraction signal

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The least mean squares adaptive filter (4)

• Zhang ? w(n1) w(n) 2 µ(n)e(n)r(n) ?? µ? ???
  ????.
• 0ltµlt1, 0 a ltlt1 ?? signal nonstarionarity? ??
  ???? ??? ??? ? ??.

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The least mean squares adaptive filter (5)

• Advantage
• Simplicity and ease of implementation
• Filter expression itself is free of
  differentiation, squaring, averaging
• Disadvantage
• Not suitable for fast-varying signals due to its
  slow convergence ? RLS Adaptive filter

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The recursive least-squares adaptive filter
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The recursive least-squares adaptive filter (1)

• Widely use in Real-time system because of its
  fase convergence
• RLS algorithm utilizes information contained in
  the input data and extends it back to the instant
  of time where the algorithm was initiated
• General scheme of the RLS filter

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The recursive least-squares adaptive filter (2)

• Performance index or objective function
• 0 lt ? 1 weighting factor(forgetting vector)
• 1 i n is the observation interval
• E(n) estimation error
• ? n-i lt 1 give more weight to the more recent
  error values.
• The normal equation in RLS
• w(n) optimal tap-weight vector for which the
  performance index is at its minimum

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The recursive least-squares adaptive filter (3)

• ?(n) M x M time averaged autocorrelation matrix
  of reference input r(i) defined as
• T(n) M x 1 time-averaged cross-correlation
  matrix between the reference input r(i) and the
  primary input x(i) defined as

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The recursive least-squares adaptive filter (4)

• Recursive techniques needed
• To obtain recursive solution, isolate the term
  corresponding to in
• And right-hand side of above equation equals the
  time-averaged and weighted autocorrelation ?(n-1)
• ?(n) ??(n-1) r(n)rT(n)

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The recursive least-squares adaptive filter (5)

• Equation 3.124 can be written as the recursive
  equation
• T(n) ? T(n-1) r(n)x(n)
• we need inverse of ?(n) to obtain tap-weight
  vector
• To determine the inverse of the correlation
  matrix ?(n), use ABCD lemma
• (ABCD)-1 A-1 A-1B(DA-1BC-1) -1DA-1
• A ??(n-1)
• B r(n)
• C 1
• D rT(n)

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The recursive least-squares adaptive filter (6)

• So we have
• ?-1(n) ?-1 ?-1(n-1)
• - ?-1 ?-1 (n-1)r(n)?-1rT(n) ?-1(n-1)r(n)1 -1
• x ?-1rT (n) ?-1(n-1)
• Since ?-1rT(n) ?-1(n-1)r(n)1 is scalar,
• For convinience
• P(n) ?-1(n)

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The recursive least-squares adaptive filter (7)

• With P(0) d-1I where d is a small constant and
  I is the identity matrix
• Then rewritten in a simpler form as
• P(n) ?-1P(n-1) - ?-1k(n)rT(n)P(n-1) - a
• From above two equation
• k(n)1 ?-1rT(n)P(n-1)r(n) ?-1 P(n-1)r(n)
• Or k(n) ?-1p(n-1)-?-1k(n)rT(n)P(n-1)r(n) -b

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The recursive least-squares adaptive filter (8)

• From a and b
• k(n) P(n)r(n)
• P(n) and k(n) have dimensions M x M and M x 1
• As weve seen
• And T(n) ? T(n-1) r(n)x(n)
• And P(n) ?-1(n)
• So recursive equation for updating the
  least-squares estimate w(n) of the tap-weight
  vector can obtained as

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The recursive least-squares adaptive filter (9)

• From P(n) ?-1P(n-1) - ?-1k(n)rT(n)P(n-1)
• Finally from k(n)P(n)r(n)
• This equation gives a recursive relationship of
  w(n)

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The recursive least-squares adaptive filter (10)

• Where w(0)0
• The quantity a(n) is often referred to as the a
  priori error , reflecting the fact that it is the
  error obtained using the old filter(filter before
  being updated)
• In the case of ANC, a(n) will be the estimated
  signal of interest v(n) after the filter has
  converged

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The recursive least-squares adaptive filter (11)

• After convergence, the primary noise estimate,
  the output of the adaptive filter y(n) is
• So we can obtain

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The recursive least-squares adaptive filter (12)

• Application
• (a) VAG signal of a normal subject. (b)
  Muscle-contraction interference.(reference) (c)
  Result of LMS filtering (d) Result of RLS
  filtering

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The recursive least-squares adaptive filter (13)

• LMS filter
• M 7, µ 0.05, a 0.98
• RLS filter
• M 7, ? 0.98
• Relatively low-frequency muscle-contraction
  interference has been removed better by the RLS
  than by the LMS filter
• LMS failed to track the nonstationarities and
  caused additional artifacts

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The recursive least-squares adaptive filter (14)

• Spectrogram of VAG in (a)

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The recursive least-squares adaptive filter (15)

• Spectrogram of the muscle-contraction
  interference signal in (b)

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The recursive least-squares adaptive filter (16)

• Spectrogram of RLS-filtered VAG in (d)
• We can see that low-frequency artifact has been
  removed by RLS filter

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Selecting an Appropriate Filter
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Selecting an Appropriate Filter (1)

• Synchronized or ensemble averaging of multiple
  realizations or copies of a signal
• Time-domain
• MA(Moving average) filtering
• Time- domain
• Frequency-domain filtering
• Optimal(Wiener) filtering
• Implemented in the time-domain or in the
  frequency-domain
• Adaptive filtering
• alter their characteristics in response to
  changes in the interferences

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Selecting an Appropriate Filter (2)

• Synchronized or ensemble averaging
• Signal is statistically stationary
• Multiple realization or copies of the signal of
  interest are available
• A trigger point or time marker is available or
  can be derived to extract and align the copies of
  the signal
• The noise is a stationary random process that is
  uncorrelated with the signal and has a zero mean
• Temporal MA filtering
• Stationary over the duration of the moving window
• Noise is a zero-mean random process
• Low frequency signal
• Fast, on-line, real-time filtering

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Selecting an Appropriate Filter (3)

• Frequency-domain fixed filtering
• Stationary signal
• Noise is a stationary random process
• Signal spectrum is limited in bandwidth compared
  to that of the noise or vice-versa
• Loss of information in the spectral band removed
  by the filter does not seriously affect the
  signal
• On-line, real-time filtering is not required
• Optimal Wiener filter
• Signal is stationary
• Noise is stationary random process
• Specific detail are available regarding the ACFs
  or the PDSs of the signal and noise

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Selecting an Appropriate Filter (4)

• Adaptive filtering
• Noise or interference is not stationary
• Noise is uncorrelated with the signal
• No information is available about the spectral
  characteristics of the signal, which may also
  overlap significantly
• Reference obtainable

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Removal of Artifacts In the ECG
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Removal of Artifacts in the ECG (1)

• ECG signal with combination of artifacts and its
  filtered versions
• Remove base line drift, high-frequency noise and
  power-line interference

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Removal of Artifacts in the ECG (2)

• Power spectra of the ECG signals before and after
  filtering and combined response of LPF/HPF/Comb
  filter

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Removal of Artifacts in the ECG (3)

• base line drift
• HPF with fc2Hz
• high-frequency noise
• LPF with fc70 Hz
• power-line interference
• Comb filter with zeros and 60, 180, 300, 420Hz

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Application Adaptive Cancellation of the
Maternal ECG to obtain the Fetal ECG
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Adaptive Cancellation of the Maternal ECG to
obtain the Fetal ECG (1)

• To obtain fetal ECG, remove the maternal ECG
• Mutiple-reference ANC, maternal ECG was obtained
  via four chest leads.
• Characteristics of the maternal ECG in the
  abdominal lead would be different from those of
  the chest-lead ECG signal used as reference input
• Optimal Wiener filter included transfer functions
  and cross-spectral vectors between the input
  source and each reference input
• (a) is chest lead ECG, the maternal ECG (b)is
  abdominal-lead ECG, combination of maternal and
  fetal ECG

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Adaptive Cancellation of the Maternal ECG to
obtain the Fetal ECG (2)

• Filter output successfully extracted the fetal
  ECG and suppressed the maternal ECG

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Application Adaptive Cancellation of
Muscle-Contraction Interference in Knee-Joint
Vibration Signals
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Adaptive Cancellation of Muscle-Contraction
Interference in Knee-Joint Vibration Signals (1)

• (a) VAG signal of a subject with Chondromalacia
  patella(??? ?? ???) (b) simultaneously recorded
  muscle-contraction interference (c) result of LMS
  filtering with M7, µ0.05, a0.98 (d) result of
  RLS filtering with M7, ?0.98

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Adaptive Cancellation of Muscle-Contraction
Interference in Knee-Joint Vibration Signals (2)

• Spectrogram of the original VAG signal

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Adaptive Cancellation of Muscle-Contraction
Interference in Knee-Joint Vibration Signals (3)

• Spectrogram of the muscle-contraction
  interference signal

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Adaptive Cancellation of Muscle-Contraction
Interference in Knee-Joint Vibration Signals (5)

• Spectrogram of the RLS-filtered VAG signal