ADAPTIVE FILTERS FOR REMOVAL OF INTERFERENCE 2004235124 김현일

ADAPTIVE FILTERS FOR REMOVAL OF INTERFERENCE

ADAPTIVE FILTERS FOR REMOVAL OF INTERFERENCE

2004235124 김현일

생체 신호 해석2004 년 10 월 1 일

목차목차

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

Adaptive Filter OverviewAdaptive Filter Overview

생체 신호 해석2004 년 10 월 1 일

Adaptive Filter Overview (1)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

생체 신호 해석2004 년 10 월 1 일

Adaptive Filter Overview (2)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.

The adaptive noise cancellerThe adaptive noise canceller

생체 신호 해석2004 년 10 월 1 일

The adaptive noise canceller (1)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’

생체 신호 해석2004 년 10 월 1 일


r(n) is uncorrelated with v(n), closely correlated with the noise m(n)

ANC 는 noise m(n) 과 가장 유사한 y(n) 을 만들기 위해 r(n) 을 filtering 하거나 수정을 가한다 .

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.

생체 신호 해석2004 년 10 월 1 일


Take the square and expectation(statistical average)

E[e2 (n)] = E[v2 (n)]+ E[{m(n) – y(n)} 2] + 2E[v(n){m(n) – y(n)}])

Since m(n) and y(n) are uncorrelated with v(n)

E[v(n){m(n) – y(n)}] = E[v(n)]E[m(n) – y(n)] = 0 rewritten

E[e2(n)] = E[v2 (n)]+ E[{m(n) – y(n)} 2]

출력을 Adaptive FIR 필터로 되돌려 주고 필터를 조정함으로 전체 시스템의 출력을 줄여 줌으로써 least-squared e(n) 를 구한다 .

생체 신호 해석2004 년 10 월 1 일


min E[e2 (n)] = E[v2 (n)]+ min E[{m(n) – y(n)} 2]

E[e2 (n)] is minimized, min E[{m(n) – y(n)} 2] is also minimized

and since e(n) – v(n) = m(n) – y(n). when E[{m(n) – y(n)} 2] minimized, E[{e(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.

생체 신호 해석2004 년 10 월 1 일


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)

생체 신호 해석2004 년 10 월 1 일


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

The least mean squares adaptive filterThe least mean squares adaptive filter

생체 신호 해석2004 년 10 월 1 일

The least mean squares adaptive filter (1)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 이 2 차 이기 떄문에 그래프는 아래가 둥근 그릇모양 (hyper-paroboloidal, bowl-like) 이 된다 . 이 그래프의 바닥에 도달하기 위해서는 ( 최소값이 되기 위해서는 ) gradient-based method of steepst descent 를 사용한다 .

In LMS algorithm w(n+1) = 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.

생체 신호 해석2004 년 10 월 1 일


The LMS algorithm approximates ∇(n) by the derivative of the squared error with respect to the tap-weight vector

w(n+1) = w(n) + 2 μe(n)r(n) ; widrow-Hoff LMS algorithm.

생체 신호 해석2004 년 10 월 1 일


Application

VAG signals recorded from the mid-patella( 슬개골 ) and the tibial( 경골 ) tuberosity( 융기 )

Reference : distal( 말초 ) rectus( 직근 ) femoris( 대퇴부 ) muscle-contraction signal

생체 신호 해석2004 년 10 월 1 일


Zhang 은 w(n+1) = w(n) + 2 μ(n)e(n)r(n) 에서 μ 를 변수로 정의했다 .

0<μ<1, 0≤ α <<1 일때 signal nonstarionarity로 인해 발생하는 문제를 해결할 수 있다 .

생체 신호 해석2004 년 10 월 1 일


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

The recursive least-squares adaptive filter

The recursive least-squares adaptive filter

생체 신호 해석2004 년 10 월 1 일

The recursive least-squares adaptive filter (1)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

생체 신호 해석2004 년 10 월 1 일


Performance index or objective function

0 < λ ≤ 1 weighting factor(forgetting vector) 1 ≤ i ≤ n is the observation interval E(n) estimation error

λ n-i < 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

생체 신호 해석2004 년 10 월 1 일


Ф(n) M x M time averaged autocorrelation matrix of reference input r(i) defined as

Θ(n) : M x 1 time-averaged cross-correlation matrix between the reference input r(i) and the primary input x(i) defined as

생체 신호 해석2004 년 10 월 1 일


Recursive techniques needed To obtain recursive solution, isolate the term

corresponding to i=n

And right-hand side of above equation equals the time-averaged and weighted autocorrelation Ф(n-1)

Ф(n) = λФ(n-1) + r(n)rT(n)

생체 신호 해석2004 년 10 월 1 일


Equation 3.124 can be written as the recursive equation

Θ(n) = λ Θ(n-1) + r(n)x(n) we need inverse of Ф(n) to obtain tap-

weight vector

To determine the inverse of the correlation matrix Ф(n), use “ABCD lemma”

(A+BCD)-1 = A-1 – A-1B(DA-1B+C-1) -1DA-1

A = λФ(n-1) B = r(n) C = 1 D = rT(n)

생체 신호 해석2004 년 10 월 1 일


So we have Ф-1(n) = λ-1 Ф-1(n-1)

- λ-1 Ф-1 (n-1)r(n)[λ-1rT(n) Ф-1(n-1)r(n)+1] -

1

x λ-1rT (n) Ф-1(n-1) Since [λ-1rT(n) Ф-1(n-1)r(n)+1] is scalar,

For convinience P(n) = Ф-1(n)

생체 신호 해석2004 년 10 월 1 일


With P(0) = δ-1I where δ is a small constant and I is the identity matrix

Then rewritten in a simpler form asP(n) = λ-1P(n-1) - λ-1k(n)rT(n)P(n-1) - a

From above two equation k(n)[1+ λ-1rT(n)P(n-1)r(n)] = λ-1 P(n-

1)r(n)Or k(n) = [λ-1p(n-1)-λ-1k(n)rT(n)P(n-1)]r(n) -b

생체 신호 해석2004 년 10 월 1 일


From a and bk(n) = P(n)r(n)

P(n) and k(n) have dimensions M x M and M x 1 As we’ve seen

And Θ(n) = λ Θ(n-1) + r(n)x(n)And P(n) = Ф-1(n) So recursive equation for updating the least-squares

estimate w(n) of the tap-weight vector can obtained as

생체 신호 해석2004 년 10 월 1 일


From P(n) = λ-1P(n-1) - λ-1k(n)rT(n)P(n-1)

Finally from k(n)=P(n)r(n)

This equation gives a recursive relationship of w(n)

생체 신호 해석2004 년 10 월 1 일


Where w(0)=0

The quantity α(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, α(n) will be the estimated signal of interest v(n) after the filter has converged

생체 신호 해석2004 년 10 월 1 일


After convergence, the primary noise estimate, the output of the adaptive filter y(n) is

So we can obtain

생체 신호 해석2004 년 10 월 1 일


Application

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

생체 신호 해석2004 년 10 월 1 일


LMS filter M = 7, μ = 0.05, α = 0.98

RLS filter M = 7, λ = 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

생체 신호 해석2004 년 10 월 1 일


Spectrogram of VAG in (a)

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

생체 신호 해석2004 년 10 월 1 일


Spectrogram of RLS-filtered VAG in (d)

We can see that low-frequency artifact has been removed by RLS filter

Selecting an Appropriate FilterSelecting an Appropriate Filter

생체 신호 해석2004 년 10 월 1 일

Selecting an Appropriate Filter (1)Selecting an Appropriate Filter (1)

1. Synchronized or ensemble averaging of multiple realizations or copies of a signal Time-domain

2. MA(Moving average) filtering Time- domain

3. Frequency-domain filtering4. Optimal(Wiener) filtering

Implemented in the time-domain or in the frequency-domain

5. Adaptive filtering alter their characteristics in response to changes in the

interferences

생체 신호 해석2004 년 10 월 1 일


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

생체 신호 해석2004 년 10 월 1 일


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

생체 신호 해석2004 년 10 월 1 일


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

Removal of Artifacts In the ECGRemoval of Artifacts In the ECG

생체 신호 해석2004 년 10 월 1 일

Removal of Artifacts in the ECG (1)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

생체 신호 해석2004 년 10 월 1 일


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

생체 신호 해석2004 년 10 월 1 일


base line drift HPF with fc=2Hz

high-frequency noise LPF with fc=70 Hz

power-line interference Comb filter with zeros and 60, 180, 300, 420Hz

Application : Adaptive Cancellation of the Maternal ECG to obtain the Fetal

ECG

Application : Adaptive Cancellation of the Maternal ECG to obtain the Fetal

ECG

생체 신호 해석2004 년 10 월 1 일

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

생체 신호 해석2004 년 10 월 1 일

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

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

Application : Adaptive Cancellation of Muscle-Contraction Interference in

Knee-Joint Vibration Signals

Application : Adaptive Cancellation of Muscle-Contraction Interference in

Knee-Joint Vibration Signals

생체 신호 해석2004 년 10 월 1 일

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


(a) VAG signal of a subject with Chondromalacia patella( 슬개골 연골 연화증 ) (b) simultaneously recorded muscle-contraction interference (c) result of LMS filtering with M=7, μ=0.05, α=0.98 (d) result of RLS filtering with M=7, λ=0.98

생체 신호 해석2004 년 10 월 1 일



Spectrogram of the original VAG signal

생체 신호 해석2004 년 10 월 1 일



Spectrogram of the muscle-contraction interference signal

생체 신호 해석2004 년 10 월 1 일



Spectrogram of the RLS-filtered VAG signal

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ADAPTIVE FILTERS FOR REMOVAL OF INTERFERENCE 2004235124 김현일