EP1565834A2

EP1565834A2 - Method and arrangement for detecting and measuring the phase of periodical biosignals

Info

Publication number: EP1565834A2
Application number: EP03767683A
Authority: EP
Inventors: Günter HENNING; Peter Husar; Alfred Pecher
Original assignee: Technische Universitaet Ilmenau
Current assignee: Cufer Asset Ltd LLC
Priority date: 2002-11-28
Filing date: 2003-11-27
Publication date: 2005-08-24
Also published as: US20060122781A1; AU2003292138A1; WO2004047610A3; WO2004047610A2; AU2003292138A8; DE10255593A1

Abstract

The invention relates to a method and an arrangement for detecting and measuring the phase of periodical biosignals. The aim of the invention is to detect and measure a causal phase response in periodical biosignals, with improved reliability and higher speed compared to conventional methods, simultaneously reducing the computing power required. According to the invention, a state observer is set up in parallel with the analysed biological system. The output variables of the biological system and the observer are evaluated in order to minimise the occurring error. The determined phase of a periodical biosignal can, for example, be used for functional diagnostic purposes.

Description

Method and arrangement for the detection and measurement of the phase of periodic biosignals

The invention relates to a method and an arrangement for real-time reliable detection and measurement of the phase of periodic physiological variables or biosignals.

Methods are known in the prior art which use an analysis window which slides or is arranged sequentially over the time profile of the biosignal to determine the phase. The signal section lying in the analysis window is based on the Fourier transform

Methods or descriptive statistics applied. For example, periodically illuminating light marks of defined intensity and sufficiently high frequency (above about 4 Hz) are used in perimetry to check the functionality of the visual system. For the functional test, the electroencephalogram (EEG) is recorded and the stimulus response to the visual stimulus is analyzed with regard to the amplitudes and the phase. The stimulus response phase is one of the crucial diagnostic parameters in functional diagnostics.

A disadvantage of the previous methods is that the statistical uncertainty of the detection or the inaccuracy of the measurement is very high. The uncertainty and inaccuracy result from the signal theory as a consequence of and in connection with the length of the analysis window. The theory states that the statistical uncertainty and thus the inaccuracy increase with decreasing length of the analysis window, which is also sufficiently proven and known in practical signal analysis. For a statistically better result The window length would have to be increased first. However, it is known from physiology that the phase can change relatively quickly and that these changes are also diagnostically relevant. With a long analysis window, the valuable information about the phase change is lost and the statistical uncertainty of the measurement result does not necessarily decrease as a result of the changes.

The invention has for its object to provide a method and an arrangement with which it is possible to detect and measure a causal phase response in periodic bio-signals with a better reliability and higher speed than conventional methods with a reduced computing effort.

According to the invention, the object is achieved in that periodic biosignals are recorded according to their physical and physiological origin, that a condition observer is set up in parallel with the analyzed biological system, that the output variables of the biological system and the observer are evaluated with the aid of a Cayman filter and for Determination of the phase to be used.

In the method according to the invention, the phase of a periodic biosignal is determined and used for functional diagnostic purposes. For example, an extended phase in comparison with healthy test objects is an important indicator of functional problems in the biological system under investigation.

In the arrangement according to the invention, the state system is parallel to the examined biological system is reproduced, a state observer is arranged, who estimates the phase state variable based on the Cayman filter in accordance with the system inodel.

The advantage here is that the phase can be estimated continuously and no sliding or sequentially applied analysis window is necessary. This makes the analysis of the phase changes over time possible. In contrast to the relatively complicated theoretical background of this phase estimator, the practical implementation is simple. Compared to conventional methods, it requires significantly less computing power, so that a phase estimation in real time is possible.

The invention is explained below with reference to the theoretical derivation and an embodiment. In the accompanying drawings:

1 shows a block diagram of an observer concept;

2 shows a system model of an arrangement according to the invention;

3 shows a basic illustration of the state observer for measuring the phase in periodic biosignals;

4 shows a profile of an estimated phase for a harmonic of the frequency 8 Hz and phase 2 rad for the static Cayman factors 2 and 20;

5 shows a curve of an estimated phase for a noisy harmonic of the frequency 8HZ and phase 2rad with an SNR from OdB (below) and dynamic Cayman factor (above);

FIG. 6 shows a phase estimate of the signal as in FIG. 5 with static Cayman factors;

7 results of the phase estimation (right) on real signals (left);

8 shows an additive superimposition of a harmonic of a frequency of 8 Hz with the noise and deliberate detuning of the analysis frequency (top) and the phase curve with rise (bottom).

A biological system that produces a periodic biosignal or responds to a periodic input signal is shown in FIG. 1 as the “real system” state model. The state equations (1) and (2) mentioned below describe this system (bold letters in large letters stand for Matrices, small for vectors):

x (t) = Ax (t) + Bu (t); x (0) = x (t ₀ ) (1) y (t) = C- (t). (2)

For the further considerations, an additive signal model is assumed, which sums up a harmonic oscillation and white, normally distributed noise:

y (t) = y-sin (ωt + φ (t)) + r _p (t) (3)

The goal is to construct a system model whose variable x (t) represents the phase φ (t) of the signal y (t) to be examined. The phase cannot be measured directly because it is an argument of a trigonometric function. An auxiliary construction is therefore required. Such a construction is a condition observer, which is arranged parallel to the system under investigation. The observer estimates the state variable by minimizing an error function that compares the outputs of the real system and the observer. In this way, after the error minimization has been completed, the phase state variable can be measured directly.

The block diagram of the observer concept is shown in FIG. 1. Since x (t) cannot be measured directly, X (t) is estimated in the observer. The inner loop in the observer minimizes the error of γ _m (t) with respect to y (t) with the aid of the correction matrix K. The equations of state (4) and (5) then result for the observer:

x _M (t) = Ax _M (t) + Bu (t) + K- [y (t) -y _M (t)}, (4)

y _M (t) = Cx _M (t). (5)

From (4) and (5) it follows:

x _M (t) = (A-KC) -x _M (t) + Bu (t) + Ky (t). (6)

It is assumed that both systems have different initial conditions. This results in the observation error:

e (t) = x (t) - x _M (t). (7) The observation error disappears iteratively with the aid of the correction matrix K, so that

e (t) = 0 for t → co. ( 8th )

The dynamics and the stability of the estimation can be described with the differential equation of the observation error (9):

e (t) = i (t) -i _M (t). (9)

Through conversion and further intermediate steps you get:

e (t) = (A-K-C) -e (t). (10)

According to the signal model (3), it can be expected that the signal under investigation is disturbed by noise. In order to reduce the influence of the noise, a caiman filter is used. Taking the noise into account, the system is described by the following state equations:

System state: x (t) = Ax (t) + Bu (t) + Mr _s (t) (11)

System output: y (t) = C • x (t) + r _p (t) (12) Observer: x _M (t) = Ax _M (t) + Bu (t) + K (t) - [y (t ) -Cx _M (t)], (13)

in which

K (t) is the correction matrix which has to achieve that e (t) = x (t) - x _M (t) → 0, e (t) is the observation error, r _s (t) is the system noise, and r _p (t) is the process noise.

To simplify the derivation, it is assumed that the noise components are broadband Gaussian zero mean processes with known covariances:

cov _rp (t ₁ , t ₂ ) = E {r _p (t ₁ ) -r _p ^τ (t ₂ )} = R _p (t ₁ ) -δ (t, -t ₂ )

Ü4) cov _rs (t ₁ , t ₂ ) = E {r _s (t ₁ ) τ _s ^τ (t ₂ )} = R _s (t ₁ ) -δ (t ₁ -t ₂ )

The noise components are independent of each other, that is

cov _rprs (t ,, t ₂ ) = 0. (15)

For a consistent estimate of x (t), the error power must be minimized using the matrix K (t):

E {e ^τ (t) -e (t)} = E {e ₁ ² (t) + e ^ (t) + - + e ² _n (t)} = f (K (t)) = Min. , (16)

Assuming the stochastic relationships with regard to the covariance, a suitable correction matrix K (t) is derived in accordance with the Kalman filter:

K (t) = cov _e (t) .C ^τ .R; ¹ (t). (17)

The formula for the error covariance cov _ε (t) can be derived from the Kalman filter:

cov _ε (t) = A-cov _ε (t) + cov _ε (t) -A ^τ -cov _ε (t) -C ^τ -R: ¹ (t) -C-cov _ε (t) + MR _s ( t) -M ^τ . (18) Estimation of the phase:

The signal under investigation is modeled according to (3) from the sum of a harmonic and noise:

y (t) = y „, (t) + r _p (t) = y ^• sin (ωt + φ (t)) + r _p (t), ω = 2πf. (19)

The phase results from the differential equation:

ψ (t) = -a-φ (t) + r _s (t) a> 0. (20)

This results in the system model in Figure 2. The phase cannot be measured directly. An observer is therefore parallel to the system, in which direct access to the estimated phase φ _M (t) is possible. However, the non-linear part y _n ι (t) in (19) is unfavorable for the observer concept. A suitable linearization yι (φ (t), t) is required to establish a linear relationship between the state variable φ _M (t) and the output y _M (t) according to (5). Based on Taylor linearization, the observer can be formulated as follows:

φ _M (t) = -a • φ _M (t) + K (t) • (y (t) -y _M (t)) (21)

y _M (= y. (φ _M (t), t). (22)

According to (21) and (22), the observer is modeled, as shown in FIG. 2. As a result of the linearization at the operating point φ _B , (22) is linked with (5). This gives the factor C, which is used in (17) to determine the correction factor K (t): K (t) = cov _e (t) -y-cos (ωt + φ _B (t)) - R _p - '(t). (23)

The phase to be determined is selected as the operating point

φ _B (t) = φ _M (t), ⁽ 24 ⁾

and the resulting differential equation for the phase

φ _M (t) = -a-φ _M (t) + cov _e (t) -y-cos (ωt + φ _M (t)) - [y (t) -y-sin (ωt + φ _M (t ))] - R ^, (t). (25)

According to (25), the phase estimator can be modeled, as shown in FIG. 3.

The error covariance must be calculated for the phase estimation in y (t). From (18) it follows:

cove (t) = -2a • cov _e (t) ^■ y ² cos ² (ωt + φ _M (t)) - cov _e ² (t) R '(t) + _s (t). (26)

Equation (26) provides a simple solution if higher-frequency components are not taken into account in the error covariance. Based on (27)

cos ² (ωt + φ _M (t)) = 1/2 + cos (2ωt + 2φ _M (t)) (27)

can (26) be simplified to:

cov _e (t) = -2a-cov. (t) -l / 2-y ² -R _p - ¹ (t) -cov _e ² (t) + R _s (t). (28)

With a suitable choice of parameter a in (28), high-frequency components as a result of temporal integration are suppressed, ie there is a low-pass behavior. Taking into account low pass can be simplified (25):

φ _M (t) = -a-φ _M (t) + cov _e (t) -y-cos (ωt + φ _M (t)) - y (t) -R _p - '(t), (29)

This simplifies the observer, shown in FIG. 3. The system proposed in (29) can be used in particular for the phase estimation of harmonics in noise.

4 shows the course of the estimated phase for a harmonic of the frequency 8 Hz and phase 2 rad for different Cayman factors. The Cayman factors are static and are 2 or 20. As can be seen in the graph, the lower the Cayman factor, the slower the estimate. Static caiman factors must be used where the time of the phase change is not known.

5 shows the curve (lower part) of the estimated phase for a noisy harmonic of the frequency 8 Hz and phase 2 wheel with an SNR (signal-to-noise ratio) of OdB and dynamic Kaiman factor (upper part). If the time of the phase change is known, the Cayman factor can be constructed so that the change is followed first and the variance of the estimate is then minimized.

For comparison, the phase estimate of the same signal as in FIG. 5 with static Cayman factors is shown in FIG. 6. 7 shows the results of a phase estimate (right column of the graphic) on real signals. To stimulate a visual system investigated by way of example, light pulse sequences with a repetition rate of 8 pulses per second were used, alternating between a pause (time from 0 to 2 of the time courses of the signals top left and bottom left) followed by light stimulation (time 2 to 5). The course of the EEG (electroencephalogram, left column of the graphic) is shown from two occipital positions after 16-fold stimulus-related averaging. Both time courses show a clear jump in the phase after the start of light stimulation.

The phase estimation becomes problematic with very noisy signals. In general, it is true that the phase is more robust against disturbances than the amplitudes, as is finally known in information technology. However, the question of the existence - i.e. the detection - of a causal phase must first be clarified in this border area, only then would the phase be estimated.

In FIG. 8, a harmonic of the frequency 8 Hz is additively superimposed on the noise starting at t = 4 s, the SNR being -10 dB (upper curve in the graphic). The amplitude of the harmonic cannot be detected in the time domain. If you use the phase estimator with a specific detuning, here with a frequency of 7.8Hz, i.e. 0.2Hz less than the frequency of the harmonics, there is an increase of 1.2rad / s in the case of a causal phase (lower graph). This increase can be used directly in combination with a discriminator to detect the signal. LIST OF REFERENCE NUMBERS

a - System parameters, selectable in the observer model

A, B, C, K - matrices in the state model of a system φ (t) - phase in the system model, quantity to be estimated φ _M (t) - phase in the observer model, measurable quantity r _p (t) - process noise r _s (t) - System noise u (t) - input variable of a system in the state model x (t) - state variable of a system in the state model

Xπ (t) - state variable of the observer in the state model y (t) - output variable of a system in the state model y _M (t) - output variable of the observer in the state model yι (φ (t), t) - linearization operator for phase

Claims

claims

1. A method for detecting and measuring the phase of response signals (y (t)) of a biosystem, comprising the following steps: a) multiplying the response signal (y (t)), the phase (φ (t)) of which is to be determined by a first factor; b) Multiplication of the product obtained from step a) by a second factor, which is represented by a trigonometric function, the argument of which results from the product of the frequency of the investigated response signal with time, added to the measured phase, the frequency of the trigonometic analysis function corresponds to the frequency at which the phase is to be determined or deviates from this frequency by a known amount, c) multiplying the measured phase by a third factor (a); d) forming the difference between the product obtained in step b) and the product obtained in step c); e) integration of the difference obtained in step d) over time, the result of this integration representing the phase of the signal to be determined f) repetition of steps a) to e) until an abort criterion is met.

2. The method according to claim 1, characterized in that the first factor is chosen to be constant or variable over time.

3. The method according to claim 1 or 2, characterized in that in the difference formation in step d) the product from step c) is subtracted from the product from step b).

4. The method according to any one of claims 1 to 3, characterized in that the response signal (y (t)) is fed to a status observer who carries out the method steps a) to f) by an estimated phase (φ. _Ι (t)) to determine, the method being terminated when the observer output signal (yιι (t)) deviates less than an error value ( _e (t)) specified by an error function (cov e (t)) from the response signal (y (t)), and wherein after termination of the method, the estimated phase (φ _M (t)) is set equal to the phase (φ (t)) of the response signal.

5. The method according to claim 4, characterized in that steps a) to d) are carried out in the observer according to the following formula:

Φ _M (0 = ~ ^{a ■} Ψ _M (0 + ∞v _e (t) ^■ y ^■ cos (ωt + φ _M (t)) ^■ y (t) ^• R; ¹ (t).

6. Arrangement for the detection and measurement of the phase of response signals of a biosystem, characterized in that the arrangement comprises a status observer, to which the response signal of the system under investigation is fed in parallel to the biosystem and which carries out the method steps according to one of claims 1 to 5.

7. Arrangement according to claim 6, characterized in that the condition observer comprises a Kalman filter with which interference signals are filtered from the response signal.