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Title: Modelling non-stationary variance in EEG time series by state space GARCH model
Article Type: Full Length Article
Section/Category:
Keywords: State space model, Kalman filter, frequency decomposition, autoregressive model, conditional
heteroscedasticity, non-stationarity, anaesthesia.
Corresponding Author: Mr. Kevin Kin Foon Wong,
Corresponding Author's Institution: Graduate University for Advanced Studies
First Author: Kevin Kin Foon Wong
Order of Authors: Kevin Kin Foon Wong; Andreas Galka; Okito Yamashita; Tohru Ozaki
Manuscript Region of Origin:
Abstract: We present a new approach to modelling nonstationarity in EEG time series by a generalised
state space approach. A given time series can be decomposed into a set of noise-driven processes, eachcorresponding to a different frequency band. Nonstationarity is modelled by allowing the variances of thedriving noises to change with time, depending on the state prediction error within the state space model. The
method is illustrated by an application to EEG data recorded during the onset of anaesthesia.

Manuscript

Modelling non-stationary variance in EEG timeseries by state space GARCH model
Kevin K.F. Wong
Graduate University for Advanced Studies, Japan
Andreas Galka
Institute of Experimental and Applied Physics,
University of Kiel, Germany;
Institute of Statistical Mathematics, Japan
Okito Yamashita
ATR Computational Neuroscience Laboratories, Japan
Tohru Ozaki
Institute of Statistical Mathematics, Japan;
Graduate University for Advanced Studies, Japan

1

State space GARCH Model on EEG
Correspondence address:
Kevin Kin Foon Wong
Minami Azabu 4-6-7,
Minato-ku, Tokyo 106-8569, Japan
Telephone: (81)-3-5421-8751
Fascimile: (81)-3-5421-8751
E-mail: wong@ism.ac.jp

Running head: State space GARCH Model on EEG

2

State space GARCH Model on EEG
Abstract
We present a new approach to modelling non-stationarity in EEG
time series by a generalized state space approach. A given time series
can be decomposed into a set of noise-driven processes, each corresponding to a diﬀerent frequency band. Non-stationarity is modelledby allowing the variances of the driving noises to change with time,depending on the state prediction error within the state space model.
The method is illustrated by an application to EEG data recorded during the onset of anaesthesia.

key words: State space model, Kalman ﬁlter, frequency decomposition,
autoregressive model, conditional heteroscedasticity, non-stationarity,
anaesthesia.

3State space GARCH Model on EEG

1

4Introduction

Brain dynamics can be analyzed by estimating the frequency spectrum of
the electroencephalogram (EEG), which can be done by parametric or nonparametric methods; Fast Fourier Transform (FFT) represents a well-known
nonparametric method (Priestley [1], Kay [2]), while ﬁtting of autoregressive (AR) models is a prominent example of a parametric method (Box and
Jenkins [3], Gersch [4]). However, in the case of the presence of pronounced
non-stationarity in the EEG, such as time-dependent changes of the power
in diﬀerent frequency bands, direct application of the FFT to the data would
be inappropriate.
Although in this case it is still possible to apply the FFT to a window
moving over the data, such approach would have the disadvantage of reduced
resolution either in time or in frequency domain; improved resolution in time
domain, desirable in order to pick out distinctive temporal characteristics
in the data, has to be paid by reduced resolution in frequency domain, andvice versa.
In contrast, parametric spectral estimation by AR models oﬀers various
advantages over the FFT, since it represents a more general and ﬂexible
framework for parsimonious dynamical modelling of time series data, which
can be readily employed for purposes such as prediction, classiﬁcation or
causality analysis of time series (Shumway and Stoﬀer [5]); in the case ofnon-stationarity, parametric spectral estimation may also be applied to aState space GARCH Model on EEG

5moving window (Ozaki and Tong [6]), but as we will show in this paper, there
is an alternative approach for this situation which avoids the introductionof a moving window.
We will model the EEG by a linear autoregressive model in a state spaceframework, which is suitable for describing the simultaneous presence of several major frequency bands; the non-stationarity will be added by employing
the Generalized Autoregressive Conditional Heteroscedasticity (GARCH)
model, as introduced by Engle [7] for the modelling of time-dependent variance, and generalized by Bollerslev [8], but in contrast to the usual usage ofthe GARCH model we will apply it within state space.
The concept of employing the GARCH model within a state space has
been introduced by Galka et al [9] in a study on the estimation of inverse
solutions from EEG time series; for simulated data they obtained improved
reconstruction of true states by this technique.
It is a property of the GARCH model that non-stationarities can bedetected with very good temporal resolution (Bollerslev [8]); therefore we
propose it as an appropriate tool for the modelling of transients and rapidchanges of spectral properties, as they are commonly observed in EEG timeseries. As an example we will discuss in this paper the case of a clinical
EEG time series displaying the transition from awake conscious state toanaesthesia.

State space GARCH Model on EEG

2

6

An example serving as motivation

The EEG time series which we will study in this paper was retained from a
recent study of John et al [10] and John [11] who have studied the changeof spectral content of clinical EEG accompanying the loss and subsequent
recovery of consciousness due to initiation and termination of anaesthesia
during surgery. The data was measured at 19 electrodes ﬁxed to the scalp
according to the international 10/20 System. The detailed experimental
procedures have been described in John et al [10] and Prichep et al [12].
Based on techniques from descriptive statistics, including FFT spectrum
estimation and computation of mean z-scores, John et al [10] found that anincrease in absolute power of the low frequency band occurs when patientslose consciousness. In this paper we will study the same topic and conﬁrm
their result by a parametric approach.
We select from their data a segment of 2048 samples from the T4 electrode (versus average reference), sampled at 100 Hz, such that the segment
extends over about 20 seconds. This data set covers the transition fromawake conscious state to anaesthesia. The data is shown in Figure 1.

Figure 1 about here

At time 0 seconds, induction of anaesthesia begins. At about 10 seconds,

State space GARCH Model on EEG

7loss of consciousness occurs. It can be seen in the ﬁgure that starting fromthis time the amplitude of low-frequency activity in the EEG displays apronounced increase.
In the lower panel of Figure 1 the spectral density according to a “conventional” moving window analysis is shown: The data was divided into 15
segments, each of length 256 samples, where consecutive segments overlap
by 128 samples. Autoregressive models of order 8 are ﬁtted to each segment,
and the resulting parametric spectra are displayed. A peak at 10 Hz can beseen persistently; at low frequency the power rises at about 10 seconds.

3

Methods

The EEG data as shown in Figure 1 appears to be a superposition of several
diﬀerent source components. We intend to decompose the data into sourcecomponents corresponding to several major spectral bands. Each component
is described by a dynamical process, and each of these processes may havea diﬀerent non-stationary behaviour of variance.
Let yt denote the observed data and xt the unobserved state. We assume
that xt depends on its past values through a linear stochastic model, containing a dynamical noise term, and that yt follows from xt through a linear
observation model, containing an observation noise term; then the following

State space GARCH Model on EEG

8state space model applies:
xt = F xt−1 + wt

(1)

yt = Hxt + t

(2)

Equations (1) and (2) are commonly known as system equation andobservation equation, respectively. wt denotes the dynamical noise termof the system equation, assumed to follow a multivariate Gaussian distribution wt ∼ N (0, Qt ), while t denotes the observation noise term of theobservation equation, assumed to follow a univariate Gaussian distribution
t ∼ N (0, R).
Kalman [13] introduced a ﬁltering technique for state space models which
can eﬃciently calculate the conditional prediction and conditional ﬁltered estimation of unobserved states. A comprehensive introduction to state spacemodels and Kalman ﬁltering has been provided by Kalman [13], Harrisonand Stevens [14], Harvey [15], Grewal and Andrews [16].
Since we aim at decomposing the data into a set of source components,
we choose a special structure for the state space model, such that pairsof elements within the state vector xt represents autoregressive models of
second order, AR(2). Each AR(2) model is capable of describing one mainfrequency found in the data. For a detailed account of the representation ofoscillations by AR(2) models see Box and Jenkins [3].
Assume that r denotes the number of source components, then the state
xt is a column vector of dimension 2r, F is a 2r × 2r matrix and H is a 2r

State space GARCH Model on EEG

9row vector. For each of the r source components there is a 2× 2 block on the
diagonal of F , all other elements of F are zero. Each 2×2 block contains two
parameters, which deﬁne frequency and damping of the corresponding AR(2)model; thereby each component contributes to modelling the spectrum of
the data.
The state vector and the two parameter matrices of the model thenbecome
2

3
(1)

2

3
(1)

6 φ1
6
6
6 1
6
6
6
6 0
6
6
6 .
.
F =6
6 .
6
6 .
6 .
6 .
6
6 .
6 .
6 .
6
4
0
»
H = c(1) 0

6 zt
7
6
7
6 (1) 76 z
7
6 t−1 7676
7
6 z (2) 7
6 t
7
6
7
6
7
(2) 7 ,
xt = 6
6 zt−1 7
6
7
6 . 7
6 . 7
6 . 7
6
7
6
7
6 (r) 7
6 zt
7
6
7
4
5
(r)
zt−1

(1)

φ2

0

···

0

0

..

0

φ1

(2)

φ2

(2)

..

..

1

0

0

..

.

0

..

.

..

.

.

..

.

φ1

···

0

c(r)

0

.

..

.

..
···
c(2)

···
0

···

···

···

0

7
7
7
7
7
7
.. 7
. 7
7
7
.. 7
. 7
7,
7
7
7
0 7
7
7
(r) 7
φ2 7
7
5
0
..
.

.
.

(r)

1
–

Qt denotes the variance matrix of the dynamical noise wt ; also thismatrix is composed of 2×2 blocks on the diagonal, each block corresponding
to one AR(2) model, but here only one element within each block is nonzero(Harvey 1989):

State space GARCH Model on EEG

2 “

Qt

=

6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4

(1)

σt

”2

10

3
0

···

0

0

..

..
.

..

.

..
.

“

···

(2)

σt

0

”2

0
0

..

.

..

..

.

..

.

.
”2
“
(r)
σt

···

0

.

..
.

The nonzero elements

···

···

.

..
.

0

···

···

···

..

0 7
7
.. 7
. 7
7
7
.. 7
. 7
7
7
.. 7
. 7
7
7
.. 7
7
. 7
7
7
7
0 7
7
5
0



(k) 2
σt
are time-dependent; according to the

GARCH approach in its general form, they are modelled by

p
q
”2
”2 X
”2 X
”2
“
“
“
“
(k)
(k)
(k)
(k)
(k)
(k)
log σt= log σ0+αi log σt−i +βj log vt−j
i=1

(3)

j=1

k = 1, 2, . . . , r



(k) 2
The ﬁrst term on the right-hand side of this equation, log σ0
represents a constant, which in general may be non-zero, but here we will set itzero in order to avoid parameter redundancy. The ﬁrst sum represents an
autoregressive model of order p for the logarithm of the variance, while thesecond sum corresponds to a moving-average (MA) dynamical noise term.
Expressing this GARCH model by the logarithm of the variance has the
beneﬁcial eﬀect of preventing the variance from becoming negative.
In the original GARCH model the dynamical noise vt−j would be given
by previous values of the prediction error of the data, thereby feeding back

State space GARCH Model on EEG

11increases of the prediction error into the variance of the dynamical noise;but in the state space approach the state prediction errors are not directly
 |yaccessible, therefore we estimate it by forming the matrix E wt−j wt−jt−j(where E(.) denotes expected value) and extract from its diagonal the ﬁrst




(1) 2
(2) 2
, the third element for log σt, etc. The detailed
element for log σt |yderivation and resulting expression for E wt−j wt−j
t−j can be found in
the Appendix.
The model parameters in Equations (1), (2) and (3) are estimated fromgiven data by the maximum-likelihood method. Given a set of parameters,
computation of the likelihood from the errors of the data prediction throughapplication of the Kalman ﬁlter is straightforward; see Mehra [17], Astrom
and Kallstrom [18] and Valdés-Sosa et al [19] for a detailed treatment.

4

Results

We choose to employ a state space model consisting of r = 4 AR(2) models,such that 4 major frequency bands can be described. By ﬁtting the modelto the data shown in Figure 1 these frequencies are found as 2.4Hz, 10.3Hz,
17.6Hz and 24.5Hz, corresponding to the delta, alpha, mid-range beta andlow-range gamma frequency bands, respectively.
(k)

We ﬁnd that the autoregressive parameters αiin Equation (3) do not

diﬀer signiﬁcantly from zero, therefore we set the AR order to p = 0, whichhelps to reduce the number of parameters to be ﬁtted. We also impose theState space GARCH Model on EEG(k)constraint β1(k)= · · · = βq12=: β (k) , such that the variances can changesmoothly, and we have a further reduction of the number of parameters.
Since the likelihood does not improve signiﬁcantly for MA order q larger than2, we set q = 2. The resulting estimate of β (k) for the 2.4Hz-component is
0.42, while for the other components values of -0.047 (10.3Hz), 0.42 (17.6Hz)
and -0.25 (24.5Hz) are found.

Figure 2 about here.

(k)

In Figure 2 we show the estimated components zt , k = 1, 2, 3, 4; thesecomponents, each corresponding to one of the four frequencies, represent a
decomposition of the original data (shown again in the bottom panel of theﬁgure), such that by summing up these components according to Equation
2, using the weights c(k) , the original data is reproduced. Note that a pronounced increase of amplitude occurs for the 2.4Hz-component at about 10
seconds; this eﬀect is solely obtained as a result of the maximum-likelihood
model ﬁt, without any input of prior knowledge concerning the change ofthe spectral composition of the data.

Figure 3 about here.

State space GARCH Model on EEG

13



(k) 2
of the compoIn Figure 3 we show explicitly the variances log σtnents as functions of time, according to the ﬁtted GARCH models. It canbe seen that for the 2.4Hz-component the variance increases at about 8 seconds to a considerably larger value than before, and maintains that larger
value within the second half of the data set. The variances of the othercomponents do not display signiﬁcant changes. This result can be readilyinterpreted by stating that the loss of consciousness at onset of anaesthesia
is reﬂected almost exclusively by an increase of power in the delta band.

5

Discussion

In this paper we have proposed a new tool for quantitative description of
non-stationarities in EEG time series. For this purpose we have introduceda model for decomposition of a given single-channel EEG time series into
components deﬁned by their main frequency, and we have shown how thevariances of the dynamical noises driving these components can be madetime-dependent by generalising the concept of GARCH modelling to thesituation of state-space modelling. As a result, changes of the distributionof power over the main spectral bands of the EEG can be traced over time.
Note that by choosing the GARCH approach for describing non-stationarityof variance we obtain a method which remains suitable for real-time monitoring, in contrast to approaches which describe non-stationarities retrospectively by ﬁtting explicitly time-dependent functions to the non-stationary

State space GARCH Model on EEG

14parameters; the GARCH model is fully compatible with predictive modelling, since it requires only information from the past.
Once a suitable model structure has been identiﬁed (with respect tothe number of components in state space and the GARCH model orders p
and q) and a corresponding set of appropriate parameters has been identiﬁed through maximum-likelihood, the Kalman ﬁlter can be applied very
eﬃciently to new data without the need of any further time-demanding
computations. This enables the application of this technique to real-timemonitoring of patients during surgery, e.g. it would be possible to monitor
the depth of anaesthesia quantitatively by the time-varying set of variancesof the relevant frequency bands. Also applications to other kinds of dataarising in the neurosciences are conceivable.
EEG time series are usually recorded not from just one electrode, butfrom a set of electrodes covering the whole scalp; in principle, the method
which we have proposed in this paper, could be applied independently toeach channel of the data, but it would be desirable to have a modelling approach capable of building a single common model from all available channels
simultaneously; thereby also the spatial information contained in the positions of electrodes could be incorporated. The generalisation of the method
to this case will be the subject of future work.

State space GARCH Model on EEG

15Appendix
2 , i.e. the estimator of the noise term
Here we will give the derivation of v̂t−j2
in Equation (3).
vt−jSince the state prediction error wt−j is not directly accessible, we derive aestimator with similar meaning. This estimator is chosen as the expectation
 , conditional on the data up to time t − j.
of the product wt−j wt−j

Let Kt−j , νt−j and Ωt−j denote the Kalman gain, residual and inverse
covariance matrix of residuals, respectively, at time t − j. These quantities
are obtained naturally through the application of the Kalman ﬁlter (fordetails see Harvey et al (1989)). Let Var(.) and Cov(.) denote variance andcovariance, respectively. Then we have
2
v̂t−j

=

`
´

E wt−j wt−j
|yt−j

=

Var (wt−j |yt−j ) + E (wt−j |yt−j ) E (wt−j |yt−j )

(4)

The expectation E (wt−j |yt−j ) is equal to Kt−j νt−j . The term Var(wt−j |yt−j )represents the conditional variance of the system noise; it can be expressedas
Var (wt−j |yt−j )=Var (wt−j |yt−j−1 ) − Cov (wt−j , νt−j |yt−j ) Var (νt−j |yt−j )−1 Cov (wt−j , νt−j |yt−j )

=Qt−j − Cov (wt−j , νt−j |yt−j ) Ωt−j Cov (wt−j , νt−j |yt−j )

State space GARCH Model on EEG

16and the covariance is obtained by
Cov (wt−j , νt−j |yt−j )=E (wt−j νt−j |yt−j ) − E (wt−j |yt−j ) E (νt−j |yt−j )

=

”
“
`
´
E wt−j yt−j − Hxt−j|t−j−1 |yt−j

=

”
“
`
˘
´¯
|yt−j
E wt−j [H (F xt−j−1 + wt−j ) + t−j ] − H F xt−j−1|t−j−1

=

”
“
`
˘
`
´¯
´
`
´

|yt−j + E wt−j wt−j
H  |yt−j + E wt−j t−j |yt−j
E wt−j HF xt−j−1 − xt−j−1|t−j−1=

Qt−1 H 






since E (νt−j |yt−j ) = 0, E wt−j HF xt−j−1 − xt−j−1|t−j−1 |yt−j = 0,
 H  |yE wt−j wt−j
t−j = Qt−1 H and E wt−j t−j |yt−j = 0. By substituting Var(wt−j |yt−j ) and Cov (wt−j , νt−j |yt−j ) into Equation (4), we get
2
v̂t−j

=



Qt−j − Qt−j H  Ωt−j HQt−j + Kt−j νt−j νt−j
Kt−j

State space GARCH Model on EEG

17

Acknowledgements
The authors would like to thank Dr Roy John and Dr Leslie Prichep forproviding the EEG data set and for dedicating their precious time to giving
the authors comments and guidance.
This work was supported by the Iwatani Naoji Foundation, the Atsumi International Scholarship Foundation, and the Japanese Society for
the Promotion of Science through fellowship no. P03059 and Kiban B no.
173000922301.

State space GARCH Model on EEG

18References[1] M.B. Priestley, Spectral Analysis and Time Series, Volumes I and II,
Academic Press, New York (1981).
[2] S.M. Kay, Modern Spectral Estimation: Theory and Application, Prentice Hall, Englewood Cliﬀs NJ (1988).
[3] G.E.P Box, G.M. Jenkins, Time series analysis, forecasting and control,
revised ed., Holden-Day, Oakland California (1970).
[4] W. Gersch, Spectral analysis of eeg’s by autoregressive decompositionof time series, Mathematical Biosciences 7, 205-222 (1970).
[5] R.H. Shumway, D.S. Stoﬀer, Time Series Analysis and Its Applications,
Springer Verlag, New York (2000).
[6] T. Ozaki, H. Tong, On the ﬁtting of non-stationary autoregressive models in time series analysis, Proceedings of the 8th Hawaii International
Conference on System Sciences, p. 224-226. Western Periodical Co.,
Hawaii (1975).
[7] R.F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inﬂation, Econometrica 50,
987-1008 (1982).
[8] T. Bollerslev, Generalized autoregressive conditional heteroscedasticity,
Journal of Econometrics, 31, 307-327 (1986).

State space GARCH Model on EEG

19[9] A. Galka, O. Yamashita, T. Ozaki. GARCH modelling of covariancein dynamical estimation of inverse solutions, Physics Letters A 333,
261-268 (2004).
[10] E.R. John, L.S. Prichep, W. Kox, P. Valdés-Sosa, J. Bosch-Bayard,
E. Aubert, M. Tom, F. DiMichele, L.D. Gugino, Invariant reversible
QEEG eﬀects of anesthetics, Consciousness and Cognition 10, 165-183(2001).
[11] E.R. John, The neurophysics of consciousness, Brain Research Reviews
39, 1-28 (2002).
[12] L.S. Prichep, L.D. Gugino, E.R. John, R.J. Chabot, B. Howard, H.
Merkin, M.L. Tom, S. Wolter, L. Rausch, W.J. Kox, The patient stateindex as an indicator of the level of hypnosis under general anaesthesia,
British Journal of Anaesthesia 92 (3) 393-399 (2004).
[13] R.E. Kalman. A new approach to linear ﬁltering and prediction problems, Transactions of the ASME Journal of Basic Engineering 82, 35-45(1960).
[14] J. Harrison, C.F. Stevens, Bayesian forecasting (with discussion), Journal of the Royal Statistical Society, Series B 38, 205-247 (1976).
[15] A.C. Harvey, Forecasting, structural time series models and the Kalmanﬁlter, Cambridge University Press, Cambridge (1989).

State space GARCH Model on EEG

20[16] M.S. Grewal, A.P. Andrews, Kalman ﬁltering: Theory and PracticeUsing MATLAB 2nd ed., Wiley, New York (2001).
[17] R.K. Mehra, Identiﬁcation of stochastic linear dynamic systems, American Institute of Aeronautics and Astronautics Journal 9, 28-31 (1971).
[18] K.J. Astrom, C.G. Kallstrom. Application of system identiﬁcation techniques to the determination of ship dynamics, P. Eykhoﬀ (ed), Identiﬁcation and system parameter estimation, p. 415-424. North-Holland,
Amsterdam (1973).
[19] P. Valdés-Sosa, J.C. Jimenez, J. Riera, R. Biscay, T. Ozaki, Nonlinear
EEG Analysis based on a neural Mass Model, Biological Cybernetics
81, 415-424 (1999).

State space GARCH Model on EEG

21

Data
50

0

−50
0

2

4

6

8

10

12

14

16

18

20

6

Power

4
2
0
−2
−4
0
10

20

20

15

30

10

40
50
Frequency/Hz

50Time/sec

Figure 1: (top) An EEG data from the T4 electrode (versus average reference) of about 20 seconds, covers the transition from awake conscious state
to anaesthesia. (bottom) A moving window spectral estimation of AR(8)models ﬁtted to 15 segments of data, each of length 256.

24.4756 Hz

17.5995 Hz

10.269 Hz

2.3893 Hz

State space GARCH Model on EEG

22

50
0
−50
50
0
−50
50
0
−50
50
0
−50

Data

50
0
−50
0

2

4

6

8

10
Time/sec

12

14

16

18

20

Figure 2: (top 4) Estimated source components of 4 diﬀerent frequencies,
scaled to the data space. (bottom) Data.

State space GARCH Model on EEG

232.3893 Hz

1

10

−1

10

−3

10

10.269 Hz

1

10

−1

10

−3

10

17.5995 Hz

1

10

−1

10

−3

10

24.4756 Hz

1

10

−1

10

−3

10

0

2

4

6

8

10
Time/sec

12

14

16

18

20

Figure 3: (top 4) Variance of prediction error of the 4 source components.

State space GARCH Model on EEG

24

Summary
In this paper, we present a new approach to modelling non-stationarity ina time series data. We do it by a generalized state space approach. We candecomposed a given electroencephalogram (EEG) time series into a set ofnoise-driven processes, each corresponding to a diﬀerent frequency band. We
show how the variances of the dynamical noises driving these components
can be made time-dependent by generalising the concept of the Generalized
Autoregressive Conditional Heteroscedastic (GARCH) modelling to the situation of state-space modelling.
Nonstationarity is modelled by allowing the variances of the driving
noises to change with time, depending on the state prediction error within
the state space model. The method is illustrated by an application to EEG
data recorded during the onset of anaesthesia.

Figure1

Data
50

0

−50
0

2

4

6

8

10

12

14

16

18

20

6

Power

4
2
0
−2
−4
0
10

20

20

15

30

10

40
50
Frequency/Hz

50Time/sec

24.4756 Hz

17.5995 Hz

10.269 Hz

2.3893 Hz

Figure2

50
0
−50
50
0
−50
50
0
−50
50
0
−50

Data

50
0
−50
0

2

4

6

8

10
Time/sec

12

14

16

18

20

Figure3

2.3893 Hz

1

10

−1

10

−3

10

10.269 Hz

1

10

−1

10

−3

10

17.5995 Hz

1

10

−1

10

−3

10

24.4756 Hz

1

10

−1

10

−3

10

0

2

4

6

8

10
Time/sec

12

14

16

18

20