An adaptive Kalman filter for ECG signal enhancement
Citation for published version (APA):
Vullings, R., Vries, de, B., & Bergmans, J. W. M. (2011). An adaptive Kalman filter for ECG signal enhancement.
IEEE Transactions on Biomedical Engineering
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58
(4), 1094-1103. https://doi.org/10.1109/TBME.2010.2099229
DOI:
10.1109/TBME.2010.2099229
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1094 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 4, APRIL 2011
An Adaptive Kalman Filter for ECG
Signal Enhancement
Rik Vullings*, Bert de Vries, and Jan W. M. Bergmans
Abstract—The ongoing trend of ECG monitoring techniques to
become more ambulatory and less obtrusive generally comes at the
expense of decreased signal quality. To enhance this quality, consec-
utive ECG complexes can be averaged triggered on the heartbeat,
exploiting the quasi-periodicity of the ECG. However, this averag-
ing constitutes a tradeoff between improvement of the SNR and
loss of clinically relevant physiological signal dynamics. Using a
Bayesian framework, in this paper, a sequential averaging filter is
developed that, in essence, adaptively varies the number of com-
plexes included in the averaging based on the characteristics of
the ECG signal. The filter has the form of an adaptive Kalman fil-
ter. The adaptive estimation of the process and measurement noise
covariances is performed by maximizing the Bayesian evidence
function of the sequential ECG estimation and by exploiting the
spatial correlation between several simultaneously recorded ECG
signals, respectively. The noise covariance estimates thus obtained
render the filter capable of ascribing more weight to newly arriv-
ing data when these data contain morphological variability, and of
reducing this weight in cases of no morphological variability. The
filter is evaluated by applying it to a variety of ECG signals. To
gauge the relevance of the adaptive noise-covariance estimation,
the performance of the filter is compared to that of a Kalman filter
with fixed, (a posteriori) optimized noise covariance. This compar-
ison demonstrates that, without using aprioriknowledge on signal
characteristics, the filter with adaptive noise estimation performs
similar to the filter with optimized fixed noise covariance, favoring
the adaptive filter in cases where no aprioriinformation is available
or where signal characteristics are expected to fluctuate.
Index Terms—Electrocardiography, Kalman filter, noise
estimation.
I. INTRODUCTION
M
ONITORING and analysis of the ECG has long been
used in clinical practice. In recent years, the application
field of ECG monitoring is expanding to areas outside the clinic.
An example of such an area is at-home monitoring of patients
with sleep apnea [1]. Also within the clinic, a transition in ECG
monitoring applications is taking place. With developments in
sensor technology (e.g., textile electrodes and capacitive elec-
Manuscript received August 26, 2010; revised November 2, 2010; accepted
December 1, 2010. Date of publication December 13, 2010; date of current
version March 18, 2011. This work was supported by the Dutch Technology
Foundation STW. Asterisk indicates corresponding author.
*R. Vullings is with the Department of Electrical Engineering, Eindhoven
University of Technology, Eindhoven, 5600 MB, The Netherlands (e-mail:
r.vullings@tue.nl).
B. de Vries and J. W. M. Bergmans are with the Department of Electrical
Engineering, Eindhoven University of Technology, Eindhoven, 5600 MB, The
Netherlands (e-mail: b.de.vries@tue.n; j.w.m.bergmans@tue.nl).
Digital Object Identifier 10.1109/TBME.2010.2099229
trodes), sensors that are incorporated in garments or the matrass
of an incubator [2] have become available.
As a result of these new sensor technologies, the comfort
of the patient is improving progressively. Whereas some years
ago the patient had to reconcile himself or herself with the dis-
comforts of the only available technology, nowadays patients
prefer the more comfortable ways of recording the ECG. How-
ever, in most cases, this increased comfort comes at the expense
of signal quality. Electrodes that are incorporated in garments
generally provide signals with a lower SNR and more artifacts
than contact electrodes that are glued to the body [3]. Another
example of ECG signals with a typically low SNR is fetal ECG
signals, either recorded invasively after membrane rupture [4]
or noninvasively from the maternal abdomen [5].
Some of the SNR and artifact problems that arise during these
recordings can be suppressed by simple, frequency-selective fil-
tering [5]–[7]. However, due to the partial overlap of signal
and noise bandwidths [8], [9], this frequency-selective filtering
can only help to some extent. Further improvement of the ECG
can be achieved by exploiting its (quasi-)periodicity. Consecu-
tive ECG complexes resemble one another and are, moreover,
in general uncorrelated with noise and artifacts. Hence, by av-
eraging several consecutive ECG complexes, the SNR can be
improved. For additive Gaussian noise, this improvement is di-
rectly related to the number of ECG complexes included in the
average [10]. The drawback of averaging multiple consecutive
ECG complexes is that, besides noise, also the physiological dy-
namics of the ECG are suppressed. That is, changes in the ECG
that originate from physiological events—for instance, changes
in the ST segment that might be associated to metabolic acido-
sis [11]—are suppressed in the averaging, complicating clinical
diagnosis.
From this, it is clear that the averaging of ECG complexes
entails a tradeoff between the pursued increase in SNR and the
time scale at which physiologically relevant changes in ECG
morphology are expected to occur. Hence, for each specific ap-
plication, the number of complexes n included in the averaging
needs to be reconsidered. If it were possible, however, to dy-
namically adapt the number of complexes in the average, based
on newly arriving data, the problem of selecting a proper value
for n could potentially be overcome. In this paper, we develop
a filter that can do exactly this.
The filter is derived using a Bayesian framework and con-
stitutes a Kalman filter in which the dynamic variations in the
ECG are modeled by a covariance matrix that is adaptively es-
timated every time new data arrive. In contrast to filters that
filter the ECG by modeling it by parametric functions [12], the
presented filter uses the actual recorded ECG as basis and infers
0018-9294/$26.00 © 2011 IEEE
VULLINGS et al.: ADAPTIVE KALMAN FILTER FOR ECG SIGNAL ENHANCEMENT 1095
Fig. 1. (a) Illustration of the state-space model that describes the evolution of
the ECG over time. The evolution of the state vectors is indicated by the dotted
box. (b) Illustration of the measurement noise estimation.
whether this ECG is corrupted by noise or dynamic variations.
As a result, unanticipated physiological anomalies in the ECG,
which cannot be easily captured by simple parametric functions,
can be accurately modeled. For parametric functions, to capture
such physiological anomalies, large families of analytical func-
tions or many function parameters need to be considered, both
inherently slowing down the filter process.
The derivation of this filter is provided in Section II. The
ECG dataset, on which the filter is evaluated, is discussed in
Section III, and the results of this evaluation are provided in
Section IV. Finally, discussion and conclusions are provided in
Sections V and VI, respectively.
II. D
ERIVATION OF ADAPTIVE KALMAN FILTER
A. Bayesian Model
Typically, ECG complexes that originate from consecutive
heartbeats are very similar but not identical. Moreover, when
recording the ECG, the signals are corrupted to some extent
by noise and artifacts. In a simplified form, both the relation
between consecutive ECG complexes and the corruption of the
recorded ECG can be described by means of a state-space model
[see also Fig. 1(a)] as follows:
x
k+1
= x
k
+ v
k
y
k+1
= x
k+1
+ w
k+1
(1)
where x
k
denotes the [T ×1] ECG complex for heartbeat k and
y
k
denotes the [T ×1] recorded signal where T is the length of
the ECG complex. The isolation of individual ECG complexes
from the recorded signals is discussed in Section III-C. Also
in this section, the choice for T and the implicit assumption
of equal lengths for all ECG complexes is discussed. The evo-
lution of the ECG complexes between heartbeats is modeled
by the [T ×1] stochastic component v
k
(also referred to as the
process noise). The measurement noise, i.e., corrupting signals,
such as electromyographic signals, movement artifacts, and in-
terferences from the powerline grid, is represented by the [T ×1]
vector w
k
.
When critically assessing (1) and Fig. 1, it is clear that based
on the state-space model alone, no clear distinction between the
process noise v
k
and the measurement noise w
k
can be made.
Therefore, a separate model [illustrated in Fig. 1(b)] is used
for estimating the measurement noise. In this model, the spa-
tial correlation between ECG signals recorded simultaneously
at different locations is exploited. This spatial correlation ren-
ders it possible to approximate a particular ECG signal by the
combination of the other, simultaneously recorded ECG signals.
The part of the ECG signal that cannot be approximated by the
combination of the other signals is subsequently assumed to be
measurement noise. The estimation of the measurement noise
will be discussed in more detail in Section II-B. With regard to
the process noise, v
k
is assumed to be zero mean with adaptive
covariance Λ
k
. Similarly, the measurement noise w
k
is assumed
to be zero mean with covariance Σ
k
. The assumption of zero
mean for both v
k
and w
k
can be justified by high-pass filtering
the ECG signals, as will be described in Section III-B.
In the state-space description of (1), the problem of enhanc-
ing the SNR of the ECG is reduced to the problem of sequen-
tially estimating the model parameter vector x
k
and the noise
covariances Σ
k
and Λ
k
. Here, sequential estimation refers to
the estimation of the relevant parameters based on the earlier
estimate and all newly arriving data.
B. Estimation of Measurement Noise
When recording several ECG signals simultaneously, these
signals are spatially correlated to some extent. Specifically,
the electrical activity of the heart can be modeled as a time-
dependent dipole that is variable in both amplitude and (3-D)
orientation. In this model, each ECG signal constitutes the pro-
jection of the electrical field generated by this dipole onto the
vector that describes the position of the recording electrode.
Hence, each ECG signal can be constructed from the linear
combination of three independent ECG signals [13]. For M
recorded ECG signals, this means that the ECG signal x
i
can
be modeled [see also Fig. 1(b)] as follows:
x
i
= X
−i
γ (2)
where X
−i
is a [T × (M − 1)] matrix, of which the M ECG
signals x
j
constitute the column vectors, and for which the ith
column is missing. The [(M −1)×1] vector γ comprises the
coefficients of the linear combination. The index k that denotes
the heartbeat in (1) is omitted from this description for clarity.
With the adopted dipole model of the heart’s electrical ac-
tivity, it can be argued that dynamical variations in the ECG
morphology are reflected in all recorded ECG signals y. Anal-
ogously, measurement noise w that does not exhibit the same
spatial correlation as the ECG is suppressed in the linear com-
bination of ECG signals. As a result, the measurement noise
vector w
i
for ECG signal i can be approximated by
ˆ
w
i
using
the estimate
ˆ
y
i
= Y
−i
γ as follows:
ˆ
w
i
= y
i
−
ˆ
y
i
(3)
also yielding an estimate for the measurement noise covariance
Σ.
1096 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 4, APRIL 2011
Fig. 2. Example of the estimation of the measurement noise in an ECG com-
plex obtained from an eight-channel, noninvasive fetal ECG recording (see
Section III). The solid line represents the recorded ECG complex and the “·-”
line represents the estimate of this ECG complex obtained by the linear com-
bination of the seven simultaneously recorded signals. The differential signal,
represented by the dotted line, constitutes an approximation of the measure-
ment noise. Note that for clarity, the measurement noise signal is depicted with
a vertical offset. The scalings of all signals are the same.
The estimates
ˆ
γ that minimize the mean-squared error (MSE)
between y
i
and its estimate
ˆ
y
i
= Y
−i
ˆ
γ can be determined by
the following:
ˆ
γ =
Y
T
−i
Y
−i
−1
Y
T
−i
y
i
. (4)
The matrix inversion in (4) exists in case the column vectors
in Y
−i
are linearly independent [14]. This condition of linear
independence is satisfied in the case of ECG signals, for one
due to the fact that each column vector is corrupted by indepen-
dent, additive noise. The estimation of the measurement noise
is illustrated in Fig. 2.
The main limitation of this method for estimating the mea-
surement noise is that, at any time, at least four ECG signals
have to be recorded: three independent ones to estimate the
fourth. For most cases exemplified in Section I, however, the
recording of multiple ECG signals is the standard procedure,
and hence, the requirement for more than three signals does not
impose a serious restriction to the applicability of the proposed
SNR enhancement method.
C. Kalman Filter for Parameter Estimation
The uncertainty in the state-space model of (1) and in the
associated noise parameters suggests the use of a probabilistic
approach for solving the parameter estimation problem [15].
In addition, the sequential nature of the estimation problem
motivates the use of a Bayesian framework in which the prior
probability distribution assigned to the unknown parameters is
updated every time new data arrive. Here, again, sequential
refers to the estimation of model parameters based on earlier
parameter estimates and on newly arriving data.
Using Bayes’ rule, the solution to the parameter estimation
problem can be described as follows:
p (x
k+1
|y
k+1
, Λ
k
, Σ
k
)
=
p (y
k+1
|x
k+1
, Λ
k
, Σ
k
) p (x
k+1
|y
k
, Λ
k
, Σ
k
)
p (y
k+1
|y
k
, Λ
k
, Σ
k
)
. (5)
The conditional probability density function p(x
k+1
|y
k+1
) is
referred to as the posterior. Since it contains all statistical in-
formation about x
k+1
, this posterior constitutes the complete
solution to the parameter estimation problem [16]. The proba-
bility density functions on the right-hand side of (5) are referred
to as the likelihood and the prior, respectively, for the numerator
and as the evidence for the denominator.
By assuming the prior and likelihood to be Gaussian, the pos-
terior and evidence are necessarily Gaussian as well. The use of
Gaussian approximations is dictated by the fact that they ren-
der the posterior describable by a limited number of parameters
and, as such, enable the estimation of the ECG in a maximum
a posteriori (MAP) fashion [15]. For applications in which the
posterior is expected to be multimodal (i.e., a function with
several peaks), a combination of Gaussians can be used, each
describing a different mode of the posterior. The fact that, here,
the posterior is assumed as a single Gaussian, implies that the
parameter vector estimate
ˆ
x
k+1
and its associated covariance
Ψ
k+1
together completely describe the posterior probability
density function and can be inferred analytically. Hence, using
(5) and the assumptions in the state-space model, the posterior
is given by [15] the following:
N (x
k+1
|
ˆ
x
k+1
, Ψ
k+1
)
=
N (y
k+1
|x
k+1
, Σ
k+1
) N (x
k+1
|
ˆ
x
k
, Ψ
k
+ Λ
k
)
N (y
k+1
|
ˆ
x
k
, Ψ
k
+ Λ
k
+ Σ
k+1
)
(6)
where N(x|y, z) denotes a Gaussian probability distribution for
x with mean y and covariance z.
By rewriting (6), the optimal Bayes estimate
ˆ
x
k+1
and its
variance Ψ
k+1
can be sequentially updated according to
ˆ
x
k+1
=
ˆ
x
k
+ K
k+1
(y
k+1
−
ˆ
x
k
) (7)
Ψ
k+1
= Ψ
k
+ Λ
k
− K
k+1
(Ψ
k
+ Λ
k
) (8)
where K
k+1
is known as the Kalman gain [17]
K
k+1
=
Ψ
k
+ Λ
k
Σ
k+1
+ Ψ
k
+ Λ
k
. (9)
Together, (7)–(9) constitute the Kalman filter equations.
D. Adaptive Process Noise Covariance Estimation
A limitation of the derived Kalman filter is its implicit as-
sumption of known aprioristatistics for the measurement noise
w
k
and process noise v
k
. Moreover, in the ECG monitoring
applications for which the filter is intended, the noise statis-
tics are expected to be nonstationary and, hence, any choice for
particular noise covariances potentially leads to large estimation
errors [18]. These estimation errors can nonetheless be restricted
by including a sequential estimation of the noise statistics in the
Kalman filter equations.
The estimation of the measurement noise statistics has been
discussed in Section II-B. The discussion in this section is hence
limited to the estimation of the process noise covariance Λ
k
.
VULLINGS et al.: ADAPTIVE KALMAN FILTER FOR ECG SIGNAL ENHANCEMENT 1097
Again using Bayes’ rule, the conditional probability density
function for Λ
k
, given the recorded signal y
k+1
is given by
p (Λ
k
|y
k+1
, Σ
k
)
=
p (y
k+1
|y
k
, Λ
k
, Σ
k
)
p (y
k+1
|y
k
)
p (Λ
k
|y
k
, Σ
k
) . (10)
It can be noted here that the likelihood of the noise covariance
p (y
k+1
|y
k
, Λ
k
, Σ
k
) is identical to the evidence function in the
parameter estimation level of (5). Hence, maximizing the evi-
dence function in this parameter estimation level is analogous
to maximizing the likelihood of Λ
k
for newly arriving data.
Maximization of the evidence function, however, yields that the
estimated noise covariance constitutes the maximum likelihood
(ML) estimate instead of the MAP estimate, implying the as-
sumption of no knowledge of the prior at the noise estimation
level [15].
When defining the model residual to be
ρ
k+1
= y
k+1
− E [y
k+1
|y
k
, Λ
k
, Σ
k
]
= y
k+1
−
ˆ
x
k
(11)
it can easily be calculated that E[ρ
k+1
|y
k
]=0 and
E[ρ
k+1
ρ
T
k+1
|y
k
]=Ψ
k
+ Λ
k
+ Σ
k+1
. Since in addition
E[ρ
T
k
ρ
l
|y
k
]=0, it follows that
p
ρ
k+1
=
exp
−
1
2
ρ
T
k+1
(Ψ
k
+ Λ
k
+ Σ
k+1
)
−1
ρ
k+1
(2π)
T/2
|Ψ
k
+ Λ
k
+ Σ
k+1
|
1/2
is equivalent to the evidence function at the parameter estimation
level given in (6). Hence, by maximizing p(ρ
k+1
) with respect
to the process noise covariance Λ
k
, the ML estimates for this
covariance can be obtained.
The maximization of p(ρ
k+1
) can be simplified, if we re-
turn to the intended purpose of the Kalman filter, to adaptively
vary the number of averages n used in the enhancement of the
ECG complexes, depending on the dynamic variations in sig-
nal morphology. From (7), it can be inferred that this purpose
means that the Kalman gain K
k
can be simplified to a scalar
matrix (i.e., a diagonal matrix with all entries equal), or even
a scalar. Specifically, by varying the scalar value of K
k
in (7),
either more or less weight can be ascribed to the newly arriving
ECG complex y
k+1
. In other words, the relative contribution of
preceding ECG complexes to the estimate
ˆ
x
k+1
varies with the
value of K
k
, essentially similar to adaptation of the number of
averages used. The scalar value for K
k
here ensures that all T
samples in y
k+1
and all T samples in
ˆ
x
k
are assigned the same
weight (K
k
for y
k+1
and (1 − K
k
) for
ˆ
x
k
), preventing distor-
tion of the ECG complexes. With the assumption of the Kalman
gain being a scalar matrix, from (9), it then follows that also Ψ
k
,
Λ
k
, and Σ
k
can be assumed scalar matrices (i.e., ψ
2
k
I, λ
2
k
I, and
σ
2
k
I, respectively, with I the [T ×T] identity matrix I), implic-
itly also assuming that both the measurement and process noise
are spatially uncorrelated. The effect of the latter assumptions
will be discussed in Section V. With the simplification of scalar
matrices, not only can each of the scalar covariance matrices
be regarded as the matrix representation of the variances of the
vectors x
k
, v
k
, and w
k
, but also does the maximization of (the
Fig. 3. Illustration of the algorithmic implementation of the developed adap-
tive Kalman filter.
logarithm of) p(ρ
k+1
) reduce to the derivative of ln p(ρ
k+1
)
with respect to λ
2
k
equated to zero
∂
∂λ
2
k
ln p
ρ
k+1
=
1
2
tr
ρ
T
k+1
ψ
2
k
I + λ
2
k
I + σ
2
k+1
I
−2
ρ
k+1
−
1
2
tr
ψ
2
k
I + λ
2
k
I + σ
2
k+1
I
−1
=0 (12)
where tr[·] denotes the trace of the matrix. The use of ln p(ρ
k+1
)
instead of the use of p(ρ
k+1
) is justified by the monotonic
behavior of the logarithm function.
Solving (12) for λ
2
k
yields an estimate for the process noise
covariance as follows:
ˆ
λ
2
k
=
1
T
ρ
T
k+1
ρ
k+1
− ψ
2
k
− σ
2
k+1
. (13)
By computing the second derivative of p(ρ
k+1
), it is straight-
forward to prove that this result indeed corresponds to a global
maximum in p(ρ
k+1
). In case the model errors
1
T
ρ
T
k+1
ρ
k+1
are
smaller than what the theoretical value of the measurement noise
σ
2
k+1
predicts, no additional process noise input is required. This
leads to the estimator as follows:
λ
2
k
=
⎧
⎨
⎩
1
T
ρ
T
k+1
ρ
k+1
− ψ
2
k
− σ
2
k+1
, if positive,
0, otherwise.
(14)
The operation of the filter can be explained as follows. In case
the model error
1
T
ρ
T
k+1
ρ
k+1
is larger than what its theoretical
value σ
2
k+1
predicts, λ
2
k
increases and this in turn leads to an
increase in the Kalman gain. Hence, more emphasis is put on
newly arriving data [15]. To improve the robustness and sta-
tistical significance of the estimator of (15), instead of a single
residual ρ
k+1
, the sample mean of N residuals will be used [18].
The effect of the chosen value for N will be evaluated in
Section IV.
Implementation of the aforedescribed methods in an algo-
rithm constitutes the sequential execution of (4) and (3) to esti-
mate the measurement noise covariance and, subsequently, (9),
(7), (8), (11), and (14) for estimation of the ECG signals and
process noise covariances. The algorithm is illustrated schemat-
ically in Fig. 3.
![Fig. 7. (a) Top panel shows a fetal ECG signal recorded from the maternal abdomen, before filtering, but after preprocessing (top graph), after filtering with the adaptive Kalman filter (center graph), and after filtering with the fixed Kalman filter (bottom graph). The ECG complexes indicated in the rectangles are shown zoomed in in the accompanying graphs. The second panel shows the estimated process noise covariance λ2 for the adaptive Kalman filter (solid line), the process noise covariance for the fixed Kalman filter (dash-dot line), and the estimated measurement noise covariance σ2 (dotted line). The bottom panel shows the Kalman gain K for both the adaptive (solid line) and fixed (dash-dot line) filters. For the estimation of λ2 and K in the adaptive Kalman filter, N is chosen equal to 10. Since the process noise covariance is often estimated as 0 [see (15)], λ2 cannot be expressed in decibel as in Fig. 6. Hence, λ2 is expressed here in an absolute sense (i.e., in V2 ). In (b), a zoom of the top panel between 42 and 55 s is depicted.](/figures/fig-7-a-top-panel-shows-a-fetal-ecg-signal-recorded-from-the-3qcmw6aw.webp)
