ECG P-WAVE SMOOTHING AND DENOISING BY QUADRATIC

VARIATION REDUCTION

Antonio Fasano

1

, Valeria Villani

1,2

, Luca Vollero

1

and Federica Censi

2

1

Universit`a Campus Bio-Medico di Roma, Rome, Italy

2

Department of Technology and Health, Italian National Institute of Health, Rome, Italy

Keywords:

ECG, P-wave, Atrial ﬁbrillation, Smoothing, Denoising, Quadratic variation, Convex optimization.

Abstract:

Atrial ﬁbrillation is the most common persistent cardiac arrhythmia and it is characterized by a disorganized

atrial electrical activity. Its occurrence can be detected, and even predicted, through P-waves time-domain and

morphological analysis in ECG tracings. Given the low signal-to-noise ratio associated to P-waves, such anal-

ysis are possible if noise and artifacts are effectively ﬁltered out from P-waves. In this paper a novel smoothing

and denoising algorithm for P-waves is proposed. The algorithm is solution to a convex optimization problem.

Smoothing and denoising are achieved reducing the quadratic variation of the measured P-waves. Simulation

results conﬁrm the effectiveness of the approach and show that the proposed algorithm is remarkably good

at smoothing and denoising P-waves. The achieved SNR gain exceeds 15 dB for input SNR below 6 dB.

Moreover the proposed algorithm has a computational complexity that is linear in the size of the vector to be

processed. This property makes it suitable also for real-time applications.

1 INTRODUCTION

Atrial ﬁbrillation (AF) is a cardiac arrhythmia charac-

terized by disorganized atrial electrical activity caus-

ing loss of effective contraction. AF is the most com-

mon persistent cardiac arrhythmia and it is also the

most common cause of arrhythmia-related hospital-

izations (Feinberg et al., 1995; Go et al., 2001). It has

an enormous social impact because of its very high

incidence and its clinical consequences. Moreover, it

is often difﬁcult to diagnose and its management is

not optimized. The incidence of AF increases with

age and given the life expectancies increasing in both

developed and developing countries, AF is projected

to become an important burden on most health care

systems (McBride et al., 2008).

AF is the most common arrhythmia in the west-

ern countries, it is responsible for 70 to 100 thousand

strokes per year in the US and is independently asso-

ciated with up to 1.9-fold increase in the risk of death

(Benjamin et al., 1998).

AF is also associated with extensive atrial struc-

tural, contractile, and electrophysiological remodel-

ing, which can sustain AF itself (Nattel et al., 2008).

Current pharmacological treatments of AF present

some limits because they can be ventricular proar-

rhythmic and not able to prevent recurrences of AF.

A great deal of research on AF has focused on the

identiﬁcation of factors that can predict its ﬁrst occur-

rence or recurrence. This could help in deﬁning the

best treatment in individual patients.

Promising results have been obtained from ECG

signal processing, particularly from the analysis of

P-waves. Electrocardiographic characteristics of AF

have proven to be helpful in identifying patients at

risk (Dilaveris and Gialafos, 2002; Platonov et al.,

2000; Carlson et al., 2001; Bayes de Luna et al.,

1999).

Prolonged and highly variable P-waves have been

observed in patients prone to AF. Time-domain and

morphological characteristics of P-waves from sur-

face ECG recordings turned out to signiﬁcantly dis-

tinguish patients at risk of AF (Bayes de Luna et al.,

1999; Perez et al., 2009).

Most of these studies rely on measurements based

on visual inspection; however, computerized auto-

matic analyses can now be performed and have been

recently introduced to estimate P-wave duration and

morphological indices (Censi et al., 2007; Censi et al.,

2008).

Due to the low signal-to-noise ratio associated

with P-waves, this portion of ECG signal is usu-

ally analyzed performing signal averaging in order

to build a P-wave template. Then, waves duration

289

Fasano A., Villani V., Vollero L. and Censi F..

ECG P-WAVE SMOOTHING AND DENOISING BY QUADRATIC VARIATION REDUCTION.

DOI: 10.5220/0003169202890294

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2011), pages 289-294

ISBN: 978-989-8425-35-5

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

and morphological features are extracted from this

template. The unavoidable drawback of this ap-

proach is that some information is lost in the aver-

aging operation. Each P-wave provides important in-

formation about the correspondingdepolarization pat-

tern throughout the atrial substrate. Thus, tracking

changes between consecutive P-waves turns out to be

extremely important in improving the understanding

of the pathophysiological mechanisms of atrial sub-

strates predisposing to AF.

However, the analysis of the temporal variabil-

ity of consecutive P-waves is possible only if reliable

beat-to-beat P-waves are available. This is attainable

only if noise and artifacts are effectively ﬁltered out

from each single P-wave.

The aim of this investigation is to propose a novel

method to smooth and denoise P-waves extracted

from high-resolution DC-coupled ECG recordings, in

order to improve the signal-to-noise ratio (SNR) of

each single P-wave.

2 RATIONALE

From the previous section, it is evident that the abil-

ity to conduct a meaningful analysis of predisposing

factors to AF strongly depends on the availability of

reliable P-waves. In this regard, P-waves are reliable

if the detrimental effects of noise and artifacts are re-

duced to an acceptable level.

In this section we propose an algorithm that is par-

ticularly effective for P-waves smoothing and denois-

ing. It is based on the following idea. The measured

P-wave is affected by noise and artifacts whose ef-

fect is to introduce additional “variability” into the

observed P-wave with respect to the true one. Thus,

provided that we introduce a suitable index of vari-

ability, smoothing and denoising can be performed by

searching for a version of the P-wave that is close, in

some sense, to the observed one, but has less “vari-

ability”. We make this idea precise in the following.

Denote by z

z

z = [z

1

···z

n

]

T

the vector collecting n

samples of a noisy P-wave extracted from an ECG

tracing. The variability of a generic vector can be

quantiﬁed introducing the following

Deﬁnition 1. Given a vector x

x

x = [x

1

···x

n

]

T

∈ R

n

, the

quadratic variation of x

x

x is deﬁned as

[x

x

x]

.

=

n−1

∑

k=1

(x

k

− x

k+1

)

2

(1)

and is denoted by [x

x

x].

The quadratic variation is a well-known property

used in the analysis of stochastic processes (Shreve,

2004). However, in this context we consider it as a

function of deterministic or random vectors.

Introducing the (n− 1) × n matrix

D

D

D =

1 −1 0 . . . 0

0 1 −1

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0

0 . . . 0 1 −1

, (2)

the quadratic variation of x

x

x can be expressed as

[x

x

x] = kD

D

Dx

x

xk

2

, (3)

where k·k denotes the Euclidean norm.

The quadratic variation is a consistent index of

variability and its use is motivated by the following

property. For vectors affected by additive noise, on

average it does not decrease and moreover it is an

increasing function of noise variances. In fact, let

x

x

x = x

x

x

0

+ w

w

w, where x

x

x

0

is a deterministic vector and

w

w

w = [w

1

···w

n

] is a zero-mean random vector with co-

variance matrix K

K

K

w

= E

w

w

ww

w

w

T

. We do not make any

assumption about the distribution of w

w

w, so the follow-

ing considerations hold regardless of it. Computing

the averaged quadratic variation of x

x

x we get

E

n

kD

D

Dx

x

xk

2

o

= kD

D

Dx

x

x

0

k

2

+ E

tr

D

D

Dw

w

ww

w

w

T

D

D

D

T

= kD

D

Dx

x

x

0

k

2

+ tr

D

D

DK

K

K

w

D

D

D

T

(4)

where, in the ﬁrst equality, we have exploited the in-

variance of the trace under cyclic permutations. Note

that tr

D

D

DK

K

K

w

D

D

D

T

≥ 0, since it is the trace of a positive

semideﬁnite matrix (Horn and Johnson, 1990), but in

all practical cases the inequality is strict. In fact, we

have

tr

D

D

DK

K

K

w

D

D

D

T

=

n−1

∑

k=1

E

n

(w

k

− w

k+1

)

2

o

=

n−1

∑

k=1

σ

2

k

+ σ

2

k+1

− 2σ

k,k+1

(5)

where σ

2

k

= E

w

2

k

and σ

k,k+1

= E {w

k

w

k+1

}. From

(5) it follows that tr

D

D

D

T

D

D

DK

K

K

w

= 0 if and only if allthe

components of the noise vector w

w

w are almost surely

equal

1

and that E

n

kD

D

Dx

x

xk

2

o

is an increasing function

of noise variances.

For example, in typical scenarios of ECG trac-

ings w

w

w = m

m

m + a

a

a, where m

m

m is due to white Gaussian

noise whereas a

a

a is due to the residual 50Hz or 60Hz

power-line noise. We may assume m

m

m ∼ N

0

0

0, σ

2

m

I

I

I

and a

a

a = [a

1

···a

n

] vector of samples from a harmonic

process, i.e., a

k

= Acos

h

2π

f

0

F

c

k+ φ

i

, with A and φ

1

That is w

1

= w

2

= · · · = w

n

with probability 1.

BIOSIGNALS 2011 - International Conference on Bio-inspired Systems and Signal Processing

290

independent, φ uniformly distributed in [0, 2π), f

0

∈

{50Hz, 60Hz} and F

c

being the sampling frequency.

Moreover m

m

m and a

a

a are independent. In this case it is

easy to verify that

tr

D

D

DK

K

K

w

D

D

D

T

=

= 2(n− 1)

σ

2

m

+ 2E

A

2

sin

2

π

f

0

F

c

=

2kx

x

x

0

k

2

(n− 1)

n

1

SNR

+

4

SIR

sin

2

π

f

0

F

c

(6)

where SNR =

kx

x

x

0

k

2

nσ

2

m

denotes the signal-to-noise ra-

tio and SIR =

2kx

x

x

0

k

2

nE

{

A

2

}

is the signal-to-interference ra-

tio, considering the power-line noise as interference.

From (6) it is evident that the average quadratic vari-

ation is a decreasing function of SNR and SIR.

In the following section we devise an efﬁcient

smoothing algorithm for P-waves exploiting the con-

cept of quadratic variation.

3 SMOOTHING P-WAVES

In this section we denote by p

p

p the vector collecting

samples from the measured P-wave, the one that is

affected by noise and artifacts, and by x

x

x the corre-

sponding vector after smoothing. Following the line

of reasoning presented in the previous section, we de-

termine x

x

x solving the following optimization problem

minimize kx

x

x− p

p

pk

2

subject to kD

D

Dx

x

xk

2

≤ a

(7)

where D

D

D is deﬁned in (2) and a is a positive con-

stant that controls the degree of smoothness for p

p

p.

Its value is chosen in accordance with the peculiar-

ity of the problem and satisﬁes a < kD

D

Dp

p

pk

2

in order

to avoid trivial solutions.

2

Note that we do not need

to know in advance the appropriate value for a in any

particular problem. In fact, as it will be clear later,

the solution to the optimization problem (7) can be

expressed in terms of a parameter that controls the

degree of smoothness, i.e., the quadratic variation of

the solution, and that is related to the value of a in (7).

In this way, smoothing can be performed without car-

ing about a in the constraint kD

D

Dx

x

xk

2

≤ a, and reducing

parametrically the quadratic variation of the solution

to the desired level. In general, the optimal value for

the controlling parameter can be found, as the one that

entails the maximum SNR gain.

2

When a ≥ kD

D

Dp

p

pk

2

the solution is x

x

x = p

p

p and no smooth-

ing is performed.

Let us consider (7) in more detail. It is a convex

optimization problem, since both the objective func-

tion and the inequality constraint are convex. As a

consequence, any locally optimal point is also glob-

ally optimal and Karush-Kuhn-Tucker (KKT) condi-

tions provide necessary and sufﬁcient conditions for

optimality (Boyd and Vandenberghe, 2004). More-

over, since the objective function is strictly convex

and the problem is feasible the solution exists and is

unique. The Lagrangian is

L (x

x

x, λ) = kx

x

x− p

p

pk

2

+ λ

kD

D

Dx

x

xk

2

− a

(8)

from the KKT conditions we get

∇L (x

x

x, λ) = 2(x

x

x− p

p

p)+ 2λD

D

D

T

D

D

Dx

x

x = 0, (9)

λ

kD

D

Dx

x

xk

2

− a

= 0, (10)

and the nonnegativity of the Lagrange multiplier λ ≥

0. However, if λ = 0 from (9) it follows that x

x

x = p

p

p,

which is infeasible since the inequality constraint is

not satisﬁed. Hence λ > 0 and from (10) it results

that the inequality constraint is active, i.e., kD

D

Dx

x

xk

2

= a.

Thus, solving (9) we get eventually

x

x

x =

I

I

I + λD

D

D

T

D

D

D

−1

p

p

p (11)

where I

I

I denotes the identity matrix, and λ is deter-

mined by

kD

D

Dx

x

xk

2

=

D

D

D

I

I

I + λD

D

D

T

D

D

D

−1

p

p

p

2

= a. (12)

Note that in (11) the inverse exists for any λ ≥

0 and when λ = 0 the solution corresponds to not

smoothing. It is interesting that the solution to (11) is

a linear operator acting on p

p

p. Moreover, the Lagrange

multiplier λ plays the role of a parameter controlling

the degree of smoothing applied to p

p

p. In fact, it is pos-

sible to prove that the quadratic variation of the solu-

tion, namely [x

x

x] = kD

D

Dx

x

xk

2

, is a continuous and strictly

decreasing function of λ for λ ∈ [0, +∞) regardless

of p

p

p, provided that it is not a constant vector.

3

This

is equivalent to say that (12), when p

p

p is not a con-

stant vector, establishes a one-to-one correspondence

between a ∈

0, kD

D

Dp

p

pk

2

i

and λ ∈ [0, +∞), with λ = 0

corresponding to a = kD

D

Dp

p

pk

2

and

lim

λ→+∞

[x

x

x] = 0. (13)

A consequence of this is that we do not need to

know in advance the value of a in (7), as smooth-

ing can be performed according to (11) and λ can be

3

If p

p

p is a constant vector [x

x

x] = [p

p

p] = 0 regardless of λ.

ECG P-WAVE SMOOTHING AND DENOISING BY QUADRATIC VARIATION REDUCTION

291

adapted to fulﬁll some performance criterion. For ex-

ample, considering the SNR gain

4

as a quality index,

λ can be chosen as the one that entails the maximum

gain.

It is important to consider the computational as-

pects related to the smoothing operation, since a ma-

trix inversion is involved in (11). If the size of the

vector p

p

p is large enough computational problems may

arise. Actually this is not an issue for the typical

length of vectors representing P-waves. However,

if the same algorithm is applied to a complete ECG

recording the computational burden, in terms of time

and memory, and the accuracy become serious issues,

even for batch processing.

It is possible to prove that due to the special struc-

ture of the matrix I

I

I + λD

D

D

T

D

D

D, which is tridiagonal,

smoothing in (11) can be performed with complex-

ity O(n), i.e., linear in the size of vector p

p

p (Golub and

Van Loan, 1996). This property is very important and

makes the proposed algorithm suitable also for real-

time applications.

Eventually, it is worthwhile noting that the algo-

rithm we propose is not limited to P-wave smoothing,

but it can be applied in very general situations, when-

ever smoothing and/or denoising are needed. This is

due to the fact that the formulation and the rationale

behind it, i.e., quadratic variation reduction, havegen-

eral validity. In this regard, we successfully applied it

also to EEG tracings denoising.

4 SIMULATION RESULTS

In order to evaluate performance of the proposed al-

gorithm, a noiseless reference model of P-wave is

needed. In this regard, we considered the P-wave

model reported in (Censi et al., 2007), obtained ﬁtting

linear combinations of Gaussian functions to mea-

sured P-waves. Such a model can be reliably used

to represent real P-waves as documented in the cited

reference.

The noiseless reference model considered is re-

ported in Figure 2. It has a duration of 200ms and

its bandwidth essentially does not exceed 100Hz. It

has been sampled at 2048Hz and the corresponding

samples have been collected in a vector denoted by

p

p

p

0

.

The noiseless P-wave reference model p

p

p

0

has been

corrupted by additive noise, denoted by w

w

w, where the

components of w

w

w are i.i.d.

5

zero mean Gaussian ran-

4

This is the ratio between the SNR after and before

smoothing.

5

Independent identically distributed.

dom variables with variance σ

2

w

. Thus, the corre-

sponding noisy P-wave is

p

p

p = p

p

p

0

+w

w

w. (14)

In order to assess performance of the proposed algo-

rithm the following quantities have been considered:

- the signal-to-noise ratio before smoothing

SNR =

kp

p

p

0

k

2

n· σ

2

w

(15)

where n is the size of vector p

p

p

0

;

- the signal-to-noise ratio after smoothing

SNR

s

=

kp

p

p

0

k

2

kx

x

x− p

p

p

0

k

2

(16)

where (x

x

x− p

p

p

0

) is the error vector with respect to

the reference model p

p

p

0

.

In deﬁnition (16), we consider as noise affecting the

smoothed vector x

x

x, both the residual Gaussian noise

and the reconstruction error.

Performances are measured in terms of SNR gain,

deﬁned as

G

snr

=

SNR

s

SNR

=

n· σ

2

w

kx

x

x− p

p

p

0

k

2

. (17)

Simulations have been carried out applying smooth-

ing in accordance with (11) and choosing for λ the

value, denoted by λ

opt

, that entails the maximumSNR

gain.

In Figure 2 the noiseless P-wave reference model

p

p

p

0

(dashed-line)is reported. Figure 1 showsthe corre-

sponding noisy version p

p

p, corrupted by additive noise

according to (14), with SNR = 0dB. Note that the

SNR in this case is quite low, nevertheless the pro-

posed algorithm is very effective in denosing p

p

p and

the resulting smoothed vector, namely x

x

x, is plotted

in Figure 2 (continuous-line) for an easy compari-

son with the reference model. In this case the re-

sulting SNR gain is quite remarkable and amounts to

G

snr

= 19.4dB. It is important to point out that we

evaluated the proposed algorithm on different models

of P-wave and the resulting gains were all consistent

with the ones reported in this work.

In order to evaluate how gain varies as input SNR

changes, in Figure 3 we report the average SNR gain

versus input SNR (bottom axis), when the reference

P-wave model of Figure 2 is corrupted by additive

noise with SNR ranging from −20dB to 40dB. The

top axis of Figure 3 represents the corresponding in-

band SNR, computed using the 100Hz bandwidth of

the P-wave reference model. The in-band SNR is

BIOSIGNALS 2011 - International Conference on Bio-inspired Systems and Signal Processing

292

about

6

10.1dB greater than the corresponding input

SNR.

For each input SNR we considered the solutions

with λ = λ

opt

for 300 noise realizations and then we

averaged the corresponding SNR gains. As Figure 3

highlights, the proposed algorithm exhibits a remark-

able ability in smoothing P-waves. It achieves consid-

erable gains over the whole range of practical input

SNRs and for input SNR ≤ 6dB the average gain ex-

ceeds 15dB. It is remarkable that even when the SNR

is quite high the algorithm exhibits considerable gain.

It is worth noting that the proposed algorithm is

able to reject both out-of-band and in-band noise. In

this regard, low-pass ﬁltering cannot reject in-band

noise without altering the signal. Indeed, an ideal

100Hz low-pass ﬁlter, in the same setting of our simu-

lation, would exhibit a constant average gain of about

10.1dB over the whole range of input SNR, as a re-

sult of rejection of the sole out-of-band noise. This

is conﬁrmed by simulation where we considered a

linear-phase FIR low-pass ﬁlter synthesized applying

the window method (Oppenheim et al., 1999) to an

ideal 100Hz low-pass ﬁlter, using a Kaiser window

and requiring 0.1dB ripple in passband and 80dB at-

tenuation in stopband. In Figure 3 we report the ﬁlter

SNR gain versus input SNR, averaged over the same

300 noise realizations used before.

Figure 3 highlights the effectiveness of the pro-

posed algorithm and shows, in particular considering

the in-band SNR, that it outperforms low-pass ﬁlter-

ing for all practical values of SNR.

0 50 100 150 200

−0.15

−0.10

−0.05

0

0.05

0.10

0.15

0.20

0.25

Time [ms]

Amplitude [mV]

Figure 1: P-wave reference model of Figure 2 corrupted by

additive Gaussian noise with SNR = 0dB.

6

Actually it is 10log(10.24)dB ≈ 10.1dB, where 10.24

is the ratio between half of the sampling frequency and

bandwidth of the P-wave reference model.

0 50 100 150 200

−0.15

−0.10

−0.05

0

0.05

0.10

0.15

0.20

0.25

Time [ms]

Amplitude [mV]

p

0

x

Figure 2: P-wave reference model p

p

p

0

(dashed-line) and

smoothed solution x

x

x (continuous-line) from the noisy P-

wave of Figure 1.

−20 −10 0 10 20 30 40

5

10

15

20

Input SNR [dB]

Average G

snr

[dB]

Proposed

algorithm

Low−pass

filter

−9.9 0.1 10.1 20.1 30.1 40.1 50.1

In−band SNR [dB]

Figure 3: Average SNR gain G

snr

versus input SNR (bottom

axis) and in-band SNR (top axis).

5 CONCLUSIONS

This work has been motivated by the need to have re-

liable beat-to-beat P-waves, as tracking changes be-

tween consecutive waves turns out to be very impor-

tant in understanding the mechanisms underlying AF.

This is attainable only if noise and artifacts are ﬁltered

out effectively from each single P-wave.

To solve this problem we have developed a

smoothing and denoising algorithm. It is based on

the notion of quadratic variation meant as a suitable

index of variability for vectors or sampled functions.

The algorithm is the closed-form solution to a con-

strained convexoptimization problem, where smooth-

ing and denoising are achieved by reducing the

quadratic variation of noisy P-waves. The compu-

tational complexity of the algorithm is linear in the

ECG P-WAVE SMOOTHING AND DENOISING BY QUADRATIC VARIATION REDUCTION

293

size of the vector to be processed, and this makes it

suitable also for real-time applications. Simulation

results conﬁrm the effectiveness of the approach and

highlight a remarkable ability to smooth and denoise

P-waves.

Eventually, it is worthwhile noting that the pro-

posed algorithm can be effectively applied to a wider

range of signals, e.g., whole ECG or EEG tracings,

whenever smoothing and/or denoising are needed.

REFERENCES

Bayes de Luna, A., Guindo, J., Vinolas, X., Martinez-

Rubio, A., Oter, R., and Bayes-Genis, A. (1999).

Third-degree inter-atrial block and supraventricular

tachyarrhythmias. Europace, 1:43–46.

Benjamin, E. J., Wolf, P. A., D’Agostino, R. B., Silbershatz,

H., Kannel, W. B., and Levy, D. (1998). Impact of

atrial ﬁbrillation on the risk of death: the Framingham

Heart Study. Circulation, 98:946–952.

Boyd, S. and Vandenberghe, L. (2004). Convex Optimiza-

tion. Cambridge University Press.

Carlson, J., Johansson, R., and Olsson, S. B. (2001). Classi-

ﬁcation of electrocardiographic P-wave morphology.

IEEE Trans Biomed Eng, 48:401–405.

Censi, F., Calcagnini, G., Ricci, C., Ricci, R. P., Santini,

M., Grammatico, A., and Bartolini, P. (2007). P-

wave morphology assessment by a gaussian functions-

based model in atrial ﬁbrillation patients. IEEE Trans

Biomed Eng, 54:663–672.

Censi, F., Ricci, C., Calcagnini, G., Triventi, M., Ricci,

R. P., Santini, M., and Bartolini, P. (2008). Time-

domain and morphological analysis of the P-wave.

Part I: Technical aspects for automatic quantiﬁca-

tion of P-wave features. Pacing Clin Electrophysiol,

31:874–883.

Clifford, G. D., Azuaje, F., and McSharry, P. (2006). Ad-

vanced Methods And Tools for ECG Data Analysis.

Artech House, Inc., Norwood, MA, USA.

Dilaveris, P. E. and Gialafos, J. E. (2002). Future concepts

in P wave morphological analyses. Card Electrophys-

iol Rev, 6:221–224.

Feinberg, W. M., Blackshear, J. L., Laupacis, A., Kron-

mal, R., and Hart, R. G. (1995). Prevalence, age

distribution, and gender of patients with atrial ﬁbril-

lation. Analysis and implications. Arch. Intern. Med.,

155:469–473.

Go, A. S., Hylek, E. M., Phillips, K. A., Chang, Y., Henault,

L. E., Selby, J. V., and Singer, D. E. (2001). Preva-

lence of diagnosed atrial ﬁbrillation in adults: na-

tional implications for rhythm management and stroke

prevention: the AnTicoagulation and Risk Factors in

Atrial Fibrillation (ATRIA) Study. JAMA, 285:2370–

2375.

Golub, G. H. and Van Loan, C. F. (1996). Matrix compu-

tations. Johns Hopkins University Press, Baltimore,

MD, USA, 3 edition.

Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis.

Cambridge University Press.

McBride, D., Mattenklotz, A. M., Willich, S. N., and

Bruggenjurgen, B. (2008). The Costs of Care in Atrial

Fibrillation and the Effect of Treatment Modalities in

Germany. Value Health.

Nattel, S., Burstein, B., and Dobrev, D. (2008). Atrial re-

modeling and atrial ﬁbrillation: mechanisms and im-

plications. Circ Arrhythm Electrophysiol, 1(1):62–73.

Oppenheim, A. V., Schafer, R. W., and Buck, J. R. (1999).

Discrete-time signal processing (2nd ed.). Prentice-

Hall, Inc.

Perez, M. V., Dewey, F. E., Marcus, R., Ashley, E. A.,

Al-Ahmad, A. A., Wang, P. J., and Froelicher, V. F.

(2009). Electrocardiographic predictors of atrial ﬁb-

rillation. Am. Heart J., 158:622–628.

Platonov, P. G., Carlson, J., Ingemansson, M. P., Roijer,

A., Hansson, A., Chireikin, L. V., and Olsson, S. B.

(2000). Detection of inter-atrial conduction defects

with unﬁltered signal-averaged P-wave ECG in pa-

tients with lone atrial ﬁbrillation. Europace, 2:32–41.

Shreve, S. E. (2004). Stochastic Calculus for Finance II:

Continuous-Time Models. Springer Finance. Springer

Science+Business Media, Inc.

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