J. Biomedical Science and Engineering, 2009, 2, 102-105
Published Online April 2009 in SciRes. http://www.scirp.org/journal/jbise JBiSE
1
The impact of frequency aliasing on spectral
method of measuring T wave alternans*
Di-Hu Chen1, Sheng Yang1
1Department of Precision Machinery & Instrumentation, University of Science and Technology of China, Hefei 230027, China.
Received Jan. 5th, 2009; revised Feb. 19th, 2009; accepted Feb. 20th, 2009.
ABSTRACT
In this paper we investigate frequency aliasing
in spectral method of measuring T wave alter-
nans, which may lead a high false positive rate.
Microvolt T wave alternans(TWA) has been
evaluated as a means of predicting occurrence
of ventricular tachyarrhythmia events and its
association with the genesis of ventricular ar-
rhythmias has been demonstrated. Nowadays,
spectral method is one of the most widely used
procedures for measurement of microvolt TWA.
In our study, based on the sampling theory, the
alternans frequency 0.5 cycles/beat, at which the
power of the spectrum is used to calculated the
Valt and K score (these two parameters indicate
the TWA), is equal to the nyquist frequency.
Thus this generates frequency aliasing which
will make the power at the alternans frequency
(P0.5) be two times of the real magnitude of the
original spectrum amplitude. With the assump-
tion that the noise spectrum follows the normal
distribution, in spectral method of measuring T
wave alternans, the measuring standard K
score>3 to consider the T wave alternans sig-
nificant is only with a p<0.133. By change the
standard to K score>6 can solve this problem
and make the p value to p<0.0027.
Keywords: TWA, Sampling, Spectral method,
Frequency Aliasing
1. INTRODUCTION
Sudden cardiac death (SCD) is the leading cause of car-
diovascular mortality in the developed countries [1].
There is no an effective diagnostic method to identify
patients at high risk for SCD. Though many non-invasive
tests related to high-risk of SCD such as frequent and
complex ventricular arrhythmias in 24-hour Holter
monitoring, ventricular late potentials in signal-average
ECG, low heart rate variability, and increased dispersion
of repolarization are introduced, the positive predictive
value of these tests is too low to consider them as suffi-
cient to make a decision about specific treatment, espe-
cial defibrillator implantation. Recently risk stratification
research has been focused on microvolt T wave alternans,
which is considered as a promising clinical marker of
arrhythmic events [2].
Microvolt T wave alternans (TWA), also called repo-
larization alternans, is a phenomenon appearing in the
electrocardiogram (ECG) as a consistent fluctuation in
the repolarization morphology on an every-other-beat
basis [3]. Microvolt TWA has been evaluated prospec-
tively in a variety of patient populations as a means of
predicting occurrence of ventricular tachyarrhythmia
events and its association with the genesis of ventricular
arrhythmias has been demonstrated [4].
In measuring of TWA, spectral method is a widely
used method. This method uses a certain measurements
taken on corresponding points of some consecutive T
wave to compute a spectrum. And then two parameters:
the alternans voltage (alt V) and alternans ratio (
K
score)
are calculated from this spectrum. These two parameters
indicate whether the TWA is significant.
This paper investigates frequency aliasing (also called
aliasing in short) in spectral method of measuring T
wave alternans, which may lead a high false positive rate.
In section II we take a brief view of the spectral method
of measuring T wave alternans. In section III, we intro-
duce the sampling theory and frequency aliasing and in
section IV, we investigate frequency aliasing in the spec-
tral method.
2. SPECTRAL METHOD OF MEASURING
T WAVE ALTERNANS
Until now, two main techniques have been applied for
measurement of microvolt TWA in clinical setting: fast
Fourier Transform (FFT) spectral method and modified
moving average (MMA) analysis method [4]. The FFT
spectral method which was developed at Massachusetts
Institute of Technology by Dr. Richard J. Cohen [5, 6, 7]
is the most widely used procedure. This technique uses
128 measurements taken on corresponding points of 128
consecutive T waves to compute a spectrum. Each T
wave is measured at the same time relative to the QRS
complex [8]. For the spectrum is created by measure-
*This work was support by the National Natural Science Foundation
of China (60571034)
SciRes Copyright © 2009

D. H. Chen et al. / J. Biomedical Science and Engineering 2 (2009) 102-105 103
SciRes Copyright © 2009 JBiSE
ments taken once per beat, its frequencies are in the units
of cycles/beat. The point on the spectrum corresponding
to exactly 0.5 cycles/beat indicates the level of alterna-
tion of T wave waveform [8].
Two measurements are obtained form the analysis: the
alternans voltage (Valt) and alternans ratio ( K score). The
Valt measured in uV, represents the square root of alter-
nans power which is defined as the difference between
the power at the alternans frequency (0.5 cycles/beat)
and the power at the noise frequency (0.44 and 0.49 cy-
cles/beat). And also, it corresponds to the root mean
square difference in the voltage between the overall
mean beat and either the odd-numbered or even- num-
bered beats. The alternans ratio K score is calculated as
the ratio of the alternans power divided by the standard
deviation of the noise in the reference frequency band.
See bellow
μ
−= 5.0 PValt (1)
σ
μ
−
=5.0
P
scoreK (2)
where P0.5 is the power at 0.5 cycles/beat,
μ
and
σ
are
the mean and standard deviation. When the alternans
power is >3 SD above the noise level (K score>3), alter-
nans is considered significant in statistic.
3. NYQUIST FREQUENCY AND FRE-
QUENCY ALIASING IN MEASURING T
WAVE ALTERNANS
Consider a continuous-time signal
()
tf . We define sam-
pling as the generation of an ordered number sequence
by taking values of
()
tf at specified instants of time
[9]. In most cases continuous-time signals are sampled at
equal increments of time. The sample increment, called
the sample period, is usually denoted as s
T. Therefore,
the sampled signal values available in the computer are
()
s
nTf , where n is an integer.
Figure 1 shows the ideal impulse sampling operation.
This is seen to be a modulation process, in which the
carrier signal
()
t
T
δ
is defined as the train of impulse
function:
()( )
∑
∞
−∞=
−=
n
ST nTtt
δδ (3)
∑
∞
−∞=
−=
n
T
nTtt )()( δδ
() tf
() tf
s
Figure 1. Impulse sampling.
The output of the modulator, denoted by
(
)
tfs is
given by
()()()() ()
∑
∞
−∞=
−==
n
ssTs nTtnTfttftf
δδ (4)
We begin to investigate the characteristics of the sam-
pling operation in Figure 1 by taking the Fourier trans-
form of
(
)
tfs. For Fourier transform, we can easy get
()( )()
∑∑ ∞
−∞=
∞
−∞=
−⎯→←−=
k
ss
F
n
sT knTtt
ωωδωδδ (5)
where
s
sT
π
ω
2
= is the sampling frequency in radi-
ans/second. The sampling frequency in hertz is giver by
S
sT
f1
=; therefore, ss f
π
ω
2
=
. For multiplication in the
time domain results in convolution in the frequency do-
main. Then from (2) and (3),
()()( )
() ()
∑
∑
∞
−∞=
∞
−∞=
−∗=
⎥
⎦
⎤
⎢
⎣
⎡−∗=
k
s
s
k
sss
kF
T
kFF
ωωδω
ωωδωω
π
ω
1
2
1
(6)
where
(
)
ω
F is the Fourier transform of
(
) tf and
(
)
ω
s
F is the Fourier transform of () tfs. Because of the
convolution property of the impulse function,
(
)
(
)( )
ss kFkF
ω
ω
ω
ω
δ
ω
−=
−
∗
(7)
Thus the Fourier transform of the impulse-modulated
signal (2) is given by
() ()
∑
∞
−∞=
−=
k
s
s
skF
T
F ωωω
1 (8)
Frequency domain characteristics of the sampling op-
eration can be derived from this result.
From (6) we see that the effect of sampling
(
)
tf is to
replicate the frequency spectrum of ()
ω
F about the
frequencies K3,2,1, ±
±
±
=
kk s
ω
. This result is show in
Figure 2(b) for the signal of Figure 2(a).
The frequency 2/
s
ω
is called the Nyquist frequency
and the Shannon sampling frequency. One of the re-
quirements for sampling is that the sampling frequency
() ω F
B
ω −
B
ω
ω
() ω
s
F
A
s
T
A
ω
B
ω −
B
ω
s
ω −
s
ω 2−
s
ω
s
ω 2
Bs
ωω +
Bs
ωω −
Figure 2. The frequency spectrum of a sampled signal.

104 D. H. Chen et al. / J. Biomedical Science and Engineering 2 (2009) 102-105
SciRes Copyright © 2009 JBiSE
must be chosen such that Ms
ω
ω
2>, where M ω is the
highest frequency in the frequency spectrum of the signal
to be sampled.
From part 2 we know, in spectral analysis of microvolt
T wave alternans, the sampling frequency is 1 cy-
cles/beat and the alternans frequency is 0.5 cycles/beat,
which is exactly 0.5 of the sampling frequency. This is
also the Nyquist frequency. In sampling theory, in-
put-signal frequencies that exceed the Nyquist frequency
are aliased. That is, they are folded back or replicated at
other positions in the spectrum above and below the
Nyquist frequency. (See Figure 3)
So in spectral method of measuring T wave alternans,
power at the alternans frequency (P0.5) which is used to
indicate the level of alternation of T wave waveform is
two times of the real magnitude of the original spectrum
at 0.5 cycles/beat.
4. DISCUSSION
In spectral method of measuring T wave alternans, as
mentioned in part 2, the T wave alternans is considered
significant when the K score is higher than 3, while the
K score represents the ratio of the alternans power and
the standard noise power deviation. In order to explain
why this rule works, let’s take a look at the normal dis-
tribution.
Figure 3. (a) Input signal frequencies exceed the Nyquist frequency
are aliased.(b) With the frequency aliasing, P0.5 is two times of
the real magnitude.
Figure 4. Probability density function of the normal distribution.
A normal distribution in a variate X with mean
μ
and variance 2 σ is a statistic distribution with probabil-
ity function
()
()
()
2
22/
2
1
σμ
πσ
−−
=x exP (9)
on the domain
(
)
∞
∞
−
∈
,x. The importance of the nor-
mal distribution as a model of quantitative phenomena in
the natural and behavioral sciences is due in part to the
central limit theorem. Many measurements, ranging from
psychological to physical phenomena can be approxi-
mated, to varying degrees, by the normal distribution. In
the spectral method of measuring T wave alternans, the
noise of the power spectrum can be assumed to be nor-
mal. In normal distribution, if
() 3>−
σ
ux, then the x is
statistically significant with the probability (0027.0
<
p)
due to chance. This can be easily calculated via (9) and
this probability is also called the false positive rate.
When there is frequency aliasing, let alt xbe the real
alternans power, and assume that the distribution of the
noise spectrum is normal no matter whether there is T
wave alternans, then
μ
+= alt xP2
5.0 and (2) can be
written as
3
22 >=
−+
=
σσ
μμ
altalt
score
xx
K (10)
This is also 5.1>
σ
alt
x, from the table of the stan-
dard normal distribution we can get that in this condition
the p value is only less than 0.133 (133.0<p), which
means a high false positive rate.
In order to solve this problem, we can change the
standard from 3>
score
K to 6>
score
K. From (10) we
can get 0027.0
<
pif consider T wave alteransn signifi-
cant when 6>
score
K.
5. CONCLUSION
In this paper study, based on the sampling theory, in
spectral method of measuring T wave alternans, the al-
ternans frequency is equal to the nyquist frequency, and
this makes frequency aliasing in the power spectrum,
which will lead the increase of the false positive rate,
from 0027.0
<
pto 133.0
<
p. By changing the standard
from 3>
score
K to 6>
score
K can effectively solve this
problem.
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