IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 47, NO. 7, JULY 2000 967
Wavelet Time Entropy, T Wave Morphology and
Myocardial Ischemia
Daniel Lemire*, Chantal Pharand, Jean-Claude Rajaonah,
Bruce Dubé, and A. Robert LeBlanc
Abstract—Using wavelets, we computed the entropy of the signal at var-
ious frequency levels (wavelet time entropy) and, thus, find an optimal mea-
sure to differentiate normal states from ischemic ones. This new indicator
is independent from the ST segment and yet provide a conclusive detection
of the ischemic states.
Index Terms—ECG, entropy, “T wave”, wavelet.
I. INTRODUCTION
In electrocardiology, myocardial ischemia is typically detected by a
shift of the ST segment as measured at the J+60 or J+80ms markers
([1]–[3] and Fig. 1). This paper examines the information content of the
complete morphology of the combined ST segment-T wave through the
fast wavelet transform (FWT) and Shannon’s entropy proposing a new
indicator of myocardial ischemia. Wavelets have already been used suc-
cessfully in various areas of electrocardiology (for examples [4] and
[5]), namely to identify the markers from a multiscale approach [6],
detect ischemia in the QRS complex [7], and characterize the time-fre-
quency signature of heart rate variability [8]. It has already been shown
that the scale entropy of the wavelet coefficients could successfully dif-
ferentiate various states in medical applications [9]. This paper, how-
ever, illustrates why the time entropy of the wavelet coefficients is more
likely to be a good indicator of myocardial ischemia.
II. SIGNAL PROCESSING THEORY
The general objective of this paper is to extract additional informa-
tion from the ST segment-T wave in order to characterize myocardial
ischemia. What is the minimal condition on a given set of functions
f
i
g
i
such as a decomposition of the signal in terms of this set con-
tains all the information of the signal? From a mathematical point of
view, the answer is that the set f
i
g
i
must be a frame. A frame is a set
of functions f
i
g
i
for which there exist A > 0 and B < 1 such
that for any square-integrable function f
Akfk
2

k
jhf j
k
ij
2
 Bkfk
2
where hf jgi is the scalar product and kfk = hf jf . Being a frame
is a weaker condition than being an orthonormal basis; however, it is
still sufficient to provide a meaningful decomposition of a signal: one
can recover the original signal from the decomposition. Indeed, for any
frame f
i
g
i
, there exists a corresponding dual frame f ~
i
g
i
such that
for any square-integrable function f; f = 
k
hf j
~

k
i
k
. Moreover,
this decomposition is the most economical in the sense that whenever
f =
k
c
k

k
k
jc
k
j
2

k
jhf jw
k
ij
2
[10] and [11].
Manuscript received April 1, 1999; revised March 2, 2000. Asterisk indicates
corresponding author.
*D. Lemire is with the Insitut de génie biomédical, Université de Montréal,
C.P. 6128 succ. Centre-Ville, Montréal, PQ, H3C 3J7 Canada (e-mail:
daniel.lemire@videotron.ca)
C. Pharand, J.-C. Rajaonah, B. Dubé, and A. R. LeBlanc are with the Insitut
de génie biomédical, Université de Montréal, Montréal, PQ, H3C 3J7 Canada.
Publisher Item Identifier S 0018-9294(00)05138-7.
Fig. 1. Direct relationship between an acute coronary artery occlusion causing
ischemia and the ST segment amplitude. Interval between R and T is the
region of interest in this study.
Wavelets are examples of frames and they do not have to be an
orthogonal basis. The scalar values hf j ~
k
i are called wavelet coeffi-
cients (see above). The wavelets must be chosen to have good time and
frequency localization. In this particular application, since signals are
short (a little over 100 sampled values per T wave), a good time local-
ization is extremely important. Moreover, since wavelet theory gener-
ally assumes that signals are infinite, special filters must be used at the
beginning and end of the signal. Otherwise most of the transformation
becomes meaningless [12]–[14]. That is, the ratio of the length of the
signal to the largest scale of interest is too small to ignore the bound-
aries. In the context of short signals and large amount of data (thou-
sands of heartbeats), the FWT [10], [11] must be used both for per-
formance considerations and mathematical modeling (the continuous
wavelet transform assumes long signals or smooth models). To make
the analysis easier, the wavelets used should be symmetrical so that
the transform is time-reversal invariant. The Cohen–Daubechies–Feau-
veau (CDF) biorthogonal (linear) spline-wavelets were used [15]. They
have compact support and are known to have optimal time localiza-
tion for a given number of null moments (frequency localization) [10],
[11]. Splines-wavelets are extremely regular and unlike other wavelets,
they are symmetric in time. The corresponding scaling functions are
B-splines which have the shortest support and best regularity for a given
number of null moments. In other words, spline-wavelets have the best
approximation properties among known wavelets for a given number
of null moments.
III. SHANNON’S ENTROPY AND CELLULAR SYNCHRONIZATION
Given a nonzero signal fx
i
g
i
, its entropy is computed by first trans-
forming it into a new array having unitary mass using the formula
"
i
= jx
i
j=
i
jx
i
j. The entropy of the signal fx
i
g
i
is defined by
the positive value 
i
"
i
ln "
i
, where we use the convention that
0  ln 0 = 0. Notice that the entropy is invariant under the multi-
plication of the signal by a constant.
Application of the entropy concept to ST segment-T wave analysis
can be stated as follows. From a cellular point of view, the T wave is
a summation of out-of-phase localized myocardial action potential. If
time dispersion of T waves is increased, as it is assumed to be the case
during ischemia, synchronization between these superposed waves is
lost. It is then hypothesized that a good indicator of synchronization
(and ischemia) would be the entropy of the position in time of each
superposed wave. Ischemia would then be detected by a significant
increase in this entropy.
Unfortunately, it is highly improbable that we will ever be able to
directly measure this entropy by nonintrusive methods because:
1) the myocardial action-potential waveforms are unknown;
0018–9294/00$10.00 © 2000 IEEE