Electrocardiography
15.24-4
ECG COMPONENT DELINEATION BY PRONY'S METHOD
NIRANJAN, U . C .
and
HURTHY, I . S . N .
Department o f Electrical Engineering
Indian Institute o f Science
Bangalore 560 012, INDIA
ABSTRACT
THEORY
A simple, non iterative method f o r
c o m p o n e n t w a v e d e l i n e a t i o n from t h e
Blectrocardiogram (ECG) is derived by
modelling it's DCT as a sum of few damped
cosinusoids. The parametex set pk [ak, 8 k #
fk' akl i.e., amplitude, phase, frequency
and damping factors of the cosinusoids are
estimated by the extended Prony method.
Based on the clusters of poles appropriate
n u m b e r o f them a r e c o m b i n e d and t h e
component waves derived thereby are in
most c a s e s
w i t h i n a percent r m s
d i f f e r e n c e ( P R D ) of 10. T h e m e t h o d
petformed well even in the presence of
different artifacts, with
a data
compression of 1 in 5.
Assume x(n) is an ECG signal, made up
of three component waves P, QRS and T.
x(n) = I: xi(")
i
BCG has been extensively used as a
diagnostic tool in hospitals, ICCUs for
the detection of many heart ailments. One
of t h e a c t i v e a r e a s of r e s e a r c h i n
a u t o m a t e d ECG s i g n a l a n a l y s i s i s t h e
delineation of the P, QRS and T waves.
While several existing methods can detect
the most prominent QRS complex, there
seems t o be no way f o r t h e u n f a i l i n g
detection of smaller P and T waves Ill. We
provide a technique for the delineation of
component waves irrespective of their
size, shape and location.
i = PI QI t
n = 0,1,2,
...,N-1
(1)
where
(n) = x(n) for 0 5 n 5 no, x (n) =
x(n) f 3 n +1
n 5 nl, xt(n) = x(nY for
nl+l i n 5 8-1 and xp(n) = xq(n) = xt (n) =
0 elsewhere.
The DCT of x(n) in eqn. (1) is, 121
x(a)
-
&
i
Xi(.)
(2ntl)RH
W-1
= & r(2/N) C, I: xi(") cos
i
n=O
2N
-
i = p, q, t
IWTRODUCTIOH
;
:
m = 0,1,2
,...,N-1
Cm = 1 for m = 0 : Cm = l/f2 for m
(2
+0
The IDCT of X(m) is
-
x(n)
i
f
N-1
(2n+l)mn
I: J(2/N) I: Cm Xi(n) cos
(3)
i
m=O
2N
p, q, t
;
n = 0,1,2
,...,N-1
EXTENDED PRONY =OD
We use the extended Prony method to
approximate the signal X(m) as a sum of r
damped cosinusoids, [3] i.e.,
METHOD
The Discrete Cosine Transform (DCT)
of the ECG signal suits well for modelling
it as a sum of damped cosinusoids. P the
p a r a m e t e r set c h a r a c t e r i z i n g t%ese
cosinusoids is estimated by the extended
P r o n y method.
The component wave
delineation then boils down to combining
appropriate number of cosinusoids based on
the clusters of model poles and computing
their Inverse Discrete Cosine Transform
(IDCT).
m = 0,1,2,
...,N-1
where N > 4r and pk is the parameter set
for the kth cosinusoid. The nonlinear
problem of estimating pk is separated
into two linear problems of detexmining
[Uk, f k ] and [ak, 8k]. T h u s x ( m ) is
expressed as
Annual International Conference of the IEEE Engineering in Medicine and BioloB Society, Vol. 13. No. 2. 1991
CH3068-4/91/oooO-0829$01.00 0 1991 IEEE
(5)
0829
others add finer correction in the form of
a small kink or bend to give proper shape
to a component wave.
where hk = a exp(jek), z k = exp(a +
j2nfk) and
senotes complex conjugaie.
X ( m ) c a n a l s o b e g e n e r a t e d by t h e
recursive difference equation
2r
X(m)+E b(j)X(m-j) = e(m); 2r
m 5 N-1 (6)
j=l
DISCUSSION
A practical problem is the choice of
number of cosinusoids for a given signal.
The order is estimated using the Akaike's
Information Criterion (AIC) and Levinson
Recursion for covariance LP. For a signal
with 300 samples, on an average about 15
damped
cosinusoids
are
needed.
Considering 4 parameters for each
cosinusoid in p
this resulted in a data
compression of %'in 5. The reconstructed
component waves differed from the actual
ones by a constant dc shift. The first DCT
coefficient is modified such that this
shift is corrected.
w h e r e e ( m ) i s t h e p r e d i c t i o n error.
Parameters b(j) obtained by solving the
covariance Linear Prediction (LP) problem
are related to the modes zl. in eqn. (5) by
The remaining unknown parameters hk, hk* ,
in eqn. (5) are estimated by minimizing
the squared error with respect to hk, hk*.
The resultinff normal fjquation is
[z zlh = z
(8)
where zH is 2r by N vandermonde matrix
obtained using the poles zk; & is 2r by 1
vector Of hk, hk*
and X 18 N by 1 data
v ctor. For the real data case the matrix
zfi z is usually invertible.
x
The peak samples of component waves
xi(") appear also as distinct peaks in the
magnitude spectrum of X(m) which in turn
are represented by the well separated
clusters of poles in the LP model. The
method gave satisfactory results even in
presence of EMG noise, mains interference
a n d b a s e l i n e wander. T h e D C T of most
abnormal ECG signals is of mixed phase and
d i d n o t p o s e any p r o b l e m in t h e i r
modelling.
No
instability
problem
associated with the covariance LP was
encountered.
RESULTS
Figs. a-d show good agreement between
the model output (whole beat shown as 0 )
and delineated component waves (solid
line) with the originals (dashed lines).
Estimated DCTs of whole beat, P, QRS and T
c o m p r i s i n g of 1 1 , 3 , 3 and 5 d a m p e d
cosinusoids respectively are shown in
Figs. e-h and their PRDs are 9.08, 24.8,
11.5 and 5.2 respectively. For small and
noisy P waves as in this example the PRD
may. g o
high
though
the
fit
is
satisfactory. Fig. i shows the distinct
and well separated pole clusters which
enable easy component delineation. On an
average P and T waves require 2 to 4 and
QRS needs 3 to 6 damped cosinusoids in
t h e i r m o d e l s , o n l y 1 or 2 of them
characterize the basic waveshape while
I
I
fl
P WAVE
REFERENCE
1.
0. Pahlm & L. Sbrnmo,
'Software QRS
detection in ambulatory
monitoring
- a review', Med & Biol Eng & Compt,
Jul 1984, pp 289-297.
N. Ahmed &
K.R.Rao,
'Orthogonal
Transforms
for Digital
Signal
Processing', Springer Verlag, New
York, 1975.
Marple S.L.Jr.,
'Digital
Spectral
Analysis', Prentice-Hall, Englewood
Cliffs, New Jersey, 1987.
2.
3.
I
I 1:
fl T WAVE
QRS
/'
'"\
,\
a
-0
3
240
i?
0
-520
0
0830
(e)
120
240 0
dct coef no.
120
(f)
240 0
120
dct coef no. (9)
240 0
120
dct coef no.
(tifa
Annual InternationalConferenceof the IEEE Engineering in Medicine and Biology Society,Vol. 13,No. 2, 1991
CH3068-4/91/0000-0830
$01.00 0 1991 IEEE