PARAMETRIC DESCRIPTION OF SPECTRAL DENSITP OF Ecc
Gopinath, R.,
k r t h y , I.S.N.
and Prasad, G.S.S.D.
Department of E l e c t r i c a l Engineering
Indian I n s t i t u t e of Science
Bangalore 560 012, INDIA
Abstract - A parametric description of t h e spectral
density of the ECG signal using i t s Discrete Cosine
Transform i s proposed. This transformation enables
modeling of t h e ECG e f f i c i e n t l y using lower order
polynomials. The Steiglitz-McBride, Shanks and the
Maxim Likelihood Estimator algorithms a r e used
for modeling. Certain parameters obtained from the
spectrum a r e used t o determine an estimate of the
autocorrelation function t h a t takes the form of a
sum of exponentials and damped sinusoids. Several
normal and abnormal ECGs have been analyzed with
An abnormal beat
highly satisfactory r e s u l t s .
required more parameters f o r i t s description than a
normal.
From eqns. ( I ) and ( 2 ) the power spectral density
(PSD) i s given by
B(z-’) B(z)
-----------
u
A(2-I) A(z)
The ACF i s obtained by taking the inverse f o u r i e r
transform of the PSD.
Thus
With the notation
INTRODUCTION
Of
the
several ways
of
representing
biosignals, spectral characterization i s commonly
a rational
used.
Zetterberg [1,2] employed
the f r e uency variable
of the type
func2ion
Q (f ) /P(f ) ,where Q ~ x and
)
P(x) a r e polynomials of
low order, t o study EEG modeled a s an ARMA process
with white noise input, under conditions
of
stationarity.
In this pper we study the ECG
signal considered as the impulse response of a
l i n e a r time invariant system. Modelin t h e d i r e c t
ECG signal requires orders of (20,207 o r higher
(30,30) [31. Instead we model the Discrete Cosine
Transform (DCT) coefficients of t h e time domain
signal. A (2,2) order w a s s u f f i c i e n t t o model the
transform of the component waves while a ( 6 , 6 )
order was
required f o r the transform of t h e
complete ECG cycle. The auto correlation function
(ACF) i s characterized by a s e t of parameters, the
resonant frequency, t h e half power bandwidth and
the power content a t the resonant frequency. While
the first two parameters a r e obtained from the
spectrum, the t h i r d one i s evaluated from the model
coefficients.
f2
(3) can be rewritten as
Using the residue technique we evaluate
the
i n t e g r a l a t the poles of the system function t o
yield
METHOD
The transformed signal i s modeled as a
zero process, whose system function
pole-
exp ( -kcti
1
H(z) = [ B ( Z ” ) / A ( X - ~ ) i] s t o be determined
Define B(2-I) = bo
+ b,z-’ +
... b9zq
!I
;
F~ exp(-ki)
=
i=l
where Fi i s the power content a t DC given by
OMS
Annual International Conferene of the IEEE Engineering in Medicine and Biology Society. Vol. 12.No. 2.1990
CY29383/9O/OO004805 $01.00 0 1990 IEEE
Hz,
______________
A(zi)
A3(zi -1
1
For the p2 complex poles zi
evaluate rk a s follows:
Denote
B(zi)
were used f o r analysis.
Of
the
three
algorithms, t h e maximum likelihood estimator gave
the best s p e c t r a l f i t i n terms of t h e normalized
r o o t mean square e r r o r (NRMSE), but had t h e highest
time complexity.
Steiglitz-McBride and
Shanks
algorithms gave similar r e s u l t s . For purposes of
i l l u s t r a t i o n we have shown t h e r e s u l t s obtained on
analyzing a normal beat and i t s component waves,
a f t e r modeling them using t h e Steiglitz-McBride
algorithm.
B(zi)
BIzi-')
F . z a
=
Bx- L jB
-,
Y
jAY-,
A(zi) = A x - +
=
B(zi
exp(-pi L j wi)
-1
=
A3(zi -1 )
B
=
'
jB
we
'
Y '
A
+ j A
3x 3y
'
'
1 ( a ) shows t h e complete ECG cycle.
Fig. F?%?gives
t h e power s p e c t r a l density (PSD)
evaluated from t h e ACFs, obtained d i r e c t l y from the
transformed s i g n a l and evaluated usin Fq. (9).
Figs.
2 ( a ) - 2 ( b ) , 3 ( a ) - 3 ( b ) and 4fa) - 4(b)
give the corresponding p l o t s f o r the component
waves.
Substituting i n ( 5 ) y i e l d s
CONCIUSIONS
Finally when q
b
E
=
=
In [ I ] , a parameter H . , defined as
the
skewness of the s p e c t r a l densiiy about the resonant
Our
frequency, i s used f o r characterization.
analysis
shows t h a t t h i s parameter
i s not
necessary. The ACF i s thus a function of only four
parameters
o . , w . , G. and F . .
Using
these
parameters
6e hhve hot only' been
able
to
reconstruct our signal but a l s o t o simulate a
whole c l a s s of ECG signals.
p , we g e t an additional term
b
-E---%
ap a.
REFEFLENCES
Hence the expression f o r the ACF takes the form
rk
=
1 . L.H.
Zetterberg, 'Experience with Analysis and
Simulation of EEG Signals with Parametric
Description of S p e c t r a ' , Automation of Clinical
Electroencephalography, 1973, 161-201
2. L.H. Zetterberg, 'Estimation of Parameters f o r a
Linear Difference Equation with Application t o
EEG Analysis , Mat. Biosciences , 1969, 227-275.
3. I.S.N.
Murthy, e t a l . , 'Homomorphic Analysis
and Modeling o f ECG Signals' , IEEE Trans. on
BME, June 1979.
E6 + P1' Fi exp(-kai)+ p2
C Gi exp(-kpi)cos(kw.)
k
i=l
i=l
where d k = 1
= 0
( 9'I
.
if q = p
otherwise.
RESULTS
Both normal and abnormal ECGs, sampled a t
I
1
500
!:I:u,hk-,
6oot
FIG.la
FIG.2a
,
,
- 50 0
100
7c
5000
____
FIG.1b
El
100
200
100
200
1 , J1
0
sample no.
I
Scale 1
0
100
I
h
:
=
0
0
sample no.
I
2x10
200
I
I
,
100
200
sample no.
I
I
0
I
,a
100
200
sample no.
FIG. 2 b
iI n
V
200
0
100
200
0
100
Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Vol. 12, No. 2, 1990
200
0806