PRELIMINARY PROCESSING OF BIOSIGNALS B.I. Yavorskyy
Ivan Pul'uj Ternopil State Technical University Rus'ka 56, Ternopil, 46001, Ukraine
E-mail: Kaf_BT@tu.edu.te.ua
Summary. Analysis of ways of biomedical signals preliminary processing under their spectral representation are given. The main reason of the filtration, log and spectrum windowing and the tapering under the spectral estimation of stationary random biosignals are appointed. At abstract viewpoint on it as a weakering of norms and metrics of the signal representation Hilbert space are considered. Metrics definition transforms under the spectral analysis of a periodical correlated and event random biosignals are established.
1. The problem. An observation of a biomedical phenomenon by the signal is provided us information by extracting of spread event or an approximately equidistant features of biosignal [1]. For example, amplitudes or other values of ECG, ERG, EEG waves and time intervals among them are what are doctors particularly used at diagnostics. All like these signal features are rare deterministic, moreover, stationary [2]. That means they have not an invariant neither to translations in the time of their observation nor to its influences on the phenomenon. As a result, they occupied too many space at visualization, and it’s by human analysis are far from success. More badly, situation are at high-resolution diagnosis, a biotechnical rehabilitation etc.
It is known that invariant for shift on their arguments, the maximum information at the minimum of their presentation has some representation of function [3].
Representations are obtained from biosignals is mostly difficult for interpretation and understanding by doctors. But certainly at present we have begun to develop and utilize these means in medical application with full understanding — as power complement utilities at analysis of biosignals, design and recognition patterns of decease are convey by signal, rehabilitation technical means. Representations are obtained by transforms similar to Fourier are the corollary of a stationary [3]. Nevertheless, necessary for this theoretical assumption is not adequate to natural signals; moreover, it is not correct for all signals. That is why for the correct determination of the representation of a signal is required some a preliminary signal processing (PSP). However, numerous reports of spectral analysis of biomedical signals have not been adequately considered, particularly, the role of PSP under the spectral analysis of random periodical correlated
or event signals. This paper deal with the given by author theory of the PSP of event, random, periodical correlated random biosignals under their spectral representation.
2. Representations of deterministic signals. We use a function(t), t, or tT,T as a mathematical model of the experimental series of dataˆ(nTd), where
Td — sample period, the "roof" denote a digital coding,n1,N . Recently was considered function expansions by a linear form
k
k
k t
t) ( , )
~(
, (1)
where k — coefficients, (k,t) — a function. It has different types of convergence's: the local in-point (by Cauchy) or point by point (by Lagrange) of their values to values of experimental data was mentioned yet or somewhere (as like by Fourier) on a time interval or the entire axes. In general, such expansions are founded on assumption of extreme properties of a functional
p
t L
t) ~( ) (
min
arg
, (2)
where — denoted a norm of function space at boundaries,
p t) L
( . (3)
(For example, to see [4], wherep 2, L — a Hilbert space, and were used for obtaining presentations and algorithms of signals processing [5]). The functional (2) has extreme when (k,t)eigenfunctions, for example, are of a shift operator
) t ( U ) s t ( :
Uts ts . (4)
When they are common with eigenfunctions of the observation operator expansion’s (1) are named as a spectral [3].
For finite functions exist an expansion as a sum of a sample function with coefficients are values of signal at the time of a sample [6-8]. Maybe Caushy was the first who studied this problem [9]. Modern heuristic extension of all these above are wavelets [10] (for details that are more interesting see J.R. Higgins [6] and A.I. Zayed [8]).
For a ring of functions with a convolution operation as a multiplication exist isomorphic representations into a field of hyperfunctions (or operators, by Mikusinski [11]) with a linear algebraic relation [12]. More a weak way of representations then isomorphic is congruence [13]. In [14] K. Steiglitz has considered the isomorphism between discreet and continuous functions what does pick up methods of the analogous signal processing for a digital signal processing.
There are need regularity, rationality and inevitability properties of an expansion
the same for the linear operator of a signal processing existing [15]. These have the spectral and samples of finite functions expansions [7].
3. Representation of stochastic stationary signals. It can be an ensemble of functions or a general samples as a stochastic function or process. Herman Wold has given the decomposition s(t) x(t)(t) for a set of such process on deterministic
) t (
x and regular (t) parts and has proposed an expansion of regular part in linear forms [16]. From the spectral decomposition of the groups of unitary operators (that in here are mean as observations and shift) Uts ejtd
he has obtained an orthogonal scattered stochastic measure(d):(s)Uts(0) ejt(d)
, where (0) — a random value with all finite moments (so, conditions like (1, 2) had been established). The power spectrum is a spectral expansion of the covariance function
(t), (s) e (d ) (d ) )
s , t (
C L j(ts) (5)
where
, L — scalar product, (s) denoted the complex conjugate for (s). It is given by the power distribution function f(,)F(d,d)((d)(d)), where F(d,d)— spectral stochastic bimeasure. For stationary processes ts and it is covariance measure of the stochastic orthogonal measure [17, 18]: the random spectrum
): (t) f( )e d
(
f j t
and the power spectrum —
d e ) ( F ) ( C : ) (
F j .
Another way for presentations of stationary processes is given by innovation integral methods (linear stochastic processes) [15, 19].
3.1. Spectral analysis doing of stationary processes. The consistent estimates
) ˆT(
f of the spectral density function may be obtained by the periodogramm iT():
,
where A() — spectral window, i i C d
T T
T
exp( )ˆ( ) 2
) 1
( ,
dt t T t
C
T
0
) ( ) 1 (
)
ˆ( (6)
or, as mathematically is equivalent, but is not for a practical design,
i C d
fˆTA( ) exp( ) ( )ˆ( ) ,
' ' ') ( ) (
)
ˆ ( A i d fTA T
where () is so called a lag window.
Sometimes are used a preliminary transformation of the data by multiplication of all the data by some numerical function k() is called a data window. Then the modification of periodogramm is obtained which were called tapering. Whereas,
T 2
0
T exp( i ) ( )d
T 2 ) 1 (
i
,
then
' ' ') ( ) (
)
( K f d
iTk Tk ,
where
2
0
) ( ) 2 exp(
) 1
(
i k d
K T
T
T k
T . In general, there is no consensus of opinions when, where and as to choice(),A(), and k(),KTk(). Nevertheless, the practical methods for a rational choice that functions are considered in many references [17].
At last, when the convolution of the data with some numerical function is used this is called a preliminary filtration: it is characteristic also, either a heuristic or an optimal are depended what a priory data are known [17].
4. Representations of non-stationary stochastic signals. Conditions (2, 3) are the background for hierarchy of representations given by Yaglom, Karhunen, Loeve, Krammer and other [3, 18]. For periodically non-stationary processes it was doing by redefinition (3) as properties of the stochastic bimeasure
) d , d (
F (7)
for so called Loeve harmonizable processes, or
) d , d ( F ) ( ) ( sup
1 ) ( , ) (
(8) for Rozanav’ harmonizable processes. However, these properties does far from being precise in the applications. This suggested the need to extend the notion of the spectrum and random spectrum beyond measures. By H.L. Hurd and G. Kallianpur has been investigated spectral and canonical expansions [20]. In [21] have presented the probabilistic concept of the spectrum by integral transforms with a distribution function kernel. In [22] have used the Wold isomorphism between an abstract stochastic process — a curve in a Hilbert space. Than an element of Hilbert space generated by the numerical sequence and have obtained decomposition of cyclostationary stochastic processes into an orthogonal sum of regular and predictable parts. Thus have stated of the regular process in terms of a linear signal processing on the sequence of random
innovations.
Another approach for obtaining of the expansion and representation of non- stationary processes and their transforms under processing's has proposed Ya.P. Dragan [3] by developing (7, 8) in
(t)2dt
R
E (9)
is defined class -processes, and in
t dt
L
L
L L
2 -
) 2 (
lim 1 E (10)
is defined class -processes, where E{} is an operator of the mathematical expectation, (t)(t)E(t) — centered values of signal samples. It had been done by using the invariant conception of like (9, 10) bilinear forms under affine transforms those are analogously to (4). For using these facts under obtaining of the random signals representation and reducing the representation of the periodically non-stationary signal to the Fourier-liked ones a rigged Hilbert spaces construction was proposed. Moreover, an isomorphic representation was considered by structures of Hilbert spaces over a Hilbert space and a reproducing kernel rigs Hilbert space. Ways for obtaining the kernel and structures for those spaces were as definitions similar to (2, 3) expressions (norms and bases). It was named as the energy theory of stochastic signals (ETSS) because of such norms. This approach as well is useful for classification of stochastic processes and design of algorithms of their processing [23].
a ) N o n s t a t i o n a r y b i o s i g n a l s w i t h a p e r i o d i c a l c o r r e l a t i o n f u n c t i o n . Its mean values m(t) E{(t)}and correlation functions
} ) ( ) ( { E ) ,
(t s t s
С have such properties that
) t ( m ) T t (
m 0 , С(tT0,sT0) С(t,s)
for some fixed T0 and every t and s. It is convenient to change the variables and use the function
} ) ( ) ( { E ) , ( ) ,
(t C t t t t
B .
Therefore B(t,) may be given as
1
0
0 ) exp(
) ( )
,
( K
k
k ik t
b t
B ,
where 0 2/T0,K T/T0,T — the sample interval and Ck() can be given by the Fourier-Stielties integral
) ( ) exp(
)
(
k
k i dF
b
,
where F0()bounded non decreasing function, Fk() is complex function of the bounded variation. It makes possible to represent the correlation function
) , ( ) ,
(t s B s t s
C as
2 1 1
1
0 2 1 2
1
0 1
0
) ( ) (
) (
exp
) ( } ) (
{ exp )
, (
d dF
k s
t i
dF s k t
i s
t C
k K
K k
k K
k
(11)
where () — Dirak delta-function.
Thus, a periodically correlated time series is harmonizable and its spectral measure F(1,2 ) is concentrated on the lines 12k0 located inside the square
1 , 2 . A spectral analysis of the sample will be made as the spectral analysis of the sequences k-components of correlation as correlation functions of a stationary process are received by the special processing of(t).
b ) R h y t h m i c e v e n t p r o c e s s . Point or event processes are fully described by the occurrence time of the event ti . The theoretical approach to the study of event processes is expect analogous to that for above, but for series are generated by biosignals we exploits rhythms. Under assumption of known rhythm , we can receive the random process
) t i i
( i
(12)
Then, in general, we serve 4 (a) for obtain of spectral characteristics.
4 . 1 . S p e c t r a l r e p r e s e n t a t i o n d o i n g o f P C R P ( p e r i o d i c a l l y c o r r e l a t e d r a n d o m p r o c e s s ) .
a ) T i m e d o m a i n . I n a c c o r d i n g w i t h ( 1 1 ) a n e s t i m a t i o n o f c o v a r i a t i o n
du u h u t u t t
С
t
) ( ) (
) ( )
, ˆ(
0
( 1 3 )
where the pulse functions h(t) is the coherent (when h(t) N1 (t nT0 ) 1
N 0 n
) or the
component (when
) 2 / u sin(
) t t (
] u 2) N 1 sin[(
) u ( h
0 0
0 1
, N1 — number of components) filter's characteristic. In general, the pulse function h(t) is design as an optimal. An
estimation of mean values we can get as m t t u h u du
t t
) ( ) ( )
( ˆ
0
0
.
b ) F r e q u e n c y d o m a i n . On the other hand, in according with (11) estimations for the correlation function and mean values are:
m , n
mk nl nm i
T l,k Dˆ l k e 0
rˆ ,
Ν1
, 1 k
k ik
T mˆ e 0
mˆ ,
where k,l, . Then estimations of the average components:
ˆ ,
2 lim 1
ˆ 0 0
ik
T k
k e
m T
where ˆk0 — are delta band pass digital filters. Estimations of symmetrical correlation components:
k
mk m i
k n u n i
nm T e u k ke
u T
D 0 ( ) 0 0 0
2 lim 1
ˆ
where: ulk.
5. Fundamentals of preliminary processing. The representation of deterministic functions in a linear form (1) was obtained as a spectral by the optimization problem (2, 3) resolution [4]. Under the spectral analysis of stationary and non-stationary random processes expression (2) was developed on the base of the new definition (3) in reason ways (7-10). In a result the spectral analysis of non-stationary processes was reduced for the famous method. Nevertheless, the main of a visible feature of bioobjects are being in norm is adaptation for environments, so their biosignals are an event, non-stationary approximately. Moreover, we can observe there on finite time intervals. However, the spectral presentation needs an invariant, periodicity property of a signal on an entire observation time, So observations of event and non-stationary biosignals need to prepare for the spectral representation in a correct way by exploit their peculiarity.
a ) D e f i n i t i o n o f p r e l i m i n a r y p r o c e s s i n g ' s . Absence a priori data about the signal a more weak conditions on its norm and metric are reached: by the PSP (a filtration, tapering) or by the addition processing of the estimate (a log or spectrum windowing). Such both signal processing are called weakering.
The specific restriction gives by the theory of periodically correlated process — the concentration of its spectral measure on lines1 2k0. The functions
,...
3 , 2 , 1 , 0 k ), (
Fk , are called spectral distribution, while fk(),k 0,1,2,3,...
are called spectral density components of the periodical correlated random-process
) t
( . We assume early that is harmonisable, i.e. is representable as Fourier-Stielties integral. Then the sequence is clearly a stationary random sequence.
General methods for approximate determinations of the spectral characteristics
) (
fk from single realization of the process (t) were given above. However, what mean the preliminary selection values of the signal throw kT0 under summation in (1) and (2)? The answer was find under notion of a topology is generated by a metric of the Hilbert space [25]. That immediately means the caution of the projective geometry of the space. It will give one sufficient freedom to express in numerical form dissimilarity between two objects (by the measure definition transform on the projective space). The PSP suggested for the spectral representation of periodical correlated process is consider as a new approach because of its property to separate the projective space on the Hilbertian subspace. A relative measures of pair objects — fk() and fi() is the invariant of the preliminary processing. Its definition based on a notion a property of signals (for a periodical correlated process it is the garmonisability).
For event biosignal (R-wave of ECG) we used (11) as measure definition preliminary processing (see also [22]).
b ) O p t i m a ' s a n d b o u n d a r i e s c o n d i t i o n . Definition of 0 recently was given as an optimization problem [26]
) ˆ ( min
arg
0
l k d f
VAR
,
where
l, k
2 l , k
d( ) f ( )
VAR is d- variation, 2, — a norm in Hilbert spaces , or
. Efficiency of representation is estimated by a meeting of an inequality bound of its variance, or by like Rayleigh quotients [27].
Conclusions. The nonparametric method of the spectral estimation under the absence of a priori data about signal and bad conditions to rich it had been used. In that case the meaning of preliminary and similarly signal processing are grow. Particularly, the abstract sense of the understanding of it will permit an optimal selection of its way as well as of its parameters. The criteria of the optimality can be ones of the quality values of the spectral estimation (the bias, consistent, effectivity). We will can now to spread a formal ways of selecting optimal PSP methods on the spectral representation of a periodical nonstationary process by proposed the new approach to PSP as to the measure definition transform in the Projective space (contrary the norm and metric weakering in the Hilbert space).
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