QUADRUPLE SYMMETRIC REAL SIGNALS SPECTRAL EVEN AND ODD DECOMPOSITION SPEKTRALNA DEKOMPOZICIJA

for the shear waves SH, μ=const is the Lame coefficient, ρ=constant represents mass density. The function w(x,t) is the anti plane (X, Y) component of the displacement vector in the direction parallel to the axis Z. On the boundary between the two neighbouring layers “i” and “i+1” the corresponding boundary conditions are satisfied. These boundary conditions represent that the unknown displacements and forces (stresses) are continuous:


INTRODUCTION
Spectral even and odd decomposition of quadruple symmetric real signals is illustrated in the case of:

SH (polarized in horizontal plane) wave
propagation through multi layered media [1,2,3,4,7] The wave propagation process, at the direction of the axis x perpendicular to the investigated multilayered media (fig.1), could be described by the following equation: The wave propagation process, at the direction of the axis x perpendicular to the investigated multilayered media (fig.1), could be described by the following equation: (1) where is the wave propagation velocity for the shear waves SH, µ=const is the Lame coefficient, ρ=constant represents mass density.The function w(x,t) is the anti plane (X, Y) component of the displacement vector in the direction parallel to the axis Z.
On the boundary between the two neighbouring layers "i" and "i+1" the corresponding boundary conditions are satisfied.These boundary conditions represent that the unknown displacements and forces (stresses) are continuous: (2) (3) where P i is the boundary force vector of the corresponding layer, is the stress tensor, and is the corresponding normal vector.The initial conditions with respect to the displacement and to the first difference are homogeneous.The both functions depend on the spatial variable x and time argument t in the initial moment t=0: (4) For the direct problem [1,2,3,4] on the bedrock the displacement boundary condition is given in the mode of the known time function: (5) The free surface stresses (force boundary conditions) are homogeneous for the investigated direct and inverse problems [1,2,3,4]: (6) On the other hand, for the inverse problem [1,2,3,4] on the free surface, the displacement boundary condition is given in the mode of the known time function:  The above formulated SH wave boundary condition problem ( 1) -( 7) could be solved by a system of differential equations, initial and boundary conditions.This differential system consists of the following elements [1,2,3,4]: • n equations in the mode (1), one differential equation for each layer, because the velocity function depending on spatial co-ordinate x is discontinued and is a terrace-like type; • 2(n -1) boundary conditions in the mode (2), (3); • surface boundary condition in the mode (6); • initial conditions in the mode (4), • and either of boundary conditions (cinematic excitation) (5) for the direct problem or (7) for the inverse problem.

Transfer function of the multi layered structure.
The connections between the signals in the algebraic system (8) could be visualized by the oriented graph.Similar oriented graph is shown in the fig.1.c.This mathematical description represents a system of 2n algebraic equations.The system of variables, the seismic signals on the both sizes of each layer boundary (X 1 , X 2 ,...,X i .,...X n , X * 1 , X * 2 ,..., X * i ,..., X * n ), and the system of coefficients, reflection and refraction layer ratios, are complex-valued.This choice of the system of variables approximates the investigated structural model to the corresponding continuous differential problem.The variables in the above mentioned system (8) (X 1 , X 2 ,...,X i .,...X n , X * 1 , X * 2 ,..., X * i ,..., X * n ) for the continuous and discrete problems are identical.The equation (7) together with the used integral transformation of the initial boundary value problem [1,2,3,4] affords the opportunity to solve direct and inverse problem of the engineering seismology [1,2,3,4] in the complex domain.In this system the differential equation ( 1) takes part only indirectly by the corresponding transfer function of the problem.The function matrix of the system ( 8) is asymmetric.Based on this fact, the common transfer function of the problem could be obtained by recurrent elimination of the system parameters.This function physically represents the quotient between images of input and output signals of the geological structure under investigation [1,2,3,4]: Substituting the analytical complex parameter "s" by the numerical imaginary parameter "jω" into system of equations ( 8), it is possible to calculate numerically the formulated direct and inverse problems [1,2,3,4] by means of Fourier integral transformation.

QUADRUPLE SYMMETRIC REAL FUNCTIONS.
The coefficients β= β(jɷ)= Re β (ɷ)+ j Im β (ɷ) in the system (8) are reflection and refraction layer ratios according to the Willebrord Snellius (1580-1626) low (see fig. 1 b, fig. 1 c).They are known complex functions of the frequency ɷ.The properties of the layers under investigation of structural model in fig. 1 [ 7,8 ] are presented in mathematical description (8) by corresponding layer transfer function signed W i (jɷ) and corresponding reflection and refraction coefficients signed β i (jɷ).By suitable selection of the real and imaginary parts of the coefficients β i (jɷ) can be obtained quadruple symmetric real functions presented in the first quadrant of the fig. 2 in a capacity of searched problem solution.In the case of real and imaginary parts of the coefficients β i according to the conditions of Theorem 1 of the present paper, the signal will be received in the second quadrant.In the case of real and imaginary parts of the coefficients β i according to the conditions of Theorem 3 of the present paper, the signal will be received in the third quadrant.The signals from third quadrant and from fourth quadrant can be obtained also in the case of real and imaginary parts of the coefficients β i according to the conditions of Theorems 2 and 4 of the present paper respectively.
The five theorems (they are published as sub conditions in the theorem 2.1 in [5] signed by * and here points) describe the Symmetry -Conjugation relation: • Theorem 1 (The phenomenon "Symmetry" in the time domain corresponds to the phenomenon "Conjugation" in the frequency domain).The complex Fourier F(jω) spectra of the symmetric real functions in the first and second quadrants are conjugated as well as.
• Theorem 2. The complex Fourier F(jω) spectra of the symmetric real functions in the third and fourth quadrants are conjugated respectively.
• Theorem 3 (The phenomenon "Anti Symmetry" in the time domain corresponds to the phenomenon "Anti Conjugation" in the frequency domain).The complex Fourier F(jω) spectra of the anti symmetric real functions in the first and third quadrant are anti conjugated as well as.• Theorem 4. The amplitudes of functions in the first and second quadrants are both positive, while the amplitudes of functions in the third and four quadrants are both negative.The functions under investigation could be of arbitrary amplitudes -negative or positive.The corresponding complex Fourier F(jω) spectra are also of arbitrary type amplitudes -negative or positive.

FOURIER TRANSFORMABLE FUNCTIONS.
The well known Dirichlet conditions are sufficient for function f(t) in the time domain to be Fourier transformable [6] pp-73.It follows: 3.1 The function f(t) is limited in the absolute value mode: (10) 3.2 f(t) has finite maxima and minima within any finite interval.
3.3 f(t) has a finite number of discontinuities within any finite interval.
The direct and inverse Fourier transformations are given by: (11)

ODD AND EVEN REPRESENTATIONS.
It is well known that any function in the time domain can be decomposed into an odd and even function [6] pp-75 as follows:

First Common Function Numerical Example:
"Rectangular Half Wave".
Three figures number 3, 4 and 5, show the process of decomposition of common function "Rectangular Half Wave" by eight samples.Figure 3 illustrates decomposition of common function in even and odd components according to [6]. Figure 4 illustrates decomposition of even component in a "Symmetric Wave Component Left" and "Symmetric Wave Component Right" according to relation (17).Figure 5 illustrates decomposition of odd component to "Anti Symmetric Wave Component Left" and "Anti Symmetric Wave Component Right" functions according to relation (23).
Table 1 and Table 2 illustrate the possibility of calculating complex Fourier spectra of the "Rectangular Half Wave" by the relation (28).The initial common function is shown in the row 2 of Table 1.The row 4 of Table 1 shows symmetric component of the initial function and the row 6 of Table 1 shows anti symmetric component of the initial function.The "Even Left Function" from the Theorem 6 is shown in the row 9 of Table 1.The "Odd Right Function" from the Theorem 6 is shown in the row 11 of Table 1.
The coefficients of the complex Fourier spectra of the "Rectangular Half Wave" are shown in the row 14 of Table 1.They are obtained by the function "fft" of MatLab program system [10].The "Complex coefficients of the ½ event left component of the rectangular half wave" are shown in the row 16 of Table 1.The "Complex coefficients of the ½ odd right component of the rectangular half wave" are shown in row 18 of Table 1.The "Complex coefficients of the ½ event left component + Complex coefficients of the ½ odd right component of the rectangular half wave" are shown in the row 20 of Table 1.The "Complex coefficients of the ½ event left component -Complex coefficients of the ½ odd right component of the rectangular half wave" are shown in the row 22 of Table 1.Finally, the "{Complex coefficients of the ½ event left component + Complex coefficients of the ½ odd right component of the rectangular half wave } + {Complex coefficients of the ½ event left component -Complex coefficients of the ½ odd right component of the rectangular half wave }" are shown in the last row 24 of Table 1.This operation closed the illustration of applying the relation (28) for the numerical example under consideration.The row 24 of Table 1 is identical of the row 14 of Table 1.This is the numerical proof of the numerical example "Rectangular Half Wave".This numerical proof shows, that the complex Fourier spectra calculated by MatLab program system [10] directly and complex Fourier spectra calculated by using the relation (28) are identical.
In addition, Table 2 showing the numerical example under investigation has illustrated the inverse composition of the initial "Rectangular Half Wave".The "Complex coefficients of the ½ event left component of the rectangular half wave" are shown in the row 27 of Table 2.The "Inverse Samples of the ½ event left component of the rectangular half wave -Time Domain" are shown in the row 29 of Table 2 2. This final numerical result from simultaneous interpretation of Table 1 and Table 2 shows, that the inverse transformation using the relation (28) restore the initial "Rectangular Half Wave" signal in the row 49 of Table 2.This numerical proof of the first numerical example illustrates the statement, that the complex Fourier spectra has been calculated directly for the "Rectangular Half Wave" by MatLab program system [10] and complex Fourier spectra using the relation (28) restores accuracy of the initial time domain signal after inverse discrete Fourier transformation.
The study of complex numerical examples like "Rectangular Half Wave" is possible only with the help of computers, fast Fourier transformation and software systems such as MatLab [10].
The next eight numerical examples are specially selected.They are only symmetric or only anti symmetric.They can be detected and studied more easily, because the appropriate equivalent symmetric or anti symmetric functions can be evaluated directly from the MstLab Command Window.

4) Rectangular Full Anti Symmetric wave
The Fourier complex spectra Y9 Octable Triangular Anti Symmetric Wave of the anti symmetric common function for this numerical example can be obtained by summarizing the Fourier complex spectra Y9 Octable Triangular Anti Symmetric Wave Left and the Fourier complex spectra Y9 Octable Triangular Anti Symmetric Wave Right.

CONCLUSIONS
The strategy of spectral even-odd decomposition of the arbitrary real function described in the paper allows constructing complex Fourier spectrum of initial signal with the length N in the time domain base on the equivalent real and imaginary spectral parts with the length N/2 in the frequency domain.The Spectral Even-Odd Decomposition of Arbitrary Real Signals is shown in the figure 8.

( 7 ) 1 . 2
Structural mathematical model of the multi layered structure.The structural model of the multilayer media is shown in fig. 1.The SH Wave Propagation Reflect -Pass Perpendicular Process is illustrated on the fig.1 a.The Block -Diagram Model of the media under investigation is shown in fig.1 b.The Flow Graph of the system signals is shown in the fig.1.c.

Theorem 6 (
The phenomenon "Symmetry" in the time domain corresponds to the phenomenon "Conjugation" in the frequency domain.The phenomenon "Anti Symmetry" in the time domain corresponds to the phenomenon "Anti Conjugation" in the frequency domain.The simultaneous operation of the Theorems 1 and 3 leads to even and odd decomposition of the Fourier complex spectrum of the common function with length N in the time domain.This result represents spectral function, composed by the equivalent nonzero real and imaginary spectral parts with length N/2 in the frequency domain).Any real common function in the time domain can be decomposed in a symmetric and anti symmetric components.The Fourier spectrum of the common function: of the above mentioned integrals in (14) and (15) and using the popular rule for zero integral from zero integrand function[9].Than the complex spectra of the common function could be obtained by the following relation: The proof of the Theorem 6 is presented analytically in the paragraph four of the present paper.This proof of the Theorem 6 can be illustrated by the following eight numerical examples.The first example represents common real function of discrete type.The next eight numerical examples are of symmetric or anti symmetric discrete type functions.All examples illustrated the process of even and odd decomposition numerically.
. The "Complex coefficients of the ½ odd right component of the rectangular half wave" are shown in the row 32 of Table 2.The "Inverse Samples of the ½ odd right component of the rectangular half wave -Time Domain" are shown in the row 34 of Table 2.The "Complex coefficients of the ½ event left component + Complex coefficients of the ½ odd right component of the rectangular half wave" are shown in the row 37 of Table 2.The "Inverse Samples of the {Complex coefficients of the ½ event left component + Complex coefficients of the ½ odd right component of the rectangular half wave} -Time Domain" are shown in the row 39 of Table 2.The "Complex coefficients of the ½ event left component -Complex coefficients of the ½ odd right component of the rectangular half wave" are shown in the row 42 of Table 2.The "Inverse Samples of the { Complex coefficients of the ½ event left component -Complex coefficients of the ½ odd right component of the rectangular half wave} -Time Domain" are shown in the row 44 of Table 2.The "Complex coefficients of the { Complex coefficients of the ½ event left component + Complex coefficients of the ½ odd right component of the rectangular half wave } + {Complex coefficients of the ½ event left component -Complex coefficients of the ½ odd right component of the rectangular half wave }" are shown in the row 46 of Table 2.The "Inverse Samples of the { Complex coefficients of the ½ event left component + Complex coefficients of the ½ odd right component of the rectangular half wave } + {Complex coefficients of the ½ event left component -Complex coefficients of the ½ odd right component of the rectangular half wave } -Time Domain" are shown in the row 49 of Table

Table 1 .
Samples and Fast Fourier Transformation Complex Coefficients of the Rectangular Half Wave and its Components

Table 2 .
Fast Fourier Transformation Complex Coefficients and Inverse Samples of the Rectangular Half Wave and its Components 16 Samples Symmetric and Anti Symmetric Numerical Examples Fig. 6.Symmetric and Anti Symmetric Numerical examples, which illustrate the proof of the Theorem 6. 6.1) Full Rectangular Wave 6.2) Rectangular Half Step Symmetric Wave 6.3) Rectangular Step Anti Symmetric Wave 6.