Numerical inversion of laplace transform using

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[4] J. B. D. Filho, G. Favier, and J. M. T. Romano, “Neural networks for blind equalization,” in Proc. IEEE GLOBECOM’96, vol. 1, London, U.K., pp. 196–200. [5] X. R. Cao and J. Zhu, “Blind equalization with a linear feedforward neural network,” in Proc. ESANN’97 , , 1997, pp. 249–254. [6] Y. Fang and T. W. S. Chow, “Blind equalization of a noisy channel by linear neural network,” IEEE Trans. Neural Networks, vol. 10, pp. 918–924, July 1999. [7] J. Karhunen, A. Cichocki, W. Kasprzak, and P. Pajunen, “On neural blind separation with noise suppression and redundancy reduction,” Int. J. Neural Syst., vol. 8, no. 2, pp. 219–237, 1997. [8] A. Cichocki, R. E. Bogner, and L. Moszczynski, “Modified Herault-Jutten algorithms for blind separation of sources,” Digital Signal Process., vol. 7, no. 7, pp. 80–93, 1997. [9] C. Jutten and J. Herault, “Blind separation of sources, Part I: An adaptive algorithm based neuromimetic architecture,” Signal Process., vol. 24, no. 1, pp. 1–20, 1991.

Fig. 1. Haar wavelet functions with

Numerical Inversion of Laplace Transform Using Haar Wavelet Operational Matrices Jiunn-Lin Wu, Chin-Hsing Chen, and Chih-Fan Chen

Abstract—In this paper, a unified derivation of the operational matrices of various orthogonal functions including the Haar wavelet is first given. Based on the derived operational matrix, this paper presents a new method for performing numerical inversion of the Laplace transform. Only matrix multiplications and ordinary algebraic operations are involved in the method. The proposed method is a much simpler as compared with the dictionary-type method and the contour-integration method. Index Terms—Haar wavelet, Laplace transform, numerical inversion, operational matrix.

I. INTRODUCTION Conventional methods of deriving the operational matrix are difficult and not unified. In this paper, we present a unified approach to deriving the operational matrices of orthogonal functions. The approach is simple and computer oriented, therefore, very useful in practice. After presenting a unified approach to the operational matrices of orthogonal functions, this paper applied the proposed approach to calculate the inverse Laplace transform, numerically. The Laplace transform approach to differential equation solving [1] comes from Heaviside’s operational method. The idea is that Heaviside truly deserves credit because the operators, his inventions, allow the reduction of differential equations into equivalent algebraic equations. Heaviside’s contribution is creative and usually, a piece of creative work is crude. Later on, the Laplace transform [1] indeed gives a rigorous foundation to the operational method. In the Heaviside time, the inverse process was established by Bromwich [2] by using contour integral which is more complicated than necessary. Since the third author and Hsiao [3], [4] extended Heaviside’s operator idea into operational matrices, many orthogonal functions are used

m = 4.

to perform the operational matrix of integration. This paper establishes a direct procedure for inversion of a transfer function of s via a general orthogonal function, such as the Haar wavelet. II. REVIEW OF HAAR WAVELETS Let us begin by briefly reviewing the Haar functions [1], [3]. They are defined in the interval [0; 1) by

h0 (t) =

p1m

hi (t) =

pm 02 ; 1

2

k01 2 k0

;

2

0;

 t < k02  t < 2k

(1)

otherwise in [0; 1)

where i = 0; 1; 2; . . . ; m 0 1; m = 2 and  is a positive integer. j and k represent the integer decomposition of the index i, i.e., i = 2j +k 01. In the construction, h0 (t) is called the scaling function and h1 (t) the mother wavelet. Fig. 1 shows the Haar wavelet functions in the case of m = 4. Any function y (t) which is square integrable in the interval 0  t < m01 1 can be expanded into Haar series by y (t) = i=0 ci hi (t) where 1 ci = 0 y(t)hi (t) dt, or in the matrix form

~yT

T

=~ c

1H

where ~ y is the discrete form of the continuous function, y(t) and ~cis called the coefficient vector, they are both column vectors, and H is the Haar wavelet matrix and is defined by

T T H = ~h0T ~h1T 1 1 1 ~hm (2) 01 T where ~h0T ; ~h1T ; . . . ; ~hm 01 are the discrete form of the Haar wavelet

bases; the discrete values are taken form the continuous curves

h0 (t); h1 (t); . . . ; hm01 (t), respectively.

III. OPERATIONAL MATRIX OF INTEGRATION Manuscript received August 23, 1999; revised May 2, 2000. This paper was recommended by Associate Editor P.K.Rajan. J. L. Wu and C. H. Chen are with the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C. C.F. Chen is with the Department of Computer Science, Met College, Boston University, Boston, MA 02215 USA. Publisher Item Identifier S 1057-7122(01)00655-9.

Although historically the idea of the operational matrix was established via the Walsh function [5], logically the one via the block pulse function is more basic [6]. The operational matrix for integration of the block pulse function matrix B , by definition, is given by

1057–7122/01$10.00 © 2001 IEEE

i

0

B ( ) d

 QB B (t) 1

(3)

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121

where 1

1

= m1 0.

..

2

QB

0

.

111

.. .

1 2

1

0

..

1

111

1 2 m2m

0

111

(4)

is called the operational matrix for integration of the block pulse function. In the following, we will present a unified approach to deriving the operational matrix of an orthogonal function set. Let 8 denote an orthogonal function set given by

8 = ~T0

T ~m  1

111

T

T ~m  01

~ i is the column vector, i = 0; 1; 2; . . . ; m 0 1. The integration where  of the orthogonal function set 8 is given by

i

8( ) d  Q8 1 8

0

(5)

where Q8 is the operational matrix for integration of 8. Since the block pulse matrix B (t) is the identity matrix with an appropriate order, the left-hand side of (5) can be expressed as

i

0

i

8( ) d =

0

8 1 B ( ) d = 8 1

From (3) and (6), we obtain

i

0

i

0

()

B  d:

8( ) d = 8 1 QB 1 B (t) = 8 1 QB :

From (5) and (7), we obtain

(6)

= 8 1 QB 1 801 :

(9)

For example, by letting 8 = H in (9), we obtain the operational matrix for integration of the Haar wavelet H given by QH

= H 1 QB 1 H 01 = H 1 QB 1 H T :

IV. INVERSION OF THE LAPLACE TRANSFORM VIA THE OPERATIONAL MATRIX To establish the procedure of inverting Laplace transforms via the operational matrix of integration, we consider the following well-known transform pair as the starting point:

i 0

( ) = Lfx(t)g 1 1 X (s)

()

x t

X s

()

x  d

s

This pair states that the integration in the time domain is corresponding to multiplication of 1=s in the s domain. From the definition of the operational matrix of integration, this pair also means that the integration in the time domain is equivalent to replacing 1=s by the operation matrix Q in the equivalent matrix. Take the following rational transfer function as an example: b

( ) = s + a:

X s

~ x

+a1

i

0

~ x dt

= b 1 ~i

(12b)

where ~ x and ~i  [1 1 1 1 1 1]T are column vectors. Assuming ~ xT = T ~ c 1 H and replacing the integral sign by the matrix of integration QH in (12b), we obtain

1 [I + a 1 QH ] = b 1 [1 1 1 1 1 1]H 01 where I is the identity matrix with the dimension m 2 m. ~cT is given ~ c

by

(8)

Thus, Q8

The discrete form of (12a) is given by

T

(7)

1 8 = 8 1 QB :

Q8

Fig. 2. The inverse Laplace transform of rational transfer function (2=s + 3) with m = 8.

(10)

= [1 1 1 1 1 1]H 01 1 [b 1 I ] 1 [I + a 1 QH ]01 = [1 1 1 1 1 1]H 01 1 QH01 1 [bQH ] 1 [I + aQH ]01 = [1 1 1 1 1 1]H 01 QH01 1 X^ (QH ): (13) 01 = H 1 Q01 1 H T . Substituting it However, from (9), we obtain QH B T

~ c

into (13) yields

= [1 1 1 1 1 1]QB01 1 H T 1 X^ (QH ) = [2m 02m 2m 02m 1 1 1 02m]12m 1 H T 1 X^ (QH ): T ~ x , the inversion of the Laplace transform X (s), is given by T cT 1 H ~ x =~ = [2m 02m 2m 02m 1 1 1 02m]12m 1 H T 1 X^ (QH ) 1 H: T

~ c

(14)

The solution given by (14) is new and much simpler compared with those from previous literature [1], [7]. The above procedures for finding ~ xT , the inverse Laplace transform of X (s), is summarized as follows. ^ (1=s). Step 2) Express X (s) in terms of 1=s, denote it as X ^ (1=s) by the operational matrix QH . Step 3) Replace each 1=s in X Step 4) Calculate ~ xT by ~ xT = [2m 02m 1 1 1 02m]12m 1 H T 1 ^ (QH ) 1 H where m is the number of segments of the Haar X wavelet matrix H . V. EXAMPLES AND RESULTS Three examples are given for demonstration.

A. Rational Transfer Function (2=s + 3) ( ) = sb 1 + as = X^ 1s : (11) ^ (1=s) = Expressing (2=s + 3) in terms of 1=s, we obtain X The transfer function X (s) in (11) can be interpreted as the solution to (2=s)=1 + (3=s). In the case of m = 8, from (14) we obtain T T ~ x = [16 016 1 1 1 016]128 1 H8 1 2QH the equation 1 (I8 + 3QH )01 1 H8 : i x+a1 x dt = b: (12a) The exact solution is x(t) = 2e03t . The results are shown in Fig. 2. 0 Equation (10) can be rewritten as X s

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TABLE I NUMERICAL INVERSION OF LAPLACE TRANSFORM IN TERMS OF OPERATIONAL MATRIX WHERE I IS THE IDENTITY MATRIX AND

~g

= [2m

02m 1 1 1 02m]

H

Fig. 3. The inverse Laplace transform of irrational transfer function s with m 8.

=

Fig. 4. The inverse Laplace transform of irrational transfer function (e =s) with m = 8.

B. Irrational Transfer Function s03=2

^ (1=s) = (1=s)3=2 . Expressing s03=2 in terms of 1=s, we obtain X In the case of m = 8, from (14) we obtain

~xT

= [16

016 1 1 1 016]128 1 H8T 1 QH 1 H8:

The exact solution is x(t) = 2 t= . The results are shown in Fig. 3.

p C. Exponential Transfer Function (e02 s =s)

p ^ (1=s) = (1=s) 1 Expressing (e02 s =s) in terms of 1=s, we obtain X 0 1=2 ]. In the case of m = 8, from(14) we obtain exp[02 1 (1=2) ~xT

016 1 1 1 016]128 1 H8T 1 QH 1 exp 02 1 QH0 1 H8 : p

= [16

The exact solution is x(t) = erfc(1= t). The results are shown in Fig. 4.

VI. CONCLUSION In this paper, a unified method for finding the operational matrix of an orthogonal function set is derived. It is simple and computer oriented. A new method for calculating the inverse Laplace transform is derived, based on the derived operational matrix of the Haar wavelets.

Instead of using contour integration or dictionary-type solution, the operational matrix for integration of an orthogonal function set is used to derive the inverse Laplace transform formula. By using the proposed method, hundred pairs of Laplace transforms of rational transfer functions can be combined into a single algorithm and the inverse of irrational and transcendental transfer functions can be calculated by the same algorithm with proper interpretation. Examples for calculating the inverses of rational, irrational and transcendental transfer functions are illustrated. A Laplace transform pair table (Table I) showing the conventional analytic inverse formula together with the new derived numerical formula by our method is included in this paper. It demonstrates that our method is simple and powerful.

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