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IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 11, NOVEMBER 2002
The Bilinear Z Transform by Pascal Matrix and Its Application in the Design of Digital Filters B. Pˇseniˇcka, F. García-Ugalde, and A. Herrera-Camacho
Abstract—In this letter, the Pascal matrix is used for transforming the normalized analog transfer function ( ) from the lowpass to the lowpass and highpass discrete transfer functions ( ). This algorithm is very simple; therefore, the transfer function ( ) can be easily transformed to the domain using an appropriate calculator. The inverse Pascal matrix can be obtained without computing the determinant of the system, and then it is very easy to obtain the associated analog transfer function ( ) if the discrete transfer function ( ) is known. Index Terms—Filter transformation.
design,
Pascal
matrix,
In order to obtain the coefficients and of the vectors and from the vectors and respectively, we must compare the numerators and denominators of the transfer functions (1) and (5). If we substitute (3) in (1), we obtain for the following expressions:
structures,
I. INTRODUCTION
T
HE ANALOG circuit is described by the transfer function
From (1), we can get the vectors
By comparing the coefficients with the same exponents of the variable , we can acquire the matrix equation (7)
(1) and
(7)
in the form
(2) and are real coefficients. where Discrete filters can be characterized by the transfer function , which has the real coefficients and , and it can be obtained using the bilineal transformation [1] (3)
A similar equation can be obtained for the vector . These equations can be represented in the following compact form
where is the Pascal matrix, and the vectors represented by (8)
where
and
are
(8) (4)
and represent the lowpass corner frequency and In (4), sampling frequency, respectively
(9)
(5) From (5), we can have the vectors
and
[2]
(6) Manuscript received May 23, 2001; revised June 14, 2002. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Geert Leus. The authors are with the Universidad Nacional Autonóma de México, Facultad de Ingeniería, Departamento de Telecomunicaciones, Ciudad Universitaria, Mexico D.F. 04510 Mexico (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LSP.2002.804420
From the classical Pascal triangle (9), we can observe that the creates the last column in the Pascal coefficient of base matrix of (7) with the exception of the even rows, which have negative constants. We conclude that the Pascal matrix can be formed by taking into account the following considerations [3]. • In the first row of the Pascal matrix the elements must be ones. • The elements of the last column can be computed by (10) where
1070-9908/02$17.00 © 2002 IEEE
.
ˇ ˇ PSENI CKA et al.: BILINEAR
TRANSFORM BY PASCAL MATRIX
• The remaining elements achieved using
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of the Pascal matrix can be (11)
where is a variant of a Pascal matrix in which the first row elements must be ones; the first column can be obtained from can be established using (10). The remaining elements
where where
Then, element of the matrix (13) can be obtained using the relation (12) (12) (13)
III. EXAMPLES In these examples, we shall transform the lowpass transfer to lowpass and highpass transfer function function using the following features specified by Hz
In the case that the transfer function has order equal to four, the Pascal matrix can be written in the form
(14)
(19)
A. Lowpass Transformation From the Domain
Domain to the
The transfer function coefficients and for can then be obtained keeping the first row as ones and using (10) and (11) in the following form:
II. TRANSFORMATION FROM LOWPASS TO HIGHPASS In order to transform the lowpass transfer function to the high, we can substitute the variable for pass transfer function in (3). Then, in this case, we have
(20)
(15) and (21) (16) In the preceding relation, represents the cut-off frequency of the highpass and the sampling frequency, respectively. In case of the substitution of (15) into (1) and comparing the numerator , we can obtain with (5) for
Then the coefficients values are
and the transfer function
takes the form
(17)
(22)
If we compare the coefficients of both sides of (17) with the same exponents, then we can get the system of equations in the matrix form
The corresponding attenuation of the lowpass filter is shown in the Fig. 1. B. Transformation of Lowpass to Highpass From the Domain to the Domain Using the Pascal matrix
or
(18)
for
, we can have
(23)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 11, NOVEMBER 2002
Fig. 2.
Fig. 1. Attenuation and phase of the lowpass filter.
Attenuation and phase of the highpass filter.
IV. CONCLUSION (24)
The highpass transfer function coefficients can be calculated using the expressions (23) and (24)
In this letter, we have used the Pascal matrix for transforming the analog lowpass transfer function to the discrete transfer functions of the lowpass and highpass filters. This algorithm is very simple, and it is very easy also to transform discrete transfer functions to the corresponding analog transfer function due to the inverse Pascal matrix. To obtain the inverse Pascal matrix, we do not need to calculate the determinant of the matrix system [2]. REFERENCES
In this way, the highpass transfer function is given by
(25) The attenuation of the highpass filter obtained from (25) is shown in Fig. 2.
[1] T. W. Parks and C. Burrus, Digital Filter Design. New York: Wiley, 1987. [2] W. Klein, Finite Systemtheorie, B. G. Teubner Stuttgart, Ed: Teubner Studienbucher, 1976. [3] V. Biolkova and D. Biolek, “Generalized pascal matrix of first order S –Z transforms,” in Proc. ICECS, Pafos, Cyprus, 1999. [4] R. Rabiner and B. Gold, Theory and Applications of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
