Z Transformation by Pascal Matrix and its Applications in the Design of IIR Filters

 

F. J. García–Ugalde*¹, B. Psenicka², M.O. Jiménez–Salinas³

 

^1,2,3 Universidad Nacional Autónoma de México. Facultad de Ingeniería, División de Ingeniería Eléctrica Circuito Exterior, Ciudad Universitaria, Coyoacán 04510, Ciudad de México, Mexico. *E–mail: fgarciau@unam.mx

 

ABSTRACT

In this work, we summarize a direct method to transform the low–pass continuous–time transfer function H(s) to several discrete–time H(z) transfer functions. Our algorithm uses the Pascal matrix that is built from the rows of a Pascal Triangle. The inverse transformation is obtained with the Pascal matrix without computing the determinant of the system, which simplifies the process to obtain the associated analog transfer function H(s) if the discrete transfer function H(z) is known. In addition, the algorithm is easy to program on a personal computer or scientific calculator because all the computations are made using matrices. The algorithm presented is illustrated with numerical examples.

Keywords: Analog filters, Digital filters, Pascal matrix, Bilinear Z–transform, IIR filters.

 

RESUMEN

En el presente trabajo, resumimos un método directo para transformar la función de transferencia paso bajas del tiempo continuo H(s) a diferentes funciones de transferencia del tiempo discreto H(z). Nuestro algoritmo utiliza la matriz de Pascal que se construye con los renglones del triángulo de Pascal. La transformación inversa se obtiene con la matriz de Pascal sin calcular el determinante del sistema, lo que simplifica el proceso para obtener la correspondiente función de transferencia analógica H(s) si la función de transferencia discreta H(z) es conocida. Además, el algoritmo es fácil de programar en una computadora personal o una calculadora científica, debido a que todos los cálculos se realizan con matrices. El algoritmo presentado es ilustrado con ejemplos numéricos.

 

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