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Mathematical Problems in Engineering
Volume 2014, Article ID 905791, 9 pages
Research Article

Transformation Matrix for Time Discretization Based on Tustin’s Method

College of Precision Instrument & Optoelectronics Engineering, Tianjin University, Tianjin 300072, China

Received 16 April 2014; Revised 27 June 2014; Accepted 10 July 2014; Published 18 August 2014

Academic Editor: Ivanka Stamova

Copyright © 2014 Yiming Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper studies rules in transformation of transfer function through time discretization. A method of using transformation matrix to realize bilinear transform (also known as Tustin’s method) is presented. This method can be described as the conversion between the coefficients of transfer functions, which are expressed as transform by certain matrix. For a polynomial of degree n, the corresponding transformation matrix of order n exists and is unique. Furthermore, the transformation matrix can be decomposed into an upper triangular matrix multiplied with another lower triangular matrix. And both have obvious regularity. The proposed method can achieve rapid bilinear transform used in automatic design of digital filter. The result of numerical simulation verifies the correctness of the theoretical results. Moreover, it also can be extended to other similar problems. Example in the last throws light on this point.