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A Reconsideration of the Causality Requirement in Proving the z-Transform of a Discrete Convolution Sum  

Chung Tae-Sang (중앙대 공대 전자전기공학과)
Lee Jae Seok (중앙대 공대 제어계측공학과)
Publication Information
The Transactions of the Korean Institute of Electrical Engineers D / v.52, no.1, 2003 , pp. 51-54 More about this Journal
Abstract
The z-transform method is a basic mathematical tool in analyzing and designing digital signal processing systems for discrete input and output signals. There are may cases where the output signal is in the form of a discrete convolution sum of an input function and a designed digital processing algorithm function. It is well known that the z-transform of the convolution sum becomes the product of the two z-transforms of the input function and the digital processing function, whose proofs require the causality of the digital signal processing function in the almost all the available references. However, not all of the convolution sum functions are based on the causality. Many digital signal processing systems such as image processing system may depend not on the time information but on the spatial information, which has nothing to do with causality requirement. Thus, the application of the causality-based z-transform theorem on the convolution sum cannot be used without difficulty in this case. This paper proves the z-transform theorem on the discrete convolution sum without causality requirement, and make it possible for the theorem to be used in analysis and desing for any cases.
Keywords
z변환;역 z변환;이산 콘볼루션 적산;인과성;
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