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Unformatted text preview: Chapter 4 DISCRETE-TIME SYSTEMS Copyright c 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, firstname.lastname@example.org. Now that we have developed a basic understanding of what discrete-time signals (or sequences) are, we move on to study discrete-time systems and some of their properties. As the reader will soon realize, this chapter includes many definitions about systems and their characterizations. While the multitude of definitions might be overwhelming at first sight, the definitions are in most cases straightforward to grasp. In addition, the reader will be able to acquire sufficient familiarity with these definitions and concepts as the discussion in the book progresses. Systems A system is defined as a mapping between an input sequence and an output sequence; it operates on an input sequence and generates an output sequence. What makes a system special is that the input sequence, x ( n ), must uniquely define the output sequence, y ( n ). In other words, there should be no ambiguity about what the output sequence will be for any given input sequence. Schematically, we write y ( n ) = S [ x ( n )] to denote a generic system. This notation means that a system S is being applied to an input sequence x ( n ) in order to generate an output sequence y ( n ). In general, each term of the output sequence y ( n ) can be a function of present, past, or future terms of the input sequence x ( n ). For the purposes of the treatment in this book, all input-output transforma- tions (or relations) S corresponding to systems will generally be described by mathematical equations. Examples of Systems Consider the transformation y 2 ( n ) = | x ( n ) | , which describes a relation between an input sequence and an output sequence. This transformation does not describe a system. This is because for any given input sequence x ( n ), the resulting output sequence y ( n ) is not uniquely defined. For instance, assume that x ( n ) = 4 u ( n ), then y ( n ) could be y ( n ) = 2 u ( n ) or y ( n ) =- 2 u ( n ). On the other hand, the following are examples of systems: 25 26 Discrete-Time Systems Chapter 4 1. y ( n ) = x ( n ). This is a very simple system. It maps the input sequence to itself. Such a system can be regarded as an model for an ideal (i.e., lossless) wire connection transmitting a signal from one point to another in a communications system. 2. y ( n ) = x ( n- 1). This is a unit delay system. It delays the input sequence by one unit of time....
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