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**Unformatted text preview: **Chapter 7 HOMOGENEOUS DIFFERENCE EQUATIONS 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, [email protected] Up to this point in the book, we have introduced several properties of discrete-time signals and systems (such as periodicity, causality, stability, linearity, and time-invariance). From now on, we shall develop tools for the analysis of discrete-time systems. These tools will enable us to answer questions such as how to determine the response of a system to an input sequence in a more systematic manner? How to characterize the behavior of a system in the frequency domain? We shall not study general discrete-time systems. Instead, we shall focus on the important subclass of systems that are described by constant-coefficient difference equations . Our objective in this chapter, and in the following one, is to describe a procedure for determining the response systems described by constant-coefficient difference equations to certain input sequences. A useful first step towards this objective is to understand how to solve homogeneous difference equations. Homogenous Equations . As a motivation, assume we are asked to identify a sequence y ( n ) that satisfies the difference equation y ( n )- ay ( n- 1) = 0 for all n and for some given real coefficient a . This is a first-order difference equation. The equation is further said to be homogeneous since its right-hand side is zero. Note that the equation, as described, does not correspond to a system since it does not specify any dependence on an input sequence x ( n ). Nevertheless, we could interpret it as defining the response of the class of systems y ( n )- ay ( n- 1) = x ( n ) to the zero input sequence, x ( n ) = 0. By determining all the sequences { y ( n ) } that satisfy the homogeneous equation y ( n )- ay ( n- 1) = 0 we are therefore determining all the possible responses of the above class of systems to the zero input sequence. 58 59 Now it is immediate to verify that the exponential sequence y ( n ) = a n satisfies the homogeneous equation, since a n- a ( a n- 1 ) = 0 . It is also easy to verify that any multiple of a n is a solution as well, i.e., y ( n ) = Ca n , for any constant C. Observe that y (0) = C so that every possible choice for the initial condition y (0) in the class of systems { y ( n )- ay ( n- 1) = x ( n ) } results in a different response to x ( n ) = 0. We therefore find that even trivial homogeneous equations of this form admit an infinite number of solutions. The same conclusion holds for higher-order homogeneous equations....

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