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**Unformatted text preview: **Chapter 8 ZERO-INPUT AND ZERO-STATE RESPONSES 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, sayed@ee.ucla.edu. Our ultimate objective from the discussions in Chapters 7 and 8 is to end up with a procedure for determining a closed-form expression for the solution of constant-coefficient difference equations over the interval n 0 say, for example, for the solution of an equation of the form y ( n )- 2 y ( n- 1) + 4 y ( n- 2) = x ( n ) , y (- 1) = 1 , y (- 2) = 0 with knowledge of both the initial conditions and the input sequence. We could in principle iterate the above equation, starting from the given initial conditions and using the given input sequence to determine the values of y ( n ) for n - 2. In general, such iterative procedures do not lead to closed-form expressions but rather to a tabular description of the response sequence y ( n ). We shall describe in this chapter a systematic procedure for determining the response of difference equations to an excitation and given certain initial conditions. Alternative procedures that employ transform techniques will be described later in the book. While the method of computation in this chapter involves only straightforward calculations, what confuses the reader is mostly the terminology. For example, by the end of this chapter, the reader will need to know what each of the following terms means: particular solution zero-input solution homogeneous solution zero-state solution complete solution forced solution transient solution unforced solution steady-state solution natural solution The steps for finding each of these solutions are simple, but only if the reader understands what each solution means. Before we proceed, we may mention that here is one important special case for which we already know how to determine the complete solution of a constant-coefficient difference equation, namely, when the equation is assumed to describe a relaxed and causal system and therefore an LTI system. In this case, we saw in Chapter 7 that we can first determine the impulse response sequence by solving a homogeneous equation and then convolve it with the given input sequence. 67 68 Zero-Input and Zero-State Responses Chapter 8 In this chapter, however, we are interested in the more general case in which the difference equation need not describe an LTI system, for example when the difference equation has nonzero initial conditions. The techniques developed in this chapter can therefore be applied to both LTI and non- LTI systems. In the case of LTI systems, they can help avoid some of the effort that goes into computing the impulse response and then the required convolution....

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