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Unformatted text preview: Chapter 5 THE IMPULSE RESPONSE SEQUENCE 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] The impulse response sequence plays a fundamental role in the characterization of linear timeinvariant (LTI) systems, so much so that knowledge of the impulse response sequence alone is all one needs in order to fully characterize the behavior of an LTI system. For instance, explicit knowledge of a mathematical model describing the inputoutput relation of the system is not even needed! This chapter expands on these remarks and emphasizes time and again the significance of the impulse response sequence. Impulse Response Sequence Consider a system described generically by the inputoutput relation y ( n ) = S [ x ( n )] The system may or may not be LTI. The impulse response sequence of S is defined as the response of the system to the unitsample sequence, x ( n ) = δ ( n ). We denote the impulse response sequence by the special symbol h ( n ) so that h ( n ) = S [ δ ( n )] As mentioned in the introductory remarks to this chapter, the sequence h ( n ) has many spe cial properties in the case of linear timeinvariant systems. For example, for such systems, knowledge of h ( n ) alone enables us to determine whether an LTI system is BIBO stable or not and whether it is causal or not. It also enables us to determine the response of the system to any input sequence x ( n ), as well as determine a mathematical inputoutput relation for the system. In later chapters, we shall see that h ( n ) also conveys information about the properties of LTI systems in the frequency domain. The Convolution Sum 38 39 We start to establish some of the above claims. We first address the fact that the response of an LTI system to any input sequence can be determined solely from knowledge of the impulse response sequence. In other words, a mathematical inputoutput relation for the system is not needed. Indeed, consider an LTI system S and let h ( n ) denote its impulse response sequence. Due to the timeinvariance of the system, its response to δ ( n k ), for any k , will be h ( n k ). That is, if the input sequence is a timeshifted unitsample sequence then the result will be an equally shifted impulse response sequence. Now, an arbitrary input sequence x ( n ) can be expressed as a linear combination of shifted unitsample sequences, namely, for any x ( n ) we can write x ( n ) = ... + x ( 1) δ ( n + 1) + x (0) δ ( n ) + x (1) δ ( n 1) + ......
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 Spring '08
 Walker
 Electrical Engineering, Digital Signal Processing, Signal Processing, LTI system theory, impulse response sequence

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