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113_1_chapter05

113_1_chapter05 - Chapter 5 THE IMPULSE RESPONSE SEQUENCE...

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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 time-invariant (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 input-output 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 input-output 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 unit-sample 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 time-invariant 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 input-output 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
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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 input-output 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 time-invariance 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 time-shifted unit-sample 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 unit-sample 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) + . . . or, more compactly, x ( n ) = X k = -∞ x ( k ) δ ( n - k ) This is a representation for x ( n ) as a linear combination of the sequences { δ ( n - k ) } . The coefficients of the linear combination are the samples of the sequence x ( n ) itself.
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