SP_II_P02_4p - Introduction to Signal Processing 56...

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Introduction to Signal Processing 5 6 Copyright © 2005-2009 – Hayder Radha D. Total Response of LTI Systems The convolution integral provided a viable tool to derive the output () yt of a Linear Time-Invariant (LTI) system, which has an impulse response ( ) ht , in response to an input ( ) x t : () () ( ) ( ) xt ht x ht d τ ττ −∞ =∗= Introduction to Signal Processing 5 7 Copyright © 2005-2009 – Hayder Radha However, it is well known that: ¾ A system can generate output in response to internal energy stored within the system, even if the (external) input to the system is zero . ¾ Internal energies stored within a system are sometimes referred to as initial conditions . These initial conditions impact the response of the system independent of the nature of the external input x t . Introduction to Signal Processing 5 8 Copyright © 2005-2009 – Hayder Radha The convolution integral does not provide the response of the system to the internal energy stored within the system; and hence we need another tool to identify the system response when the input is zero 0 xt = , and when the system generates an output in response to some internal energy. This type of response is known as the Zero-Input Response . Introduction to Signal Processing 5 9 Copyright © 2005-2009 – Hayder Radha Definition Zero-input Response (ZIR) ZIR is the response of the system when no input is applied, i.e. () 0 x t = . The response of the system is then the result of internal energy storages, and it is independent of any external inputs.
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Introduction to Signal Processing 6 0 Copyright © 2005-2009 – Hayder Radha Definition Zero-state Response (ZSR) The response of the system when the system is in zero state, meaning the absence of all internal energy storages; i.e. all initial conditions are zero . In other words, the ZSR is the response of the system to (external) input signals. Introduction to Signal Processing 6 1 Copyright © 2005-2009 – Hayder Radha The response provided by the convolution integral assumes that the system has zero energy, or as commonly known, the system is in zero state . In other words, convolution provides the Zero-State Response of an LTI system: () () ()( ) x th t xh t d τ ττ −∞ =∗= Zero State Response Introduction to Signal Processing 6 2 Copyright © 2005-2009 – Hayder Radha Since an LTI system adheres to the superposition principle, the total response of the system is the summation of the zero-state response and the zero-input response: Total Response Input =+ Total Response Zero State Response Zero Response Hence, x t h t Input =∗+ TotalResponse Zero Response Introduction to Signal Processing 6 3 Copyright © 2005-2009 – Hayder Radha Therefore, a key question is how to derive the zero- input response of an LTI system. To answer this question, we focus on a class of LTI systems that are known as linear differential systems . These are LTI systems that satisfy linear differential equations. To motivate the study of linear differential systems, we look at a familiar application from linear circuits.
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This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

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SP_II_P02_4p - Introduction to Signal Processing 56...

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