Using linearity we have where we can deduce that the ROC of x t is Res 2 which

# Using linearity we have where we can deduce that the

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---- Using linearity, we have where we can deduce that the ROC of x ( t ) is Re{s}>-2, which contains the intersection of the ROCs of X 1 ( s ) and X 2 ( s ). Also note that the pole at s = -1 is canceled by the zero at s = -1. 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 40

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Examples Example 11 Example 12 Example 13 Example 14 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 41
Examples (Solution to Example 11) (Solution to Example 12) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 42

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Examples (Solution to Example 13) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 43
Examples (Solution to Example 14) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 44

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¾ In an LTI system with impulse response function h ( t ), the Laplace transform H ( s ) is referred to the system function or the transfer function . Recall the causality condition from time domain: h ( t ) = 0 for any t < 0. [A necessary condition of causality] This implies that the ROC of H ( s ) must contain the right-half plane. (Note, the converse may not be true.) [A sufficient condition of causality] If H ( s ) is rational and its ROC is the right-half plane to the right of the rightmost pole, then the system is causal. Example 15 Causality from the System Function 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 45 h(t)
Causality from the System Function Solution to Example 15 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 46

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Stability from the System Function Recall the stability condition from time domain: which corresponds to the existence condition of the Fourier transform of h ( t ). This means that the ROC of H ( s ) should include the j -axis . Using the causality conditions, we can deduce the following result: A causal system with rational H ( s ) is stable if an only if all the poles of H ( s ) lie in the left-half of the s -plane, i.e. , all of the poles have negative real part. Example 16 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 47 (Hint: There are two poles, leading to 3 possibilities of ROC patterns. We should discuss one by one.)
Causality and Stability from the System Function Long discussion of Example 16 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 48

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Causality and Stability from the System Function (continued) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 49
System Functions of LTI Systems ¾ What to do: For an LTI system described by a linear constant-coefficient differential equation, we can obtain the system function from the Laplace transform. We can further characterize the system behavior by analyzing the system function. ¾ First-order differential equations: Suppose the input-output of an LTI system is described as follows Applying the Laplace transform which yields , implying a transfer function Note: Here, the ROC is not specified. There is one pole at s = - a , corresponding to two possibilities of the ROCs. Such an ambiguity can be resolved by incorparating the causality and stability conditions. 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 50

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Relate System Function to System Behavior 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 51 - a s -plane s = j In this example, amplitude = reciprocal of the length of the pole vector, phase angle = - The same method can be applied to analyze higher order systems.
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