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

This preview shows page 39 - 51 out of 55 pages.

---- 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
Image of page 39

Subscribe to view the full document.

Examples Example 11 Example 12 Example 13 Example 14 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 41
Image of page 40
Examples (Solution to Example 11) (Solution to Example 12) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 42
Image of page 41

Subscribe to view the full document.

Examples (Solution to Example 13) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 43
Image of page 42
Examples (Solution to Example 14) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 44
Image of page 43

Subscribe to view the full document.

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)
Image of page 44
Causality from the System Function Solution to Example 15 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 46
Image of page 45

Subscribe to view the full document.

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.)
Image of page 46
Causality and Stability from the System Function Long discussion of Example 16 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 48
Image of page 47

Subscribe to view the full document.

Causality and Stability from the System Function (continued) 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 49
Image of page 48
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
Image of page 49

Subscribe to view the full document.

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.
Image of page 50
Image of page 51
  • Fall '13

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes