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Leo Lam © Signals and Systems EE235
Todays menu Leo Lam © Fourier Transform table posted Laplace Transform
Leo Lam © Focus on: –Doing (Definitions and properties) –Understanding its possibilities (ROC) –Poles and zeroes (overlap with EE233)
Laplace Transform Definition: Where Inverse: Leo Lam © Good news: We dont need to do this, just use the tables. Inverse Laplace expresses f(t) as sum of exponentials with fixed has specific requirements
Region of Convergence Example: Find the Laplace Transform of: Leo Lam © We have a problem: the first term for t= doesnt always vanish!
Region of Convergence Example: Continuing… In general: for In our case if: then Leo Lam © For what value of s does: Pole at s=-3. Remember this result for now!
Region of Convergence A very similar example: Find Laplace Transform of: For what value does: This time: if then Same result as before! Leo Lam ©
Region of Convergence Comparing the two: Leo Lam © ROC -3 ROC -3 s-plane Laplace transform not uniquely invertible without region of convergence Casual, Right-sided Non-casual, Left-sided
Finding ROC Example Example: Find the Laplace Transform of: From table: Leo Lam © ROC: Re(s)>-6 ROC: Re(s)>-2 Combined: ROC: Re(s)>-2
Laplace Example No Laplace Example: Leo Lam © ROC: Re(s)>-1 ROC: Re(s)<-3 Combined: ROC: None!
Laplace and Fourier If the ROC includes the j–axis, then the signal has a Fourier Transform where s= j Caution: If the ROC doesn't quite include the j-axis (if poles lie on the jw-axis), then it might still have a Fourier transform, but it is not given by s=j. Leo Lam © σ jw ROC –a–a
Laplace and Fourier No Fourier Transform Example: ROC exists: Laplace ok ROC does not include j-axis, Fourier Transform is not F(j). (In fact, here it does not exist). Leo Lam © ROC: Re(s)>-3 ROC: Re(s)<-1 Combined: -3
Finding ROC Example Example: Find the Laplace Transform of: From table: Thus: With ROC: Leo Lam © ROC: Re(s)<-2 ROC: Re(s)>-3 Combined: ROC: -3
Poles and Zeros (the Xs and Os) H(s) is almost always rational for a physical system: Rational = Can be expressed as a polynomial ZEROs = where H(s)=0, which is POLES = where H(s)=, which is Example: Leo Lam ©
Plotting Poles and Zeros Leo Lam © H(s) is almost always rational for a physical system: Plot is in the s-plane (complex plane) σ jωjω x x o
Plotting Poles and Zeros Leo Lam © What does it look like?
ROC Properties (Summary) All ROCs are parallel to the j–axis Casual signal right-sided ROC and vice versa Two-sided signals appear either as a strip or no ROC exist (no Laplace). For a rational Laplace Transform, the ROC is bounded by poles or. If ROC includes the j-axis, Fourier Transform of the signal exists = F(j). Leo Lam ©
Laplace and Fourier Very similar (Fourier for Signal Analysis, Laplace for Control, Circuits and System Designs) ROC includes the j-axis, then Fourier Transform = Laplace Transform (with s=j) If ROC does NOT include j-axis but with poles on the j-axis, FT can still exist! Example: But Fourier Transform still exists: No Fourier Transform if ROC is Re(s)<0 (left of j-axis) Leo Lam © ROC: Re(s) > 0 Not including jw-axis
Ambiguous? Define it away! Bilateral Laplace Transform: Unilateral Laplace Transform (for causal system/signal): For EE, its mostly unilateral Laplace (any signal with u(t) is causal) Not all functions have a Laplace Transform (no ROC) Leo Lam ©
Inverse Laplace Example, find f(t) (assuming causal): Table: What if the exact expression is not in the table? –Hire a mathematician –Make it look like something in the table (partial fraction etc.) Leo Lam ©
Laplace properties (unilateral) Leo Lam © Linearity: f(t) + g(t) F(s) + G(s) Time-shifting: Frequency Shifting: Differentiation: and Time-scaling
Laplace properties (unilateral) Leo Lam © Multiplication in time Convolution in Laplace Convolution in time Multiplication in Laplace Initial value Final value Final value result Only works if All poles of sF(s) in LHP
Another Inverse Example Leo Lam © Example, find h(t) (assuming causal): Using linearity and partial fraction:
Another Inverse Example Leo Lam © Here is the reason:
Summary Laplace intro Region of Convergence Causality Existence of Fourier Transform & relationships Leo Lam ©
Leo Lam © Signals and Systems EE235 Lecture 30.
Leo Lam © Signals and Systems EE235. Leo Lam © Happy Tuesday! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).
Signal and System I The inverse z-transform Any path in the ROC to make the integral converge Example ROC |z|>1/3.
The z-Transform: Introduction Why z-Transform? 1.Many of signals (such as x(n)=u(n), x(n) = (0.5) n u(- n), x(n) = sin(nω) etc. ) do not have a DTFT. 2.Advantages.
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Department of Computer Eng. Sharif University of Technology Discrete-time signal processing Chapter 3: THE Z-TRANSFORM Content and Figures are from Discrete-Time.
EE-2027 SaS, L14 1/14 Lecture 14: Laplace Transform Properties 5 Laplace transform (3 lectures): Laplace transform as Fourier transform with convergence.
Leo Lam © Signals and Systems EE235. Leo Lam © Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's.
Leo Lam © Signals and Systems EE235 Lecture 31.
Signals and Systems Fall 2003 Lecture #17 4 November Motivation and Definition of the (Bilateral) Laplace Transform 2. Examples of Laplace Transforms.
EE-2027 SaS, L13 1/13 Lecture 13: Inverse Laplace Transform 5 Laplace transform (3 lectures): Laplace transform as Fourier transform with convergence factor.
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ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Familiar Properties Initial and Final Value Theorems Unilateral Laplace.
Leo Lam © Signals and Systems EE235 Lecture 25.
بسم الله الرحمن الرحيم University of Khartoum Department of Electrical and Electronic Engineering Third Year Dr. Iman AbuelMaaly Abdelrahman.
Leo Lam © Signals and Systems EE235. Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.
The Z-Transform Quote of the Day Such is the advantage of a well-constructed language that its simplified notation often becomes the source of profound.
System analysis in the time domain involves ( finding y(t)): Solving the differential equation Using the convolution integral OR Both Techniques can results.
Signals and Systems Using MATLAB Luis F. Chaparro.
EE-2027 SaS, L12 1/13 Lecture 12: Laplace Transform 5 Laplace transform (3 lectures): Laplace transform as Fourier transform with convergence factor. Properties.
Lecture 20 Outline: Laplace Transforms Announcements: Reading: “6: The Laplace Transform” pp. 1-9 HW 7 posted, due next Wednesday My OHs Monday cancelled,
Signals and Systems Fall 2003 Lecture #18 6 November 2003 Inverse Laplace Transforms Laplace Transform Properties The System Function of an LTI System.
Leo Lam © Signals and Systems EE235. Leo Lam © Futile Q: What did the monserous voltage source say to the chunk of wire? A: "YOUR.
Leo Lam © Signals and Systems EE235. Leo Lam © Laplace Examples A bunch of them.
Signals and Systems Fall 2003 Lecture #22 2 December Properties of the ROC of the z-Transform 2.Inverse z-Transform 3.Examples 4.Properties of the.
Leo Lam © Signals and Systems EE235. Courtesy of Phillip Leo Lam ©
Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.
Leo Lam © Signals and Systems EE235 Leo Lam.
Leo Lam © Signals and Systems EE235. Today’s menu Leo Lam © Laplace Transform.
Leo Lam © Signals and Systems EE235. Today’s menu Leo Lam © Almost done! Laplace Transform.
THE LAPLACE TRANSFORM LEARNING GOALS Definition The transform maps a function of time into a function of a complex variable Two important singularity functions.
Meiling chensignals & systems1 Lecture #07 Z-Transform.
9.0 Laplace Transform 9.1 General Principles of Laplace Transform linear time-invariant Laplace Transform Eigenfunction Property y(t) = H(s)e st h(t)h(t)
10.0 Z-Transform 10.1 General Principles of Z-Transform linear, time-invariant Z-Transform Eigenfunction Property y[n] = H(z)z n h[n]h[n] x[n] = z n.
1 Laplace Transform. CHAPTER 4 School of Computer and Communication Engineering, UniMAP Hasliza A Samsuddin EKT 230.
Laplace Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University.
Leo Lam © Signals and Systems EE235. Fourier Transform: Leo Lam © Fourier Formulas: Inverse Fourier Transform: Fourier Transform:
Laplace Transform (2) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University.
The Laplace Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1.
Signal and Systems Prof. H. Sameti Chapter 9: Laplace Transform Motivatio n and Definition of the (Bilateral) Laplace Transform Examples of Laplace.
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Leo Lam © Signals and Systems EE235. Leo Lam © x squared equals 9 x squared plus 1 equals y Find value of y.
Leo Lam © Signals and Systems EE235. Leo Lam © Today’s menu Homework 2 due now Convolution!
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