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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. Leo Lam © Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's.
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Leo Lam © Signals and Systems EE235. Courtesy of Phillip Leo Lam ©
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Leo Lam © Signals and Systems EE235 Leo Lam.
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The. of and a to in is you that it he for.
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