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Leo Lam © Signals and Systems EE235

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Todays menu Leo Lam © Fourier Transform table posted Laplace Transform

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Leo Lam © Focus on: –Doing (Definitions and properties) –Understanding its possibilities (ROC) –Poles and zeroes (overlap with EE233)

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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

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Region of Convergence Example: Find the Laplace Transform of: Leo Lam © We have a problem: the first term for t= doesnt always vanish!

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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!

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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 ©

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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

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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

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Laplace Example No Laplace Example: Leo Lam © ROC: Re(s)>-1 ROC: Re(s)<-3 Combined: ROC: None!

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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

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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

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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

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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 ©

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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

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Plotting Poles and Zeros Leo Lam © What does it look like?

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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 ©

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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

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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 ©

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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 ©

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Laplace properties (unilateral) Leo Lam © Linearity: f(t) + g(t) F(s) + G(s) Time-shifting: Frequency Shifting: Differentiation: and Time-scaling

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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

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Another Inverse Example Leo Lam © Example, find h(t) (assuming causal): Using linearity and partial fraction:

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Another Inverse Example Leo Lam © Here is the reason:

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Summary Laplace intro Region of Convergence Causality Existence of Fourier Transform & relationships Leo Lam ©

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