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Signals and Systems EE235 Leo Lam ©

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**Today’s menu Fourier Transform table posted Laplace Transform**

Leo Lam ©

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**Laplace Transform Focus on: Doing (Definitions and properties)**

Understanding its possibilities (ROC) Poles and zeroes (overlap with EE233) Leo Lam ©

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**Laplace Transform Definition: Where Inverse:**

Good news: We don’t need to do this, just use the tables. Fourier Series coefficient dk differs from its F(w) equivalent by 2pi Inverse Laplace expresses f(t) as sum of exponentials with fixed s has specific requirements Leo Lam ©

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**Region of Convergence Example: Find the Laplace Transform of:**

We have a problem: the first term for t=∞ doesn’t always vanish! Leo Lam ©

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**Region of Convergence Example: Continuing… In general: for**

In our case if: then For what value of s does: Pole at s=-3. Remember this result for now! Leo Lam ©

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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! Note that both cases have the region dissected at s=-3, which is the ROOT of the Laplace Transform. Leo Lam ©

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**Region of Convergence Laplace transform not uniquely**

Comparing the two: Laplace transform not uniquely invertible without region of convergence ROC -3 ROC -3 Non-casual, Left-sided Casual, Right-sided Laplace transform not uniquely invertible without region of convergence s-plane Leo Lam ©

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**Finding ROC Example Example: Find the Laplace Transform of:**

From table: ROC: Re(s)>-6 ROC: Re(s)>-2 Combined: ROC: Re(s)>-2 Causal signal: Right-sided ROC (at the roots). Leo Lam ©

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**Laplace Example No Laplace Example: ROC: Re(s)>-1 ROC: Re(s)<-3**

Combined: ROC: None! No Laplace Transform since there is no overlapped ROC! Leo Lam ©

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Laplace and Fourier If the ROC includes the jw –axis, then the signal has a Fourier Transform where s= jw Caution: If the ROC doesn't quite include the jw-axis (if poles lie on the jw-axis), then it might still have a Fourier transform, but it is not given by s=jw. σ jw ROC –a No Laplace Transform since there is no overlapped ROC! Leo Lam ©

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**Laplace and Fourier No Fourier Transform Example:**

ROC exists: Laplace ok ROC does not include jw-axis, Fourier Transform is not F(jw). (In fact, here it does not exist). ROC: Re(s)>-3 ROC: Re(s)<-1 Combined: -3<ROC<-1 No Laplace Transform since there is no overlapped ROC! Leo Lam ©

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**Finding ROC Example Example: Find the Laplace Transform of:**

From table: Thus: With ROC: ROC: Re(s)<-2 ROC: Re(s)>-3 Combined: ROC: -3<Re(s)<-2 x o Causal signal: Right-sided ROC (at the roots). Leo Lam ©

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**Poles and Zeros (the X’s and O’s)**

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**

H(s) is almost always rational for a physical system: Plot is in the s-plane (complex plane) σ jω x o Leo Lam ©

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**Plotting Poles and Zeros**

What does it look like? Leo Lam ©

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**ROC Properties (Summary)**

All ROCs are parallel to the jw –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 jw-axis, Fourier Transform of the signal exists = F(jw). If it has poles on the jw-axis, FT can still exist. However, it is no longer s=jw, almost always something else. 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 jw-axis, then Fourier Transform = Laplace Transform (with s=jw) If ROC does NOT include jw-axis but with poles on the jw-axis, FT can still exist! Example: But Fourier Transform still exists: No Fourier Transform if ROC is Re(s)<0 (left of jw-axis) ROC: Re(s) > 0 Not including jw-axis If it has poles on the jw-axis, FT can still exist. However, it is no longer s=jw, almost always something else. Leo Lam ©

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**Ambiguous? Define it away!**

Bilateral Laplace Transform: Unilateral Laplace Transform (for causal system/signal): For EE, it’s mostly unilateral Laplace (any signal with u(t) is causal) Not all functions have a Laplace Transform (no ROC) Laplace transform not uniquely invertible without region of convergence 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.) Hire a Mathematician! Or write F(s) in recognisable terms and use the table (using Laplace Properties) Leo Lam ©

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**Laplace properties (unilateral)**

Linearity: f(t) + g(t) F(s) + G(s) Time-shifting: Frequency Shifting: Differentiation: and Hire a Mathematician! Or write F(s) in recognisable terms and use the table (using Laplace Properties) Time-scaling Leo Lam ©

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**Laplace properties (unilateral)**

Multiplication in Laplace Convolution in time Multiplication in time Convolution in Laplace Initial value Final value Final value result Only works if All poles of sF(s) in LHP Leo Lam ©

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**Another Inverse Example**

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

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**Another Inverse Example**

Here is the reason: Leo Lam ©

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**Summary Laplace intro Region of Convergence Causality**

Existence of Fourier Transform & relationships Leo Lam ©

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