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Signals and Systems EE235 Leo Lam ©
Today’s menu Fourier Transform table posted Laplace TransformLeo Lam ©
Laplace Transform Focus on: Doing (Definitions and properties)Understanding its possibilities (ROC) Poles and zeroes (overlap with EE233) Leo Lam ©
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 ©
Region of Convergence Example: Find the Laplace Transform of:We have a problem: the first term for t=∞ doesn’t always vanish! Leo Lam ©
Region of Convergence Example: Continuing… In general: forIn our case if: then For what value of s does: Pole at s=-3. Remember this result for now! Leo Lam ©
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 ©
Region of Convergence Laplace transform not uniquelyComparing 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 ©
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 ©
Laplace Example No Laplace Example: ROC: Re(s)>-1 ROC: Re(s)<-3Combined: ROC: None! No Laplace Transform since there is no overlapped ROC! Leo Lam ©
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 ©
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 ©
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 ©
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 ©
Plotting Poles and ZerosH(s) is almost always rational for a physical system: Plot is in the s-plane (complex plane) σ jω x o Leo Lam ©
Plotting Poles and ZerosWhat does it look like? Leo Lam ©
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 ©
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 ©
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 ©
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 ©
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 ©
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 ©
Another Inverse ExampleExample, find h(t) (assuming causal): Using linearity and partial fraction: Leo Lam ©
Another Inverse ExampleHere is the reason: Leo Lam ©
Summary Laplace intro Region of Convergence CausalityExistence of Fourier Transform & relationships Leo Lam ©
Z-Plane Analysis DR. Wajiha Shah. Content Introduction z-Transform Zeros and Poles Region of Convergence Important z-Transform Pairs Inverse z-Transform.
Signals and Systems Fall 2003 Lecture #18 6 November 2003 Inverse Laplace Transforms Laplace Transform Properties The System Function of an LTI System.
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