Signals and Systems EE235 Leo Lam © 2010-2012

Today’s menu Fourier Transform table posted Laplace Transform Leo Lam © 2010-2012

Laplace Transform Definition: Where Inverse: Leo Lam © 2010-2012

Region of Convergence Example: Find the Laplace Transform of: We have a problem: the first term for t=∞ doesn’t always vanish! Leo Lam © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

Plotting Poles and Zeros What does it look like? Leo Lam © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2011

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 © 2010-2011

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 © 2010-2011

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 © 2010-2011

Another Inverse Example Example, find h(t) (assuming causal): Using linearity and partial fraction: Leo Lam © 2010-2011

Another Inverse Example Here is the reason: Leo Lam © 2010-2011

Summary Laplace intro Region of Convergence Causality Existence of Fourier Transform & relationships Leo Lam © 2010-2012