2 Definition of Bilateral Laplace Transform (b for bilateral or two-sided transform)Let s=σ+jωConsider the two sided Laplace transform as the Fourier transform of f(t)e-σt. That is the Fourier transform of an exponentially windowed signal.Note also that if you set the evaluate the Laplace transform F(s) at s= jω, you have the Fourier transform (F(ω))
3 Unilateral Laplace Transform (Implemented in Mathematica)
4 Difference Between the Unilateral Laplace Transform and Bilateral Laplace transform Unilateral transform is used when we choose t=0 as the time during which significant event occurs, such as switching in an electrical circuit.The bilateral Laplace transform are needed for negative time as well as for positive time.
5 Laplace Transform Convergence The Laplace transform does not converge to a finite value for all signals and all values of sThe values of s for which Laplace transform converges is called the Region Of Convergence (ROC)Always include ROC in your solution!Example:0+ indicates greater than zero valuesRemember: e^jw is sinusoidal; Thus, only the real part is important!
8 Example – RCO may not always exist! Note that there is no common ROC Laplace Transform can not be applied!
9 Laplace Transform & Fourier Transform Laplace transform is more general than Fourier TransformFourier Transform: F(ω). (t→ ω)Laplace Transform: F(s=σ+jω) (t→ σ+jω, a complex plane)
10 How is Laplace Transform Used (Building block of anegative feedback system)This system becomes unstable if βH(s) is -1. If you subsittuteds by jω, you can use Bode plot to evaluate the stability ofthe negative feedback system.
11 Understand Stability of a system using Fourier Transform (Bode Plot) (unstable)
12 Understand Stability of a System Using Laplace Transform Look at the roots of Y(s)/X(s)
13 Laplace TransformWe use the following notations for Laplace Transform pairs – Refer to the table!
23 Application i=CdV/dt (assume initial voltage is 0) Integrate i/C with respect to t, will get you I/(sC), which is the voltage in Laplace domainV=Ldi/dt (assume initial condition is 0)Integrate V/L with respect to t, get you V/(sL), which is current in Laplace domain.
30 Applications of Laplace Transform Easier than solving differential equationsUsed to describe system behaviorWe assume LTI systemsUses S-domain instead of frequency domainApplications of Laplace Transforms/Circuit analysisProvides the general solution to any arbitrary wave (not just LRC)TransientSinusoidal steady-state-response (Phasors)Signal processingCommunicationsDefinitely useful for Interviews!
31 Example of Bilateral Version Find F(s):ROCS-planeRe(s)<aaFind F(s):RememberThese!Note that Laplace can also be found for periodic functions