Lecture 20 Outline: Laplace Transforms Announcements: Reading: “6: The Laplace Transform” pp. 1-9 HW 7 posted, due next Wednesday My OHs Monday cancelled, available by appointment next W-F Guest lecture Monday: Paul Nuyujukian, “Using signal processing to decode brain signals”. Motivation for Laplace Transforms Bilateral Laplace Transform S-Plane and Region of Convergence Example: Right-sided Exponential

Review of Last Lecture Linear Convolution from Circular with Zero Padding Block-by-block linear convolution Breaks x[n] into shorter blocks; computes y[n] block-by-block. Overlap-add method: breaks x[n] into non-overlapping segments Segments computed by circular method: FFT/IFFT has complexity.5Nlog2N (vs. N 2 for DFT/IDFT) Digital Spectral Analysis exploits DSP to obtain CTFTs x 1 [n] * x 2 [n]= L=4 P=6 M=L+P-1=9

Motivation for Laplace Transforms Why do we need another transform? We have the CTFT, DTFT, DFT, FFT Most signals don’t have a Fourier Transform Requires that the Fourier integral converges: true if Need a more general transform to study signals and systems whose Fourier transform doesn’t exist Laplace transform x(t)  X(s) has similar properties as CTFT In general there is no Fourier Transform for power signals x(t) h(t) x(t)*h(t) X(s) H(s) Y(s)=X(s)H(s) Holds even when Fourier transforms don’t exit

Bilateral Laplace Transforms (Continuous Time) Definition: Relation with Fourier Transform: If we set  =0 then L[x(t)]=F[x(t)] : The bilateral Laplace transform exists if

S-Plane and Region of Convergence Definition of Region of Convergence (ROC) for Laplace transform L[x(t)]=X(s)=X(  +j  Defined as all values of s=  +j  such that L[x(t)] exists Convergence depends only on , not j , as it requires: s-Plane: Plot of  +j  with  on real (x) axis, j  on imaginary (y) axis. Show the ROC (shaded region) for L[x(t)] on this plane s-plane Values of Real Axis Imaginary Axis Smallest  : X(s) exists ROC consists of strips along j  axis

Example: Right-Sided Real Exponential This converges if Under this condition Special cases:, a real, i.e., if ROC

Main Points Laplace transform allows us to analyze signals and systems for which their Fourier transforms do not converge Laplace transform X(s) has similar properties as the Fourier Transform X(j  ) and equals X(j  ) when s=j  Laplace transform is defined over a range of s=  +j  values for which the transform converges The set of s=  +j  values for which the Laplace transform exists is called its Region of Convergence (RoC) RoC plotted on the S-plane Real axis for , imaginary axis for j 