Signals and Systems

Laplace Transform

Support Material: Transform table for reference

TABLE 9.1 PROPERTIES OF THE LAPLACE TRANSFORM Laplace Section Property Signal Transform ROC x(1) X(s) R XI (t) X1 (s) x2 (t ) X2(s) R2 9.5.1 Linearity ax, (t ) + bx2(t) ax, (s) + bX2(s) At least R, n R2 9.5.2 Time shifting x(t - to) e-sto X (s) R 9.5.3 Shifting in the s-Domain eso' x(t) X(s - So) Shifted version of R (i.e., s is in the ROC if s - So is in R) 9.5.4 Time scaling x(at) Scaled ROC (i.e., s is in the ROC if s/a is in R) 9.5.5 Conjugation x' (t) X' (s' ) R 9.5.6 Convolution X1 (1) * X2(1) X1 (s) X2(s) At least R, n R2 9.5.7 Differentiation in the -x(1) SX(s) At least R Time Domain 9.5.8 Differentiation in the -tx(t) d x (s ) R ds s-Domain 9.5.9 Integration in the Time x (7)d (T) -X (S ) At least R n {Refs} > 0} Domain Initial- and Final-Value Theorems 9.5.10 If x(t) = 0 for t , then lim x(t) = lim sX(s) 0C + 1 S-+0Q1. Laplace Transformation. a. Fill in the blanks. (10 points) Consider a signal x(t) = e-at cos(bt]] u(t), for which Laplace Transform, X(s), is already listed on the table from the lecture slides. Let a be real. Lets derive this. Firstly, we rewrite the terms in x(t) into the following using x(t) = e-at (_ _Ju(t) x(t) =[e-(a+bit + e-(a-bilt ] u(t) Now, Laplace transform of x(t) is given by X(s) = [ x(t) e-st dt X(s) = + (write the two integrals here, with correct limits) Now solving the integral X(s) = + [List final answer here after adding the two terms, check: compare with the table] The above expression for X(s) should be in the N(s)/D(s) form. b. Poles and Zeros (3 points). For a = 2 and b = 4 plot the poles and zeros of X(s) on the 's-plane' with appropriate labeling. c. Provide ROC for above X(s) with reasoning. (2 points)