Signals and Systems - Electrical Engineering

164 C H A P T E R 2: Continuous-Time Systems Let the input to the envelope detector be x(t)=[p(t)+P] cos( 0 t) wherePis the min ...

CHAPTER 3 The Laplace Transform............................................................................ What we know is not ...

166 C H A P T E R 3: The Laplace Transform and of transient and steady-state responses. This is a significant reason to study th ...

3.2 The Two-Sided Laplace Transform 167 LTI System H(s) x(t)=es^0 t y(t)=x(t) H(s 0 ) FIGURE 3.1 Eigenfunction property of LTI s ...

168 C H A P T E R 3: The Laplace Transform An inputx(t)=es^0 t,s 0 =σ 0 +j 0 , is called an eigenfunction of an LTI system with ...

3.2 The Two-Sided Laplace Transform 169 The two-sided Laplace transform of a continuous-time functionf(t)is F(s)=L[f(t)]= ∫∞ −∞ ...

170 C H A P T E R 3: The Laplace Transform nExample 3.1 A problem in wireless communications is the so-calledmultipath effecton ...

3.2 The Two-Sided Laplace Transform 171 giving as the system function for the channel, H(s)=α 0 e−st^0 +···+αNe−stN Notice that ...

172 C H A P T E R 3: The Laplace Transform For the Laplace transform off(t)to exist we need that ∣∣ ∣∣ ∣∣ ∫∞ −∞ f(t)e−stdt ∣∣ ∣∣ ...

3.2 The Two-Sided Laplace Transform 173 100 50 0 − 2 0 − 2 2 1 − 1 − 3 0 Damping Frequency − 50 − 100 FIGURE 3.4 Three-dimension ...

174 C H A P T E R 3: The Laplace Transform the Laplace transform to test for convergence, we lets=σ+jand the term|ej|=1. Thus, ...

3.2 The Two-Sided Laplace Transform 175 Indeed, the integral defining the Laplace transform is bounded for any value ofσ6=0. IfA ...

176 C H A P T E R 3: The Laplace Transform Att=0 we assume thatu( 0 )=0.5 to getf( 0 )from the sumfc( 0 )+fac( 0 ). The Laplace ...

3.3 The One-Sided Laplace Transform 177 Solution Even thoughδ(t)is not a regular signal, its Laplace transform can be easily obt ...

178 C H A P T E R 3: The Laplace Transform u = sym(’Heaviside(t)’) U=laplace(u) % Delta function d = sym(’Dirac(t)’) D = laplace ...

3.3 The One-Sided Laplace Transform 179 = −ejθ s−j 0 e−σt−j(−^0 )t|∞t= 0 = ejθ s−j 0 ROC: σ > 0 According to Euler’s iden ...

180 C H A P T E R 3: The Laplace Transform FIGURE 3.6 Location of the poles and zeros of cos( 2 t+θ)u(t)for (a)θ= 0 and for (b) ...

3.3 The One-Sided Laplace Transform 181 0 2 4 − 1 −0.5 0 0.5 1 t y(t)=cos(10t) exp(−t)u(t) − 2 0 2 − 10 − 5 0 5 10 σ jΩ (a) (b) ...

182 C H A P T E R 3: The Laplace Transform The Laplace transform forcc(t)=e−atu(t), as seen before, is Cc(s)= 1 s+a with a regio ...

3.3 The One-Sided Laplace Transform 183 FIGURE 3.8 Poles (top right) of the Laplace transform of the autocorrelation c(t)=e−^2 | ...