Signals and Systems - Electrical Engineering

184 C H A P T E R 3: The Laplace Transform nExample 3.7 Compute the Laplace transform of the ramp functionr(t)=tu(t)and use it t ...

3.3 The One-Sided Laplace Transform 185 o o o o 1 t − 1 1 s-plane w(t) j 4 π j 2 π −j 2 π −j 4 π jΩ σ FIGURE 3.9 The Laplace tra ...

186 C H A P T E R 3: The Laplace Transform f(t)=Ae−atu(t)whereain general can be a complex number is F(s)= A s+a ROC:σ >−a Th ...

3.3 The One-Sided Laplace Transform 187 which are located on thejaxis. The farther away from the origin of thejaxis the poles ...

188 C H A P T E R 3: The Laplace Transform 0 5 10 0 0.5 1 t u(t) 0 5 10 − 1 0 1 t cos(5t)u(t) 0 5 10 0 0.5 1 t exp(−0.5t)u(t) 0 ...

3.3 The One-Sided Laplace Transform 189 L [ d^2 f(t) dt^2 u(t) ] =s^2 F(s)−sf( 0 −)− df(t) dt ∣ ∣t= 0 − (3.12) In general, iff(N ...

190 C H A P T E R 3: The Laplace Transform Remarks n The derivative property for a signal x(t)defined for all t is ∫∞ −∞ dx(t) d ...

3.3 The One-Sided Laplace Transform 191 which is a first-order linear differential equation with constant coefficients, zero ini ...

192 C H A P T E R 3: The Laplace Transform The above results can be explained by looking for a dual of the derivative property. ...

3.3 The One-Sided Laplace Transform 193 =− 0 sin( 0 t)u(t)+cos( 0 t)δ(t) =− 0 sin( 0 t)u(t)+δ(t) so that the Laplace transf ...

194 C H A P T E R 3: The Laplace Transform and so L [ df(t) dt ] =sF(s)−f( 0 ) =Y(s) sincef( 0 )=0 (the integral over a point), ...

3.3 The One-Sided Laplace Transform 195 nExample 3.12 Suppose we wish to find the Laplace transform of the causal sequence of pu ...

196 C H A P T E R 3: The Laplace Transform FIGURE 3.13 Full-wave rectified causal signal. − 1 0 1 2 3 4 5 6 7 −0.2 0 0.2 0.4 0.6 ...

3.4 Inverse Laplace Transform 197 If the input of an LTI system is the causal signalx(t)and the impulse response of the system i ...

198 C H A P T E R 3: The Laplace Transform whereσmaxis the maximum of the real parts of the poles ofX(s). Since in this section ...

3.4 Inverse Laplace Transform 199 Table 3.1One-Sided Laplace Transforms Function of Time Function ofs, ROC δ(t) 1, wholes-plan ...

200 C H A P T E R 3: The Laplace Transform where the{pk}are simple real poles ofX(s), its partial fraction expansion and its inv ...

3.4 Inverse Laplace Transform 201 and A 2 =X(s)(s+ 2 )|s=− 2 = 3 s+ 5 s+ 1 |s=− 2 = 1 Therefore, X(s)= 2 s+ 1 + 1 s+ 2 and as su ...

202 C H A P T E R 3: The Laplace Transform Simple Complex Conjugate Poles The partial fraction expansion of a proper rational fu ...

3.4 Inverse Laplace Transform 203 Remarks n An equivalent partial fraction expansion consists in expressing the numerator N(s)of ...