Signals and Systems - Electrical Engineering

204 C H A P T E R 3: The Laplace Transform nExample 3.15 Consider the Laplace function X(s)= 2 s+ 3 s^2 + 2 s+ 4 = 2 s+ 3 (s+ 1 ...

3.4 Inverse Laplace Transform 205 − 2 0 2 −2.5 − 2 −1.5 − 1 −0.5 0 0.5 1 1.5 2 2.5 σ jΩ −0.5 0 5 10 0 0.5 1 1.5 2 2.5 t x(t) (a) ...

206 C H A P T E R 3: The Laplace Transform When we have double real poles we need to express the numeratorN(s)as a first-order p ...

3.4 Inverse Laplace Transform 207 FIGURE 3.15 Inverse Laplace transform of X(s)= 4 /(s(s+ 2 )^2 ): (a) poles and zeros and (b)x( ...

208 C H A P T E R 3: The Laplace Transform Solution The partial fraction expansion is X(s)= A s+ 1 −j √ 3 + A∗ s+ 1 +j √ 3 + B s ...

3.4 Inverse Laplace Transform 209 function pfeLaplace(num,den) % disp(’>>>>>Zeros<<<<<’) z = roots ...

210 C H A P T E R 3: The Laplace Transform FIGURE 3.16 Inverse Laplace transform of X(s)=( 3 s^2 + 2 s− 5 )/ (s^3 + 6 s^2 + 11 s ...

3.4 Inverse Laplace Transform 211 is obtained by first considering the term 1/s, which hasu(t)as inverse, and then using the inf ...

212 C H A P T E R 3: The Laplace Transform where F(s)= 1 −e−s s+ 1 The inverse ofF(s)is f(t)=e−tu(t)−e−(t−^1 )u(t− 1 ) and the i ...

3.4 Inverse Laplace Transform 213 convergenceRe(s) <2. That this is so is confirmed by the intersection of these two regions ...

214 C H A P T E R 3: The Laplace Transform n If the system is BIBO stable and causal, then the region of convergence includes th ...

3.5 Analysis of LTI Systems 215 I(s)= ∑N k= 1 ak   k∑− 1 m= 0 sk−m−^1 y(m)( 0 )   That is,I(s)depends on the initial conditi ...

216 C H A P T E R 3: The Laplace Transform whereh(t)=L−^1 [H(s)]andh 1 (t)=L−^1 [H 1 (s)], and i(t)=L−^1 [I(s)]= ∑N k= 1 ak   ...

3.5 Analysis of LTI Systems 217 In fact, for any real poles=−α,α >0, of multiplicitym≥1, we have that L−^1 [ N(s) (s+α)m ] = ...

218 C H A P T E R 3: The Laplace Transform Assume the above equation represents a system with inputx(t)and outputy(t). Find the ...

3.5 Analysis of LTI Systems 219 s= 0 ; if the system input is x 1 (t)=u(t)so that X 1 (s)= 1 /s, then Y 1 (s)= 1 /(s^2 (s+ 1 )). ...

220 C H A P T E R 3: The Laplace Transform after replacingX(s)= 1 /s. We find thatB 1 = 1 /2,B 2 =1, andB 3 =− 1 /2, so that the ...

3.5 Analysis of LTI Systems 221 and substitutingX(s)=1, then H(s)=Y(s)= 1 sT [1−e−sT] The impulse response is then h(t)= 1 T [u( ...

222 C H A P T E R 3: The Laplace Transform Its inverse is y(t)=r(t)−r(t− 1 ) wherer(t)is the ramp signal. This result coincides ...

3.5 Analysis of LTI Systems 223 withy( 0 )the initial voltage in the capacitor andi(t)the current through the resistor, inductor ...