University:
Michigan State University
Course:
ECE 201 | Circuits and Systems I
Academic year:

2019
Views:

33

Pages:

33
Author:

Hallie Randolph

Chapter 6 6.1-1 (a) f(t) = u(t) = u(t - 1) F(s) = integrate e ^ (- s * t) dt from 0 to 1 =- e^ -st s | 0 ^ 1 = - 1/s * [e ^ (- s) - 1] = 1/s * [1 - e ^ (- s)] Note that the result is valid for all values of ; hence the region of convergence is the entire s-plane. The abscissa. of convergence is sigma_{0} = - ∞ (b) f(t) = t * c ^ (- t) * u(t) F(s) = integrate t * e ^ (- t) * e ^ (- s * t) dt from 0 to ∞ = integrate t * e ^ (- (s + 1)) dt from 0 to ∞ = - (e ^ (- (s + 1) * t))/((s + 1) ^ 2) [-(s+1)t-1] 0 ^ infty = 1/((s + 1) ^ 2) f(t)=t cos omega 0 tu(t) provided that e ^ (- (n + 1) * m) = 0 or Ite (s + 1) > emptyset. Hence the abscissa of convergence is Re ( )> - 1 or sigma_{0} > - 1 F(s)= int 0 ^ infty t cos omega 0 t * e ^ (- s * t) * dt = 1/2 * \{integrate [t * e ^ ([j*omega_{0} - s) * t) + t * e ^ (- (j*omega*p + s) * t)] dt from 0 to ∞\} = 1/2 * [1/((s - j*omega_{0}) ^ 2) + 1/((s + j*omega_{0}) ^ 2)] = (s ^ 2 - omega_{0} ^ 2)/((s ^ 2 + omega_{0} ^ 2) ^ 2) Re(a) > 0 (c) (d) f(t) = (e ^ (2t) - 2e ^ (- t)) * u(t) F(s) = integrate (e ^ (2t) - 2e ^ (- t)) * e ^ (- s * t) dt from 0 to ∞ = integrate e ^ (2t) * e ^ (- s * t) dt from 0 to ∞ - 2 * integrate e ^ (- t) * e ^ (- s * t) dt from 0 to ∞ = integrate e ^ (- (s - 2) * t) dt from 0 to ∞ - 2 * integrate e ^ (- (s + 1) * t) dt from 0 to ∞ = 1/(s - 2) - 2/(s + 1) 99 We get the first term only if Re a > 2 and we get the second term only if Re (s) > - 1 Both conditions will be satisfied if Re (s) > 2 or sigma_{0} > 2 Hence: F(s) = 1/(s - 2) - 2/(s + 1) for sigma_{0} > 2 (e) f(t)=cos omega 1 t cos omega 2 tu[t) = [1/2 * cos(omega_{1} + omega_{2}) * t + 1/2 * cos(omega_{1} - omega_{2}) * t] * u(t) F(s) = 1/2 * integrate cos(omega_{1} + omega_{2}) * t * e ^ (- s * t) dt from 0 to ∞ + 1/2 * integrate cos(omega_{1} - omega_{2}) * t * e ^ (- s * t) dt from 0 to ∞ = 1/2 * [s/(s ^ 2 + (omega_{1} + omega_{2}) ^ 2) + s/(s ^ 2 + (omega_{1} - omega_{2}) ^ 2)] f(t) = cosh(at) * u(t) F(epsilon) = 1/2 * [integrate e ^ (zt) * e ^ (- s * t) dt from 0 to ∞ + integrate e ^ (- alpha * t) * e ^ (- s * t) dt from 0 to ∞] = 1/2 * [integrate e ^ (- (s - a) * t) dt from 0 to ∞ + integrate e ^ (- (s + a) * t) dt from 0 to ∞] = s/(s ^ 2 - a ^ 2) Res >|n|| provided that Re (s) > 0 (۲) (g) f(t) = sinh(at) * u(t) F(s) = 1/2 * [integrate e ^ (- (s - u) * t) dt from 0 to ∞ - integrate e ^ (- (s + u) * t) dt from 0 to ∞] = s/(s ^ 2 - a ^ 2) s > |a| (h) f(t) = e ^ (-2t) * cos(5t + theta) * u(t) = 1/2 * [e ^ (-2tj(3i + theta)) + e ^ (- 2t - zeta(5t + theta))] = 1/2 * e ^ (j*theta) * e ^ (- (2 - j * 3) * epsilon) + 1/2 * e ^ (- j * theta) * e ^ (- (2 + j*delta) * t) F(s)= 1/2 * e ^ (j*vartheta) * (1/(s + 2 - j * 5)) + 1 2 e^ -j theta ( 1 s+2+j5 ) Hence This is valid if Re (s) > - 2 for both terms; hence F(s) = ((s + 2) * cos theta - 5sin theta)/(s ^ 3 + 4s + 29) 6.1-2 (a) F(s)= int 0 ^ 1 te^ -st dt= e^ -st s (-st-1)| 0 ^ 1 = 1/(s ^ 2) * (1 - e ^ (- s) - s * e ^ (- s)) F(s)= int 0 ^ pi sin t e^ -st dt= (e ^ (- s * t))/(s ^ 2 + 1) * (- s * sin t - cos t) | 0 ^ pi = 1+e^ - pi s s^ 2 +1 67 (b)