4.8 The laplace transform: excercises

Exercises

1. Find the Laplace Transform of the following signals, for each case indicate the Laplace transform property that was used:
   1. $x\left(t\right)=4e^{-0.2t}u\left(t\right)$
   2. $x\left(t\right)=4t{e}^{-0.2t}u\left(t\right)$
   3. $x\left(t\right)=4e^{-0.2(t-10)}u(t-10)$
   4. $x\left(t\right)=\delta (t-5)$
   5. $x\left(t\right)=10tu\left(t\right)$
   6. $x\left(t\right)=sin\left(10\pi t\right)u\left(t\right)$
   7. $x\left(t\right)=e^{-3t}sin\left(10\pi t\right)u\left(t\right)$
   8. $x\left(t\right)=\text{rect}(t-0.5,1)$
2. Suppose that two filters having impulse responses $h_1\left(t\right)$ and $h_2\left(t\right)$ are cascaded (i.e. connected in series). Find the transfer function of the equivalent filter assuming $h_1\left(t\right)=10e^{-10t}u\left(t\right)$ and $h_2\left(t\right)=5e^{-5t}u\left(t\right)$ .
3. Find the inverse Laplace transforms of the following:
   1. $X\left(s\right)=\frac{e^{-2s}}{s+5}$
   2. $X\left(s\right)=\frac{s{e}^{-s}}{s^2+9}$
   3. $X\left(s\right)=\frac{1}{{(s+3)}^2}$
   4. $X\left(s\right)=10$
   5. $X\left(s\right)=\frac{10}{s^2}$
   6. $X\left(s\right)=\frac{e^{-s}}{s}$
4. Use partial fraction expansions to find the inverse Laplace transforms of the following:
   1. $X\left(s\right)=\frac{s+2}{(s+5)(s+2)(s+1)}$
   2. $X\left(s\right)=\frac{s+1}{{(s+2)}^3(s+3)}$
   3. $X\left(s\right)=\frac{s}{(s^2+9)(s+2)}$
   4. $X\left(s\right)=\frac{s^2-3s+1}{(s+1)(s+2)}$
5. Consider a filter having impulse response $h\left(t\right)=e^{-2t}u\left(t\right)$ . Use Laplace transforms to find the output of the filter when the input is given by:
   1. $x\left(t\right)=u\left(t\right)$
   2. $x\left(t\right)=tu\left(t\right)$
   3. $x\left(t\right)=e^{-4t}u\left(t\right)$
   4. $x\left(t\right)=cos\left(10t\right)u\left(t\right)$
6. Indicate whether the following impulse responses correspond to stable or unstable filters:
   1. $h\left(t\right)=u\left(t\right)$
   2. $h\left(t\right)=e^{-3t}u\left(t\right)$
   3. $h\left(t\right)=e^{-3t}cos\mathrm{(4}\mathrm{t)}u\left(t\right)$
   4. $h\left(t\right)=cos\left(10t\right)u\left(t\right)$
7. Use Laplace transform tables to find the impulse response of the second-order lowpass filter in terms of $\zeta$ and ${\Omega }_n$ for the overdamped, critically damped, and underdamped case.
8. Use a series RLC circuit to design a critically damped second-order lowpass filter with a corner frequency of 100 rad/sec. Use a $R=6.8$ k $\Omega$ resistor in your design.
9. Using a 10 k $\Omega$ resistor, design a critically damped bandpass filter, having a center frequency of 100 rad/sec and indicate the resulting bandwidth of the filter. What is the quality factor of the filter?
10. Use bode plots to find the magnitude and phase response of the following filters
    1. $$H\left(s\right)=\frac{1}{{(s+1)}^3}$$
    2. $$H\left(s\right)=\frac{{10}^{3}s}{\left(s,+,10\right)\left(s,+,{10}^3\right)}$$