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# PROBLEMS 79 PROBLEMS Section 2-1 and 2-2: LTI Systems 2.1 For each of the following systems, specify whether or not the system is: (i) linear and/or (ii) time-invariant. (a) $y(t)=3 x(t)+1$ *(b) $y(t)=3 \sin (t) x(t)$ (c) $\frac{d y}{d t}+t y(t)=x(t)$ (d) $\frac{d y}{d t}+2 y(t)=3 \frac{d x}{d t}$ (e) $y(t)=\int_{-\infty}^{t} x(\tau) d \tau$ (f) $y(t)=\int_{0}^{t} x(\tau) d \tau$ (g) $y(t)=\int_{t-1}^{t+1} x(\tau) d \tau$ 2.2 For each of the following systems, specify whether or not the system is: (i) linear and/or (ii) time-invariant. (a) $y(t)=3 x(t-1)$ (b) $y(t)=t x(t)$ (c) $\frac{d y}{d t}+y(t-1)=x(t)$ (d) $\frac{d y}{d t}+2 y(t)=\int_{-\infty}^{t} x(\tau) d \tau$ (e) $y(t)=x(t) u(t)$ (f) $y(t)=\int_{t}^{\infty} x(\tau) d \tau$ (g) $y(t)=\int_{t}^{2 t} x(\tau) d \tau$ 2.3 Compute the impulse response of the LTI system whose step response is 2.4 Compute the impulse response of the LTI system whose step response is 2.5 The step response of an LTI system is Compute the response of the system to the following inputs. * Answer(s) in Appendix F. (a) *(b) (c) (d) 2.6 Compute the response $y(t)$ of an initially uncharged RC circuit to a pulse $x(t)$ of duration $\epsilon$, height $\frac{1}{\epsilon}$, and area $\epsilon \frac{1}{\epsilon}=1$ for $\epsilon \ll 1$ (Fig. P2.6). Figure P2.6: Circuit and input pulse. The power series for $e^{a x}$ truncated to two terms is $e^{a x} \approx 1+a x$ and is valid for $a x \ll 1$. Set $a=\frac{\epsilon}{R C}$ and substitute the result in your answer. Show that $y(t)$ simplifies to Eq. (2.17). 2.7 Plot the response of the RC circuit shown in Fig. P2.6 to the input shown in Fig. P2.7, given that $R C=1 \mathrm{~s}$. Figure P2.7: Input pulse for Problem 2.7. 2.8 For the RC circuit shown in Fig. 2-5(a), apply the superposition principle to obtain the response $y(t)$ to input: (a) $x_{1}(t)$ in Fig. P1.23(a) (in Chapter 1) (b) $x_{2}(t)$ in Fig. P1.23(b) (in Chapter 1) (c) $x_{3}(t)$ in Fig. P1.23(c) (in Chapter 1) 2.9 For the RC circuit shown in Fig. 2-5(a), obtain the response $y(t)$ to input:Please Answer Question 2.5 A/B/C/D and show steps, Thank you!  