Question

2.6.1-2.6.6

2.6.1 Consider a causal contimuous-time LTI system described by the differential equation

$$ y^{\prime \prime}(t)+y(t)=x(t) $$

(a) Find the transfer function \(H(s)\), its \(R O C\), and its poles.

(b) Find the impulse response \(h(t)\).

(c) Classify the system as stable/unstable.

(d) Find the step response of the system.

2.6.2 Given the impulse response of a continuous-time LTI system, find the transfer function \(H(s),\) the \(\mathrm{ROC}\) of \(H(s)\), and the poles of the system. Also find the differential equation describing each system.

(a) \(h(t)=\sin (3 t) u(t)\)

(b) \(h(t)=\mathrm{e}^{-t / 2} \sin (3 t) u(t)\)

(c) \(h(t)=e^{-t} u(t)+e^{-t / 2} \cos (3 t) u(t)\)

2.6.3 A causal continuous-time LTI system is described by the equation

$$ y^{\prime \prime}(t)+2 y^{\prime}(t)+5 y(t)=x(t) $$

where \(x\) is the input signal, and \(y\) is the output signal.

(a) Find the impulse response of the system.

(b) Accurately sketch the pole-zero diagram.

(c) What is the de gain of the system?

(d) Classify the system as either stable or unstable.

(e) Write down the form of the step response of the system, as far as it can be determined without actually calculating the resides. (You do not need to complete the partial fraction expansion).

2.6.4 Given a causal LTI system described by the differential equation find \(H(s),\) the \(R O C\) of \(H(s),\) and the impulse response \(h(t)\) of the system. Classify the system as stable/unstable. List the poles of \(H(s)\). You should the Matlab residue command for this problem.

(a) \(y^{\prime \prime \prime}+3 y^{\prime \prime}+2 y^{\prime}=x^{\prime \prime}+6 x^{\prime}+6 x\)

(b) \(y^{\prime \prime \prime}+8 y^{\prime \prime}+46 y^{\prime}+68 y=10 x^{\prime \prime}+53 x^{\prime}+144 x\)

2.6.5 It is observed of some continuous-time LTI system that the input signal

$$ x(t)=\mathrm{e}^{-2 t} u(t) $$

produces the output signal

$$ y(t)=0.5 \mathrm{e}^{-2 t} u(t)+2 \mathrm{e}^{-3 t} \cos (2 \pi t) u(t) $$

What ean be concluded about the pole positions of the LTI system?

2.6.6 A causal continuous-time LTI system is described by the equation

$$ y^{\prime \prime}(t)+4 y^{\prime}(t)+5 y(t)=x^{\prime}(t)+2 x(t) $$

where \(x\) is the input signal, and \(y\) is the output signal.

(a) Find the impulse response of the system.

(b) Accurately sketch the pole-zero diagram.

(c) What is the de gain of the system?

Answer 1