(Solved): 2 (a) Consider a causal, linear, time-invariant, continuous-time system with frequency response, \( ...

2 (a) Consider a causal, linear, time-invariant, continuous-time system with frequency response, \( H(j \omega)=e^{-j \omega} \). The input to this system is the signal \( x(t)=\cos (8 \pi t) \). Determine the output signal, \( y(t) \). [5 marks] (b) Let \( x_{1}(t)=t[u(t)-u(t-2)] \), where \( u(t) \) is the unit step function. Sketch and label each of the following signals: (i) \( x_{2}(t)=x_{1}(2 t) \) [2 marks] (ii) \( x_{3}(t)=\frac{d x_{1}(t)}{d t} \) [3 marks] (c) Evaluate the integral \[ \int_{0}^{\infty}\left(t^{2}-1\right) \delta(t-3) d t \] where \( \delta(t) \) is the unit impulse function. [3 marks] (d) Consider a linear, time-invariant system with impulse response \[ h(t)=e^{-t} u(t) \] where \( u(t) \) is the unit step function. (i) Is the system stable in the bounded-input bounded-output sense? Justify your answer. (ii) Determine and sketch the unit step response of the system, \( s(t) \).