The impulse response h(t) of a linear time invariant system is given by h(t) = $${e^{ - 2t}}u(t),$$ where u(t) denotes the unit step function.

The frequency response H(ω) of the system in terms of angular frequency 'ω' is given by h( ω)

Group I gives two possible choices for the impedance Z in the diagram. The circuit elements in Z satisfy the condition $${R_2}{C_2} > {R_1}{C_{1.}}$$ The transfer $${\textstyle{{{V_0}} \over {{V_1}}}}$$ function represents a kind of controller. Match the impedances in Group I with the types of controllers in Group II. Group - I

Group - II
1. PID controller
2. Lead compensator
3. Lag compensator

The impulse response h(t) of a linear time invariant system is given by h(t) = $${e^{ - 2t}}u(t),$$ where u(t) denotes the unit step function.

The output of this system to the sinusoidal input x(t) = 2cos(t) for all time 't' is

The logic function implemented by the following circuit at the terminal OUT is