A continuous-time LTI system has the input x(t) and the impulse response h(t) as shown below. Solve for and sketch the system output y(t) for all time. An automobile with poor shock absorbers is observed bouncing along after striking a speed bump. The height of the front bumper gives the impulse response h(t) as shown below. This graph shows that when an impulse arrives at a system initially at rest, the output begins to oscillate with decreasing amplitude. Eventually the vehicle drives smoothly again, until another bump is encountered. Give an expression for the system output h2(t) in terms of h(t) if a second identical bump is encountered at time t = 5 s. Sketch h2(t).

A linear system has an unknown response function h(t). A unit rectangle rect(t) input to the system leads to the output as shown below. Find h(t).

Consider the system below with h(t) = e-αt

u(t). Assume that the system is BIBO stable. The
input signal is x(t) = sin(t) + cos(3t). Find y(t).
Find the Fourier series in complex exponential form for the signal shown below.
Find the Fourier series in complex exponential form for the signal shown below. 
Calculate the frequency response of the circuit shown below. The values are R = 1 ohm, L= 0.1 mH and C = 4 × 10-3 F. Determine what type of ideal filter is approximated by the
circuit.

8. Calculate the frequency response of the circuit shown below. The values are R = 1 ohm, L= 0.1 mH and C = 4 × 10-3 F. Determine what type of ideal filter is approximated by the
circuit.

The circuit shown below is a Butterworth filter.
(a) Determine the order of the filter.
(b) What is the 3-dB bandwidth of the filter if R = 5 k? and C = 5 nF?
(c) Design a low-pass Butterworth filter with a cut-off frequency of 1 kHz by modifying the circuit.
(d) Design a high-pass Butterworth filter with a cut-off frequency of 10 kHz by modifying the circuit and choosing appropriate component values.