Explain what this system does and is the system bibo stable

Question

Slution to Q 3, 4 and 5 only

Question 1:
Show that
a) _-∞∫^∞ δ(t)e^-jωt dt = 1

b) _-∞∫^∞ δ(t -2)cos(Πt/4)dt = 0.

c) _-∞∫^∞ δ(2 - t)e^-2(x-t) dt = e^-2(x-2)

Question 2:

a) If x_e(t) and x₀(t) are even and odd components of a real signal x(t), then show that: _-∞∫^∞ x_e(t)x_o(t)dt = 0

b) Show that _-∞∫^∞ x(t)dt = _-∞∫^∞x_e(t)dt

Question 3:

Given a continuous-time system: y(t)= 0.55_-∞∫^∞ x(τ)[δ(t-τ)-δ(t+τ))dτ

a) Explain what this system does

b) Is the system BIBO stable? Justify your answer

c) Is the system linear? Justify your answer

d) Is the system memoryless? Justify your answer

e) Is the system causal? Justify your answer

f) Is the system time variant? Justify your answer