Use block diagrams to illustrate the following systems and make appropriate labeling:
(a) y(t) = x(t) ∗ [h1(t) ∗ h2(t)] + h3(t);
(b) y(t) = h1(t) ∗ x(t) ∗ [h2(t) + h3(t)];
(c) y1(t) = [x(t) - y2(t)] ∗ h1(t)
y2(t) = y1(t) ∗ h2(t)
y(t) = y1(t)
(d) h(t) = [h1(t) ∗ h2(t) + h3(t)] ∗ h4(t);
(e) y(t) = [x1(t) ∗ h1(t) · x2(t)] ∗ h2(t), " · " = , i.e., a multiplier.
Problem 2 [24 pts.]: Prove (proof steps needed, if applicable) whether the following systems
are
• Memoryless,
• Invertible,
• Causal,
• Stable,
• Time-invariant,
• Linear.
(a) y(t) = 2cos(2t - 1)x(t);
(b) A system with h(t) = u(t - t0).
(c) A system with h(t) = δ(t - 1).
Problem 3: Consider a capacitor with capacitance C as a system, an input current signal i(t) = e -atu(t), a > 0 is applied to the system; the system output is the voltage across the capacitor, designated as v(t). Answer the following questions:
(a) What is the system's h(t)?
(b) From Problem 2, what can you deduce about this system? No proof needed, just answer "yes" or "no" for the 6 properties.
(c) Calculate the output of the system v(t);
(d) Calculate the system's unit step response s(t).
Problem 4: Define S : y(t) = x(t) - 2x(t - 1),
(a) Find out h(t), sketch and label it.
(b) Find out s(t), sketch and label it.
(c) Is this system LTI?
(d) Is this system stable? Why?
(e) Is this system causal? Why?
Problem 5: An LTI system has its input x(t) and unit impulse response h(t) defined as below:
x(t) = (
-t + 2T , 0 < t < 2T
0 , t = otherwise
h(t) = (
1 , 0 < t < T
0 , t = otherwise
Find its output y(t) = x(t) ∗ h(t) and sketch it.