Question 1 -

Determine the forward Fourier Transform of the following signals. As usual, "*" is the convolution operator. Avoid explicitly using the Fourier transform integral by exploiting known Fourier transforms and properites.

(a) x(t) = 7 cos²(13πt) Hint: First use a common trig identity for cos²(·).

(b) x(t) = δ(t - 3) - 2e^-5(t-3)u(t - 3)

(c) x(t) = cos((25/3)πt) [u(t - 7) - u(t - 11)]

(d) x(t) = cos((25/3)πt) * [u(t - 7) - u(t - 11)]

Question 2 -

The frequency responses of two ideal LTI systems are

H₁(jω) = {u(ω + 20π) - u(ω - 20π} e^-j2ω/3

H₂(jω) = {u(ω + 10π) - u(ω - 10π)} e^-j2ω/3

(a) Create a new filter by "subtracting" the two LTI systems defined above:

H₃(jω) = H₁(jω) - 0.95H₂(jω).

Make a sketch of the magnitude and phase of H₃(jω).

(b) Describe the type of filter in the plot of the previous part (LPF, HPF, or BPF).

(c) Using the filter from part (a), determine the output of the systems when the input signal is

x(t) = 3 cos(11πt + π/3) + (7sin(9πt)/πt).

Question 3 -

A long transmission line has a 50 Ω impedance, with a 2 Volt DC source on one end and a resistor load. The line is in steady state. You measure the voltage and current in the middle of the transmission line and find them to be 1 V and +0.01 A, respectively.

a. Your friend finds it strange that the ratio of voltage to current that you measured is not Z₀. Explain to him briefly why that is the case.

b. What is the forward going voltage (V⁺) and the backward-going voltage (V^-)?

c. What are the generator and load resistances (R_G and R_L)?

d. If the load resistor were suddenly yanked out of the circuit board, what would be the voltage at the load end of the transmission line right afterward?

e. If this transmission line were part of a circuit connecting the output of one logic gate (such as an inverter) to the input of another (identical) logic gate, as in the below diagram, name a significant problem with the circuit. Assuming that RG and RL cannot change since they are part of the inverter design, propose at least one solution to fix that problem.

This question about Fourier transform questions. Can you explain detail?