Exam 2 Practice

Question 1

What is a Linear Constant-Coefficient Differential Equation (LCCDE) for an LTI system?

Answer 1

An equation of the form ∑ a_k d^k y / dt^k = ∑ b_m d^m x / dt^m, describing the relationship between input x(t) and output y(t).

Question 2

What is the frequency response H(ω) for an LCCDE?

Answer 2

H(ω) = [∑{i=0}^m b{m-i} (jω)^i] / [∑{i=0}^n a{n-i} (jω)^i].

Question 3

How to find the steady-state output for a sinusoidal input x(t) = A cos(ω t + φ) to an LTI system?

Answer 3

y_ss(t) = A |H(ω)| cos(ω t + φ + ∠H(ω)).

Question 4

For an impulse response h(t) = ∑ c_k e^{-a_k t} u(t), what is H(ω)?

Answer 4

H(ω) = ∑ c_k / (a_k + jω).

Question 5

If H(ω_0) = 0 for input cos(ω_0 t + φ), what is the output?

Answer 5

The steady-state output is 0.

Question 6

How to find the input x(t) given output y(t) for an LTI system?

Answer 6

For frequency components, X(ω) = Y(ω) / H(ω); For DC, X(0) = Y(0) / H(0).

Question 7

What are the Fourier series coefficients a_0, a_n, b_n for a periodic signal x(t) with period T_0?

Answer 7

a_0 = (1/T_0) ∫ x(t) dt over one period; a_n = (2/T_0) ∫ x(t) cos(n ω_0 t) dt; b_n = (2/T_0) ∫ x(t) sin(n ω_0 t) dt, where ω_0 = 2π / T_0.

Question 8

What is the complex Fourier series coefficient c_n?

Answer 8

c_n = (1/T_0) ∫ x(t) e^{-j n ω_0 t} dt; |c_n|^2 = a_n^2 + b_n^2 for n>0, c_0 = a_0 / 2.

Question 9

State Parseval’s theorem for Fourier series.

Answer 9

The average power P = (1/T_0) ∫ |x(t)|^2 dt = (a_0^2)/2 + (1/2) ∑ (a_n^2 + b_n^2) = c_0^2 + (1/2) ∑ |c_n|^2 (sum over n=1 to ∞).

Question 10

How to compute the percentage of power in the fundamental component?

Answer 10

(Power in n=1) / Total power * 100%, where power in n=1 is (1/2) |c_1|^2.

Question 11

What is the Fourier transform F(ω) of a signal f(t)?

Answer 11

F(ω) = ∫_{-∞}^∞ f(t) e^{-j ω t} dt.

Question 12

What is the scaling property of the Fourier transform?

Answer 12

If F(ω) is FT of f(t), then FT of f(a t) = (1/|a|) F(ω / a).

Question 13

What is the modulation property of the Fourier transform?

Answer 13

FT of f(t) cos(ω_0 t) = (1/2) [F(ω - ω_0) + F(ω + ω_0)].

Question 14

What is the differentiation property of the Fourier transform?

Answer 14

FT of d^n f / dt^n = (j ω)^n F(ω).

Question 15

State Parseval’s theorem for the Fourier transform.

Answer 15

Energy E = ∫{-∞}^∞ |x(t)|^2 dt = (1/(2π)) ∫{-∞}^∞ |X(ω)|^2 dω.

Question 16

For X(ω) = 1 / (a + j ω), what is x(t) and its energy?

Answer 16

x(t) = e^{-a t} u(t); Energy = 1 / (2 a).

Question 17

What is the frequency response of a series RL circuit with V_out across the inductor?

Answer 17

H(ω) = j ω L / (R + j ω L); |H(ω)| = ω L / √(R^2 + (ω L)^2).

Question 18

Describe the magnitude plot for a high-pass RL filter.

Answer 18

Low frequencies: |H| ≈ 0; High frequencies: |H| ≈ 1; Corner frequency at ω = R/L, increasing at 20 dB/decade.