ECE102, Spring 2024 Signals & Systems Practice Final Exam

发布时间：2024-06-12

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

ECE102, Spring 2024

Signals & Systems

Practice Final Exam

1. Signal and Systems Basics (21 points)

(a) (12 points) System properties. For each of the following systems, determine (with reasoning) if they are linear, time invariant, causal and stable.

i. (4 points) y(t) = x(3t + 2) + 5

ii. (4 points) y(t) = sin(dx(dt t))

iii. (4 points) y(t) = y(t) = e x 2 (t)

(b) (9 points) LTI System Analysis. Consider an LTI system with input x(t), output y(t) and impulse response h(t). The Fourier transforms X(jω) and H(jω) are as shown below.

Evaluate y(t).

2. Fourier transform (29 points)

(a) (12 points) A signal x(t) has has the following the Fourier Transform.

Plot the magnitude and phase plots for the Fourier Transform of the following signals:

i. (4 points) x(t/2)

ii. (4 points) x(t − 3)

iii. (4 points) Re(x(t))

(b) (9 points) Evaluate the Fourier Transforms of the following signals:

i. (4 points) x(t) = e −2|t−1|

ii. (5 points) x(t) = te−atcos(ω0t)u(t), a > 0

i. (4 points)

ii. (4 points) X(jω) = cos(2ω + π6)

3. Fourier Series (15 points)

(a) (5 points) Evaluate the Fourier Transform of the following signal x(t).

(b) (5 points) Using your solution from part (a), evaluate the fourier series of the following signal ˜x(t).

What is the output when ˜x(t) is passed through this system?

4. Frequency domain understanding (20 points)

(a) Identify if the following statements are ‘True’ or ‘False’ with appropriately detailed reasoning.

i. (1 points) Sampling at a frequency greater than Nyquist frequency is a necessary condition for perfect reconstruction, for every signal.

ii. (2 points) If we have two bandlimited signals, x1(t) with a bandwidth B1 and x2(t) with a bandwidth B2, the signal y(t) = x1(t)x2(t) has a bandwidth max{B1, B2}. (max{a, b} is equal to the maximum value among a and b)

iii. (2 points) Consider a periodic function x(t) with a fundamental period T. If x(t) is an odd function, the sum of all its Fourier series coefficients (Σ∞k=−∞ck) is zero for any odd x(t).

(b) Let F(jω) = j2πωe−2|ω| . Without computing f(t) answer the following questions with appropriate reasoning.

i. (2 points) Is f(t) real/imaginary/complex?

ii. (2 points) Is f(t) odd/even/neither?

iii. (1 points) What is f(0)?

i. (7 points) Let x(t) = 4 4+t 2 . Evaluate the Fourier transform X(jω). (Hint: use the duality property)

ii. (3 points) Using the Fourier transform from the previous part, evaluate the energy of x(t).

5. Sampling (15 points)

(a) (4 points) The sampling theorem says that for a bandlimited signal, a signal must be sampled at a frequency greater than its Nyquist frequency to guarantee perfect reconstruction. Identify the Nyquist frequency for the following signals:

i. (2 points) x(t) = cos(3000πt) − sin(2000πt)

ii. (2 points) x(t) = sin(2000πt)/πt

(b) (6 points) Consider a signal x(t) with a Nyquist frequency ω0. Determine the Nyquist frequency for the following signals:

i. (2 points) x 2 (t)

ii. (2 points) x(t)cos(ω1t)

iii. (2 points) dx(dt t)

(c) (5 points) We know that ideal sampling is carried out by multiplying the time domain analog signal with an impulse train. Consider a modified sampling regime, where we multiply with the following signal p(t):

The sampling process for a signal x(t) is shown in the following figure:

Let x(t) be bandlimited with a one-sided bandwidth of ωm, with the following fourier transform:

i. (2 points) If ∆ < ω π m, draw the Fourier transform of xp(t).

ii. (3 points) If ∆ < ω π m, determine a system to recover x(t) from xp(t).

6. Laplace transform ( (15 points)

A casual LTI system can be described by the following differential equation:

y ′′(t) + 4y ′ (t) + 4y(t) = 4x(t) + 1x ′′(t).

You may assume resting initial conditions (y(0)= 0, y’(0) =0, y”(0) = 0)

(a) (8 points) Find the transfer function H(s).

(b) (7 points) What is the impulse response h(t) of this system?