EE 353
Final Exam 4 May 2020
Instructions:
_______________________________________ _______________________________________
Last Name (Print):
First Name (Print):
PSUUserID(e.g.map244): _______________________________________
Problem
Weight
Score
1
12
2
9
3
9
4
9
5
9
6
9
7
9
8
9
9
9
10
16
Bonus
10
Total
100
Complete the solutions in the space provided. If you need additional space, simply include an extra page in your exam packet with the work; place it immediately following the problem. Circle or otherwise enclose your final solutions. The clarity of your mathematical analysis is an important part of your work; unclear analysis or missing intermediate steps can lead to loss of credit even if your solution is correct. This exam is open book and open notes. You are free to use a calculator.
In the exam, you will occasionally see words in bold, underlined text telling you the way you MUST solve the problem to receive credit. I expect all your answers to be supported mathematically.
Academic Integrity Statement:
The work on this exam represents my efforts. From the start of the exam to the end of the exam, I did not communicate with any student or outside party. I understand that exams without a signed academic integrity statement will receive a grade of zero.
Signature: _________________________________________________
Problem 1 (12 points)
1. (6 points) A discrete-time signal is created by sampling the continuous-time function
𝜋𝜋
𝑓𝑓(𝑡𝑡) = 5 cos(2𝑡𝑡) − 3 cos�6𝑡𝑡 + 3� + 6sin(14𝑡𝑡)
every 100 ms. The resulting discrete-time signal is 𝜋𝜋
𝑓𝑓[𝑘𝑘] = 5 cos[0.2𝑘𝑘] − 3 cos �0.6𝑘𝑘 + 3� + 6 sin[1.4𝑘𝑘]
If possible, determine the period of both 𝑓𝑓(𝑡𝑡) and 𝑓𝑓[𝑘𝑘]. If it is not possible, explain why based on your mathematical analysis.
2. (6 points) Given that 𝑓𝑓[𝑘𝑘] = sinc[𝑘𝑘] rect �𝑘𝑘8�, determine if the signal is a power signal, an energy signal, or neither. If it is a power or energy signal, determine the value of the metric.
Problem 2 (9 points)
A periodic function is given by the Fourier Series
1∞1
𝑓𝑓(𝑡𝑡) = 2 + � 100𝑛𝑛𝜋𝜋2 cos(200𝜋𝜋𝑛𝑛𝑡𝑡)
𝑛𝑛=1
The function has a fundamental frequency of 𝑓𝑓0 = 100 Hz. Assume that the signal is sampled with a sampling frequency 𝑓𝑓𝑠𝑠 = 2 kHz. Is it possible to recover the signal 𝑓𝑓(𝑡𝑡) from its samples or will aliasing interfere? Explain your answer in words and mathematically.
Problem 3 (9 points)
Giventhesystemmodel 𝑦𝑦[𝑘𝑘]−12𝑦𝑦[𝑘𝑘−1]+14𝑦𝑦[𝑘𝑘−2]=𝑓𝑓[𝑘𝑘]−14𝑓𝑓[𝑘𝑘−1]
where 𝑦𝑦[−1] = −2 and 𝑦𝑦[−2] = 8. Assume 𝑓𝑓[𝑘𝑘] = 𝑢𝑢[𝑘𝑘]; solve recursively for 𝑦𝑦[𝑘𝑘] for 0 ≤ 𝑘𝑘 ≤ 2.
Problem 4 (9 points)
A system has the impulse response
h[𝑘𝑘] = 0.5𝑘𝑘 cos �𝜋𝜋3 𝑘𝑘� 𝑢𝑢[𝑘𝑘]
Using a convolution method, determine the system’s zero state response, 𝑦𝑦[𝑘𝑘], to the input signal 𝑓𝑓[𝑘𝑘] = �16�𝑘𝑘 𝑢𝑢[𝑘𝑘].
Problem 5 (9 points) Given the system model
6𝑦𝑦[𝑘𝑘] − 5𝑦𝑦[𝑘𝑘 − 1] = 18𝑓𝑓[𝑘𝑘]
where 𝑦𝑦[0] = 2. Assume 𝑓𝑓[𝑘𝑘] = �− 25�𝑘𝑘 𝑢𝑢[𝑘𝑘]. Use the classical solution method to solve the linear difference equation to find 𝑦𝑦𝜙𝜙[𝑘𝑘], the particular solution, and 𝑦𝑦𝑛𝑛[𝑘𝑘], the natural solution, for 𝑘𝑘 > 0.
Problem 6 (9 points)
Demonstrate sliding tape convolution to find
𝑦𝑦[𝑘𝑘] = h[𝑘𝑘] ∗ 𝑓𝑓[𝑘𝑘]
Assume that h[𝑘𝑘] = 7𝛿𝛿[𝑘𝑘 + 1 ] + 4𝛿𝛿[𝑘𝑘] and 𝑓𝑓[𝑘𝑘] = 2𝛿𝛿[𝑘𝑘 ] + 9𝛿𝛿[𝑘𝑘 − 2]
Let 𝑓𝑓[𝑘𝑘] = 9𝛿𝛿[𝑘𝑘 ] − 6𝛿𝛿[𝑘𝑘 − 1] + 3𝛿𝛿[𝑘𝑘 − 2 ]. Use the definition of the Discrete Fourier Transform to find 𝐹𝐹𝑟𝑟, the spectrum of the signal. In your solution, express complex numbers in polar form using degrees for your angles.
Problem 7 (9 points)
Problem 8 (8 points)
A system has the impulse response h[𝑘𝑘] = �0.2𝑘𝑘 + (−0.3)𝑘𝑘�𝑢𝑢[𝑘𝑘]
Use the summation definition of the Z-transform to find the transfer function of the system. Express your solutions as a ratio of polynomials in z with the denominator in standard form.
Problem 9 (9 points) Given the system model
𝑦𝑦[𝑘𝑘] − 0.3𝑦𝑦[𝑘𝑘 − 1] + 0.02𝑦𝑦[𝑘𝑘 − 2] = 2𝑓𝑓[𝑘𝑘] − 3𝑓𝑓[𝑘𝑘 − 1]
where 𝑦𝑦[−1] = 2 and 𝑦𝑦[−2] = 1. Assume 𝑓𝑓[𝑘𝑘] = �23�𝑘𝑘 𝑢𝑢[𝑘𝑘]. Solve the difference equation using the
delay property of the Z-transform to determine the zero state solution 𝑌𝑌𝑧𝑧𝑠𝑠[𝑧𝑧] and the zero input solution 𝑌𝑌𝑧𝑧𝑧𝑧[𝑧𝑧]. Express your solutions as a ratio of polynomials in z with the denominator in standard form. Do not solve for 𝑦𝑦[𝑘𝑘].
Problem 10 (16 points)
The Z-transform of the output of a system is given by
6𝑧𝑧 − 6.6𝑧𝑧 + 1.67𝑧𝑧 + 0.105𝑧𝑧 𝑌𝑌[𝑧𝑧]=4 3 2
(𝑧𝑧 + 0.3)(𝑧𝑧 − 0.6)(𝑧𝑧 − 0.5)2
Use the method of partial fraction expansion to determine the inverse Z-transform of 𝑌𝑌[𝑧𝑧].
Bonus Problem (10 points)
Figure 1 shows a system block diagram.
Figure 1. The system block diagram for the bonus problem.
Assume the proportional gain is 𝐾𝐾𝑝𝑝 = 2, the plant transfer function is 𝐺𝐺𝑝𝑝[𝑧𝑧] = 𝑧𝑧 and the sensor
𝑧𝑧+2
transfer function is 𝐻𝐻[𝑧𝑧] = 𝑧𝑧 . Find the overall transfer function of the system, 𝐻𝐻𝑠𝑠𝑠𝑠𝑠𝑠[𝑧𝑧], so that
𝑧𝑧+0.3 𝑌𝑌[𝑧𝑧] = 𝐻𝐻𝑠𝑠𝑠𝑠𝑠𝑠[𝑧𝑧]𝐹𝐹[𝑧𝑧]
Express your solutions as a ratio of polynomials in z with the denominator in standard form. Determine mathematically if the resulting system is asymptotically stable.