# EE313 Linear Systems and Signals - Midterm #2

Each midterm exam will be an open book, open notes, open laptop exam that is scheduled to last the entire period. The laptop must have all external networking connections disabled. Because each midterm is an open book, notes and laptop exam, you'll likely need to move quickly through the exam but also need to think deeply about possible ways to solve the problems. To this end, having a week of regular sleep, eating, and exercise will be very helpful.

Midterm #2 for the Fall 2017 semester will be on Thursday, Nov. 16th, during lecture time (12:30pm to 2:00pm) but in the following rooms:

• EER 1.516 if your last name starts with A-L
• WRW 102 if your last name starts with M-Z
WRW 102 seats 123, and EER 1.516 seats 122. (WRW is the Aerospace Engineering Bldg.) With 64 students in the class, you should have several empty seats or an aisle on either side of you. The extra space will help you be more comfortable to arrange your books, notes, and laptop for the midterm exam.

## Previous Tests

Here are several example midterm #2 exams:
Here are the questions from the previous exams that are related to the material to be covered on midterm #2 this semester.
• Midterm #1 Questions
• Midterm #1, Summer 2016, Problem 4. Continuous-Time Convolution. "Zero state" means that the system state is zero; i.e., the initial conditions are zero. So, the systems in this problem are linear and time-invariant. In problem 4(b), please note that δ(t) is the Dirac delta even though the plot of h(t) seems to imply otherwise.
• Midterm #1, Summer 2016, Problem 5. Discrete-Time Convolution. "Zero state" means that the system state is zero; i.e., the initial conditions are zero. So, the systems in this problem are linear and time-invariant.
• Midterm #1, Fall 2010, Problem 1.2. Convolution.
• Midterm #1, Fall 2010, Problem 1.3. Continuous-Time System Properties.
• Midterm #1, Fall 2010, Problem 1.4. Discrete-Time Convolution.
• Midterm #1, Fall 2010, Problem 1.5. Discrete-time systems part only. We haven’t covered most of the continuous-time systems mentioned in the answer.
• Midterm #1, Spring 2009, Problem 1.2. Continuous-Time Convolution.
• Midterm #1, Spring 2009, Problem 1.3. Continuous-Time Tapped Delay Line.
• Midterm #1, Spring 2009, Problem 1.4. Potpourri. Signal Properties, System Properties.
• Midterm #1, Fall 2005, Problem 1.3. Continuous-Time Tapped Delay Line.
• Midterm #1, Fall 2005, Problem 1.4. Continuous-Time System Properties.
• Midterm #1, Fall 2005, Problem 1.5. Potpourri. We haven’t covered 1.5(b) part ii yet.
• Midterm #1, Fall 2003, Problem 1.2. Discrete-Time System Response (Convolution).
• Midterm #1, Fall 2003, Problem 1.3. Continuous-Time Tapped Delay Line.
• Midterm #1, Fall 2003, Problem 1.4. Differentiator.
• Midterm #1, Fall 2003, Problem 1.5. Potpourri. Convolution, resonators, and oscillators.
• Midterm #1, Fall 2001, Problem 1.2. Continuous-Time Convolution.
• Midterm #1, Fall 2001, Problem 1.3. Continuous-Time System Properties.
• Midterm #1, Spring 2001, Problem 1.2. Continuous-Time Convolution.
• Midterm #1, Spring 2001, Problem 1.3. Discrete-Time Tapped Delay Line.
• Midterm #1, Spring 2001, Problem 1.4. Step Response.
• Midterm #1, Fall 1999, Problem 1.2. Continuous-Time Convolution.
• Midterm #2 Questions
• Midterm #2, Summer 2016, Problem 2.3, Z-Transforms
• Midterm #2, Summer 2016, Problem 2.4, Discrete-Time LTI Oscillator with Infinite Impulse Response
• Midterm #2, Spring 2009, Problem 2.1, Difference Equation
• Midterm #2, Spring 2009, Problem 2.2, Discrete-Time Convolution
• Midterm #2, Spring 2009, Problem 2.3, Discrete-Time Tapped Delay Line
• Midterm #2, Spring 2009, Problem 2.5a, Discrete-Time Convolution
• Midterm #2, Fall 2003, Problem 2.2, Z-Transforms
• Midterm #2, Fall 2003, Problem 2.4, Transfer Functions and Frequency Responses for a Discrete-Time LTI System
• Midterm #2, Spring 2001, Problem 2.1, Solving a Difference Equation
• Midterm #2, Spring 2001, Problem 2.2, Discrete-Time Impulse and Step Responses
• Midterm #2, Spring 2001, Problem 2.5(a) and (b), Discrete-Time Filter Design
• Midterm #2, Fall 1999, Problem 2.3, Discrete-Time Tapped Delay Line
• Midterm #2, Fall 1999, Problem 2.4, Discrete-Time Transfer Functions
• Final Exam Questions
• Final Exam, Summer 2016, Problem 3, Discrete-Time Fourier Transform
• Final Exam, Summer 2016, Problem 4, Discrete-Time Frequency Response
• Final Exam, Summer 2016, Problem 5, Discrete-Time Filter Design
• Final Exam, Summer 2016, Problem 7, Continuous-Time Convolution
• Final Exam, Summer 2016, Problem 8, Discrete-Time Averaging Filters
• Final Exam, Fall 2010, Problem 4, Discrete-Time Stability
• Final Exam, Fall 2010, Problem 6, Discrete-Time Filter Analysis
• Final Exam, Fall 2010, Problem 7, Discrete-Time Filter Design
• Final Exam, Spring 2009, Problem 3, Discrete-Time Convolution and Continuous-Time Convolution
• Final Exam, Spring 2009, Problem 6, Discrete-Time Filter Analysis
• Final Exam, Spring 2009, Problem 7, Discrete-Time Filter Design
• Final Exam, Fall 2005, Problem 4a, Discrete-Time Convolution
• Final Exam, Fall 2005, Problem 6, Discrete-Time Filter Analysis
• Final Exam, Fall 2005, Problem 7, Discrete-Time Filter Design
• Final Exam, Fall 2003, Problem 4, Z-Transforms
• Final Exam, Fall 1999, Problem 1, Difference Equations
• Final Exam, Fall 1999, Problem 2, Discrete-Time Convolution
• Final Exam, Fall 1999, Problem 8e, Discrete-Time Filter Design

## Coverage

A review of midterm #2 material is available from a previous offering of the course. We haven't covered the material on the Laplace transform yet.

For midterm #2 in EE 313, you will be responsible for the following sections of Signal Processing First book by McClellan, Schafer and Yoder:

• Chapter 5: FIR Filters
• Chapter 6: Frequency Response of FIR Filters
• Chapter 7: z-Transforms
• Chapter 8: IIR Filters
• Chapter 9: Continuous-Time Signals and LTI Systems
You will also be responsible for the material in
• Slides for lectures 7-13
• Presentations and discussions for lecture 7-13 slides
• Homework 4-7 assignments and their solutions
• Mini-project #2 assigment and its solution
• Handouts C-J
• Canvas announcements

There will likely be five questions on midterm #2. There will be no questions about Matlab.

## Other Questions

Is the region of convergence affected when there is a time shift?

The region of convergence is not affected by a time shift. A time shift simply causes the Laplace transform to have a multiplication term in the form on exp(-s t0) where t0 is the time shift.

Also, if it is a unilateral Laplace transform, do we need to worry about how to find the region of convergence (in lecture slide 11-10 it states that there is no need to specify a region of convergence)?

The region of convergence is not necessary if one is computing the inverse Laplace transform and the time-domain signal that will result is causal. However, for other reasons, such as BIBO stability checking for transfer functions and converting transfer functions to frequency responses, knowing the region of convergence is critical.

Problem 2 on Fall 1999 Midterm #2
This problem gives the step response of an LTI system, which we'll call ystep(t). It then asks you to find the response to a new signal x(t). Here are two different approaches for solving this problem:

• Special case: We can write x(t) as 5 u(t) - 5 u(t-2). Using linear and time-invariant properties, the output is

y(t) = 5 ystep(t) - 5 ystep(t - 2).

• General case: From the given information, we can find the transfer function of the LTI system:

H(s) = Ystep(s) / U(s)

where U(s) is the Laplace transform of the unit step function, i.e. U(s) = 1 / s. Once we know the transfer function H(s),

Y(s) = H(s) X(s)

Finally, we can take the inverse Laplace transform of Y(s) to obtain y(t).

Last updated 11/19/17. Send comments to bevans@ece.utexas.edu.