Midterm #1 for the Fall 2018 semester will be on Thursday, Oct. 4th, during lecture time (12:30pm to 1:45pm) but in larger rooms. The extra space will help you be more comfortable to arrange your books, notes, and laptop for the midterm exam. You will be assigned to one of the following rooms shortly:

- EER 1.528 (64 seats; 1 table per student) last names beginning with A-E
- EER 1.504 (64 seats; 1 table per student) last names beginning with F-L
- EER 1.516 (122 seats; lecture room) last names beginning with M-Z

- Chapter 1: Introduction
- Chapter 2: Sinusoids
- Chapter 3: Spectrum Representation
- Chapter 4: Sampling and Aliasing

You will also be responsible for the material in

- Slides for lectures 1-6
- Presentations and discussions for lectures 1-6
- Handout D: Discrete-Time Periodicity
- Handout N: Derivation of the Fourier Series
- Homework 1-3 assignments and their solutions
- Mini-project #1 assigment and its solution
- Canvas announcements

There will likely be five questions on Midterm #1. There will be no questions about Matlab commands or syntax on the midterm.

The Web site that accompanies
the *Signal Processing First* book has dozens of worked homework
problems from chapters 2-4 of *DSP First*.
Chapters 2-4 of *DSP First* cover identical material as chapters
2-4 of *Signal Processing First*.
You can access the homework problems and their solutions via the Homework
menu at the top of the Web page, or by using the Homework link in the
supplemental material for each chapter.

- Fall 2018 without solutions and with solutions
- Fall 2017 without solutions and with solutions
- Summer 2016 without solutions and with solutions
- Fall 2010 without solutions and with solutions
- Spring 2009 without solutions and with solutions
- Fall 2005 without solutions and with solutions
- Fall 2003 without solutions and with solutions
- Fall 2001 without solutions and with solutions
- Spring 2001 without solutions and with solutions
- Fall 1999
without solutions and
with solutions

(The solutions appear at the end of the file.)

Here are the questions from the previous exams that are related to the material to be covered on midterm #1 this semester:

**Midterm 1 Questions:**Summer 2016, problem 1.1; Spring 2009, problem 1.4, parts (a) and (b); Fall 2005, problem 1.5(b)**Midterm 2 Questions:**Summer 2016, problem 2.1(c); Fall 2010, problem 2.5**Final Exam Questions:**Fall 2017, problem 1; Fall 2010, problem 1; Fall 1999, problem 6; Fall 1999, problem 8(c)

**Problem 2:** You can solve this problem with using only one
addition (1+1) and one multiplication (1 times 1).

(a) Ideas used in finding the answer without doing any calculations:

- Convolving two pulses of the same extent produces a triangle (see the handout on convolution in the reader)
- When convolving two signals of finite extent, the extent of the convolution result is equal to the sum of the extents of the two signals being convolved.
- The maximum value of the convolution result occurs when the product of the two functions has the largest area.

(b) Ideas used to find the answer without any calculations:

- The problem answer to convolve two flipped unit step functions,
so the result should be a flipped version of convolving
*u*(*t*) with itself - The convolution of
*u*(*t*) with itself produces a ramp*t u*(*t*)

**Problem 3:** This problem involves very little math. It is meant
to test concepts. (As a side note: if the description of a system response
involves a summation, it does not necessarily mean that the system is a
discrete-time system. Conversely, if a description of a system response
involves an integral, it does not necessarily mean that the system is a
continuous-time system.)

(a) Finite impulse response

(b) The impulse response is the system response when an impulse is
input. Since the system is continuous time, use a Dirac delta
functional *d*(*t*) for the impulse:
*x*(*t*) = *d*(*t*).
So, in the summation for *y*(*t*),
replace *x* with *d*.

(c) The step response is the system response when a step function is
input. So, in the summation for *y*(*t*),
replace *x* with *u*.

(d) For *N* = 3, the step response is

- 0 for
*t*< 0 *a*_{0}for 0 <=*t*<*T**a*_{0}+*a*_{1}for*T*<=*t*< 2*T**a*_{0}+*a*_{1}+*a*_{2}for*t*>= 2*T*

(e) (*N* - 1) *T*

**Problem 4:** The solution to part (b) gets to very tedious.
In the future, I would try to not assign a problem this tedious on a midterm.

(a) The characteristic equation is
1 - 3/2 *D*^{-1} + *K* *D*^{-2} = 0.
So, there are two roots:

*r*_{0}= 3/4 + 1/4 sqrt(9 - 16*K*)*r*_{1}= 3/4 - 1/4 sqrt(9 - 16*K*)

(b) The roots need to be inside the unit circle. So,
| *r*_{0} | < 1 and | *r*_{1} | < 1.
Solve for *K*. This is the tedious part.
The answer is something like 1/2 < *K* < 1.

It means that if there are non-zero initial conditions, the system will output a weighted combination of its characteristic modes. That output would be sustained for all time from time 0 to time infinity.

**Question 2:** *I get confused about what types of systems can
be used for what types of filters, and i can't really find a specific
section in the book about it, can you direct me toward where I might
find a better understanding of what systems make what filters and what
applications they can have?*

This notion will be more clear after we learn more about frequency responses (Laplace and Fourier transforms) in the second part of the course.

As far as midterm #1, we have seen two examples of a lowpass filter (integrator in continuous-time and an averager in discrete-time) and a highpass filter (differentiator in continuous-time and a first-order difference in discrete-time). We have also seen one example of an all-pass filter (homework problem 3.3).

The Mandrill (Baboon) demonstration uses a cascade of a lowpass and a highpass filter, and the cascade has a bandpass response.

I have not presented any examples of bandstop filters yet.

**Question 3:** *Finally, in my differential equations class, we
didn't really use the e ^{j t} for complex roots, we just made it as
sin t + cos t. I'm used to solving it this way, so the book method
of using e^{j t} or cos(t + theta) is kind of confusing. Should I use
my time to learn this method or would the other way be ok on the test?*

In terms of solving a differential equation, what matters is getting the right answer with a mathematically correct method.

In terms of understanding the behavior of systems governed by
differential equations, it is important to know how the roots of
the characteristic polynomial are mapped into characteristic modes.
This is where the e^{lambda t} and
*t ^{k}* e

Last updated 10/08/18. Send comments to Prof. Brian L. Evans at bevans@ece.utexas.edu.