# EE313 Linear Systems and Signals - Midterm #1

Midterm #1 for Fall 2010 will take place on Thursday, Oct. 7th, during lecture time from 11:00 am to 12:30 pm:
• 21 Students with last names starting with A-K report to ENS 127.
• 21 Students with last names starting with L-Z report to ENS 314.
With about 40 students enrolled, you should have at least two empty seats or an aisle on either side of you.

## Coverage

For midterm #1 in EE 313, you will be responsible for the following sections of Roberts' book:
• Chapter 1: all sections except 1.4
• Chapter 2: all sections except 2.14
• Chapter 3: all sections
• Appendix A, D, H and I

You will also be responsible for the material covered in homework assignments 1-5 and the solution sets for those homework assignments.

In the reader, you will be responsible for

• Slides for lectures 1-8
• All handouts in the back of the reader, except for Appendix H Modulation Example, Appendix I Modulation Summary and Appendix L Sample Midterm #2 exams.
You are responsible for what I said during lecture and what is on the course Blackboard site (e.g. homework hints).

There will likely be five questions on Midterm #1. There will be no analog circuits or analog electronics on the midterm. There will be no questions about either Matlab or Mathematica on the midterm.

If you have decided to buy the optional text DSP First, then the first two chapters would be relevant to what we have studied so far in EE 313.

## Hints on Sample Quiz #1 from Fall 1999

Problem 1: The differential equation has a double root. The form of the solution is C1 e-t + C2 t e-t.

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.
The answer is a triangle that starts at t = 0 and ends at t = 2. The maximum value of 1 occurs at t = 1.

(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)
The convolution is a ramp heading in the negative time direction, i.e., -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

a0 u(t) + a1 u(t - T) + a2 u(t - 2 T)
So, the step response can be described as a piecewise continuous function as follows:
• 0 for t < 0
• a0 for 0 <= t < T
• a0 + a1 for T <= t < 2 T
• a0 + a1 + a2 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:

• r0 = 3/4 + 1/4 sqrt(9 - 16 K)
• r1 = 3/4 - 1/4 sqrt(9 - 16 K)

(b) The roots need to be inside the unit circle. So, | r0 | < 1 and | r1 | < 1. Solve for K. This is the tedious part. The answer is something like 1/2 < K < 1.

## Other Questions

Question 1: What does it mean when it says any combination of characteristic modes can be sustained by the system alone without requiring an external input?

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 ej 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 ej 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 elambda t and tk elambda t forms arise.

Last updated 10/28/10. Send comments to bevans@ece.utexas.edu