# EE313 Linear Systems and Signals - Midterm #2

## Coverage

For midterm #2 in EE 313, you will be responsible for the
following sections of Roberts' book:
- Chapter 3: sections 3.5 and 3.7
- Chapter 4: sections 4.1, 4.2 and 4.3
- Chapter 5: sections 5.1, 5.2, 5.3, 5.4 and 5.5
- Chapter 6: sections 6.1, 6.2, 6.3 and 6.9
- Chapter 7: section 7.1, 7.2 and 7.3
- Chapter 9: all sections
- Chapter 10: sections 10.1, 10.2, 10.3 and 10.4
- Appendix A, E and F

You will also be responsible for the material covered in homework
assignments 6-10 and the solution sets for those homework assignments.
In the reader, you will be responsible for

- Slides for lectures 8-13 and 16-20
- All appendices

You are responsible for what we said during lecture and
what is on the course Blackboard site (e.g. homework hints).
There will probably be five questions on Midterm #2.
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 chapters 1-6 would be relevant to what we have studied so
far in EE 313.

## 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 *y*_{step}(*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 *y*_{step}(*t*) -
5 *y*_{step}(*t* - 2).

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

*H*(*s*) =
*Y*_{step}(*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 07/28/16.
Send comments to
bevans@ece.utexas.edu