ELEC 320, Fall 2006

Prof. Rich Kozick

Prof. Rich Kozick

**Date Assigned:** Wednesday, October 18, 2006
**Date Due:** Friday, October 20 (items 2 and 3) and Wednesday,
October 25, 2006 (item 4)

**Reading:**
Please study Chapter 3 in the Lathi text on Fourier series.
We will not have time to discuss the details in Section 3.2 on
correlation, but you should read it on your own.
The topic is very important in signal processing applications
such as time-delay estimation and digital communication.

**Exam 2** will be on Monday, October 23.
The topics will include the following.

- Chapter 2 on the time-domain analysis of LTIC systems (impulse response, step response, ZSR, convolution, ZIR, characteristic modes, etc.)
- Beginning concepts for Fourier series, such as items 2 and 3 on this assignment.

- Please experiment with the
Simulink demonstration
of Fourier series that we used in class.
You can access the demonstration from the course home page
under
Class Notes and Demonstrations.
You do not have to submit anything for this part.

- In the Simulink demonstration
`fsgen.mdl`, the three individual sine waves that are added together are

Compute the ``inner products'' of pairs of these sine waves over the interval . That is, compute the following integrals:

Here are some useful identities:

You may find it useful to sketch the integrand in each case. - What does it mean geometrically when two
*vectors*have an inner (or dot) product of zero? Draw a picture of this situation in two dimensions. - Prove that for a continuous-time system that
is
*linear*and*time-invariant*, the zero-state response (ZSR) of the system to a sinusoidal input is a sine wave with the same frequency as the input wave, but a different amplitude and phase shift. Also, find an expression for the*frequency response*of the system in terms of the*impulse response*.

An outline of the approach follows.

- Please explain why it is true that the ZSR of any
linear, time-invariant (LTI) system is completely described by
the impulse response of the system.
(Are there any LTI systems for which this is not true?)
If the impulse response is known, then the system
output due to any input is given by

- Now consider a particular input
that is applied to a LTI system with impulse response .
Put this into the convolution integral,
and look at the resulting .
You should be able to recognize that is a sine
wave with the same frequency , but with
a different amplitude and phase shift.
The trigonometric identities at the bottom of the page
will be helpful.
- In terms of the frequency response of the system
, recall that we expect that the system output
has the form

Use your result from item (b) to relate the frequency response of the system to the impulse response . This provides a mathematical connection between the frequency domain and time domain descriptions of a system. - You now understand the very important result
that a sine wave input to a LTI system produces a
sine wave output with the same frequency but different
amplitude and phase shift!

Here are some useful identities:

where and . - Please explain why it is true that the ZSR of any
linear, time-invariant (LTI) system is completely described by
the impulse response of the system.
(Are there any LTI systems for which this is not true?)
If the impulse response is known, then the system
output due to any input is given by