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Lecture 15: Continuous-Time Fourier Transforms
EE 313 Linear Systems and Signals - Lecture 15
Lecture by Prof. Brian L. Evans
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Before Lecture
- "What is 5G?",
"The Daily Planet", Discovery Channel, April 12, 2018,
featuring Dr. Arun Ghosh (AT&T Labs, Austin, TX) and
Mr. Sean McShane (Discovery Channel).
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- Unique mobile phone subscribers worldwide increased by 100M
from 4.96B in April 2017 to 5.06B in April 2018 according to
GSMA Intelligence.
This increase matches the increase in
global population
during this time from 7.5B to 7.6B.
(Population by year.)
How to increase the number of devices on the cellular network?
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-
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- Since 2006, capabilities of cellular and Wi-Fi systems have dramatically
increased to meet the exponential growth in the demand for global mobile data
traffic.
Global mobile data traffic, which accounts for about half of the Web traffic today,
increased 4000x from 2006 to 2016, or 2.3x per year
(News story) and
7x from 2017 to 2022 (projected), or 1.46x per year
(Forecast).
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Announcements
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- Midterm #2 exams returned.
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Solution Set.
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- Homework #8
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- Midterm #2 exams returned.
- Homework #9
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is due on Friday, Dec. 1st, by 12:30pm, through Canvas submission.
- Final exam is on Monday, Dec. 18th, 9:00am-12:00pm,
in EERC 1.516 and EERC 1.518.
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is due on Friday, Nov. 30th, by 5:00pm, through Canvas submission.
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is due on Friday, Nov. 30th, by 10:59pm, through Canvas submission.
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- Final exam is TBA.
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- Final exam is Wednesday, Dec. 19th, 2:00-5:00pm.
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- Additional office hours for Prof. Evans on Wednesdays & Fridays 9:00-10:00am, EER 6.882,
through December 7th.
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- Homework #8
is due on Friday, Nov. 12th, by 5:00pm, through Canvas submission.
- Midterm #2
is on Tuesday, Nov. 16th, 11:00am-12:30pm
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- Homework #9
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is due on Friday, Dec. 3rd, by 11:59pm, through Canvas submission.
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is due on Friday, Dec. 1st, by 11:59pm, through Canvas submission.
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is due on Friday, Dec. 6th by 11:59pm, through GradeScope submission.
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- Lecture slides on Continuous-Time Fourier Transforms in
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PowerPoint format.
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- Notes by Mr. Houshang Salimian (TA):
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PowerPoint format.
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Fall 2021
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- Fall 2018 Notes by Mr. Houshang Salimian (TA):
Part 1 -
Part 2
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- Fall 2023
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- Fall 2021
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- Fall 2018 Notes by Mr. Houshang Salimian (TA):
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Part 1 -
Part 2
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- "The more general uncertainty principle, beyond quantum",
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3Blue1Brown, Feb. 24, 2018,
describes the interplay between time observations and frequency resolution
by using examples from audio and radar systems.
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3Blue1Brown, Feb. 24, 2018.
The video provides audio and radar examples.
Here are some of the connections made by the video to the idea that
the longer we observe a signal, the more accurately we know its frequency content:
- We can model the amount of time in our observation as a rectangular pulse of
width tau.
Its frequency content is a sinc pulse whose passband is centered at
0 rad/s and is 2 pi / tau rad/s wide (lecture slide 15-6).
We determine the passband width by measuring the width where the magnitude
response is at half of the maximum value.
As the amount of observation time increases, the passband width decreases and hence the resolution increases.
- An extreme case is a Dirac delta in time that contains all frequencies
with equal strength (lecture slide 11-4).
- Another extreme case is a Dirac delta in the frequency domain is a
constant signal over all time (lecture slide 15-7).
- The Scaling Property of the Fourier Transform (lecture slide 15-10) also shows the idea.
- In practice, we can only observe a signal x(t) for a finite amount of time,
T seconds.
Our observed signal is then x(t) rect((t - T/2)/T) where rect((t - T/2)/T) is
a rectangular pulse that lasts from 0 to T seconds.
Multiplication in the time domain means convolution in the frequency domain.
So, the frequency content of the signal x(t) under finite duration observation
will be a distorted version of the frequency content of x(t).
This is an example how observing a signal disturbs the signal.
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Last updated %G%.
Send comments to Prof. Evans at
bevans@ece.utexas.edu
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