# EE 313 Linear Systems and Signals - Lecture 15

Lecture by Prof. Brian L. Evans

## 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).
• 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?

## Announcements

• Midterm #2 exams returned.
• Homework #9 is due on Friday, Nov. 30th, by 10:59pm, through Canvas submission.
• Final exam is Wednesday, Dec. 19th, 2:00-5:00pm.
• Additional office hours for Prof. Evans on Wednesdays & Fridays 9:00-10:00am, EER 6.882, through December 7th.

## Lecture

• Lecture slides on Continuous-Time Fourier Transforms in PowerPoint format.
• Notes by Mr. Houshang Salimian (TA): Part 1 - Part 2
• "The more general uncertainty principle, beyond quantum", 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.

Last updated 12/06/18. Send comments to Prof. Evans at bevans@ece.utexas.edu