EE 313 Linear Systems and Signals - Lecture 15

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).
• 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).

Announcements

• Homework #9 is due on Friday, Dec. 6th by 11:59pm, through GradeScope submission.

Lecture

• Lecture slides on Continuous-Time Fourier Transforms in PowerPoint format.
  
  • Review from last lecture
  • Fourier transform of a Dirac delta
  • Handout: Visual Dictionary of Fourier transform pairs from M. Roberts Fundamentals of Signals and Systems (available on Canvas only)

• Fall 2018 Notes by Mr. Houshang Salimian (TA): Part 1 - Part 2

Supplemental Information

• Fall 2023
  • Shift in time gives shift in phase
  • Fourier transform modulation property
• Fall 2021
  • Fourier transform of Dirac delta
  • Dividing complex numbers
  • Rectangular pulse
  • Fourier transform as a linear system
  • Inverse Fourier transform of Dirac delta
  • Fourier Series and Transform for Cosine
  • Duality in Fourier transform pairs and properties
  • Modulation and demodulation
• "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 12/05/24. Send comments to Prof. Evans at bevans@ece.utexas.edu