Signals and Systems

Topics

Overview of Signals and Systems

Individual Subjects

• Linear Time Invariant Systems
  • "The Game"
• Orthogonal Functions
  • Douglas Harder Lecture on Functions as Vectors
• Finding the Energy in a Signal
  • Rayleigh's Theorem
• Fourier Series
• Fourier Transforms
  • Discrete Fourier transform
• Sampling
• FIR Filter Example
• Relationship between e, sin and cos

Some Useful Links to Suppliment or Substitute for a Textbook

Books on Signal Processing

• Spectral Audio Signal Processing, by Julius O. Smith III
• The Scientist and Engineer's Guide to Digital Signal Processing by Steven W. Smith, Ph.D. The professor likes this one.
• Discrete Time Signals & Systems
• The Fast Fourier Transform by E. O. Brigham This was one of the professor's textbooks when he took this class.
• Mathematics of Signal Processing: A First Course, by Charles L. Byrne Good chapters on Computer Aided Tomography (CAT Scan) as well as a lot of other interesting things.
• Signals & Systems, with MATLAB/Simulink, Karris

Fourier Series

• Interactive Mathematics (like a textbook with some examples)
• Mathworld
• Wikipedia
• MIT handout on Fourier Series, Fourier Transform, and Laplace Transform
• Fourier Theory B..M..N.. Clarke

Dirac Delta Function and Convolution

• MIT handout on Dirac Delta Function and Convolution

Multi-rate Filtering

Multirate Filters Introduction

Slides from a Presentation on Polyphase Decimation and Interpolation by Mark Fowler

FIR Filters

• This is a very easy-to-understand summary of FIR basics, properties, design, and implementation

• Another easy-to-understand article about decimation

• Another easy-to-understand article about interpolation

• Fast Convolution Based on the FFT This reference shows how end effects are dealt with. To use the FFT for convolution, you need to do it in blocks, which leads to end effects, and more latency, but if your blocks are big enough, it speeds up the convolution.

• The truncated Fourier series or DFT method of FIR filter design gives the best approximation of the desired frequency response in a least squared error sense, but it is not very good near discontinuities. The Parks-McClellan (Remez) iterative algorithm gives equal ripple in the pass band and stop band. This is the best in terms of minimizing the maximum error in each area, which is usually better in the real world. MATLAB/octave have automated functions to do that. See remez() in the Signals package of octave.
• A nice detailed description of the Parks McClellan algorithm with MATLAB/octave code.

Adaptive FIR Filters

Introduction to Adaptive Filters, Simon Haykin

Simon Haykin's book chapter

Adaptive LMS in an FPGA

Scott Douglas' chapter on Adaptive Filters has some interesting applications, among them linear predictive coding (LPC) used in speech codecs.

Adaptive Filters In the Frequency Domain

Adaptive Filter Echo Cancellation

Adaptive Filters and Applications (Mathworks)

More Applications of Adaptive Filters

Constant Modulus Algorythm

Using the CMA on antenna arrays

Math Editors

• SageMath uses sage and latex to make your document. It has a nice preview that shows you what you get with your LaTex code. It works well with the online LaTex editor below.
  • Online LaTeX Editor
• Lurch (a math editor for your desktop that will check your logic)
• wxMaxima is a document based GUI for the Maxima computer algebra system.
• PTC Mathcad works for Windows and costs something.

Course Pages

2005-2006 Assignments

2006-2007 Assignments

2008-2009 Assignments

2009-2010 Assignments

Class notes for Signals & Systems

Articles

Octave Tutorials

A nice list of MATLAB tutorials from Duke University

Installing Octave on a Mac (Chris Lau)

Octave and Scilab on a Mac (Ben Henry)

Installing Octave (with the GUI) from source on Ubuntu

ASN2 - Octave Tutorial (Jodi S. Hodge)

A u(t) function example

FIR Filter Example Code for Octave

FIR Filter Design and Testing Using the DFT

Leakage Example Octave Script

Interpolation using the DFT Example Script

An example showing how convolution is just a dot product

Tuner Upper Removal Demonstration

Airplane Noise Removal Demonstration

Final Project (2011)

Matlab/Octave deMorse.m

morse.m This is the one from mathworks

Homework Assignments

Please put your name next to the assignment, linking it to your submission

• HW #1 - Make a personal page on this wiki (Chris Lau)(Jodi S. Hodge)(Chris Wills)(Victor Shepherd)
• HW #2 - Write a tutorial about installing and/or using Octave (Chris Lau)(Jodi S. Hodge)(Victor Shepherd)
• HW #3 - Show graphically that $\int _{-\infty }^{\infty }e^{j2\pi f(t-u)}\,df=\delta (t-u)$ (Chris Lau)(Jodi S. Hodge)(Chris Wills)(Victor Shepherd)

• HW #4 - Given a linear time-invariant system where $\ u(t)$ produces an output $\ w(t)$, find the output due to any function $\ x(t)$ (Chris Lau)
• HW #5: (Chris Lau)
  • Part 1 - Find ${\mathcal {F}}[e^{-\sigma t}x(t)u(t)]$ and relate it to the Laplace Transform. Derive the Inverse Laplace Transform of this from the inverse Fourier Transform.
  • Part 2 -
    

• HW #6 - Pick a property of the Fourier Transform & present it on the Wiki. Make a table with all your properties. Interpret your property. (Ben Henry)(Chris Lau)(Victor Shepherd)
• HW #7 - Finish the practice tests
• HW #8 - Make a page about interpolating FIR filters. Note how many multiply/add operations.(Jodi S. Hodge)(Chris Lau)(Victor Shepherd)
• HW #9 - Add to #8 writeup how to do a decimating filter and figure out how many multiply & adds are needed for a n/2 decimating low pass filter.(Jodi S. Hodge)(Chris Lau)(Victor Shepherd)
• HW #10 - Use Octave (or Mathlab or Silab) to plot the frequency response of low pass filters with cut off frequencies of 1/32T, 1/8T, and 1/4T and compare how many coeffficients are needed with an eye to answer the question "Is it less calculation to decimate and then filter, or better to put the filter in the pre-decimation filter?" (Jodi S. Hodge)(Victor Shepherd)
• HW #11 - Is our method the same as Mark Fowler's? See

Wiki. Same # multiply and adds? See Notes 11/3/10. (Jodi S. Hodge)(Victor Shepherd)

• HW #12 - Experiment with a variety of signals having a 3Khz bandwidth to determine the resolution you can get when doing a cross correlation $\ r(m)=\displaystyle \sum \limits _{n=0}^{N-1}x(n)x(n+m)$. You can generate the signals randomly and filter them to obtain the band-limited signals. (Jodi S. Hodge)
• HW #13 - Derive the following realtions:
  • a) $DFT(x(k-l))\!$
  • b) $DFT(e^{j2\pi lk/N}x(k)\!$
  • c) $\sum \limits _{k=0}^{N-1}x(k)y(k)^{*}=c\sum \limits _{k=0}^{N-1}X(n)Y(n)^{*}$ (Victor Shepherd)

• HW #14 - Come up with a use for an adaptiveFIR filter and make an Octave script to demonstrate it. (Jodi S. Hodge)(Victor Shepherd)
• HW #15 - Do Practice Exam II (Victor Shepherd)
• CW-Robot Octave Simulation

People Involved with this Wiki

2013-2014 Contributors

Brian Lenz

Brian Clark

Daniel Liwag

Alex Haesche

2012-2013 Contributors

Brian Haddad

Kurt Hildebrand

Denver Lodge

Michael von Pohle

2011-2012 Contributors

Matthew Blaire

Cody Lorenz

2010-2011 Contributors

Ben Henry

Christopher Garrison Lau I

Chris Wills

Jodi S. Hodge

Luke Chilson

Victor Shepherd

2009-2010 Contributors

Nick Christman

Joshua Sarris

Kevin Starkey

Max Woesner

Jodi Hodge

Corneliu Turturica

2008-2009 Contributors

Eric Clay

Chuck Yang

Elton Zebron

Luke Chilson

Brandon Price

Greg Fong

2007-2008 contributors

Baldwin Britton

Denver Harris

Mark Priddy

Chris Rasmussen

Michael Roth

Sally Roth

2006-2007 contributors

Ryan J Smith

Nathan Ferch

Andrew Lopez

Nathan Sherman

Chris Adkins

2005-2006 contributors

Gabriela Valdivia

Raymond Betz

Jenni Christensen

Jeffrey Wonoprabowo

Paul Wilson

Instructor: Rob Frohne

2004-2005 contributors

Sam Barnes

Shawn Santana

Aric Goe

Todd Caswell

David Anderson

Anthony Guenterberg