Lesson Title: Fast Fourier Transform Overview

Dale R. ThompsonComputer Science and Computer Engineering Dept.

University of Arkansas

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This material is based upon work supported by the National Science Foundation under Grant No. DUE-0736741.

Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Copyright © 2008, 2009 by Dale R. Thompson {d.r.thompson@ieee.org}

Fourier Analysis

• Do you remember in Differential Equations about transforming a problem using the Laplace Transform and then solving the problem algebraically?– This is analogous to Fourier Analysis– In Fourier Analysis, we decompose the signals into

sinusoids.

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Why sinusoids?

• Sinusoidal fidelity– Only amplitude and phase change– Frequency and shape stay the same

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Aperiodic-Continuous Signal

• Requires Fourier Transform

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Periodic-Continuous Signal

• Requires Fourier Series

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Aperiodic-Discrete Signal

• Requires Discrete Time Fourier Transform

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Periodic-Discrete Signal

• Requires Discrete Fourier Transform

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Math Issues• All signals are from -∞ to +∞• How do we apply a transform to go from time domain to frequency

domain?– Assume left and right of signal are zero

• Therefore, use Discrete Time Fourier Transform• Problem: An aperiodic signal requires ∞ number of sinusoids!

– Assume left and right of signal are duplicates. Make finite data look like infinite length signal.• Use Discrete Fourier Transform• Advantages

– A periodic signal requires a finite number of sinusoids– Efficient implementation

• Disadvantages?

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Main Point

• A time domain signal can be represented by a combination of sine and cosine functions at different magnitudes and frequencies

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Fast Fourier Transform (FFT) Math Introduction

• FFT is an efficient implementation• Each sample point represented by a complex number• We use the complex number to represent magnitude and phase

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FFT High-Level View

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Fast Fourier Transform (FFT) Output

• However, Octave starts indexing at 1 instead of zero! Therefore, add one to indexes below.

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FFT in Octave

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Time domain signal

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Frequency domain signalPositive and negative frequency

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Frequency domain signalPositive frequencies only

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Sampling Theorem

• Must sample at twice the rate as the highest frequency component. For example, to sample a 1 KHz sine wave, we must sample at 2 KHz.

• Maximum k = N/2• Engineering rule of thumb is to sample at four

or five times the highest frequency component.

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Contact InformationDale R. Thompson, Ph.D., P.E.Associate ProfessorComputer Science and Computer Engineering Dept.JBHT – CSCE 5041 University of ArkansasFayetteville, Arkansas 72701-1201

Phone: +1 (479) 575-5090FAX: +1 (479) 575-5339E-mail: d.r.thompson@ieee.orgWWW: http://comp.uark.edu/~drt/

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Copyright Notice, Acknowledgment, and Liability Release

• Copyright Notice– This material is Copyright © 2008, 2009 by Dale R. Thompson. It may be freely redistributed in its

entirety provided that this copyright notice is not removed. It may not be sold for profit or incorporated in commercial documents without the written permission of the copyright holder.

• Acknowledgment– These materials were developed through a grant from the National Science Foundation at the

University of Arkansas. Any opinions, findings, and recommendations or conclusions expressed in these materials are those of the author(s) and do not necessarily reflect those of the National Science Foundation or the University of Arkansas.

• Liability Release– The curriculum activities and lessons have been designed to be safe and engaging learning

experiences and have been field-tested with university students. However, due to the numerous variables that exist, the author(s) does not assume any liability for the use of this product. These curriculum activities and lessons are provided as is without any express or implied warranty. The user is responsible and liable for following all stated and generally accepted safety guidelines and practices.

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