Lesson Title: Fast Fourier Transform Overview Dale R. Thompson Computer Science and Computer Engineering Dept. University of Arkansas 1 This material is based upon work supported by the National Science Foundation under Grant No. DUE 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

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. 2

Why sinusoids? Sinusoidal fidelity – Only amplitude and phase change – Frequency and shape stay the same 3

Aperiodic-Continuous Signal Requires Fourier Transform 4

Periodic-Continuous Signal Requires Fourier Series 5

Aperiodic-Discrete Signal Requires Discrete Time Fourier Transform 6

Periodic-Discrete Signal Requires Discrete Fourier Transform 7

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? 8

Main Point A time domain signal can be represented by a combination of sine and cosine functions at different magnitudes and frequencies 9

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 10

FFT High-Level View 11

Fast Fourier Transform (FFT) Output However, Octave starts indexing at 1 instead of zero! Therefore, add one to indexes below. 12

FFT in Octave 13

Time domain signal 14

Frequency domain signal Positive and negative frequency 15

Frequency domain signal Positive frequencies only 16

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. 17

Contact Information Dale R. Thompson, Ph.D., P.E. Associate Professor Computer Science and Computer Engineering Dept. JBHT – CSCE University of Arkansas Fayetteville, Arkansas Phone: +1 (479) FAX: +1 (479) WWW:

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. 19