lec17 - Introduction to Algorithms 6.046J/18.401 Lecture 17...

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Introduction to Algorithms 6.046J/18.401 Lecture 17 Prof. Piotr Indyk

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Introduction to Algorithms April 17, 2003 L17.2 © 2003 by Piotr Indyk Fast Fourier Transform • Discrete Fourier Transform (DFT): – Given: coefficients of a polynomial a(x)=a 0 +a 1 x+… +a n-1 x n-1 – Goal: compute a( n 0 ), a( n 1 ) … a( n n-1 ) , n is the “principal n-th root of unity” • Challenge: Perform DFT in O(n log n) time.
Introduction to Algorithms April 17, 2003 L17.3 © 2003 by Piotr Indyk Motivation I: 6.003 FFT is essential for digital signal processing – a 0 , a 1 , … , a n-1 : signal in the “time domain” – a( n 0 ), a( n 1 ) … a( n n-1 ) : signal in the “frequency domain” – FFT enables quick conversion from one domain to the other Used in Compact Disks, Digital Cameras, Synthesizers, etc, etc.

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Introduction to Algorithms April 17, 2003 L17.4 © 2003 by Piotr Indyk Example application: SETI Searching For Extraterrestial Intelligence (SETI): “At each drift rate, the client searches for signals at one or more bandwidths between 0.075 and 1,221 Hz. This is accomplished by using FFTs of length 2 n ( n = 3, 4, . .., 17) to transform the data into a number of time-ordered power spectra.”
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lec17 - Introduction to Algorithms 6.046J/18.401 Lecture 17...

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