sparse fourier transform
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Sparse Fourier Transform (MIT) 2012
AARTHI RAGHAVENDRAJEREMY COXPIYALI DAS
RUPEN MITRA
Fourier Transform - Review
𝑓መሺሻ=න 𝑓ሺ𝑥ሻ∞−∞ 𝑒−2𝜋𝑖𝑥𝑑𝑥
Time Domain Frequency Domain
sign
al
sign
al
t f
1963: Fast Fourier Transform
• Found a way to do a (discrete) FT in O (n log n) time– Classic divide and conquer; the base operation is the ‘fancy
math’• Revolutionary, as FT applies to many problems
• Can we beat that? I mean, O (n log n) rules– Consider that n is the number of points in your waveform
sample– Base operation is expensive– EEK!
So saying O ( n log n) is only half the story?!?
Divide and Conquer
Past 15 years
• What if I don’t want to find ALL the frequencies, just the k biggest ones?
• This is the “Sparse FT”• Surely I can improve on O (n log n)
• 2005: Complex FT– O( k logc n ), c 4– k < n / log3n
• About 20 papers have contributed tricks for improvement
2012: New SFT
• O( k log n log(n/k) )• Achieved with
– Dolph-Chebyshev filters work best– k-appropriate size sampling of data
• Convolution – typical algo to calculate a FT– F{} is Fourier transform– f is unknown signal, g is a known function such as
Dolph-Chebyshev filter
known unknown known (predetermined by filter choice)
New SFT example with Boxcar filter
• FFT
• FFT on B points
• SFT(B points) Algorithm or
Heuristic?
Data correction
• Problem: no filter is perfect; you will miss signal in “bad” or “blind” parts of domain.
• Use randomized permutation to pick points, to prevent sampling bias or “leakage”
• Can now use sparse matrix operations
Random Sampling
Applications
• Solving a polynomial’s roots use FFT
• SFT cannot help, all roots are equally important
Applications: Faster GPS (MIT) 2012
• GPS locks on to satellites using FFT– Pick the satellite out of noise
• Analyzing n frequency vs. time samples:– Old: O(n log n), may have to take more samples– New: uses p subsamples and SFT to divide and
conquer search for principle signal• New algorithm adaptive:– With high SNR, subsampling reduced; O(n)
Applications
• Image Compression– Take 8x8 grid of pixels– Convert RGB channels into 3 discrete graphs– Identify k dominant frequencies (usually 7)– (best case) 192 bytes 21 bytes
• MRI processing speed (2013)• Speech recognition (2013)
Applications (Our Ideas)
• Seismic Reflection Method (‘Radar’) for finding gas and oil
• Communication: determine maximum frequency without distortion
• Cutting through noise is radar signals, such as cloud cover
• Calculating refraction constants in optics
In Conclusion
• It’s brand new: an efficient Sparse Fourier Transform
• Know what it is and when to apply it, and you will be the next hero at your job
• FFT is everywhere, you are bound to bump into a good use for SFT
Credits• Background image
http://nextbigfuture.com/2012/01/faster-than-fast-fourier-transform.html• FT demohttp://en.wikipedia.org/wiki/Fourier_transform• Images compliments of original author
http://people.csail.mit.edu/indyk/fourier-gsip.pdf• http://groups.csail.mit.edu/netmit/sFFT/paper.html
– Nearly Optimal Sparse Fourier Transform [SLIDES] Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price. STOC, May 2012.
Works Cited• Simple and Practical Algorithm for Sparse Fourier Transform
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price. SODA, January 2012.
• Nearly Optimal Sparse Fourier TransformHaitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price. STOC, May 2012.
• Faster GPS Via the Sparse Fourier TransformHaitham Hassanieh, Fadel Adib, Dina Katabi, and Piotr Indyk ACM MOBICOM, August 2012.
• PPT of author talk:http://people.csail.mit.edu/indyk/fourier-gsip.pdf
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