Sparse Fourier Transform (MIT) 2012

AARTHI RAGHAVENDRAJEREMY COXPIYALI DAS

RUPEN MITRA

Fourier Transform - Review

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Time Domain Frequency Domain

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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