This ppt gives an overview of the recent MIT paper on Sparse Fourier transform using which data can be processed 10 or 100 times faster than the traditional Fast Fourier Transform. This is possible as the FFT has a complexity of O(nlogn), where the sparse FT has a potentially lower complexity of O(klogn) in the sparse spectrum case.

1. Sparse Fourier
Transform (MIT) 2012
AARTHI RAGHAVENDRA
JEREMY COX
PIYALI DAS
RUPEN MITRA

2. Fourier Transform - Review
𝑓  = 𝑓 𝑥
∞
−∞
𝑒−2𝜋𝑖 𝑥
𝑑𝑥
Time Domain Frequency Domain

3. 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

4. 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

5. 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)

6. New SFT
example with Boxcar filter
• FFT
• FFT on
B points
• SFT
(B points) Algorithm or
Heuristic?

7. 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

8. Random Sampling

9. Applications
• Solving a polynomial’s roots use FFT
• SFT cannot help, all roots are equally
important

10. 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)

11. 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)

12. 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

13. 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

14. Credits
• Background image
http://nextbigfuture.com/2012/01/faster-than-fast-fourier-transform.html
• FT demo
http://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.

15. 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 Transform
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price.
STOC, May 2012.
• Faster GPS Via the Sparse Fourier Transform
Haitham 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