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**The Sparse FFT: From Theory to Practice**

Dina Katabi O. Abari, E. Adalsteinsson, A. Adam, F. adib, A. Agarwal, O. C. Andronesi, Arvind, A. Chandrakasan, F. Durand, E. Hamed, H. Hassanieh, P. Indyk, B. Ghazi, E. Price, L. Shi, V. Stojanovik

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**Ongoing sFFT Projects (Beyond Theory)**

Spectrum Sharing GPS Medical Imaging sFFT Chip Light Field Photography

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**Spectrum Crisis Spectrum Sharing Sense to find unused bands; Use them!**

How do you capture GHz of spectrum? The FCC predicts a spectrum crunch starting 2013 But at any time, most of the spectrum is unused Seattle January 7, 2013

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**Challenges in Sparse GHz Acquisition**

GHz sampling is expensive and high-power Tens of MHz ADC < a dollar Low-power A Few GHz ADC Hundreds of dollars 10x more power Compressive sensing using GHz analog mixing is expensive, and requires heavy computation

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**Recap of sFFT 1- Bucketize 2- Estimate**

Hash the spectrum into a few buckets Can ignore empty bucket 2- Estimate Estimate the large coefficient in each non-empty bucket

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**Spectrum Sensing & Decoding with sFFT**

Bucketize Estimate

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**Spectrum Sensing & Decoding with sFFT**

Bucketize Estimate Sub-sampling time Aliasing the frequencies

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**Spectrum Sensing & Decoding with sFFT**

Bucketize Estimate Hash freqs. using multiple co-prime aliasing filters Same frequencies don’t collide in two filters Identify isolated freq. in one filter and subtract them from the other; and iterate … Low-speed ADCs, which are cheap and low-power

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**Spectrum Sensing & Decoding with sFFT**

Bucketize Estimate Estimate frequency by repeating the bucketization with a time shift ∆T ∆Phase =2𝜋𝑓 ∆𝑇

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**BigBand: Low-Power GHz Receiver**

Built a 0.9 GHz receiver using three 50 MHz software radios First off-the-shelf receiver that captures a sparse signal larger than its own digital bandwidth

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**Concurrent Senders Hopping in 0.9 GHz**

Number of MHz Senders Randomly Hopping gin in 0.9 GHz

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**Realtime GHz Spectrum Sensing**

Cambridge, MA January 2013 sFFT enables a GHz low-power receiver using only a few MHz ADCs

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**Probability of Declaring a Used Frequency as Unused**

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**Ongoing sFFT Projects (Beyond Theory)**

Spectrum Sharing GPS Medical Imaging sFFT Chip Light Field Photography

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**Magnetic Resonance Spectroscopy**

Analyses the chemical making of a brain voxel Disease Bio-markers

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**Challenges Long acquisition time**

patient is in the machine for 40min to hours Artifacts due to acquisition window

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**Windowing Artifacts Fourier transform of a window is a sinc**

(Inverse) Fourier Transform Acquisition Window Convolution with a sinc

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**Windowing Artifacts 𝛿(𝑥) 𝛿(𝑥−0.5) Discretization Convolve**

Tail

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**Challenges with In-Vivo Brain MRS**

clutter due to sinc tail hours in machine Can sparse recovery help?

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**Compressive Sensing + 30% data**

Lost some Biomarkers

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**Non-Integer Sparse FFT**

Problem and Model Sparse in the continuous case The railings are because of non-integer frequencies Algorithm Use original sparse FFT to estimate integer frequencies Use gradient descent algorithm to find the non-integer frequencies to minimize the residue of our estimation over the samples

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**Challenges with In-Vivo Brain MRS**

clutter due to sinc tail hours in machine Can sparse recovery help?

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**sFFT provides clearer images while reducing the acquisition time by 3x**

Sparse FFT + 30% of data Removed Clutter without losing Biomarkers sFFT provides clearer images while reducing the acquisition time by 3x

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**Light-Field Photography**

Generate depth and perspective using images from a 2D camera array Images are correlated 4D frequencies are sparse Goal: Same performance but with fewer images

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**Reconstructed with 11% of data**

Original

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Conclusion Many applications are sparse in the frequency domain and hence can benefit from sFFT We showed that sFFT enables GHz low-power spectrum sensing and decoding, and improves MRS medical imaging and 4D light-filed capture We just scratched the surface and expect more applications soon

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