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Sparse Fourier Transform. By: Yanglet Date: 2013/3/6. Outline. Frequency-Sparsity Application Example: MobiCom 2012 Understanding of the DFT The FFT Algorithm Sparse Fourier Transform Possible Issues. Frequency-Sparsity. “frequency-sparsity” almost everywhere! newwu.jpg
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Sparse Fourier Transform By: Yanglet Date: 2013/3/6
Outline Frequency-Sparsity Application Example: MobiCom 2012 Understanding of the DFT The FFT Algorithm Sparse Fourier Transform Possible Issues
Frequency-Sparsity • “frequency-sparsity” almost everywhere! • newwu.jpg • size: 1440 1152 3 1440 * 1152 pixels; RGB. • Pictures are sparse in the frequency domain. • (Just an example!) • original picture frequencies (reshaped to 1D-plot)
Frequency-Sparsity • MobiCom 2012: Faster GPS via the Sparse Fourier Transform
Frequency-Sparsity • MobiCom 2012: Faster GPS via the Sparse Fourier Transform
Understanding of the DFT • Discrete Fourier transform (DFT) • DFT by matrix multiplication • How to understand DFT?
The FFT Algorithm • Why it is ever possible? • 由于W具有周期性和对称性 u=0 u=1 u=2 u=3
The FFT Algorithm • The calculation flow:
The FFT Algorithm • Optimal? • 1. • 2. For the “exact” case. • How to improve it? • Why we want to improve it? • 1. To reduce runtime for big data (signal), e.g. real-time app., (the GPS sys.) • 2. Sparse is everywhere, there is no need calculate all n-frequency in engineering tasks? • 3. To save energy byusing less calculations. • 4. ubiquitous applications • “you don’t really study the Fourier transform for what it is,” says Laurent Demanet, an assistant professor of applied mathematics at MIT. “You take a class in signal processing, and there it is. You don’t have any choice.”
A simple trial • My way of improvement.
sFFT & sIFFT • Piotr Indyk, Dina Katabi, Eric Price, Haitham Hassanieh • Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, "Nearly Optimal Sparse Fourier Transform," STOC, 2012. • Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, "Simple and Practical Algorithm for Sparse Fourier Transform," SODA, 2012.
TheMedianOperator • Median Original one Noised one Denoised/Recovered one Parameter d=0.2. Using median filter
Near-Optimal? • Compressive Sensing Approach • Y is the random linear encoding results of K-sparse vector X • Results • We need only to recovery X
Joint Sparsity Models • JSM-1: Common Sparse Supports • Same support set, but with different coefficients. • JSM-2: Sparse Common component + sparse innovations • JSM-3: Nonsparse common component + sparse innovations
Information Theoretic Framework • Sparsity: • Sparsity: • Joint sparsity: calculate collaboratively • Conditional sparsity: