Algorithm is faster than FFT for all sparse signals

Feb. 9, 2012
Researchers at the Massachusetts Institute of technology (Cambridge, MA) have developed two algorithms that are faster than the fast Fourier transform (FFT) for all sparse signals.

Being able to compute the Fourier transform of an input signal is crucial in photonics—for example, in determining the spatial frequencies in an image. The standard method of computing a discrete Fourier transform (DFT) is by using the fast Fourier transform (FFT) algorithm. However, algorithms faster than the FFT would be desirable. Researchers at the Massachusetts Institute of technology (Cambridge, MA) have developed two algorithms that are faster than the FFT for all sparse signals. (A sparse signal is one in which some of its Fourier coefficients are near enough to zero that they can be ignored.) While other algorithms have previously been developed to improve on the FFT for sparse signals, none of them have improved on the FFT’s runtime for the whole range of sparse signals.

For a signal with k nonzero Fourier coefficients, and a length n of the input signal that is a power of 2, the researchers show two new DFT algorithms. The first is an O(k log n)-time algorithm for the exactly k-sparse case (where k is small). (O means “on the order of.”) The second is an O(k log n log(n/k))-time algorithm for the general case. In contrast, the FFT computes the DFT in O(n log n) time. Contact Haitham Hassanieh at [email protected].

About the Author

John Wallace | Senior Technical Editor (1998-2022)

John Wallace was with Laser Focus World for nearly 25 years, retiring in late June 2022. He obtained a bachelor's degree in mechanical engineering and physics at Rutgers University and a master's in optical engineering at the University of Rochester. Before becoming an editor, John worked as an engineer at RCA, Exxon, Eastman Kodak, and GCA Corporation.

Sponsored Recommendations

Achromatic Lenses: High-Quality Custom Optics

March 13, 2025
Ensure clarity and accuracy in your optics systems with Lacroix’s achromatic lenses. Explore how our custom solutions minimize chromatic aberration for perfect results.

Manufacturing Considerations for Tolerancing Aspheres

March 13, 2025
Understand the critical factors in manufacturing aspheres and how Lacroix Optics ensures precise tolerancing in every optical component.

Explore Our Videos: Insights into Precision Optics

March 13, 2025
Get an inside look at Lacroix Optics with our collection of informative videos showcasing our capabilities, innovations, and processes.

Optical Assemblies: Reliable and Precise Solutions

March 13, 2025
Ensure your optical system works seamlessly with Lacroix Optics' custom optical assemblies. Discover the precision and reliability we bring to every project.

Voice your opinion!

To join the conversation, and become an exclusive member of Laser Focus World, create an account today!