# MIT: The faster-than-fast Fourier transform

January 18, 2012The Fourier transform is one of the most fundamental concepts in the information sciences. It's a method for representing an irregular signal -- such as the voltage fluctuations in the wire that connects an MP3 player to a loudspeaker -- as a combination of pure frequencies. It's universal in signal processing, but it can also be used to compress image and audio files, solve differential equations and price stock options, among other things.

The reason the Fourier transform is so prevalent is an algorithm called the fast Fourier transform (FFT), devised in the mid-1960s, which made it practical to calculate Fourier transforms on the fly. Ever since the FFT was proposed, however, people have wondered whether an even faster algorithm could be found.

At the Association for Computing Machinery's Symposium on Discrete Algorithms (SODA) this week, a group of MIT researchers will present a new algorithm that, in a large range of practically important cases, improves on the fast Fourier transform. Under some circumstances, the improvement can be dramatic -- a tenfold increase in speed. The new algorithm could be particularly useful for image compression, enabling, say, smartphones to wirelessly transmit large video files without draining their batteries or consuming their monthly bandwidth allotments.

Like the FFT, the new algorithm works on digital signals. A digital signal is just a series of numbers -- discrete samples of an analog signal, such as the sound of a musical instrument. The FFT takes a digital signal containing a certain number of samples and expresses it as the weighted sum of an equivalent number of frequencies.

"Weighted" means that some of those frequencies count more toward the total than others. Indeed, many of the frequencies may have such low weights that they can be safely disregarded. That's why the Fourier transform is useful for compression. An eight-by-eight block of pixels can be thought of as a 64-sample signal, and thus as the sum of 64 different frequencies. But as the researchers point out in their new paper, empirical studies show that on average, 57 of those frequencies can be discarded with minimal loss of image quality.

**Heavyweight division**

Signals whose Fourier transforms include a relatively small number of heavily weighted frequencies are called "sparse." The new algorithm determines the weights of a signal's most heavily weighted frequencies; the sparser the signal, the greater the speedup the algorithm provides. Indeed, if the signal is sparse enough, the algorithm can simply sample it randomly rather than reading it in its entirety.

"In nature, most of the normal signals are sparse," says Dina Katabi, one of the developers of the new algorithm. Consider, for instance, a recording of a piece of chamber music: The composite signal consists of only a few instruments each playing only one note at a time. A recording, on the other hand, of all possible instruments each playing all possible notes at once wouldn't be sparse -- but neither would it be a signal that anyone cares about.

The new algorithm -- which associate professor Katabi and professor Piotr Indyk, both of MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL), developed together with their students Eric Price and Haitham Hassanieh -- relies on two key ideas. The first is to divide a signal into narrower slices of bandwidth, sized so that a slice will generally contain only one frequency with a heavy weight.

In signal processing, the basic tool for isolating particular frequencies is a filter. But filters tend to have blurry boundaries: One range of frequencies will pass through the filter more or less intact; frequencies just outside that range will be somewhat attenuated; frequencies outside that range will be attenuated still more; and so on, until you reach the frequencies that are filtered out almost perfectly.

If it so happens that the one frequency with a heavy weight is at the edge of the filter, however, it could end up so attenuated that it can't be identified. So the researchers' first contribution was to find a computationally efficient way to combine filters so that they overlap, ensuring that no frequencies inside the target range will be unduly attenuated, but that the boundaries between slices of spectrum are still fairly sharp.

**Zeroing in**

Once they've isolated a slice of spectrum, however, the researchers still have to identify the most heavily weighted frequency in that slice. In the SODA paper, they do this by repeatedly cutting the slice of spectrum into smaller pieces and keeping only those in which most of the signal power is concentrated. But in an as-yet-unpublished paper, they describe a much more efficient technique, which borrows a signal-processing strategy from 4G cellular networks. Frequencies are generally represented as up-and-down squiggles, but they can also be though of as oscillations; by sampling the same slice of bandwidth at different times, the researchers can determine where the dominant frequency is in its oscillatory cycle.

-end-

Massachusetts Institute of Technology

## Related Algorithm Articles from Brightsurf:

CCNY & partners in quantum algorithm breakthrough

Researchers led by City College of New York physicist Pouyan Ghaemi report the development of a quantum algorithm with the potential to study a class of many-electron quantums system using quantum computers.

Machine learning algorithm could provide Soldiers feedback

A new machine learning algorithm, developed with Army funding, can isolate patterns in brain signals that relate to a specific behavior and then decode it, potentially providing Soldiers with behavioral-based feedback.

New algorithm predicts likelihood of acute kidney injury

In a recent study, a new algorithm outperformed the standard method for predicting which hospitalized patients will develop acute kidney injury.

New algorithm could unleash the power of quantum computers

A new algorithm that fast forwards simulations could bring greater use ability to current and near-term quantum computers, opening the way for applications to run past strict time limits that hamper many quantum calculations.

QUT algorithm could quash Twitter abuse of women

Online abuse targeting women, including threats of harm or sexual violence, has proliferated across all social media platforms but QUT researchers have developed a sophisticated statistical model to identify misogynistic content and help drum it out of the Twittersphere.

New learning algorithm should significantly expand the possible applications of AI

The e-prop learning method developed at Graz University of Technology forms the basis for drastically more energy-efficient hardware implementations of Artificial Intelligence.

Algorithm predicts risk for PTSD after traumatic injury

With high precision, a new algorithm predicts which patients treated for traumatic injuries in the emergency department will later develop posttraumatic stress disorder.

New algorithm uses artificial intelligence to help manage type 1 diabetes

Researchers and physicians at Oregon Health & Science University have designed a method to help people with type 1 diabetes better manage their glucose levels.

A new algorithm predicts the difficulty in fighting fire

The tool completes previous studies with new variables and could improve the ability to respond to forest fires.

New algorithm predicts optimal materials among all possible compounds

Skoltech researchers have offered a solution to the problem of searching for materials with required properties among all possible combinations of chemical elements.

Read More: Algorithm News and Algorithm Current Events

Researchers led by City College of New York physicist Pouyan Ghaemi report the development of a quantum algorithm with the potential to study a class of many-electron quantums system using quantum computers.

Machine learning algorithm could provide Soldiers feedback

A new machine learning algorithm, developed with Army funding, can isolate patterns in brain signals that relate to a specific behavior and then decode it, potentially providing Soldiers with behavioral-based feedback.

New algorithm predicts likelihood of acute kidney injury

In a recent study, a new algorithm outperformed the standard method for predicting which hospitalized patients will develop acute kidney injury.

New algorithm could unleash the power of quantum computers

A new algorithm that fast forwards simulations could bring greater use ability to current and near-term quantum computers, opening the way for applications to run past strict time limits that hamper many quantum calculations.

QUT algorithm could quash Twitter abuse of women

Online abuse targeting women, including threats of harm or sexual violence, has proliferated across all social media platforms but QUT researchers have developed a sophisticated statistical model to identify misogynistic content and help drum it out of the Twittersphere.

New learning algorithm should significantly expand the possible applications of AI

The e-prop learning method developed at Graz University of Technology forms the basis for drastically more energy-efficient hardware implementations of Artificial Intelligence.

Algorithm predicts risk for PTSD after traumatic injury

With high precision, a new algorithm predicts which patients treated for traumatic injuries in the emergency department will later develop posttraumatic stress disorder.

New algorithm uses artificial intelligence to help manage type 1 diabetes

Researchers and physicians at Oregon Health & Science University have designed a method to help people with type 1 diabetes better manage their glucose levels.

A new algorithm predicts the difficulty in fighting fire

The tool completes previous studies with new variables and could improve the ability to respond to forest fires.

New algorithm predicts optimal materials among all possible compounds

Skoltech researchers have offered a solution to the problem of searching for materials with required properties among all possible combinations of chemical elements.

Read More: Algorithm News and Algorithm Current Events

Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.