# Sampling Algorithm for the Reproduction of Complex Digital Signals

Summary

This technology utilizes a set of mathematical algorithms that permit fast and robust reconstruction of signals from digital samples without the need to resort to band limitations or uniform sampling. This allows a signal to be digitized in a more sophisticated way, sampling it at frequent intervals when it is rapidly changing, and less often when it is changing slowly. Band limits imposed by uniform sampling are eliminated.

Description

This technology may revolutionize analog-to-digital conversion, a foundation of the digital revolution currently based on mathematics done at Bell Labs in the 1950's. With the successful application to vertical markets (for example, oil and gas exploration, defense electronics, medical imaging and music), the range of long-term opportunities is considerable.

Dr. Akram Aldroubi of Vanderbilt University and Dr. Karlheinz Gröchenig of the University of Connecticut have developed a new method of sampling analogue signals that might overcome limitations of the traditional method first developed in 1948 by Claude Shannon, a mathematician at Bell Labs in New Jersey. The conversion of analogue to digital was the start of the information revolution, and Shannon's method for doing so -- a technique known as uniform sampling -- is still in use and remains largely unchanged today. The new method devised by Dr. Aldroubi and Dr. Gröchenig has the potential to revolutionize fields such as body scanning that need to process relatively large amounts of information.

Analogue signals are, in effect, waves. Uniform sampling measures the height of the wave at regular intervals and stores the results as binary numbers. When a CD is recorded, 44,100 samples of the original signal are taken every second. When the record- ing is played back, those numbers are used to reconstruct something close enough to the original that the human ear cannot detect much difference.

One problem with this type of sampling is that the original analogue signal has to be "band limited." This means that the signal must stay within pre-defined limits (in the case of music, within a particular range of frequencies); otherwise the sampling is inaccurate. Most signals, however, are rarely band limited. Musical recordings get around this problem because human hearing is itself band limited: notes above and below certain frequencies are inaudible. Other applications, though, are more sensitive. Magnetic resonance imaging (MRI), a common body-scanning technique, generates so much data that imposing any band limit inevitably results in a loss of information, to the detriment of the patient.

This technology utilizes a set of mathematical algorithms devised by Dr. Aldroubi and Dr. Gröchenig, and among other things, allows fast and robust reconstruction of signals from digital samples without the need to resort to band limitations or uniform sampling. Any wave, no matter how complex, can be described as a function. The aim of systems such as CD and MRI, which turn digital data back into analogue so that people can interpret the result, is to reconstruct this function from the digital data. Dr. Aldroubi and Dr. Gröchenig are able to do this from data that are the result of non-uniform sampling. That allows a signal to be digitized in a more sophisticated way -- sampling it at frequent intervals when it is rapidly changing, and less often when it is changing slowly. Band limits imposed by uniform sampling are eliminated.

The method is not done in a single step, as is the old method. The new method employs a series of iterations. First, the original sample is analyzed using one of the algorithms. This initial iteration, in Dr. Aldroubi's words, makes a "very poor man's approximation of what the function is like." It does not describe the function's actual shape, but roughly fits the "mathematical space" in which the function lies. This approximation is then compared with the sample data, and the errors between the two are calculated by the algorithm in order to eliminate them. That process is repeated, each iteration reducing the discrepancy between function and data, until an adequate and efficient match is reached. The final version is then ready for use.

Potential Market Size

This technology has a number of potential applications that require the development of new or the refinement of existing engineering applications. The following examples indicate some of these opportunities for nonuniform sampling technology:

Examples of Potential Applications:

Data Processing related to communication:

When data from a uniformly sampled signal (function) is lost, the result is generally a sequence of nonuniform samples. This scenario is usually referred to as a missing data problem. Often, missing samples are due to the partial destruction of storage devices (e.g., scratches on a CD). A nonuniform sampling technology could solve these problems or minimize the disruption.

Medical imaging:

Computerized tomography (CT) and magnetic resonance imaging (MRI frequently use the nonuniform polar and spiral sampling sets.

Astronomical measurements:

The measurement of star luminosity gives rise to extremely nonuniformly sampled time series. Daylight periods and adverse nighttime weather conditions prevent regular data collection.

Music or data compression:

Improved methods for compressing and decompressing music and/or data files could be developed which would permit the more rapid transfer of larger music or data files with more accuracy.

Intellectual Property Status

A utility patent application entitled "System and Methods of Nonuniform Data Sampling and Data Reconstruction in Shift Invariant and Wavelet Spaces" was filed in the U.S. Patent and Trademark Office on June 10, 2003. It claims priority to a U.S. Provisional Application Serial No. 60/389,852, which was filed on Jun. 18, 2002. The patent application has been assigned an Application Serial No. 10/458,475 and has been published by the U.S. Patent and Trademark Office on March 4, 2004, with a Publication No. 20040044715.

US Patent # 7,496,619: System and methods of nonuniform data sampling and data reconstruction in Shift Invariant and Wavelet Spaces

Inventors:
Akram Aldroubi
Licensing manager:
Peter Rousos

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