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# Systolic array implementation of the Jacobi decomposition

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Although we find it an efficient method to observe characteristics of the signal surface, on-line eigenanalysis is still not commonly in use in digital signal processing applications. By performing eigendecomposition on a signal correlation matrix, several DSP problems such as spectral estimation and adaptive filtering can be solved.

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## Citations

... Equations 10 and 11 are the key to the parallelism inherent in Jacobi's algorithm. The time parallel aspects have been well explored [2] [6], although in this study we show that there are still some improvements to be made in the details. What has not been fully explored is the utility of the space parallelism of Jacobi's method. ...

... Equations 10 and 11 are the key to the parallelism inherent in Jacobi's algorithm. The time parallel aspects have been well explored [2, 6], although in this study we show that there are still some improvements to be made in the details. What has not been fully explored is the utility of the space parallelism of Jacobi's method. ...

Matrix diagonalization is an important component of many aspects of computational science. There are a vari- ety of algorithms to accomplish this task. Jacobi's algo- rithm is a good choice for parallel environments. Jacobi's algorithm consists of a series of matrix plane rotations, the ordering of which can dramatically affect performance. We show a new ordering which cuts the number of necessary operations approximately in half. Additionally, Jacobi's algorithm can be made parallel in space as well as time. This is an advantage when deal- ing with very large matrices, such as those found in quan- tum chemistry.

... For all Jacobi variants, the convergence is asymptotically quadratic. Our decision to use Jacobi methods is their inherent parallelism, and because very similar solutions can be used for both of the procedures proposed in [10]. ...

Although we find it an efficient method to observe characteristics of the signal surface, on-line eigenanalysis is still not commonly in use in digital signal processing applications. By performing eigendecomposition on a signal correlation matrix, several DSP problems such as spectral estimation and adaptive filtering can be solved. Calculating eigenvalues and eigenvectors of the Signal Autocorrelation Matrix is, compared to other DSP methods, a numerically complex procedure. This problem can be solved in real time by using parallel processor arrays. Considering characteristics of the signal correlation matrix, the use of Symmetric Parallel Jacobi Method on a 2dimensional processor array is a suggested method. Both parallel solutions of the transformation and notch filter simulation results for single and multiple interference sinusoids are presented. #. The Filtering Problem The Least Mean Square adaptive algorithm is the most widely used real time filtering algorithm due to its computing requirements. As VLSI processing units became cheaper the more effective algorithms with more computational requirements can be built in the processor. Such methods have faster convergence rates. An important algorithm with near optimal convergence rate is the Recursive Least Squares algorithm. Those algorithms estimate the inverse of the input signal autocorrelation matrix R # - . Unfortunately, the computational requirements for the RLS algorithm are large and it is not easily carried out in real time. Many users in adaptive filtering only know when to use a transversal filter rather than a recursive one and why they prefer LMS gradient-based descent technique. They do not spend a lot of time to analysing the known algorithms from linear algebra, choosing the appropriate one ...

... It has been shown [17] that the cyclic by rows ordering and condition (3.3) ensure convergence of the Jacobi method applied to A T A and convergence of the cyclic by rows Hestenes method follows. ...

In this paper, some basic image restoration problems are presented to introduce the importance of fast linear algebra algorithms. Then, an overview to some basic systolic arrays algorithms and mapping principles of these algorithms into systolic arrays is shown. Special attention has been done to the adaptive image filtering techniques. Moreover, the Singular Value Decomposition has been applied in a two-dimensional adaptive FIR filtering technique. However, a two-dimensional adaptive algorithm based on a Singular Value Decomposition (SVD) method will be presented using systolic arrays that is applied in the area of image processing.

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