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Scaling of numbers in residue arithmetic with the flexible selection of scaling factor

Ulman Z., Czyżak M., Smyk R.

Poznan University of Technology Academic Journals. Electrical Engineering

2013

No. 76

175--179

A scaling technique of numbers in residue arithmetic with the flexible selection of the scaling factor is presented. The required scaling factor can be selected from the set of moduli products of the Residue Number System (RNS) base. By permutation of moduli of the number system base it is possible to create many auxiliary Mixed-Radix Systems (MRS). They serve as the intermediate systems in the scaling process. All MRS's are associated with the given RNS with respect to the base, but they have different sets of weights. For the scaling factor value resulting from the requirements of the given signal processing algorithm, the suitable MRS can be chosen that allows to obtain the scaling result in most simple manner.

Pipelined scaling of signed residue numbers with the mixed-radix conversion in the programmable gate array

Czyżak M., Smyk R., Ulman Z.

Poznan University of Technology Academic Journals. Electrical Engineering

2013

No. 76

89--99

In this work a scaling technique of signed residue numbers is proposed. The method is based on conversion to the Mixed-Radix System(MRS) adapted for the FPGA implementation. The scaling factor is assumed to be a moduli product from the Residue Number System (RNS) base. Scaling is performed by scaling of terms of the mixed-radix expansion, generation of residue representations of scaled terms, binary addition of these representations and generation of residues for all moduli. The sign is detected on the basis of the value of the most significant coefficient of the MRS representation. For negative numbers their residues are adequately corrected. The basic blocks of the scaler are realized in the form of the modified two-operand modulo adders with included additional multiply and modulo reduction operations. The pipelined realization of the scaler in the Xilinx environment is shown and analyzed with respect to hardware amount and maximum pipelining frequency. The design is based on the LUTs(26x 1) that simulate small RAMs serving as the main component for the look-up realization.

A scaling out-of-kilter algorithm for minimum cost flow

Ciupala L.

Control and Cybernetics

2005

Vol. 34, no 4

1169-1174

The out-of-kilter algorithm is one of the basic algorithms that solve the minimum cost flow problem. Its drawback is that it can improve the objective function at each iteration by only a small value. Consequently, it runs in pseudo-polynomial time. In this paper, we describe a new out-of-kilter algorithm for minimum cost flow that runs in polynomial time. Our algorithm is a scaling algorithm and improves the objective function at each time by a "sufficiently large" value.