Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

Improvement of Radix-\$2^k\$ Signed-Digit Number for High Speed Circuit

Masaaki Niimura
Shinshu University, Nagano
Yasushi Fuwa
Shinshu University, Nagano

Summary.

In this article, a new radix-\$2^k\$ signed-digit number (Radix-\$2^k\$ sub signed-digit number) is defined and its properties for hardware realization are discussed. \par Until now, high speed calculation method with Radix-\$2^k\$ signed-digit numbers is proposed, but this method used ``Compares With 2" to calculate carry. ``Compares with 2'' is a very simple method, but it needs very complicated hardware especially when the value of \$k\$ becomes large. In this article, we propose a subset of Radix-\$2^k\$ signed-digit, named Radix-\$2^k\$ sub signed-digit numbers. Radix-\$2^k\$ sub signed-digit was designed so that the carry calculation use ``bit compare'' to hardware-realization simplifies more.\par In the first section of this article, we defined the concept of Radix-\$2^k\$ sub signed-digit numbers and proved some of their properties. In the second section, we defined the new carry calculation method in consideration of hardware-realization, and proved some of their properties. In the third section, we provide some functions for generating Radix-\$2^k\$ sub signed-digit numbers from Radix-\$2^k\$ signed-digit numbers. In the last section, we defined some functions for generation natural numbers from Radix-\$2^k\$ sub signed-digit, and we clarified its correctness.

The terminology and notation used in this paper have been introduced in the following articles [8] [11] [9] [1] [4] [3] [10] [7] [2] [6] [5]

Contents (PDF format)

1. Definition for Radix-\$2^k\$ Sub Signed-Digit Number
2. Definition for New Carry Calculation Method
3. Definition for Translation from Radix-\$2^k\$ Signed-Digit Number
4. Definiton for Translation from Radix-\$2^k\$ Sub Signed-Digit Number to INT

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