@MISC{RADIX_3.MIZ,
  AUTHOR = {Niimura, Masaaki and Fuwa, Yasushi},
  TITLE = {Improvement of Radix-$2^k$ Signed-Digit Number for High Speed Circuit},
  SECTION1 = {Definition for Radix-$2^k$ Sub Signed-Digit Number},
  SECTION2 = {Definition for New Carry Calculation Method},
  SECTION3 = {Definition for Translation from Radix-$2^k$ Signed-Digit Number},
  SECTION4 = {Definiton for Translation from Radix-$2^k$ Sub Signed-Digit Number to INT},
  DAY = {03},
  MONTH = {January},
  YEAR = {2003},
  ADDRESS1 = {Shinshu University\\Nagano},
  ADDRESS2 = {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.}}

