@MISC{RADIX_4.MIZ,
  AUTHOR = {Niimura, Masaaki and Fuwa, Yasushi},
  TITLE = {High Speed Adder Algorithm with Radix-$2^k$ Sub Signed-Digit Number},
  SECTION1 = {Preliminaries},
  SECTION2 = {Carry Operation at $n+1$ Digits Radix-$2^k$ Sub Signed-Digit Number},
  SECTION3 = {Definition for Adder Operation on Radix-$2^k$ Sub Signed-Digit Number},
  DAY = {03},
  MONTH = {January},
  YEAR = {2003},
  ADDRESS1 = {Shinshu University\\Nagano},
  ADDRESS2 = {Shinshu University\\Nagano},
  SUMMARY = {In this article, a new adder algorithm using Radix-$2^k$ sub signed-digit
 numbers is defined and properties for the hardware-realization is
 discussed.\par Until now, we proposed Radix-$2^k$ sub signed-digit numbers
 in consideration of the hardware realization. In this article,
 we proposed High Speed Adder Algorithm using this Radix-$2^k$ sub signed-digit
 numbers. This method has two ways to speed up at hardware-realization.
 One is 'bit compare' at carry calculation, it is proposed
 in another article. Other is carry calculation between two numbers.
 We proposed that $n$ digits Radix-$2^k$ signed-digit numbers is expressed
 in $n+1$ digits Radix-$2^k$ sub signed-digit numbers, and addition result
 of two $n+1$ digits Radix-$2^k$ sub signed-digit numbers is expressed in $n+1$ digits.
 In this way, carry operation between two Radix-$2^k$ sub signed-digit numbers
 can be processed at $n+1$ digit adder circuit and additional circuit to
 operate carry is not needed.\par
 In the first section of this article, we prepared some useful
 theorems for operation of Radix-$2^k$ numbers.
 In the second section, we proved some properties about carry on
 Radix-$2^k$ sub signed-digit numbers.
 In the last section, we defined the new addition operation using
 Radix-$2^k$ sub signed-digit numbers, and we clarified its correctness.}}

