电子种越女学 University of Electroale Science and Technelery of China 986 Chapter 14 Redundant Arithmetic Xiang LING National Key Lab of Science and Technology on Communications
Chapter 14 Redundant Arithmetic Xiang LING National Key Lab of Science and Technology on Communications
数值表征系统 /96 ■数值表征是一个古老的数学问题 ■源于先祖的”结绳计事” ■革命性的突破在于印度人发明的“阿拉伯数字” ■数值表征的基本问题 ■表示 ·计算 ■数值表征的研究方法 ■经典代数 ■近世代数 2
2 数值表征系统 数值表征是一个古老的数学问题 源于先祖的”结绳计事” 革命性的突破在于印度人发明的“阿拉伯数字” 数值表征的基本问题 表示 计算 数值表征的研究方法 经典代数 近世代数
43 数值表征系统 /966 流水，并行， 奇偶检测、溢出 展开，折叠， 检测、大小比较 重定时等 结构方法 、缩放问题 CPA,CLA,CSA, 前缀计算 BOOTH,ARRAY, ELMMA, Bit level 计算方法 端回进位 2的补码， 1的补码，BCD, 表示方法 已有典型的余 移码，格雷码， 数值 结构 数余数基 冗余数值，etc. TCS 数值表征系统 RNS 3
3 数值表征系统 表示方法 数值 结构 计算方法 结构方法 数值表征系统 流水,并行, 展开,折叠, 重定时等 CPA,CLA,CSA, BOOTH, ARRAY, Bit_level 2的补码, 1的补码, BCD, 移码, 格雷码, 冗余数值, etc. 前缀计算， ELMMA， 端回进位 TCS RNS 已有典型的余 数余数基 奇偶检测、溢出 检测、大小比较 、缩放问题
14.1 Redundant and non redundant /96 A non-redundant radix-r number has digits from the seto,1,...r-1}and all numbers can be represented in a unique way. A radix-r redundant signed-digit number system is based on digit set S={β，（β-1）.1,0,1,2，.，a, where the notation x denotes-x,1≤β，a≤(r-l) The digit set s contains more than r values multiple representations for any number in signed digit format.Hence,the name redundant. ■A symmetric signed digit has a=β. 4
4 14.1 Redundant and non redundant A non-redundant radix-r number has digits from the set{0, 1, … , r-1} and all numbers can be represented in a unique way. A radix-r redundant signed-digit number system is based on digit set S＝{β,(β-1),…1,0,1,2,…,α}, where the notation x denotes –x, 1≤ β, α≤(r-1). The digit set S contains more than r values → multiple representations for any number in signed digit format. Hence, the name redundant. A symmetric signed digit has α=β
14.1 Redundant and non redundant /986 Carry-free addition is an attractive property of redundant signed-digit numbers. This allows most significant digit (msd)first redundant arithmetic,also called on-line arithmetic. 5
5 Carry-free addition is an attractive property of redundant signed-digit numbers. This allows most significant digit (msd) first redundant arithmetic, also called on-line arithmetic. 14.1 Redundant and non redundant
14.2 Redundant number representations 96 A symmetric signed-digit representation uses the digit set Dr.=a1,0,1,..a}where r is the radix and a is the largest digit in the set. A number in this representation is written as W- Xe=xr-1xw-3Xw-3…xn=∑xw-广 i=0 The sign of the number is given by the sign of the most significant non-zero digit. ·Digit set D={1,0,1, ■3=00110r0101 6
6 14.2 Redundant number representations A symmetric signed-digit representation uses the digit set D={α, …, 1, 0, 1, …, α }, where r is the radix and α is the largest digit in the set. A number in this representation is written as : The sign of the number is given by the sign of the most significant non-zero digit. Digit set D={1,0,1}, 3=0011 or 0101 1 0 , 1 2 3 0 1 . .... W i i r W W W W i X x x x x x r
14.2 Redundant number representations /966 Minimally redundant and maximally redundant Digit Set D.r.d 0 Redundancy Factor p r-1 Incomplete 是 Minimally redundant =「r/2 and r-1 >1 The greater the redundancy factor,the greater the redundancy of the representation. D2.1>is also referred to as signed binary digit (SBD), redundant signed digit(RSD),and borrow save(BS) representations. 7
7 Minimally redundant and maximally redundant The greater the redundancy factor, the greater the redundancy of the representation. D is also referred to as signed binary digit (SBD), redundant signed digit (RSD), and borrow save (BS) representations. r 1 14.2 Redundant number representations
14.3 Carry-free Radix-2 Addition and /96 subtraction Redundant number representations limit the carry propagation to a few bit-positions,which is usually independent of the word length W. The algorithm to carry out signed binary digit addition is not unique,and therefore its logic implementation can be diverse. A radix-2 signed digit number is coded using 2 unsigned binary numbers,1 positive and 1 negative,as X=X+-X-. 8
8 14.3 Carry-free Radix-2 Addition and subtraction Redundant number representations limit the carry propagation to a few bit-positions, which is usually independent of the word length W. The algorithm to carry out signed binary digit addition is not unique, and therefore its logic implementation can be diverse. A radix-2 signed digit number is coded using 2 unsigned binary numbers, 1 positive and 1 negative, as X=X+-X -
14.3 Carry-free Radix-2 Addition and /96 subtraction In a hybrid operation,1st input operand and the output operand are in redundant signed digit representation,and the 2nd input operand is conventional unsigned number. A signed digit addition can be viewed as a concatenation of one hybrid addition and one hybrid subtraction. 9
9 In a hybrid operation, 1st input operand and the output operand are in redundant signed digit representation, and the 2nd input operand is conventional unsigned number. A signed digit addition can be viewed as a concatenation of one hybrid addition and one hybrid subtraction. 14.3 Carry-free Radix-2 Addition and subtraction
/966 Appendix:PPM operator In b b d=anbvancvbac PPM c=a⊕b⊕c d Symbol cquations schematic 10
10 In Appendix: PPM operator