Overview of Residue Number System (RNS) for Advanced VLSI Design and VLSI Signal Processing NTUEE 吳安宇

Outline History Why RNS is needed ? Fundamental concepts in RNS. Conversion between Decimal and Residue. Conventional complex RNS Difficult arithmetic operations Applications

History The ancient study of the residue numbering system begins with a verse from a third- centry book, Suan-Ching, by Sun Tzu. We have things of which we do not know the number, If we count them by three, the remainder is 2, If we count them by five, the remainder is 3, If we count them by seven, the remainder is 2, How many things are there?

History We commemorate this contribution as the Chinese Remainder Theorem, or CRT. This theorem, as well as RNS, was set forth in the 19th by Carl Friedrich Gauss in his celebrated Disquisitiones Arithmetical.

Why RNS? It is a “carry-free” system that performs addition, substraction, and multiplication as parallel operations. FA HAFA

Definition The RNS is defined in terms of a set of rela- tively prime moduli. If P denotes the moduli set, then The dynamic range M is

Any integer in the residue class Z M has a unique L-tuple representation given by where X i =X mod p i and is called the ith residue. Definition

X1X X2X → → → → → → → → → → → → → → → → → → X1X X2X X1X X2X X X X Example

X → → → → → → → → → → → → → X1X X2X X3X X → → → → → → → → → → → → → X3X X → → → → → → → → → → → → → X3X X → → → → → → → → → → → → → X3X X → → → → → → → → → → → → → X3X X1X X1X X1X X1X X2X X2X X2X X2X Example

Signed Representation For signed RNS, any integer in (-M/2, M/2], has a unique RNS L-tuple representation where x i =(X mod p i ) if X>0, and (M-|X|) mod p i otherwise. X1X X2X → → → → → → → → → → → → → → → X1X X2X X1X X2X X X X

Operation

7  (1, 3, 2 ) +3  (0, 3, 3 ) 10  (1 mod 3, 6 mod 4, 5 mod 5) = (1,2,0) 7  (1, 3, 2 ) - 3  (0, 3, 3 ) 4  (1 mod 3, 0 mod 4, -1 mod 5) = (1,0,4) 7  (1, 3, 2 ) *3  (0, 3, 3 ) 21  (0 mod 3, 9 mod 4, 6 mod 5) = (0,1,1) Operation (Example)

X → → → → → → → → → → → → → X1X X2X X3X X → → → → → → → → → → → → → X3X X → → → → → → → → → → → → → X3X X → → → → → → → → → → → → → X3X X → → → → → → → → → → → → → X3X X1X X1X X1X X1X X2X X2X X2X X2X (3,4,5) Modulo Set

Efficient and rapid implementation of the operation (X i o Y i ) mod p i must be found. Only for the case where p i =2^n can be easily implemented in conventional system. RAM or ROM Table Lookup Z i = (X i o Y i ) mod p i X i (n bits) Y i (n bits) Z i (n bits) Conversion

To be competitive system, the speed of data acquisition and the accompanying decimal- to-residue conversion must be equally fast. Decimal-to-residue Conversion

2 j mod p i j=4 i=1 i=2 j=3j=2j=1j= p i -(2 5 mod p i ) i=1 i=2 3 3

Generally speaking, the speed limitation of residue-to-decimal conversion is a problem in residue number system. The R/D problem can be solved in one of the following two ways: –Mixed radix conversion (MRC) –Chinese Remainder Theorem (CRT) Residue-to-decimal Conversion

Multiplicative Inverse The multiplicative inverse of g of modulo p i is denoted as g i -1 and satisfies Example: P i =5 g g i -1 0  g P i =6 0   3  4  5 5 g g i -1 P i =7 0 

Mixed Radix Conversion RNS MRC The MRC representation is given by where v 1 =1, v 2 =p 1, v 3 =p 1 p 2, v 4 =p 1 p 2 p 3, …

Mixed radix conversion

Mixed Radix Conversion Use nesting subtractions and multiplicative inverse to make sequential conversion.

Block Diagram of MRC p p M M p 1L -1 M p M p 2L -1 M p L-1,L -1 M

Chinese Remainder Theorem where and s i -1 is the multiplicative inverse of s i mod p i, so that

Chinese Remainder Theorem X 1 s 1 -1 mod p 1 X 2 s 2 -1 mod p 2 X L s L -1 mod p L mod M sLsL s2s2 s1s1

Chinese Remainder Theorem 1*2mod 3 0*3mod 4 4*3 mod mod 60

Conventional Complex RNS A complex number Z is defined to be

Difficult Arithmetic Operations Magnitude comparison / Sign detection Division Base extension Scaling / Rounding / Truncation Overflow detection

Magnitude Comparison Sign Detection Not a weighted number system. Every digits are equally important. The problem makes the RNS-based signal processors inefficient.

Division Blend of nested subtractions and magnitude comparisons, so it is more difficult in the RNS domain. RNS is not closed under division, since the RNS is an integer system. It is an slow, high-overhead operation, and should be avoided in RNS.

Base Extension Increase the dynamic range Increase the resolution. R/D and D/R conversions are needed. R/D conversion L - tuple RNS D/R conversion K - tuple RNS Moduli extension

Scaling / Rounding / Truncation Prevent dynamic-range overflow. A special form of division. The scaling operation can be easily to implemented when scaling factor is: –Power of two (2 n ) in 2’complement. –Product of one or more of the moduli in RNS.

Overflow Detection A special form of magnitude comparison. Can be avoided by using scaling operation during run-time.

Applications High-speed systems (**) FPGA-based solution Filtering (FIR, IIR) systems Linear transformations Adaptive systems

References [1] A good tutorial paper of RNS: Fred J. Taylor, "Residue Arithmetic: A Tutorial with Examples", IEEE Trans. on Computer, pp. 50~62, May [2] A good paper collections for RNS: M. A. Soderstrand, W. K. Jenkins, G. A. Jullien, F. J. Taylor (eds.), Residue Number System Arithmetic: Modern Applications in digital Signal Processing, IEEE Press, New York, 1986.