6 residue number system (color) 11-30-2002

7/30/2019 6 Residue Number System (Color) 11-30-2002

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Overview ofResidue Number System (RNS)

forAdvanced VLSI Design and VLSI Signal

Processing

11-30-2002

NTUEE


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Outline

History

Why RNS is needed ?

Fundamental concepts in RNS.

Conversion between Decimal and Residue.

Conventional complex RNS Difficult arithmetic operations

Applications


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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?


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History

We commemorate this contribution as the

Chinese Remainder Theorem, orCRT.

This theorem, as well as RNS, was set forth

in the 19th by Carl Friedrich Gauss in his

celebratedDisquisitiones Arithmetical.


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Why RNS?

It is a carry-free system that performsaddition, substraction, and multiplication as

parallel operations.

FA FA FA HAFA

00ba11ba22ba33ba44ba

0s

1s2s3s4soutc


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Definition

The RNS is defined in terms ofa set of rela-

tively prime moduli.

IfP denotes the moduli set, then

Lppp ,,, 21 P

jiforpp ji ,1,GCD The dynamic rangeMis

LpppM 21


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Any integer in the residue class ZM has a

uniqueL-tuple representation given by

LRNS

XXXX ,,, 21

whereXi=Xmodpi and is called the ith residue.

Definition


8/34

5,3, 21 ppP 1553 M

X1

0

1

2

0

1

X2

0

1

2

3

4

X1

2

0

1

2

0

X2

0

1

2

3

4

X1

1

2

0

1

2

X2

0

1

2

3

4

X

0

1

2

3

4

5

6

7

8

9

X

10

11

12

13

14

X

Example


9/34

4

5

6

7

8

9

10

11

0

1

2

3

X

X1

0

1

2

0

1

2

0

1

2

0

1

2

X2

0

1

2

3

0

1

2

3

0

1

2

3

X3

0

1

2

3

4

0

1

2

3

4

0

1

16

17

18

19

20

21

22

23

12

13

14

15

X

X3

2

3

4

0

1

2

3

4

0

1

2

3

28

29

30

31

32

33

34

35

24

25

26

27

X

X3

4

0

1

2

3

4

0

1

2

3

4

0

40

41

42

43

44

45

46

47

36

37

38

39

X

X3

1

2

3

4

0

1

2

3

4

0

1

2

52

53

54

55

56

57

58

59

48

49

50

51

X

X3

3

4

0

1

2

3

4

0

1

2

3

4

X1

0

1

2

0

1

2

0

1

2

0

1

2

X1

0

1

2

0

1

2

0

1

2

0

1

2

X1

0

1

2

0

1

2

0

1

2

0

1

2

X1

0

1

2

0

1

2

0

1

2

0

1

2

X2

0

1

2

3

0

1

2

3

0

1

2

3

X2

0

1

2

3

0

1

2

3

0

1

2

3

X2

0

1

2

3

0

1

2

3

0

1

2

3

X2

0

1

2

3

0

1

2

3

0

1

2

3

5,4,3,, 321 pppP 60543 M

Example


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Signed Representation

Forsigned RNS, any integer in (-M/2,M/2],

has a unique RNS L-tuple representation

where xi=(X mod pi) ifX>0, and (M-|X|)modpi otherwise.

4,3, 21 ppPX1

0

1

2

0

X2

2

3

0

1

X1

1

2

0

1

X2

2

3

0

1

X1

2

0

1

2

X2

2

3

0

1

X

-6

-5

-4

-3

-2

-1

0

1

X

2

3

4

5

X


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LRNS

L

RNS

YYYY

XXXX

,,,

,,,

21

21

YXZ

LRNS ZZZZ ,,, 21

tionmultiplicaandnsubtractioaddition,present

,mod 111 pYX ,mod 222 pYX LLL pYX mod,

Operation


12/34

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)


13/34

4

5

6

7

8

9

10

11

0

1

2

3

X

X1

0

1

2

0

1

2

0

1

2

0

1

2

X2

0

1

2

3

0

1

2

3

0

1

2

3

X3

0

1

2

3

4

0

1

2

3

4

0

1

16

17

18

19

20

21

22

23

12

13

14

15

X

X3

2

3

4

0

1

2

3

4

0

1

2

3

28

29

30

31

32

33

34

35

24

25

26

27

X

X3

4

0

1

2

3

4

0

1

2

3

4

0

40

41

42

43

44

45

46

47

36

37

38

39

X

X3

1

2

3

4

0

1

2

3

4

0

1

2

52

53

54

55

56

57

58

59

48

49

50

51

X

X3

3

4

0

1

2

3

4

0

1

2

3

4

X1

0

1

2

0

1

2

0

1

2

0

1

2

X1

0

1

2

0

1

2

0

1

2

0

1

2

X1

0

1

2

0

1

2

0

1

2

0

1

2

X1

0

1

2

0

1

2

0

1

2

0

1

2

X2

0

1

2

3

0

1

2

3

0

1

2

3

X2

0

1

2

3

0

1

2

3

0

1

2

3

X2

0

1

2

3

0

1

2

3

0

1

2

3

X2

0

1

2

3

0

1

2

3

0

1

2

3

5,4,3,, 321 pppP 60543 M

(3,4,5) Modulo Set


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Efficient and rapid implementation of the

operation (Xi o Yi) modpi must be found.

Only for the case where pi=2^n can beeasily implemented in conventional system.

RAM or ROMTable Lookup

Zi = (Xi o Yi) modpi

Xi (n bits)

Yi (n bits)

Zi (n bits)

Conversion


15/34

To be competitive system, the speed ofdata

acquisition and the accompanying decimal-

to-residue conversion must be equally fast.

Decimal-to-residue Conversion

1

0

0121'2 22N

j

j

j

N

NNNcomps bbbbbbbX

iN

j

i

j

ji

N

iN

ii

ppbppb

pXX

modmod2mod2

mod

1

0


16/34

Decimal-to-residue Conversion

binarydecimalX 01110129 7,5, 21 ppP

45mod1120413111301 X

17mod1120411121302

X

2j modpi j=4

i=1

i=2

j=3 j=2 j=1 j=0

1 3 4 2 1

2 1 4 2 1

pi-(25modpi)

i=1

i=2

3

3

1

0mod2mod2

N

j

i

j

ji

N

iN pbppb


17/34

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


18/34

Multiplicative Inverse

The multiplicative inverse ofgof modulopi

is denoted asgi-1 and satisfies

1mod1

ii pgg

Example:

Pi=5g

gi-1

0

1

1

2

3

3

2

4

4

g

gi-1

Pi=60

1

1

2

3

4

5

5

g

gi-1

Pi=70

1

1

2

4

3

5

4

2

5

3

6

6


19/34

Mixed Radix Conversion

X LXXX ,,, 21 RNS

LXXX

,,

,

21

MRC

The MRC representation is given by

MXvXvXvX LL mod 2211 where v1=1, v2=p1, v3=p1p2, v4=p1p2p3,


20/34

Mixed radix conversion

LL

LL

XpppXppXpX

XvXvXvXvX

121321211

332211

111mod XXpX

LLLL

XppXpXXXp

XpppXppXpXX

123221

1

1

121321211

221

1

2,1mod XpXXp

221212,1 mod XpXXp


21/34

Mixed Radix Conversion

33211

3,1

1

3,2

1343321

1

1

1

2

124323221

1

1

mod

XpXXXpp

XppXpXXXXpp

XppXppXpXXXp

LL

LL

33213

1

3,1

1

3,2mod XpXXXpp

Use nesting subtractions and multiplicative

inverse to makesequential conversion.


22/34

Block Diagram of MRC

p12-1

p13-1 M

M

p1L-1 M

p23-1 M

p2L-1 M pL-1,L

-1 M

1X

2X

3X

LX

1X

2X

3X

LX


23/34

Chinese Remainder Theorem

MpsXsXL

i

iiii modmod

1

1

where

andsi

-1 is the multiplicative

inverse ofsi modpi, so that

1mod1 iii pss

Liii

i ppppppMs 1121


24/34


MpsXsXL

i

iiii modmod1

1

X1s1-1 modp1

X2s2-1 modp2

XLsL-1 modpL

modM

1X

2X

LX

X

sL

s2

s1


25/34


5,4,3,, 321 pppP 60543 M

1mod..3125

1mod..3154

1mod..2203

33

1

3

1

333

221

21

222

11

1

1

1

111

psstsssp

psstsssp

psstsssp

4,0,1X

1*2 mod 3

0*3 mod 4

4*3 mod 5

20

15

12

mod 604X


26/34

Conventional Complex RNS

MMLRNS ZYZXZZZjYXZ ,;,,, 21

A complex numberZis defined to be

iii pYjpXZ modmod

LZZZZZZ 33231213 ,,,

iiiii YYjXXZ 21213

LZZZZZZ 33231213 ,,, iiiiiiiii YXYXjYYXXZ 122121213


27/34

Difficult Arithmetic Operations

Magnitude comparison / Sign detection

Division

Base extension

Scaling / Rounding / Truncation

Overflow detection


28/34

Magnitude Comparison

Sign Detection

Not a weighted number system.

Every digits are equally important.

3,3,30,1,13,3,2 ZYX

?6?0? ZYYX

The problem makes the RNS-based signal

processors inefficient.


29/34

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, andshould be avoided in RNS.


30/34

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


31/34

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 (2n) in 2complement.

Product of one or more of the moduli in RNS.


32/34

Overflow Detection

A special form ofmagnitude comparison.

Can be avoided by using scaling operation

during run-time.


33/34

Applications

High-speed systems

(**) FPGA-based solution

Filtering (FIR, IIR) systems

Linear transformations

Adaptive systems


34/34

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 1984.[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.

6 residue number system (color) 11-30-2002

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