ETTC'2003 - SEE

Currently various avenues are explored in order to develop a high bandwidth special
numerical processor for applications such as
High bandwidth digital down converter
Neural network
Adaptive signal processing
OFDM
High speed imaging
Cryptography
for reconfigurable computing platforms.
1.1. RNS background
The RNS [1] is a non-positional number systems. Due to this feature, one can
implement fast additions and multiplications. The theoretical basis for the
representation comes from the CRT (Chinese Remainder Theorem).
The RNS arithmetic [1] is defined in terms of a basis set { }
m1, m2,... m L of relatively
prime integers. The dynamic range of an RNS system is given by M ,
L
where: M = ∏ m ; if M is odd, the dynamic range of an RNS becomes [-(M-1)/2,
i=
1
i
(M-1)/2); if it is even, it is [-M/2, (M/2-1)]. Each natural integer X in the dynamic
range is mapped onto the legitimate range and represented as an L-tuple of residue
digits.
The residue arithmetic is defined by ( x1, x2,..., xL) ( yi, y2,..., yL) = ( z1, z2,..., zL)
,
where zi = xi yi
, I=1,…L and denotes one of addition, subtraction or
mi
( x , x ,..., x ) .
multiplication. 1 2
n
The residues of a positive number and a negative number are evaluated as follows:
1.1.1. Choosing the RNS moduli
xi = X modmixi
X ≥ 0
x = ( m − X mod m ) X < 0.
i i i
The set of moduli controls the dynamic range of the application. In a set, the largest
modulus controls the speed of arithmetic operations. One tries to select all the moduli
comparable [2] in magnitude to the largest one. This strategy allows to select the RNS
moduli with the smallest number of bits for the largest modulus, thus maximizing the
speed of the RNS arithmetic. In here, there is a trade-off to do the speed and the cost
does not just depend on the width of the residues but also on the moduli chosen.
n− 1 n n+
1
Selecting the moduli in the form of ( 2 ,2 ,2 ) offers considerable ease in forward
and reverse conversion, scaling and other operations [3].