ETTC'2003 - SEE

1.1.2. Complex signal processing
This can be handled by RNS in a similar way to standard complex number [4]. The
basic difference between the RNS and Complex RNS [4] is the fact that in the
complex RNS one needs to deal with cross-product terms. However under certain
special cases of choice of moduli (type of prime 4k+1), the decoupling of computation
from real and imaginary parts of the result is feasible. This is known as QRNS. The
primary disadvantage of the QRNS is the restriction on the type of moduli. The
modified QRNS (MQRNS) accepts any modulus at the expense of multiplication.
Conceptual view of the complex digital signal processing in RNS
2S/Ms Ms/RNS Alogorithm
RNS/Ms MS/2s
1.1.3. RNS Division
RN S/QRNS QRNS/RN S
Algorithms must be designed to control the growth of intermediate results (Example
FFT) and a periodic scaling is necessary. In RNS, the division is classified in two
groups: scaling and division. Division by a constant is denoted as scaling which is a
special form of multiplication. In a weighted binary system, the scaling is done by left
or right shift by power of 2. In RNS, the scaling factor is composed of one or more
inverse moduli. Various scaling methods exist for implementation based on either
CRT or MRC (Mixed Radix Conversion). Many algorithms using the RNS system
require an extension or a change of basis. The change from one basis to another is
illustrated as follows: the basis changes for X, represented by RNS base
n
t
Bm = ( m1, m2,..., mn)
, towards the base of another RNS B = ( p1, p2,..., p ) with
n t
i and P= i
i= 1 i=
1
M m p
= ∏ ∏
p t
Various base extension algorithms exist but almost all are derived from two basic
algorithms [5,6] known as
1. Szabo and Tanaka base extension method
2. Shenoy and Kumaresan method