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A REVERSE CONVERTER FOR THE FIVE COPRIME MODULI SET
{
}
Prof. Mohammed I Daabo, Valentine Aveyom, Gabriel Kofi Armah (PhD)
School of Computing and Information Sciences, CKT-University of Technology and Applied Sciences P.O. Box 24,
Navrongo.
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Abstract
Residue to binary conversion is presented for the five
moduli set {
} in this paper. A novel converter for the moduli set
using modular adders, multipliers, and carry save adders
is proposed using a cyclic jump method. The binary
representation, hardware implementation and
comparison with a state-of- the- art scheme put the
proposed converter ahead. The moduli set is carefully
selected to provide for larger dynamic range needed for
digital signal processing.
Keywords: Residue Number system; Moduli set;
Dynamic Range; Cyclic Jump Technique
i. Introduction
Residue Number System (RNS) is an emerging area of
research. This is because of its suitability for the
implementation of high-speed digital signal processing
devices and its inherent parallelism, modularity, fault
tolerance and carry free propagation properties [7].
Arithmetic operations such as addition and
multiplication are performed more easily and efficiently
in RNS than conventional two’s complement number
systems [6].
The traditional moduli set {
}, has been
one of the most popularly studied in RNS.
The moduli set{
} which shares a common factor of 2
between the third, fourth and fifth moduli has been
applied. This moduli set offers consecutiveness and
allows for equal width adders and multipliers in
hardware design. This gives it high study significance
than the traditional moduli sets [5].
ii. Fundamentals of Residue Number
System(RNS)
RNS is presented using relatively prime moduli set
{
}
such that, the greatest common
divisor (
) for , while
∏
, is the dynamic range. The residues of a
decimal number is obtained as
||
.A decimal
number X can therefore be represented in RNS as
,
. This representation
is unique for any integer [ ]. ||
is the
modulo operation of with respect to
[1],[4].
Two standard conversion techniques exist for
performing reversion conversions in RNS. They are The
Chinese Remainder Theorem and the Mixed Radix
Conversion Method. However, other derived versions for
performing backward conversion also exist.
iii. Mixed Radix Conversion
The Mixed Radix Conversion (MRC) approach is an
alternative method to the CRT for performing reverse
conversion. This method does not involve the use of the
large modulo-M computation as is required by the CRT.
This method is used to perform `residue to binary
conversion of
based on the moduli set
{
} as follows;
=
(2)
Where
(MRDs) which
can be computed below as shown in [2],[3],[8] ;
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iv. The Cyclic Jump Technique
A cyclic jump approach to reverse conversion is
presented in this paper. The technique uses the first
residue as an initial position and then jumps to new
locations until a final point is reached. The various
jumps are then summed when all residues turn to zero,
to arrive at the decimal number . This technique is an
MRC based approach.
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 09 Issue: 08 | Aug 2022 www.irjet.net p-ISSN: 2395-0072