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19I16 IJAET0916949 v6 iss4 1615to1621.pdf


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International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963
represented by N mixed radix digits. The digits can be generated sequentially through residue
subtraction. In [9] the design of residue number system (RNS) to binary converters was considered.
The moduli set uses moduli of uniform word length (n to n+1bits). It is derived from a previously
investigated four-moduli set. Three RNS-to-binary converters were proposed for this moduli set: one
using mixed radix conversion and the other two using Chinese Remainder Theorem. In [10] various
FFT processors, such as pipelined or memory-based architectures, have been proposed for different
applications have been shown. Authors designed Radix-2 FFT Algorithm and Radix-4 FFT Algorithm
of a system by using FFT controller. Authors in [11] present about how the 128-point mixed radix
FFT/IFFT algorithm implemented to power consumption and hardware cost also be saved. Memory
based FFT architecture decompose a larger FFT computation into several cascaded smaller FFTs and
utilizes a single FFT core to reduce the hardware cost. Section II gives developing Mixed Radix
system for any two integers using two radices. In Section III results and discussion for all possible
ways in which the mixed radix Multiplication takes place has been discussed. Section IV presents
conclusion and scope for the future work. Section V gives the applications in Computers.

II.

MIXED RADIX SYSTEM

Considering N1 and N 2 as any integers consisting of radices r1 , r2 , then general representation of any
integer is given below as:

N1  r2 r1m2  r1m1  m0
N2  r2 r1n2  r1n1  n0
N1  N2   r2 r1m2  r1m1  m0    r2r1n2  r1n1  n0 

(1)
(2)
(3)

log  N1  N2   log   r2 r1m2  r1m1  m0    r2 r1n2  r1n1  n0  

log  N1  N2   log  r2 r1m2  r1m1  m0   log  r2 r1n2  r1n1  n0 

(4)

Applying antilog to the both sides of (4), the value of N1  N 2 is obtained.
In Mixed Radix the combination can be

r1  2; r2  3 then m1  0,1; m2  0,1, 2
r1  2; r2  5 then m1  0,1, 2; m2  0,1, 2,3
r1  2; r2  7 then m1  0,1, 2,3; m2  0,1, 2,3, 4
and so on as shown in [1].
There can be four cases for the integers in the mixed radix form represented:
Case 1: If N1  even; N2  even, the output of results with all possible values of m1 , m2 , n1 , n2 are
considered under the criteria of Mixed Radix applicability with the help of a code generated in
MATLAB and obtained the possible values of N1 and N2 , along with N1 * N2 .
% N1 is even and N2 is even (multiplication)
syms r1 r2 m1 m2 n1 n2 m0 n0 N1 N2;
x=1;
r1=2;r2=3;
for m1=0:1:(r1-1)
for n1=0:1:(r1-1)
for m2=0:1:(r2-1)
for n2=0:1:(r2-1)
n0=0;
m0=0;
N1=r1.*r2.*m2+r1.*m1+m0;
N2=r1.*r2.*n2+r1.*n1+n0;
fprintf('\n m2
m1
m0
n2
n1
n0
N1
N2
N1*N2\n')
fprintf('\n %3.4f %3.4f %3.4f %3.4f %3.4f %3.4f %3.4f %3.4f
%3.4f\n',m2,m1,m0,n2,n1,
n0, N1, N2, N1*N2)

1616

Vol. 6, Issue 4, pp. 1615-1621