Radix-3 Algorithm for Realization of Discrete Fourier Transform

M.Narayan Murty. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 7, ( Part -5) July 2016, pp.47-51
www.ijera.com 47|P a g e
Radix-3 Algorithm for Realization of Discrete Fourier Transform
M.Narayan Murty*
and H.Panda**
*
(Visiting Faculty, Department of Physics, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam –
515134, Andhra Pradesh, India)
**
(Department of physics, S.K.C.G. Autonomous College, Paralakhemundi -761200, Odisha, India)
ABSTRACT
In this paper, a new radix-3 algorithm for realization of discrete Fourier transform (DFT) of length N = 3m
(m =
1, 2, 3,...) is presented. The DFT of length N can be realized from three DFT sequences, each of length N/3. If
the input signal has length N, direct calculation of DFT requires O (N2
) complex multiplications (4N2
real
multiplications) and some additions. This radix-3 algorithm reduces the number of multiplications required for
realizing DFT. For example, the number of complex multiplications required for realizing 9-point DFT using the
proposed radix-3 algorithm is 60. Thus, saving in time can be achieved in the realization of proposed algorithm.
Keywords: Discrete Fourier transform, fast Fourier transform, radix-3 algorithm
I. INTRODUCTION
Discrete transforms play a significant role
in digital signal processing. Among all the discrete
transforms, the discrete Fourier transform (DFT) is
the most popular transform, and it is mainly due to
its usefulness in very large number of applications
in different areas of science and technology. The
DFT plays a key role in various digital signal
processing and image processing applications [1,
2]. Not only it is frequently encountered in many
different applications, but also it is computation-
intensive. Since DFT is highly computation-
intensive, algorithms and architectures are
suggested for implementation of the DFT in
dedicated very-large-scale integration circuit [3].
DFT and inverse discrete Fourier transform (IDFT)
have been regarded as the key technologies for
signal processing in orthogonal frequency division
multiplexing (OFDM) communication systems.
The fast Fourier transform (FFT) is an algorithm
that computes the DFT using much less operations
than a direct realization of the DFT. The fast
realization approach of DFT [4] is known as FFT.
FFT algorithms [5, 6] are used for efficient
computation of DFT.
In this paper, a new radix-3 algorithm for
realization of DFT of length N = 3m
(m = 1, 2, 3,...)
is presented. The DFT of length N can be realized
from three DFT sequences, each of length N/3. If
the input signal has length N, direct calculation of
DFT requires O (N2
) complex multiplications (4N2
real multiplications) and some additions. This
radix-3 algorithm reduces the number of
multiplications. For example, the number of
complex multiplications required for realizing 9-
point DFT using the proposed radix-3 algorithm is
60. Therefore, the hardware complexity and
execution time for implementing radix-3 DFT
algorithm can be reduced.
The rest of the paper is organized as
follows. The proposed radix-3 algorithm for DFT is
presented in Section-II. An example for
implementation of DFT of length N = 9 is
presented Section-III. Conclusion is given in
Section-IV.
II. PROPOSED RADIX-3 ALGORITHM FOR DFT
The 1-D DFT of input sequence {x (n); n = 0, 1, 2,...., N-1} is defined as
21
0
( ) ( )
j k nN
N
n
Y k x n e



 
0,1, 2, ..., 1for k N  (1)
Where 1j  
The ( )Y k values represent the transformed data.
Taking
2j k n
k n
N
NW e


 , (1) can be written as
1
0
( ) ( )
N
k n
N
n
Y k x n W


 
(2)
The term kn
NW is known as twiddle factor.
RESEARCH ARTICLE OPEN ACCESS

Taking ( 1, 2 , 3, ...)3
m
mN  , the N output components ( 0 ), (1), ( 2 ), ..., ( 1)Y Y Y Y N  are arranged
in three groups, namely  3 , ( 3 1) ( 3 1),Y r Y r a n d Y N r   where 0, 1, 2 , ..., 1
3
N
r   .
The following expressions can be derived from (2).
1
3
0
2
(3 ) ( )
3 3
N
k n
N
n
N N
Y r x n x n n W


    
        
    

(3)
Where 3k r and 0, 1, 2 , ..., 1 .
3
N
r  
1 2
3
3 3
0
2
(3 1) ( )
3 3
N
N N
k k
k n
N N N
n
N N
Y r x n x n nW W W
    
   
   

    
         
     

(4)
Where 3 1k r  and 0, 1, 2 , ..., 1 .
3
N
r  
1 2
3
3 3
0
2
( 3 1) ( )
3 3
N
N N
k k
k n
N N N
n
N N
Y N r x n x n nW W W
    
   
   

    
          
     

(5)
Where ( 3 1)k N r   and 0, 1, 2 , ..., 1 .
3
N
r  
The output components  ( ); 0,1, 2, ..., 1Y k k N  can be realized using (3), (4) and (5) as shown in the
butterfly structure of Fig. 1.
0,1, 2, ...., 1 ;
3
N
n   0,1, 2, ..., 1
3
N
r  
Fig.1. Butterfly structure for realizing DFT of length N
III. EXAMPLE FOR REALIZATION OF DFT OF LENGTH N = 9
Substituting N = 9 and k = 0 in (1), we have
 
8 2
0 0
( 0 ) ( ) ( ) ( 3 ) ( 6 )
n n
Y x n x n x n x n
 
      
 
2 2
9
0
( 0 ) ( ) ( 3) ( 6 )
j k n
n
Y x n x n x n e



      fo r 0k  (6)
Substituting N = 9 and k = 3 in (1), we get

2 38
9
0
( 3 ) ( )
j n
n
Y x n e
 


 
2 3 2 6
9 9
( 0 ) (1) ( 2 )
j j
x x e x e
  
  
   
 
2 3 2 6
2
9 9
( 3 ) ( 4 ) (5)
j j
j
x x e x e e
 

 
   
   
 
2 3 2 6
4
9 9
( 6 ) (7 ) (8)
j j
j
x x e x e e
 

 
   
   
 
(7)
Since 2
1
j
e

 and 4
1
j
e

 , (7) can be expressed as
 
2 2
9
0
( 3) ( ) ( 3) ( 6 )
j k n
n
Y x n x n x n e



      fo r 3k  (8)
2 68
9
0
( 6 ) ( )
j n
n
Y x n e
 


 
2 6 2 1 2
9 9
( 0 ) (1) ( 2 )
j j
x x e x e
  
  
   
 
2 6 2 1 2
4
9 9
( 3 ) ( 4 ) (5)
j j
j
x x e x e e
 

 
   
   
 
2 6 2 1 2
8
9 9
( 6 ) (7 ) (8 )
j j
j
x x e x e e
 

 
   
   
 
(9)
Since 4
1
j
e

 and 8
1
j
e

 , (9) can be written as
 
2 2
9
0
( 6 ) ( ) ( 3) ( 6 )
j k n
n
Y x n x n x n e



      fo r 6k  (10)
In general (6), (8) and (10) can be expressed as
 
2 2
9
0
( 3 ) ( ) ( 3) ( 6 )
j k n
n
Y r x n x n x n e



     fo r 3k r 0,1 & 2 .a n d r 
Since
2j kn
k n
N
NW e


 , the above expression can be expressed as
 
2
( 3 )
9
0
( 3 ) ( ) ( 3) ( 6 )
r n
n
Y r x n x n x n W

      fo r 3k r 0,1 & 2 .a n d r  (11)
The above expression is same as (3) for N = 9, 3k r and 0,1 & 2r  .
Substituting N = 9 and k = 1 in (1), we obtain
28
9
0
(1 ) ( )
j n
n
Y x n e



 
2 4
9 9
( 0 ) (1) ( 2 )
j j
x x e x e
 
  
   
 
2 4 6
9 9 9( 3 ) ( 4 ) (5)
j j j
x x e x e e
     
   
 
2 4 1 2
9 9 9( 6 ) (7 ) (8 )
j j j
x x e x e e
     
   
 
Since 6
9
1 3
2
j
j
e

  

and 1 2
9
1 3
2
j
j
e

  

, the above expression can be written as
2 2
9
0
1 3 1 3
(1) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y x n x n x n e



       
         
     
 fo r 1k  (12)
2 48
9
0
( 4 ) ( )
j n
n
Y x n e
 


 
8 1 6
9 9
( 0 ) (1) ( 2 )
j j
x x e x e
 
  
   
 
8 1 6 2 4
9 9 9( 3 ) ( 4 ) (5)
j j j
x x e x e e
     
   
 
8 1 6 4 8
9 9 9( 6 ) (7 ) (8)
j j j
x x e x e e
     
   
 

Since 2 4
9
1 3
2
j
j
e

  

and 4 8
9
1 3
2
j
j
e

  

2 2
9
0
1 3 1 3
( 4 ) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y x n x n x n e



       
         
     
 fo r 4k  (13)
2 78
9
0
( 7 ) ( )
j n
n
Y x n e
 


 
1 4 2 8
9 9
( 0 ) (1) ( 2 )
j j
x x e x e
 
  
   
 
1 4 2 8 4 2
9 9 9( 3 ) ( 4 ) (5)
j j j
x x e x e e
     
   
 
1 4 2 8 8 4
9 9 9( 6 ) (7 ) (8 )
j j j
x x e x e e
     
   
 
Since 4 2
9
1 3
2
j
j
e

  

and 8 4
9
1 3
2
j
j
e

  

2 2
9
0
1 3 1 3
( 7 ) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y x n x n x n e



       
         
     
 fo r 7k  (14)
2 2
9
0
1 3 1 3
( 3 1) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y r x n x n x n e



       
         
     

fo r 3 1k r  0,1 & 2 .a n d r 
Since
2j kn
k n
N
NW e


 , the above expression can be expressed as
2
( 3 1 )
9
0
1 3 1 3
( 3 1) ( ) ( 3) ( 6 )
2 2
r n
n
j j
Y r x n x n x n W


       
          
     

fo r 3 1k r  0,1 & 2 .a n d r  (15)
The above relation is same as (4) for N = 9, 3 1k r  and 0,1 & 2r  .
Substituting N = 9 and k = 2 in (1), we obtain
4 8
9 9
( 2 ) ( 0 ) (1) ( 2 )
j j
Y x x e x e
 
  
   
 
4 8 1 2
9 9 9( 3 ) ( 4 ) (5)
j j j
x x e x e e
     
   
 
4 8 2 4
9 9 9( 6 ) (7 ) (8 )
j j j
x x e x e e
     
   
 
2 2
9
0
1 3 1 3
( 2 ) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y x n x n x n e



       
         
     
 fo r 2k  (16)
1 0 2 0
9 9
( 5 ) ( 0 ) (1) ( 2 )
j j
Y x x e x e
 
  
   
 
1 0 2 0 3 0
9 9 9( 3 ) ( 4 ) (5)
j j j
x x e x e e
     
   
 
1 0 2 0 6 0
9 9 9( 6 ) (7 ) (8 )
j j j
x x e x e e
     
   
 
2 2
9
0
1 3 1 3
(5 ) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y x n x n x n e



       
         
     
 fo r 5k  (17)
1 6 3 2
9 9
(8 ) ( 0 ) (1) ( 2 )
j j
Y x x e x e
 
  
   
 
1 6 3 2 4 8
9 9 9( 3 ) ( 4 ) (5)
j j j
x x e x e e
     
   
 
1 6 3 2 9 6
9 9 9( 6 ) (7 ) (8 )
j j j
x x e x e e
     
   
 
2 2
9
0
1 3 1 3
(8 ) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y x n x n x n e



       
         
     
 fo r 8k  (18)

2 2
9
0
1 3 1 3
( 9 3 1) ( ) ( 3) ( 6 )
2 2
j k n
n
j j
Y r x n x n x n e



       
          
     

fo r 9 3 1k r   0,1 & 2 .a n d r 
2
( 9 3 1 )
9
0
1 3 1 3
( 9 3 1) ( ) ( 3) ( 6 )
2 2
r n
n
j j
Y r x n x n x n W
 

       
           
     

fo r 9 3 1k r   0,1 & 2 .a n d r  (19)
The above relation is same as (5) for N = 9, 9 3 1k r   and 0,1 & 2r  .
The output components (0 ), (1), ( 2 ), ...., (8)Y Y Y Y can be realized using (11), (15) and (19) as shown in Fig. 2.
Fig.2.Signal flow graph for realization of DFT of length N = 9
IV. CONCLUSION
A new radix-3 algorithm for realization of
DFT of length N = 3m
(m = 1, 2, 3,...) has been
proposed. The DFT of length N is realized from
three DFT sequences, each of length N/3. A
butterfly structure for realizing the radix-3
algorithm for DFT of length N is shown and the
signal flow graph for DFT of length N = 9 is given
to clarify the proposal. Direct calculation of DFT
of length N requires O (N2
) complex multiplications
(4N2
real multiplications) and some additions. This
radix-3 algorithm reduces the number of
multiplications required for realizing DFT. For
example, the number of complex multiplications
required for realizing 9-point DFT using the
proposed radix-3 algorithm is 60. Therefore, the
hardware complexity and execution time for
implementing radix-3 DFT algorithm can be
reduced.
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edition ( New York: McGraw-
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applications ( Prentice-Hall, 1996).
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York: Wiley, 1999).
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machine computation of complex Fourier
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Radix-3 Algorithm for Realization of Discrete Fourier Transform

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