Mapping between Discrete Cosine Transform of Type-VI/VII and Discrete Fourier Transform

M.N.Murty. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 4, (Part - 3) April 2016, pp.60-62
www.ijera.com 60|P a g e
Mapping between Discrete Cosine Transform of Type-VI/VII and
Discrete Fourier Transform
M.N.Murty1
and H.Panda2
1
(Visiting Faculty, Department Of Physics, Sri Sathya Sai Institute Of Higher Learning, Prasanthi Nilayam –
515134, Andhra Pradesh, India)
2
(Department Of Physics, S.K.C.G. Autonomous College, Paralakhemundi -761200, Odisha, India)
ABSTRACT
In this paper, the mapping between discrete cosine transform of types VI and VII (DCT-VI/VII) of even length
N and (2N – 1)-point one dimensional discrete Fourier transform (1D-DFT) is presented. The technique used in
this paper is the mapping between the real-valued data sequence to an intermediate sequence used as an input to
DFT.
Keywords – Discrete Fourier transform, discrete cosine transform, discrete sine transform, mapping.
I. INTRODUCTION
Discrete transforms play a significant role
in digital signal processing. Discrete cosine
transform (DCT) and discrete sine transform
(DST) are used as key functions in many signal and
image processing applications. There are eight
types of DCT and DST. Of these, the DCT-II,
DST-II, DCT-IV, and DST-IV have gained
popularity. The DCT and DST transform of types I,
II, III and IV, form a group of so-called “even”
sinusoidal transforms. Much less known is group of
so-called “odd” sinusoidal transforms: DCT and
DST of types V, VI, VII and VIII.
The original definition of the DCT
introduced by Ahmed et al. in 1974 [1] was one-
dimensional (1-D) and suitable for 1-D digital
signal processing. The DCT has found wide
applications in speech and image processing as
well as telecommunication signal processing for the
purpose of data compression, feature extraction,
image reconstruction, and filtering. Thus, many
algorithms and VLSI architectures for the fast
computation of DCT have been proposed [2]-[7].
Among those algorithms [6] and [7] are believed to
be most efficient two-dimensional DCT algorithms
in the sense of minimizing any measure of
computational complexity. DCT of types I, II, III
and IV are extensively used in coding image, video,
speech and audio. The so-called “odd-type” DCTs
of types V, VI, VII and VIII, however, have not
been widely used for purposes of coding image,
video, speech and audio data.
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 [8,9]. Not only it
is frequently encountered in many different
applications, but also it is computation-intensive.
Recently DST of types VI and VII have
surfaced as useful tool in image and video coding.
Fast algorithms for computing DST-VII of lengths
4 and 8 have been proposed [10] and these
algorithms are currently under consideration for
inclusion in ISO/IEC/ITU-T High Efficiency Video
Coding (HEVC) standard.
In this paper, the mapping between DCT-
VI/VII of even length N and (2N – 1) -point DFT
has been established. The procedure used in this
paper is the mapping between the real-valued data
sequence to an intermediate sequence used as an
input to DFT. This mapping technique may find
application in image and video coding.
The rest of the paper is organized as
follows. The proposed mapping between DCT-VII
and DFT is presented in Section-II. Section-III
establishes the mapping between DCT-VI and
DFT. Conclusion is given in Section-IV
II. MAPPING BETWEEN DCT-VII
AND DFT
The 1-D DFT of input sequence { xn ; n =
0,1,2,....,N-1} is defined by
21
0
j k nN
N
k n
n
exY



 
0,1, 2, ..., 1for k N  (1)
Where 1j  
The DCT of type-VII is defined as
1
0
( 2 1)
0,1, 2 , ..., 1
2 1
N
V II
k n
n
k n
C O S fo r k N
N
xX


 
  
  

(2)
The scale factors have been omitted in (1) and (2).
From (1), the real component of DST is written as
RESEARCH ARTICLE OPEN ACCESS

1
0
2
N
k n
n
k n
R C O S
N
xY


 
  
   
 

(3)
Replacing k by 2 1k  and N by( 2 1)N  , (3)
can be written as
2 ( 1 )
2 1
0
2 ( 2 1)
2 1
N
k n
n
k n
R C O S
N
xY




 
  
    

(4)
For N even or odd, (4) can be expressed as
1
2 1 2 1
1
2 (2 1)
2 1
N
k o n N n
n
k n
R C O S
N
x x xY


  

 
     
      

(5)
For mapping between DCT-VII and DFT, define
2 1
0
1, 2, ..., 1
o
n N n
fo r n N
x
x x  


  
(6)
Substituting (6) in (5), we obtain
1
2 1
1
2 (2 1)
2 2 1
N
k
n
n
R k n
C O S
N
Y
x




    

  

(7)
Since from (6) 0
ox  , (7) can be written as
1
2 1
0
2 (2 1)
2 2 1
N
k
n
n
R k n
C O S
N
Y
x




    

  

(8)
Taking N as even, (8) can be expressed as
1 1
2 2
2 1
2
0 0
(2 1)2
(2 1) 2 2
2 2 1 2 1
N N
k
Nn n
n n
N
k n
R k n
COS COS
N N
Y
x x


 


 
  
         
  
     
 
 
 
(9)
Define another sequence u2n and uN+2n given by
2
2
2
0,1, 2, ..., 1
2
0,1, 2, ..., 1
2
n n
N n N n
N
fo r n
N
fo r n
x u
x u 

  


   

(10)
Substituting (10) in (9), we get
1 1
2 2
2 1
2 2
0 0
(2 1) 2 (2 1)( 2 )
2 2 1 2 1
N N
k
n N n
n n
R k n k N n
COS COS
N N
Y
u u
 
 


 
        
 
       
 
(11)
For even N , the DCT-VII defined in (2) can be
expressed as
1 1
2 2
2
0 0
(2 1)
(2 1) 2
2 1 2 1
N N
V II
Nk n n
n n
N
k n
k n
C O S C O S
N N
x xX


 

 
  
      
  
     
 
 
 
(12)
The above expression shows that the RHS of the
real part of 1D-DFT given by (11) is in the form of
DCT-VII. Hence,
2 1
2
V IIk
k
R Y
X

 
 

(13)
The above expression shows the mapping between
DCT-VII and DFT.
III. MAPPING BETWEEN DCT-VI AND
DFT
The DCT of type-VI for input sequence
{xn ; n = 0,1,2,....,N-1} is defined by
1
0
( 2 1)
0,1, 2, ..., 1
2 1
N
V I
k n
n
k n
C O S fo r k N
N
xX


 
  
  

(14)
The scale factor has been omitted in (14).
Replacing N by 2 1N  in (3), we obtain
2 ( 1 )
0
2
2 1
N
k n
n
k n
R C O S
N
xY



 
  
    

(15)
For N even or odd, (15) can be written as
1
2 1
1
2
2 1
N
k o n N n
n
k n
R C O S
N
x x xY

 

 
     
      

(16)
For mapping between DCT-VI and DFT, define
2 1
0
1, 2, ..., 1
o
n N n
fo r n N
x
x x  


  
(17)
Substituting (17) in (16), we get
1
0
2
2 2 1
N
k
n
n
R k n
C O S
N
Y
x



    

  

(18)
For even N , define another sequence
( 1 )
0,1, 2, ..., 1n N n
fo r n Nvx  
  
(19)

Substituting (19) in RHS of (18), we get
1
0
( 2 1)
2 2 1
N
k
n
n
R k n
C O S
N
Y
v



    

  

(20)
Since the RHS of the above expression is in the
form of DCT-VI defined by (14), we have
2
V Ik
k
R Y
X
 
 

(21)
As the LHS of (21) represents the real part of DFT,
the mapping between DCT-VI and DFT is
established by this equation.
IV. CONCLUSION
In this paper, the mapping between DCT-
VI/VII of even length N and (2N – 1) -point DFT
has been established using an intermediate
sequence. The performance DST can be compared
to that of DCT and therefore, DST may be
considered as a viable alternative to DCT. Recently
DST of types VI and VII have surfaced as useful
tool in image and video coding. Fast algorithms for
computing DST-VII of lengths 4 and 8 have been
proposed [10] and these algorithms are currently
under consideration for inclusion in ISO/IEC/ITU-
T High Efficiency Video Coding (HEVC) standard.
Like DST-VI/VII, the proposed mapping technique
for DCT-VI/VII may find application in image and
video coding.
REFERENCES
[1]. N. Ahmed, T.Natarajan, and K.R.Rao,
Discrete cosine transform, IEEE Trans.
Comput., vol. C-23, 1974, 90-93.
[2]. R.K.Rao and P.Yip, Discrete cosine
transform: algorithm, advantages, and
applications (New York: Academic,
1990).
[3]. M.J.Narasimha and A.M.Peterson, On the
computation of the discrete cosine
transform, IEEE Trans. Communications,
vol.COM-26, no.6, June 1978, 934-936.
[4]. H.S.Hou, A fast recursive algorithms for
computing the discrete cosine transform,
IEEE Trans. Acoust., Speech, Signal
Processing, vol. ASSP-35(10), Oct.1987,
1455-1461.
[5]. PeiZong Lee and Fang-Yu Huang,
Reconstructed recursive DCT and DST
algorithms, IEEE Trans. Signal
Processing, vol.42, no.7, Jul.1994, 1600-
1609.
[6]. P.Duhamel and C.Guillemot, Polynomial
transform computation of 2-D DCT, in
Proc. ICASSP’90, Apr.1990, 1515-1518.
[7]. E.Feig and S.Winograd, Fast algorithms
for the discrete cosine transform, IEEE
Trans. Signal Processing, vol.40, no.9,
Sept.1992, 2174-2193.
[8]. R.N.Bracewell, The Fourier transform and
its applications, 3rd
edition (New York:
McGraw-Hill, 2000).
[9]. J.G.Proakis and D.G.Manolakis, Digital
signal processing: principles, algorithms
and applications (Prentice-Hall, 1996).
[10]. Ravi K. Chivukula and Yuriy A. Reznik,
Fast computing of cosine and sine
transforms of types VI and VII,
Proceedings of the SPIE, vol.8135, 2011.

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