A Novel Split-Radix Fast Algorithm for 2-D Discrete Hartley Transform

TLDR: This paper presents a fast split-radix- (2×2)/(8-times;8) algorithm for computing the 2-D discrete Hartley transform (DHT) of length N ×N with N = q · 2 m, where q is an odd integer.

Abstract: This paper presents a fast split-radix- (2t2)/(8t8) algorithm for computing the 2-D discrete Hartley transform (DHT) of length N ×N with N = q · 2 m, where q is an odd integer. The proposed algorithm decomposes an N × N DHT into one N /2 × N /2 DHT and 48 N /8 × N /8 DHTs. It achieves an efficient reduction on the number of arithmetic operations, data transfers and twiddle factors compared to the split-radix-(2×2)/(4×4) algorithm. Moreover, the characteristic of expression in simple matrices leads to an easy implementation of the algorithm. If implementing the above two algorithms with fully parallel structure in hardware, it seems that the proposed algorithm can decrease the area complexity compared to the split-radix-(2×2)/(4×4) algorithm, but requires a little more time complexity. An application of the proposed algorithm to 2-D medical image compression is also provided.

Figures

Table V Comparison of Twiddle Factors for q = 1 (Saving1 and Saving2 denote respectively the saving of twiddle factors compared to radix algorithm in [29] and

Table II Comparison of Arithmetic Complexities for q = 3 (Saving denotes the saving of arithmetic complexity compared to the algorithm in [29] and [30]).

Table I Comparison of Arithmetic Complexities for q = 1 (Saving denotes the saving of arithmetic complexity compared to the algorithm in [29] and [30]).

Table IV Comparison of Data Transfers for q = 3(Saving1 and Saving2 denote respectively the saving of data transfers compared to the algorithm in [29] and [30],

Table III Comparison of Data Transfers for q = 1 (Saving1 and Saving2 denote respectively the saving of data transfers compared to the algorithm in [29] and [30], and the row-column algorithm based on [3]).

Fig. 4. Flowgraph of a length-3×3 DHT

Full text

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A Novel Split-Radix Fast Algorithm for 2-D Discrete
Hartley Transform
Longyu Jiang, Huazhong Shu, Jiasong Wu, Lu Wang, Lot Senhadji
To cite this version:
Longyu Jiang, Huazhong Shu, Jiasong Wu, Lu Wang, Lot Senhadji. A Novel Split-Radix Fast
Algorithm for 2-D Discrete Hartley Transform. IEEE Transactions on Circuits and Systems Part 1
Fundamental Theory and Applications, Institute of Electrical and Electronics Engineers (IEEE), 2010,
57 (4), pp.911-924. �10.1109/TCSI.2009.2028639�. �inserm-00405223�

1
Abstract—This paper presents a fast split-radix-(2×2)/(8×8)
algorithm for computing the two-dimensional (2-D) discrete
Hartley transform (DHT) of length N×N with N = q*2
m
, where q is
an odd integer. The proposed algorithm decomposes an N×N DHT
into one N/2×N/2 DHT and forty-eight N/8×N/8 DHTs. It achieves
an efficient reduction on the number of arithmetic operations,
data transfers and twiddle factors compared to the
split-radix-(2×2)/(4×4) algorithm. Moreover, the characteristic of
expression in simple matrices leads to an easy implementation of
the algorithm. If implementing the above two algorithms with
fully parallel structure in hardware, it seems that the proposed
algorithm can decrease the area complexity compared to the
split-radix-(2×2)/(4×4) algorithm, but requires a little more time
complexity. An application of the proposed algorithm to 2-D
medical image compression is also provided.
Index Terms—Two-dimensional (2-D) discrete Hartley
transform (DHT), split-radix, fast algorithm
I. INTRODUCTION
he discrete Hartley transform (DHT) is widely used in
signal and image processing applications. The advantage
of the DHT over the discrete Fourier transform (DFT) is that it
can be used to avoid complex operations when the input
sequence is real. Moreover, the forward and inverse DHTs
differ from each other in their form only in the scaling factor.
Owing to these properties, the DHT is now finding an
increasing interest in the signal processing community. In the
Manuscript received November 23, 2008. This work was supported by the
National Natural Science Foundation of China under Grant 60873048, the
Program for Changjiang Scholars and Innovative Research Team in University
and the Natural Science Foundation of Jiangsu Province of China under Grant
BK2008279.
L. Jiang is with the Laboratory of Image Science and Technology, School
of Computer Science and Engineering, Southeast University, Nanjing 210096,
China (e-mail: jianglongyu01412@yahoo.com.cn).
H. Shu and L. Wang are with the Laboratory of Image Science and
Technology, School of Computer Science and Engineering, Southeast
University, Nanjing 210096, China, and also with the Centre de Recherche en
Information Biomédicale Sino-Français (CRIBs), Nanjing 210096, China
(e-mail: shu.list@seu.edu.cn; wanglu@seu.edu.cn).
J. Wu is with the Laboratory of Image Science and Technology, School of
Biological Science and Medical Engineering, Southeast University, Nanjing
210096, China, and with the Centre de Recherche en Information Biomédicale
Sino-Français (CRIBs), Nanjing 210096, China, and with INSERM, U 642,
35000 Rennes, France, and with the Laboratoire Traitement du Signal et de
l’Image (LTSI), Université de Rennes 1, 35000 Rennes, France, and also with
the Centre de Recherche en Information Biomédicale Sino–Français (CRIBs),
35000 Rennes, France (e-mail: jswu@seu.edu.cn).
L. Senhadji is with INSERM, U 642, 35000 Rennes, France, and with the
Laboratoire Traitement du Signal et de l’Image (LTSI), Université de Rennes 1,
35000 Rennes, France, and also with the Centre de Recherche en Information
Biomédicale Sino–Français (CRIBs), 35000 Rennes, France (e-mail:
lotfi.senhadji@univ-rennes1.fr).
past decades, fast algorithms and implementations of
one-dimensional (1-D) DHT and DFT have been extensively
investigated [1]-[21]. Meantime, special attention has also been
paid on the two-dimensional (2-D) and three-dimensional
(3-D) DHT [22]-[39], this is due to the growing interest in
applications involving multi-dimensional (M-D) signals. In this
paper, fast algorithm means lower computational complexity in
terms of the number of arithmetic operations, data transfers and
twiddle factors.
The algorithms proposed for fast computing the 2-D DHT
can be classified into four categories: i) the row-column
method; ii) the vector-radix fast Hartley transform (FHT)
algorithms [22]-[24]; iii) the split-radix FHT algorithm
[25]-[31]; and iv) the polynomial transform FHT algorithm
[32]-[34]. The row-column method computes the 2-D DHT by
taking the 1-D FHT sequentially along each dimension of the
input data while in the vector-radix algorithm, the 2-D DHT is
decomposed into many smaller ones until the trivial sequence
length is reached. The vector-radix method reduces the number
of arithmetic operations over the row-column algorithm and
possesses the desirable properties such as regular structure and
low implementation cost. This approach was then extended to
3-D DHT [35]-[37] and M-D DHT [23]. In [39], a
vector-radix-3×3 algorithm was developed for computing the
2-D DHT of sequence whose length is 3
m
×3
m
. The polynomial
transform based FHT algorithms for M-D DHT have been
reported in [32] and [34], which lead to a great reduction of the
arithmetic operations at the expense of very complicated
structure. The split-radix 2-D DHT algorithm is more efficient
than the vector-radix algorithm in terms of arithmetic
complexity and it is easy to implement. All the split-radix
algorithms for 2-D DHT reported so far are based on a mixture
of radix-2×2 and radix-4×4 index maps.
Huang et al. [25] applied a radix-2×2 decomposition to the
even-even, even-odd, odd-even indexed samples and a
radix-4×4 decomposition to the odd-odd indexed samples.
Thus, an N×N DHT is decomposed into three N/2×N/2 DHTs
and four N/4×N/4 DHTs. By using a radix-4×4 decomposition
to even-odd, odd-even and odd-odd indexed terms, an
improved split-radix algorithm for 2-D DHT was further
derived [28], which decomposes an N×N 2-D DHT into one
N/2×N/2 DHT and twelve N/4×N/4 DHTs. The split-radix
algorithms for the 2-D DHT have been presented using
decimation-in-frequency (DIF) [29] and decimation-in-time
(DIT) [30]. It seems that the algorithms reported in [29] and [30]
are the most efficient ones among all the existing split-radix
algorithms in terms of the arithmetic complexity.
A Novel Split-Radix Fast Algorithm for 2-D
Discrete Hartley Transform
Longyu Jiang, Huazhong Shu, Senior Member, IEEE, Jiasong Wu, Lu Wang and Lotfi Senhadji,
Senior Member, IEEE
T

2
Moreover, these two algorithms support various sequence
lengths. Specifically, the block size can be chosen as
q*2
m
×q*2
m
, where q is an odd integer. In [31], the radix-2/4
approach has been generalized to the M-D DHT. In particular,
for the case of 2-D DHT, it has the same arithmetic complexity
as that of the algorithms presented in [29] and [30].
Among all the algorithms mentioned above, the split-radix
algorithms based on radix-2/4 are the most attractive ones
because they provide a good comprise between the arithmetic
and structural complexities. Recently, Bouguezel et al. [3]
proposed a new split-radix fast algorithm based on a mixture of
radix-2 and radix-8 index maps for 1-D DHT of sequences
whose length is q×2
m
, where q is an odd integer. This algorithm
is more efficient than the conventional split radix-2/4 FHT
algorithm in terms of the number of data transfers and twiddle
factor evaluations, which also contribute significantly to the
execution time of FHT algorithms. Inspired by the algorithm
presented in [3], we propose a split-radix-(2×2)/(8×8)
algorithm for computing the 2-D DHT of sequences with
length-q*2
m
×q*2
m
, which consists of decomposing an N×N
DHT into one N/2×N/2 DHT and forty-eight N/8×N/8 DHTs.
Besides, the split radix-2/8 algorithm has been already used for
computing the 2-D DFT [40], [41].
The rest of the paper is organized as follows. Section II
presents the derivation of the algorithm. In Section III, the
computational complexity and the hardware area and time
complexity of the proposed algorithm are analyzed, and the
comparison with some existing algorithms is also provided.
Section IV presents the result of software implementation of
the proposed and some existing algorithms. Section V
concludes the work.
II. P
ROPOSED RADIX-(2×2)/(8×8) ALGORITHM
The 2-D DHT X(k
1
, k
2
) of real valued sequence, x(n
1
, n
2
),
for 0 ≤ n
1
, n
2
≤ N – 1, is defined by
12
12
11 2
12 12
00 1
(, )
2
(, )cas , 0 , 1,
NN
ii
nn i
Xk k
xn n nk k k N
N
π
−−
== =
⎛⎞
=≤≤−
⎜⎟
⎝⎠
∑∑ ∑
(1)
where
cas( ) cos( ) sin( ).
θ
θθ
=+ The sequence length N is
assumed to be q×2
m
, where q is an odd integer and m > 0.
Let us first consider the case when m = 1, that is, N = 2q.
A. The case m = 1, i.e., N = 2q
In this case, the radix-2×2 algorithm is used to decompose a
length-2q×2q DHT. The even-even indexed outputs are
obtained by
12
12
11
2
00 1 2 1 2
00 1
(2 ,2 )
2
( , )cas , 0 , 1.
qq
ii
nn i
Xk k
ynn nk kk q
q
π
−−
== =
⎛⎞
=≤≤−
⎜⎟
⎝⎠
∑∑ ∑
(2)
The even-odd, odd-even, and odd-odd indexed outputs can
be computed by
11 2 2
12
12
11 2 2
11
2
()
12
00 1
(2 ,2 )
2
( 1) ( , )cas ,
qq
np n p
pp i i
nn i
Xk pqk pq
ynn nk
q
π
−−
+
== =
++
⎛⎞
=−
⎜⎟
⎝⎠
∑∑ ∑
(3)
where
12 12 12
,0,1,(,)(0,0), 0, 1.pp pp kk q
=
≠≤≤−
The sequences
12
,12
(, )
pp
y
nn
for p
1
, p
2
= 0, 1, in (2) and (3) are
obtained from the original input sequence as
()
()
00 1 2 01 1 2 10 1 2 11 1 2
221212 1212
(, ), (, ), (, ), (, )
( ) ( , ), ( , ), ( , ), ( , )
T
T
ynnynnynnynn
H H xnnxnn qxnqnxnqn q=⊗ + + + +
(4)
where T denotes the transpose,
⎥
⎦
⎤
⎢
⎣
⎡
−
=
11
11
2
H , and “
⊗
” is
the Kronecker product [42]. Fig. 1 shows the implementation of
(4).
B. The case m = 2, i.e., N = 4q
When m = 2, the decomposition of (1) for the even-even
indexed outputs is given by
12
12
2121
2
2/4
00 1 2 1 2
00 1
(2 ,2 )
2
( , )cas , 0 , 2 1,
2
qq
ii
nn i
Xk k
ynn nk kk q
q
π
−−
== =
⎛⎞
=
≤≤−
⎜⎟
⎝⎠
∑∑ ∑
(5)
where
2/4
00 12 12 12
1212
(, ) [(, ) (, 2)]
[( 2, ) ( 2, 2)].
ynn xnn xnn q
x
nqnxnqnq
=++
++ ++ +
(6)
The even-odd, odd-even and odd-odd indexed outputs are
obtained as follows
12
12 12
11 2 2
11 2 2
12
00 1 1
2/4 2/4
, 12 , 12 12
(4 , 4 )
2
( , )cas
2
(, ) (, ), 0 , 1,
NN
ii i i
nn i i
pp pp
Xk pqk pq
xn n nk np
q
F kk G kk kk q
ππ
−−
== = =
±±
⎛⎞
=±
⎜⎟
⎝⎠
=
±≤≤−
∑∑ ∑ ∑
(7)
where
12
12
2/4
,12
11 2 2
12
00 1 1
(, )
2
(, )cos cas ,
2
pp
NN
ii ii
nn i i
Fkk
x
nn np nk
q
ππ
−−
== = =
⎛⎞
⎛⎞
=
⎜⎟
⎜⎟
⎝⎠
⎝⎠
∑∑ ∑ ∑
(8)
12
12
2/4
,12
11 2 2
12
00 1 1
(, )
2
( , )sin cas .
2
pp
NN
ii ii
nn i i
Gkk
x
nn np nk
q
ππ
−−
== = =
⎛⎞
⎛⎞
=−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
∑∑ ∑ ∑
(9)
Using the matrix representation, (7) can be expressed as
12
12
11 2 2
11 2 2
2/4
,12
2
2/4
,12
((4 )mod4 ,(4 )mod4 )
(( 4 )mod4 ,( 4 )mod4 )
(, )
.
(, )
pp
pp
Xkpq qk pq q
X
Nkpq qNkpq q
Fkk
H
Gkk
++
⎡
⎤
⎢
⎥
+− +−
⎣
⎦
⎡⎤
=
⎢⎥
⎢⎥
⎣⎦
(10)

3
1) Even-odd output terms (p
1
= 0, 2; p
2
= 1)
12
2/4
,12
(, )
pp
F
kk and
12
2/4
,12
(, )
pp
Gkk defined by (8) and (9) can be
further decomposed as follows.
12
12
12
2/4
,12
2121
12 1 2
00
2
12 1 2
1
2
1
2121
22
2/4
01 1 2
00 1 1
(, )
{[ ( , ) ( 2 , )]
[( , 2) ( 2, 2)]}cos
2
2
cas
2
( , ) cos cas
2
pp
qq
nn
ii
i
ii
i
qq
ii ii
nn i i
Fkk
xn n xn qn
x
nn q xn qn q np
nk
q
ynn np nk
q
π
π
ππ
−−
==
=
=
−−
== = =
=++
⎛⎞
−++++
⎜⎟
⎝⎠
⎛⎞
×
⎜⎟
⎝⎠
⎛
⎛⎞
=
⎜
⎜⎟
⎝⎠
⎝
∑∑
∑
∑
∑∑ ∑ ∑
12
12
11
2
2/4
,12
00 1
2
( , )cas ,
qq
pp ii
nn i
fnn nk
q
π
−−
== =
⎞
⎟
⎠
⎛⎞
=
⎜⎟
⎝⎠
∑∑ ∑
(11)
where
2/4
01 12 12 12
1212
(, ) [(, ) (, 2)]
[( 2, ) ( 2, 2)],
ynn xnn xnn q
x
nqnxnqn q
=−+
++ −+ +
(12)
1
12
1
/2
2/4 2/4 2/4
,12 0112 011 2
2
(1)2 2/4
01 1 2
1
2
/2
2/4
01 1 2
1
(,) (,)(1) ( ,)
cos ( 1) [ ( , )
2
(1) ( , )]sin .
2
p
pp
q
ii
i
p
ii
i
fnn ynn ynqn
np y n n q
ynqnq np
π
π
+
=
=
⎡⎤
=+−+
⎣⎦
⎛⎞
×+− +
⎜⎟
⎝⎠
⎛⎞
+− + +
⎜⎟
⎝⎠
∑
∑
(13)
The decomposition of
12
2/4
,12
(, )
pp
Gkk(p
1
= 0, 2; p
2
= 1) can be
done in a similar way.
12
12
12
12
2/4
,12
2121
22
2/4
01 1 2
00 1 1
11
2
2/4
,12
00 1
( , )
2
( , )sin cas
2
2
( , )cas
pp
qq
ii ii
nn i i
qq
pp ii
nn i
Gkk
ynn np nk
q
gnn nk
q
ππ
π
−−
== = =
−−
== =
⎛⎞
⎛⎞
=−
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎛⎞
=−
⎜⎟
⎝⎠
∑∑ ∑ ∑
∑∑ ∑
(14)
where
1
12
1
/2
2/4 2/4 2/4
,12 0112 011 2
2
(1)2 2/4
01 1 2
1
2
/2
2/4
01 1 2
1
(,) (,)(1) ( ,)
sin ( 1) ( , )
2
( 1) ( , ) cos
2
p
pp
q
ii
i
p
ii
i
gnn ynn ynqn
np y n n q
ynqnq np
π
π
−
=
=
⎡⎤
=+−+
⎣⎦
⎛⎞
⎡
×+− +
⎜⎟
⎣
⎝⎠
⎛⎞
⎤
+− + +
⎜⎟
⎦
⎝⎠
∑
∑
(15)
12
2/4
,12
(, )
pp
f
nn and
12
2/4
,12
(, )
pp
g
nn (p
1
= 0, 2; p
2
= 1) defined by
(13) and (15) can be expressed in matrix form as
11
11
2/4
0,1 1 2
2/4
2,1 1 2
2/4
0,1 1 2
2/4
2,1 1 2
22
1
22
22
22
(1)
(, )
(, )
(, )
(, )
cos 0 sin 0
22
0(1)cos 0 (1)sin
22
sin 0 cos 0
22
0(1)sin 0 (1)cos
22
10 0 0
01 0 0
00( 1)
nn
nn
q
fnn
fnn
gnn
gnn
nn
nn
nn
nn
ππ
ππ
ππ
ππ
+
−
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
⎡⎤
−
⎢⎥
⎢⎥
−−
⎢⎥
⎢⎥
=
⎢⎥
⎢⎥
⎢⎥
−−
⎢⎥
⎣⎦
×
−
2/4
01 1 2
2/4
01 1 2
2
2/4
01 1 2
(1)2
2/4
01 1 2
(, )
10 1 0
(, )
10 1 0
01 0 1
0
(,)
01 0 1
000(1)
(, )
q
ynn
ynnq
ynqn
ynqnq
−
⎡⎤
⎡⎤
⎡⎤
⎢⎥
⎢⎥
+
−
⎢⎥
⎢⎥
⎢⎥
⎢⎥
+
⎢⎥
⎢⎥
⎢⎥
−
−
⎣⎦
++
⎣⎦
⎣⎦
(16)
The above equation can be rewritten as
12
12
2/4
2/4 2/4
,
2
(1)/2
2/4 2/4
2/4
2
,
2/4
2/4
111
01
2/4
12 2
(1)
,
pp
eo eo
q
eo eo
pp
−
⎡⎤
⎡⎤
−
⎡⎤
=
⎢⎥
⎢⎥
⎢⎥
−
⎢⎥
⎣⎦
⎣⎦
⎣⎦
⎡⎤
×
⎢⎥
⎣⎦
f
I0
CS
0I
SC
g
JRJ
y
RJ J
(17)
where
2
I is the identity matrix, and
12 12
2/4 2/4
0,1 1 2 0,1 1 2
2/4 2/4
,,
2/4 2/4
2,1 1 2 2,1 1 2
(, ) (, )
, ,
(, ) (, )
pp pp
fnn gnn
fnn gnn
⎡
⎤⎡⎤
==
⎢
⎥⎢⎥
⎢
⎥⎢⎥
⎣
⎦⎣⎦
fg
(18)
1
1
01
2/4
01
01
2/4
01 2
01
cos 0
,
0(1)cos
sin 0
, ( )2,
0(1)sin
eo
n
eo
n
n
α
α
α
απ
α
⎡⎤
=
⎢⎥
−
⎣⎦
⎡⎤
==
⎢⎥
−
⎣⎦
C
S
(19)
2/4
12 1
10 0 1
, , diag(1, 1),
10 0 1
⎡⎤ ⎡ ⎤
=
==−
⎢⎥ ⎢ ⎥
−
⎣⎦ ⎣ ⎦
JJ R (20)
2/4 2/4 2/4
01 01 1 2 01 1 2
2/4 2/4
01 1 2 01 1 2
((,) (, )
( , ) ( , )) .
T
ynn ynnq
ynqn ynqnq
=+
+++
y
(21)
Fig. 2 shows the implementation of (17).
2) Odd-even output terms (p
1
= 1; p
2
= 0, 2)
As for the previous case, the odd-even output terms can be
obtained as
12
12
2/4
2/4 2/4
,
22
2/4
10
(1)/2
2/4 2/4
2/4
22
,
,
(1)
pp
oe oe
q
oe oe
pp
−
⎡⎤
⎡⎤
−
⎡⎤⎡⎤
=
⎢⎥
⎢⎥
⎢⎥⎢⎥
−
⎢⎥
⎣⎦⎣⎦
⎣⎦
⎣⎦
f
I0H0
CS
y
0I0H
SC
g
(22)
where
12 12
2/4 2/4
1,0 1 2 1,0 1 2
2/4 2/4
,,
2/4 2/4
1,2 1 2 1,2 1 2
(, ) (, )
, ,
(, ) (, )
pp pp
fnn gnn
fnn gnn
⎡
⎤⎡ ⎤
==
⎢
⎥⎢ ⎥
⎢
⎥⎢ ⎥
⎣
⎦⎣ ⎦
fg
(23)

4
2
2
10
2/4
10
10
2/4
10 1
10
cos 0
,
0(1)cos
sin 0
, ( )2,
0(1)sin
oe
n
oe
n
n
α
α
α
απ
α
⎡⎤
=
⎢⎥
−
⎣⎦
⎡⎤
==
⎢⎥
−
⎣⎦
C
S
(24)
2/4 2/4 2/4
10 10 1 2 10 1 2
2/4 2/4
10121012
((,) (, )
( , ) ( , )) ,
T
ynn ynnq
ynqn ynqnq
=+
+++
y
(25)
2/4
10 12 12 12
1212
(, ) [(, ) (, 2)]
[ ( 2 , ) ( 2 , 2 )].
ynn xnn xnn q
x
nqnxnqnq
=++
−+ ++ +
(26)
3) Odd-odd output terms (
p
1
= 1, –1; p
2
= 1)
We have the following decomposition for the odd-odd output
terms
12
12
2/4
2/4 2/4
,
212
2/4
11
( 1)/2 2/4 2/4
2/4 2/4
2/4
21221
,
(1)
pp
oo oo
q
oo oo
pp
−
⎡⎤
⎡⎤
−
⎡⎤⎡⎤
=
⎢⎥
⎢⎥
⎢⎥⎢⎥
−
⎢⎥
⎣⎦⎣⎦
⎣⎦
⎣⎦
f
I0 JJ
CS
y
0 I RJ RJ
SC
g
(27)
where
12 12
2/4 2/4
1,1 1 2 1,1 1 2
2/4 2/4
,,
2/4 2/4
1,1 1 2 1,1 1 2
2/4
2
(, ) (, )
,,
(, ) (, )
diag( 1,1),
pp pp
fnn gnn
fnn gnn
−−
⎡⎤⎡⎤
==
⎢⎥⎢⎥
⎢⎥⎢⎥
⎣⎦⎣⎦
=−
fg
R
(28)
11 11
2/4 2/4
11 11
11 2 1 11 1 2
cos 0 sin 0
, ,
0cos 0sin
[( )]2, [( )]2,
oo oo
nn nn
αα
αα
απαπ
⎡⎤⎡⎤
==
⎢⎥⎢⎥
′′
⎣⎦⎣⎦
′
=− =+
CS
(29)
2/4 2/4 2/4
11 11 1 2 11 1 2
2/4 2/4
11 1 2 11 1 2
((,) (, )
( , ) ( , )) ,
T
ynn ynnq
ynqn ynqnq
=+
+++
y
(30)
2/4
11 12 12 12
1212
(, ) [(, ) (, 2)]
[ ( 2 , ) ( 2 , 2 )].
ynn xnn xnn q
x
nqnxnqn q
=−+
−+ −+ +
(31)
C. The case m ≥ 3
By introducing a mixture of radix-2×2 and radix-8×8 index
maps, we propose a novel decomposition of (1). The even-even
output terms can be computed by
12
12
21 21
2
2/8
00 1 2 1 2
00 1
(2 , 2 )
2
( , )cas , 0 , 2 1,
2
NN
ii
nn i
Xk k
ynn nk kkN
N
π
−−
== =
⎛⎞
=≤≤−
⎜⎟
⎝⎠
∑∑ ∑
(32)
where
2/8
00 12 12 12
1212
(, ) [(, ) (, 2)]
[ ( 2, ) ( 2, 2)].
ynn xnn xnnN
xn N n xn N n N
=++
++ ++ +
(33)
The even-odd, odd-even, and odd-odd output terms can be
derived as follows.
12
12 12
11 2 2
11 2 2
12
00 1 1
2/8 2/8
, 12 , 12 12
(8 ,8 )
22
( , )cas
8
(, ) (, ), 0 , 81,
NN
ii i i
nn i i
pp pp
Xk pqk pq
xn n nk np
NNq
FkkGkk kkN
ππ
−−
== = =
±±
⎛⎞
=±
⎜⎟
⎝⎠
=± ≤≤−
∑∑ ∑ ∑
(34)
where
12
12
2/8
,12
11 2 2
12
00 1 1
(, )
22
( , ) cos cas ,
8
pp
NN
ii ii
nn i i
Fkk
x
nn np nk
Nq N
ππ
−−
== = =
⎛⎞⎛⎞
=
⎜⎟⎜⎟
⎝⎠⎝⎠
∑∑ ∑ ∑
(35)
12
12
2/8
,12
11 2 2
12
00 1 1
(, )
22
( , )sin cas .
8
pp
NN
ii ii
nn i i
Gkk
x
nn np nk
Nq N
ππ
−−
== = =
⎛⎞⎛ ⎞
=−
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
∑∑ ∑ ∑
(36)
Equation (34) can be written in matrix form as
12
12
11 2 2
11 2 2
2/8
,12
2
2/8
,12
((8 ) mod , (8 ) mod )
(( 8 ) mod ,( 8 ) mod )
(, )
(, )
pp
pp
Xkpq Nk pq N
X
Nkpq NNkpq N
Fkk
H
Gkk
++
⎡
⎤
⎢
⎥
+− +−
⎣
⎦
⎡⎤
=
⎢⎥
⎢⎥
⎣⎦
(37)
The input data sequences
12
2/8
,12
(, )
pp
F
kk and
12
2/8
,12
(, )
pp
Gkk are
determined as follows.
1) Even-odd output terms (
p
1
= 0, 2, 4, 6; p
2
= 1, 3)
Equation (35) can be decomposed as
12
12
12
12
2/8
,12
21 21
22
2/8
01 1 2
00 1 1
/8 1 /8 1 2
2/8
,12
00 1
(, )
22
( , )cos cas
8
2
( , )cas ,
8
pp
NN
ii ii
nn i i
NN
pp ii
nn i
Fkk
ynn np nk
Nq N
fnn nk
N
ππ
π
−−
== = =
−−
== =
⎛⎞⎛⎞
=
⎜⎟⎜⎟
⎝⎠⎝⎠
⎛⎞
=
⎜⎟
⎝⎠
∑∑ ∑ ∑
∑∑ ∑
(38)
where
[
]
[]
2/8
01 12 12 12
1212
(, ) (, ) (, 2)
( 2, ) ( 2, 2) ,
ynn xnn xnnN
xn N n xn N n N
=−+
++ −+ +
(39)
12
12
33
2/8 2/8
12
,12 011 2
00
2
11 2 2
1
(, ) ,
88
2
cos ( ) .
4
pp
ll
ii
i
lN l N
fnn yn n
np q pl pl
Nq
ππ
==
=
⎛⎞
=++
⎜⎟
⎝⎠
⎛⎞
×++
⎜⎟
⎝⎠
∑∑
∑
(40)
Equation (36) can be decomposed in a similar manner as
12 12
12
/8 1 /8 1 2
2/8 2/8
,12 ,12
00 1
2
( , ) ( , )cas
8
NN
pp pp ii
nn i
Gkk gnn nk
N
π
−−
== =
⎛⎞
=−
⎜⎟
⎝⎠
∑∑ ∑
(41)
where
12
12
33
2/8 2/8
12
,12 011 2
00
2
11 2 2
1
(, ) ,
88
2
sin ( ) .
4
pp
ll
ii
i
lN l N
gnn yn n
np q pl pl
Nq
ππ
==
=
⎛⎞
=++
⎜⎟
⎝⎠
⎛⎞
×++
⎜⎟
⎝⎠
∑∑
∑
(42)
We need to use the following lemma, which was stated in [3].
Lemma 1: Let
q
cπβ 22))4(cos( =
and
q
sπβ 22))4(sin( = , where
β
is an odd integer. Then the
following is true
i) For
qβ
=
,
2)1(
)1(
−
−=
q
qq
sc .
ii) For
qβ 3
=
,
qq
cc
−
=
3
and
qq
ss =
3
.
Letting
2
01
1
2
ii
i
np
Nq
π
γ
=
=
∑
(43)