Discrete fractional hadamard transform - National Taiwan University
DISCRETE FRACTIONAL HADAMARD TRANSFORM Soo-Chang Pei' and Min-Hung Yeh2
Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R. 0.C. 2Department of Computer Information Science, Tfirnsui Oxford University College, Tamsui, Taipei, Taiwan, R. 0. C.
Abstract
[3][4],the N-point DFRFT kernels are computed as follows: N-I
Hadamard transform is an important tool in discrete signal processing. In this paper, we define the discrete fractional Hadamard transform which is a generalized one. The development of discrete fractional Hadamard is based upon the same spirit of discrete fractional Fourier transform.
1. Introduction Fractional Fourier transform(FRFT) is a generalization of Fourier transform, and its output can have the mixed time and frequency components of signal[l]. Because of thc importance of FRFT, the discrete fractional Fourier transform (DFRFT) has become an important issue in rewnt ycnrs. In [2], a DFRFT has been proposed by B. Santhanam and J. H. McClellan. Unfortunately, this DFRFT can not have similar results as those of continuous case. In 1996, Pei and Yeh have found that the DFRFTs with DFT Hermite eigenvectors can provide similar results as continuous case[3][4]. On the other hand, many orthogonal transforms have been successfully used in signal processing. Some typical ones are discrete cosine transform (DCT)[5], discrete Hartley transform (DHT)[5] and Hadamard transform. Until now, the fractional version of DFT[4] and DHT[6] have also been investigated and successfully used in signal processing. But the discrete fractional Hadamard transform was still absent. The goal of this paper is give a definition for the discrete fractional Hadamard transform.
where V = [ v o l v l l . . . IVN-11. Vk is the the k-th order DFT Hermite eigenvector. cy indicates the rotation angle of transform in the time-frequency plane. The methods for finding the k-th order DFT Hermite eigenvectors have been shown in [3] and [4]. A Hadamard matrix is a symmetric matrix whose elements are the real numbers 1 and -1[5]. The rows(and columns) of a Hadamard matrix are mutually orthogonal. The normalized Hadamard matrices of order 2", denoted by H,, can be recursively b,v defining
(4)
In [SI,a method for finding the eigenvalues and eigenvectors of Hadamard transform has been proposed. Here we will state it briefly.
Proposition 1 If vn,k is an eigenvector of H, corresponding t o the eigenvalue A,
2. Preliminary (5)
For the factor of normalization, The DFT kernel is defined as the following way:
[:
:::
1
wC-2
1
w,"-'
1
will be an eigenvector of Hn+l, and at corresponds to the eigenvalue A. Proof : See [8]. Proposition 1 gives us a method for finding the eigenvectors from order 2" to order 2"+'. The initial eigenvector that is an eigenvector of H1 is show as follows:
[ fil-
11
where WN =
In the history of DFRFT, many
DFRFTs have been proposed.
The DFRFT in [2] can-
not have the similar results[7]. The DFRFT concerned in this paper is based on the eigen decomposirwn rnvthod proposed by Pei and Yeh(31 [4]. The methods in [3][l) use the DFT Hermite eigenvectors to construct DFRFI' kcr uel matrix. ,and have similar outputs as the coutinuous results. In
Using Proposition 1 and Eq.(6), only one eigenvector of order 2" can be computed. In [8], a sequence of matrices E, are defined to generate a set of orthogonal and complete eigenvectors.
0-7803-5471-0/99/$10.0001999IEEE
111- 179
Besides E k , a matrix Pnk is also defined by the direct sum of 2"-' copies of &'s: Pnk=EkfE'Ek@..*fE'Ek
Proof : To begin with, the orthogonality in will be proved.
Then the orthogonal and complete Hadamard eigenvectors can be obtained:
=Qi-lYi-1
25i 6 , - 1 + 4 t
[3-2\/2,
1
1 - 4 , I-&,
v2.3 v2.2
1
Table 2. An example for the Hadamard eigenvectors, N=4
e2.3
Eigenvalue 1 -1 -1 1 1 -1 -1
+2,3
1
+2,0 +2,0
G2.l G2.1 c2,2
+2,2
Order 2"
Eigenvector 1, -1+Jz, -I+&, ' 1 - 4 ,
-3+2&,
'1-4, -3+2&, '3-2fi,
-7+5fi,
'3-24,
I-&,
-
3-24, -3+2&,
7-54,
1, - I + & , 1 - 4 ,
'3-2&,
-3+2\/2,
- 1 + 4 ,
7-54,
3-24,
- 7 + 5 4
-1+&
3 - 2 4
-I+&,
1 - 6 , -3+2&,
1 - 4 , 1, - 7 + 5 4 ,
1, - 3 + 2 4 , 1 - a ,
1, -1+&
-3+2fi,
[7-5&,3-2&,3-24,
- 1 - 4 ,
-1+4,3-22JZ,
-3+24,
-3+24,
1 - 4 , 3-24,1-Jz, -3+24,
-?'+Sa, -3+2fi,
-1+fi,
3 - 2 4 1, - 1 + f i
-14-4
1 - 4 ,
13
v3.7
7-54? 3 - 2 4
1 - 4 , 1 , -3-f-24,
-14-fi
Table 3. An example for the Hadamard eigenvectors, N=8 01
a = d16
,
C i = d 071
, oq
i
Figure 1. The discrete fractional Hadamard transform of an impulse signal
111- 182
a-d4
0 5,
a - a
Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R. 0.C. 2Department of Computer Information Science, Tfirnsui Oxford University College, Tamsui, Taipei, Taiwan, R. 0. C.
Abstract
[3][4],the N-point DFRFT kernels are computed as follows: N-I
Hadamard transform is an important tool in discrete signal processing. In this paper, we define the discrete fractional Hadamard transform which is a generalized one. The development of discrete fractional Hadamard is based upon the same spirit of discrete fractional Fourier transform.
1. Introduction Fractional Fourier transform(FRFT) is a generalization of Fourier transform, and its output can have the mixed time and frequency components of signal[l]. Because of thc importance of FRFT, the discrete fractional Fourier transform (DFRFT) has become an important issue in rewnt ycnrs. In [2], a DFRFT has been proposed by B. Santhanam and J. H. McClellan. Unfortunately, this DFRFT can not have similar results as those of continuous case. In 1996, Pei and Yeh have found that the DFRFTs with DFT Hermite eigenvectors can provide similar results as continuous case[3][4]. On the other hand, many orthogonal transforms have been successfully used in signal processing. Some typical ones are discrete cosine transform (DCT)[5], discrete Hartley transform (DHT)[5] and Hadamard transform. Until now, the fractional version of DFT[4] and DHT[6] have also been investigated and successfully used in signal processing. But the discrete fractional Hadamard transform was still absent. The goal of this paper is give a definition for the discrete fractional Hadamard transform.
where V = [ v o l v l l . . . IVN-11. Vk is the the k-th order DFT Hermite eigenvector. cy indicates the rotation angle of transform in the time-frequency plane. The methods for finding the k-th order DFT Hermite eigenvectors have been shown in [3] and [4]. A Hadamard matrix is a symmetric matrix whose elements are the real numbers 1 and -1[5]. The rows(and columns) of a Hadamard matrix are mutually orthogonal. The normalized Hadamard matrices of order 2", denoted by H,, can be recursively b,v defining
(4)
In [SI,a method for finding the eigenvalues and eigenvectors of Hadamard transform has been proposed. Here we will state it briefly.
Proposition 1 If vn,k is an eigenvector of H, corresponding t o the eigenvalue A,
2. Preliminary (5)
For the factor of normalization, The DFT kernel is defined as the following way:
[:
:::
1
wC-2
1
w,"-'
1
will be an eigenvector of Hn+l, and at corresponds to the eigenvalue A. Proof : See [8]. Proposition 1 gives us a method for finding the eigenvectors from order 2" to order 2"+'. The initial eigenvector that is an eigenvector of H1 is show as follows:
[ fil-
11
where WN =
In the history of DFRFT, many
DFRFTs have been proposed.
The DFRFT in [2] can-
not have the similar results[7]. The DFRFT concerned in this paper is based on the eigen decomposirwn rnvthod proposed by Pei and Yeh(31 [4]. The methods in [3][l) use the DFT Hermite eigenvectors to construct DFRFI' kcr uel matrix. ,and have similar outputs as the coutinuous results. In
Using Proposition 1 and Eq.(6), only one eigenvector of order 2" can be computed. In [8], a sequence of matrices E, are defined to generate a set of orthogonal and complete eigenvectors.
0-7803-5471-0/99/$10.0001999IEEE
111- 179
Besides E k , a matrix Pnk is also defined by the direct sum of 2"-' copies of &'s: Pnk=EkfE'Ek@..*fE'Ek
Proof : To begin with, the orthogonality in will be proved.
Then the orthogonal and complete Hadamard eigenvectors can be obtained:
=Qi-lYi-1
25i 6 , - 1 + 4 t
[3-2\/2,
1
1 - 4 , I-&,
v2.3 v2.2
1
Table 2. An example for the Hadamard eigenvectors, N=4
e2.3
Eigenvalue 1 -1 -1 1 1 -1 -1
+2,3
1
+2,0 +2,0
G2.l G2.1 c2,2
+2,2
Order 2"
Eigenvector 1, -1+Jz, -I+&, ' 1 - 4 ,
-3+2&,
'1-4, -3+2&, '3-2fi,
-7+5fi,
'3-24,
I-&,
-
3-24, -3+2&,
7-54,
1, - I + & , 1 - 4 ,
'3-2&,
-3+2\/2,
- 1 + 4 ,
7-54,
3-24,
- 7 + 5 4
-1+&
3 - 2 4
-I+&,
1 - 6 , -3+2&,
1 - 4 , 1, - 7 + 5 4 ,
1, - 3 + 2 4 , 1 - a ,
1, -1+&
-3+2fi,
[7-5&,3-2&,3-24,
- 1 - 4 ,
-1+4,3-22JZ,
-3+24,
-3+24,
1 - 4 , 3-24,1-Jz, -3+24,
-?'+Sa, -3+2fi,
-1+fi,
3 - 2 4 1, - 1 + f i
-14-4
1 - 4 ,
13
v3.7
7-54? 3 - 2 4
1 - 4 , 1 , -3-f-24,
-14-fi
Table 3. An example for the Hadamard eigenvectors, N=8 01
a = d16
,
C i = d 071
, oq
i
Figure 1. The discrete fractional Hadamard transform of an impulse signal
111- 182
a-d4
0 5,
a - a
