on the fast fractional jacket transform - 上海交通...

$: On the Fast Fractional Jacket Transform - 上海交通 …math.sjtu.edu.cn/faculty/xiaodong/paper/2012/IEEETCS59...proved algorithm for the fast fractional jacket transform (FRJT)$
Circuits Syst Signal Process (2014) 33:1491–1505DOI 10.1007/s00034-013-9699-8

On the Fast Fractional Jacket Transform

Yun Mao · Jun Peng · Ying Guo · Moon Ho Lee

Received: 31 March 2013 / Revised: 19 October 2013 / Published online: 28 November 2013© Springer Science+Business Media New York 2013

Abstract Motivated by the center weighted Hadamard matrix, we propose an im-proved algorithm for the fast fractional jacket transform (FRJT) based on eigende-composition of the fractional jacket matrix (FRJM). Employing a matrix diagonal-ization transformation that decomposes a matrix of large size into products of thematrices composed of eigenvectors and eigenvalues, an FRJM of large size can befast factored into products of several sparse matrices in a recursive fashion. To gener-ate an FRJM of large size, an algorithm for the factorable FRJM can be convenientlydesignated with a reduced computational complexity in terms of additions and mul-tiplications. Since the proposed FRJM itself concerns interpretation as a suitable ro-tation in the time-frequency domain, it is applicable for optics and signal processing.

Keywords Fractional jacket transform · Jacket matrix · Matrix decomposition ·Fractional Hadamard transform · Hadamard transform · Signal processing

1 Introduction

The Hadamard matrix and its generalizations are orthogonal matrices with many ap-plications in signal transforms and data processing [1]. The jacket matrix, motivated

Y. Mao · J. Peng · Y. Guo (B)School of Information Science & Engineering, Central South University, Changsha 410083, Chinae-mail: [email protected]

Y. Maoe-mail: [email protected]

J. Penge-mail: [email protected]

M.H. LeeInstitution of Information & Communication Engineering, Chonbuk National University,Chonju 561-756, Koreae-mail: [email protected]

mailto:[email protected]




1492 Circuits Syst Signal Process (2014) 33:1491–1505

by the center weighted Hadamard matrix with an inverse constraint, is a special ma-trix with its inverse matrix being determined by the element-wise inverse of the ma-trix [5]. In particular, several interesting matrices, such as the Hadamard matrix, theFourier matrix, and the slanted matrix, belong to the jacket matrix family [6]. Sincethe inverse jacket matrix can be determined easily, the jacket matrix and its transfor-mations have been extensively investigated [6–8, 14]. In addition, the jacket matrix isrelated to many useful matrices, such as the unitary matrix and the Hermitian matrix,that can be potentially applied in signal processing, data compression, cryptography,orthogonal code design, and so on [4].

Recently, eigen-decomposition has been used for the discrete fractional Fouriertransform (DFRFT) according to the relationship of eigenvalues and eigenvectors oftransform matrices; the DFRFT is a generalization of the discrete Fourier transform(DFT) [2]. It is a case of the linear canonical transform that provides a fitting approachto interpret a rotation of the mixed time-frequency component of the discrete signals.The DFRFT has become an important issue, since it is useful for signal processingand orthogonal code designing. For example, the eigen-decomposition of the DFRFThas been consolidated in the fields of practical optics and signal processing [3, 12].Based on the DFRFT, several related fractional signal transforms have been inves-tigated, such as the discrete fractional Hartley transform [9], the discrete fractionalHadamard transform [10], and the discrete fractional cosine and sine transforms [13].

The aforementioned fractional transforms have attracted a considerable amountof attention, resulting in many practical application in optics and signal processing.Unfortunately, a fractional version of the jacket transform has been absent until now;that is, a satisfactory definition of the fractional jacket transform (FRJT) that is con-sistent with the jacket transform has been lacking. The purpose of this paper is toconsolidate a definition of the FRJT with an improved algorithm based on the fasteigen-decomposition of the jacket matrix. This definition has several essential prop-erties, such as unitarity, index additivity, periodical rotations, consistency with thejacket matrix, and so on. It exhibits internal consistency with an analytical elegancethat we take for granted with the ordinary transforms while decomposing the originaljacket matrix in a fractional fashion with respect to a particular set of eigenvectors,which is the major motivation for defining the FRJT in the first place. The proposedFRJT would be very desirable in order to provide as many operational properties ofthe potential transform as possible in the advent of modern signal processing.

This paper is organized as follows. In Sect. 2, the structure of the fractional jacketmatrix (FRJM) is described with the known jacket matrix in a specific structure.The eigen-decomposition of the FRJM based on the relationship of eigenvectors andeigenvalues is also investigated with a recursive structure. In Sect. 3, we develop afast algorithm for decomposition of the FRJT based on an eigen-decomposition ofthe Hadamard matrix with a given rotation parameter. Then a generalized FRJT oflarge size is investigated with different rotation parameters by using the derived re-cursive relationship based on the Kronecker product of the small-size matrices. Thisalgorithm can be implemented with a smaller computational complexity compared tothe direct computation approach. Our conclusions are presented in Sect. 4.

Circuits Syst Signal Process (2014) 33:1491–1505 1493

2 Structure of the FRJM

In many problems of signal processing related to matrix transformation, one shouldknow exactly the structure of the resulting matrix. In this section, we introduce thedefinition of the FRJM with several potential properties in signal processing.

Definition 1 Let matrix A = (ai,j )m×n be of size m × n and matrix B = (bs,t )k×l ofsize k × l, respectively. The Kronecker product A and B , which is denoted by A ⊗ B

[11], is a matrix of size mk × nl, i.e.,

A ⊗ B =

⎛⎜⎜⎜⎝

a1,1B a1,2B · · · a1,nB

a2,1B a2,2B · · · a2,nB...

.... . .

...

am,1B am,2B · · · am,nB

⎞⎟⎟⎟⎠ . (1)

Recently, the center weighted Hadamard matrix has been discovered as a typicalcase of the jacket matrix whose inverse matrix is only achieved from the element-wiseinverse of the initial matrix [6]. In what follows we extend this idea and introduce thedefinition of the jacket matrix.

Definition 2 Let a square matrix of size N × N be denoted by JN = (js,t )N×N .The matrix JN is a jacket matrix if its inverse matrix can be simply obtained by itselement-wise inverse, i.e., for 1 ≤ s, t ≤ N , we obtain

J −1N = 1

c(1/js,t )

TN×N, (2)

where c is a normalized constant such that J −1N JN = JNJ −1

N = IN and the super-script T denotes the matrix transposition operation. In detail, we have

JN =

⎛⎜⎜⎜⎝

j1,1 j1,2 · · · j1,N

j2,1 j2,2 · · · j2,N

......

...

jN,1 jN,2 · · · jN,N

⎞⎟⎟⎟⎠ , (3)

and its inverse matrix given by

J −1N = 1

N

⎛⎜⎜⎜⎜⎝

j−11,1 j−1

2,1 · · · j−1N,1

j−11,2 j−1

2,2 · · · j−1N,2

......

...

j−11,N j−1

2,N · · · j−1N,N

⎞⎟⎟⎟⎟⎠

. (4)

For simplicity of the description, we ignore the normalized constant c without causingconfusion when deriving the inverse jacket matrix throughout this paper.

It is necessary to note that the inverse matrix of the jacket matrix JN can besimply calculated in an algebraic way due to this interesting structure according tothe relation expressed in Eqs. (3) and (4). Namely, the afore-mentioned jacket matrix


JN has the specific property that its inverse matrix J −1N can be conveniently achieved

from its element-wise inverse, as shown in Eq. (4). It has the elegant general purposeof the conventional jacket, where the obverse and reverse are more likely to be similarin appearance. Moreover, it can be employed handily for many practical applicationsdue to the two-sided function. That is a motivation for which we derived the so-calledjacket matrix several decades ago [6, 7].

Currently, the fractional transform has been applied in a wide range of subjects. Inthe mathematics literature, the FRJM is a generalized jacket matrix with an inverseconstraint. For clarity of the FRJT, we should mathematically define the FRJM basedon an eigen-decomposition of the jacket matrix.

Definition 3 A nonsingular jacket matrix is an FRJM if it can be decomposed intoproducts of several matrices composed of the eigenvectors and eigenvalues for a givenrotation parameter α, i.e.,

J (α)N = ZNΛ

(α)N ZT

N, (5)

where Λ(α)N is the diagonal matrix given by

Λ(α)N = diag

{λ

(α)0 , λ

(α)1 , . . . , λ

(α)N−1

}

=

⎛⎜⎜⎜⎜⎝

λ(α)0 0 · · · 00 λ

(α)1 · · · 0

......

. . ....

0 0 · · · λ(α)N−1

⎞⎟⎟⎟⎟⎠

, (6)

whose nonzero elements are eigenvalues {λ(α)k = e−jkα : 0 ≤ k ≤ N − 1}, and ZN is

an eigenvector-based matrix given by (z0, z1, . . . , zN−1) whose columns zl are nor-malized vectors, l ∈ {0,1, . . . ,N − 1}. It is obvious that the eigenvector-based matrixZN is orthogonal. In addition, the FRJM J (α)

N is continuous for the given rotationangle α. For α = 0 it becomes an identity matrix, and for α = π it is transformedinto a jacket matrix [6]. Although potentially useful for practical optics and signalprocessing, the FRJM appears to have remained largely unknown to the data pro-cessing community. We are actually concerned with the interpretation as a rotation inthe time-frequency domain. From the FRJM with any rotation angles, as described inEq. (5), one obtains some properties as follows:

1. Zero rotation: J (0)N = J (2π)

N = I ;

2. Periodical rotation: J (α+2π)N = J (α)

N ;

3. Consistency with the jacket matrix: J (π)N = JN ;

4. Symmetric matrix: (J (α)N )T = J (α)

N ;

5. Additivity of rotations: J (α)N J (β)

N = J (α+β)N ;

6. Unitarity: (J (α)N )∗ = J (−α)

N = (J (α)N )−1.

Here the superscript ∗ denotes the complex conjugation operation.


In the eigenvalue-based matrix Λ(α)N in (6), the nonzero elements λ

(α)k are eigen-

values of the FRJM, i.e., λ(α)k = e−jkα . Compared to the eigen-decomposition of the

standard jacket matrix [8], i.e., JN = ZNΛNZTN , where ΛN = diag{λ1, . . . , λN }, the

diagonal elements of the eigenvalue-based matrix Λ(α)N of the FRJM are substituted

with the complex values depending greatly on the nonzero rotation angle α. In addi-tion, since the inverse matrix of the FRJM is easily obtained from its element-wiseinverse, the FRJM may have potential applications in, e.g., signal processing andmodern communications.

3 Construction of the Eigendecomposition-Based FRJM

In order to implement a task of FRJT-based signal processing, one should know ex-actly the structure of the FRJM for a given rotation parameter α. In this section wepropose an FRJM of large size with an elegant recursive relation based on the eigen-decomposition of the jacket matrix [11].

Theorem 1 For a given sequence of matrices {Zpk : k ∈ {1,2, . . . , l}} with a recur-sive relation Zpk = Zp ⊗Zpk−1 , the induced matrix Zpl can be factorized as follows:

Zpl =l∏

k=1

(Ipl−k ⊗ Zp ⊗ Ipk−1), (7)

where It denotes an identity matrix of size t × t , and Zp denotes an arbitrary squarematrix of size p × p given by

Zp =

⎛⎜⎜⎜⎝

z11 z12 · · · z1p

z21 z22 · · · z2p

......

. . ....

zp1 zp2 · · · zpp

⎞⎟⎟⎟⎠ . (8)

Proof Let Z(k)

pl be a sequence of matrices defined by

Z(k)

pl = Ipl−k ⊗ Zpk . (9)

Then we have Z(l)

pl = Zpl and

Z(k)

pl = R(k)

pl Z(k−1)

pl , (10)

where R(k)

pl is an order-l quasi-diagonal matrix given by

R(k)

pl = Ipl−k ⊗ Zp ⊗ Ipk−1 . (11)

Subsequently, we show the correctness of the decomposition Z(k)

pl = ∏kt=1 R

(t)

pl by aninduction method. First of all, for k = 1, we obtain

Z(1)

pl = R(1)

pl = Ipl−1 ⊗ Zp. (12)


Secondly, with the induction hypothesis we achieve

Z(k−1)

pl =k−1∏t=1

R(t)

pl . (13)

Then, from Eq. (10) we have

R(k)

pl

(k−1∏t=1

R(t)

pl

)= R

(k)

pl Z(k−1)

pl = Z(k)

pl . (14)

Taking k = l in Eq. (14) completes the proof of this theorem. �

Theorem 1 suggests an algorithm for the fast decomposition or construction of amatrix of large size in a recursive fashion. In what follows, it can be employed for theconstruction of an FRJM of a large size from one of a small size.

3.1 The Achieved FRJM from the Hadamard Matrix

The standard Hadamard matrix H2l for any non-negative integer l, which has beenproved to be a special jacket matrix, can be generated recursively, i.e.,

H2l = H2 ⊗H2l−1 , (15)

where H2 is the conventional Hadamard matrix given by

H2 =(

1 11 −1

). (16)

Direct calculation of eigenvalues and eigenvectors of the matrix H2l yields the fol-lowing eigen-decomposition:

H2l = Z2lΛ2l ZT2l

= V2l Λ2l VT2l (17)

where Λ2l and Λ2l are the equivalent eigenvalue-based matrices given by

Λ2l = diag{λk = (−1)k : k ∈ {

0,1, . . . ,2l − 1}}

, (18)

Λ2l = η−l diag{λk = (−1)k : k ∈ {

0,1, . . . ,2l − 1}}

, (19)

while Z2l and V2l are the eigenvector-based matrices given by

Z2l = (z0, z1, . . . , z2l−1), (20)

V2l = (v0,v1, . . . ,v2l−1), (21)

with zk = η−l/2vk and η = 1 + (√

2 − 1)2. It is easy to prove that matrix V2l can berepresented as

V2l = V2l P2l , (22)


where V2l is generated recursively from

V2l = V2 ⊗ V2l−1 , (23)

and P2n is a permutation matrix generated recursively from the constraints

P2l (2i, j) = P2l−1(i, j),

P2l

(2i,2l−1 + j

) = P2l (2i + 1, j) = 0, (24)

P2l

(2i + 1,2l−1 + j

) = P2l−1

(i,2l−1 − j − 1

),

∀i, j ∈ {0,1, . . . ,2l−1 − 1}, with the initial matrices V2 and P2 given by

V2 =(

1 −a

a 1

), P2 =

(1 00 1

). (25)

It is obvious that V2 = V2. According to Theorem 1, matrix V2l and its transposematrix V T

2l can be factorized, i.e.,

V2l =l∏

k=1

(I2l−k ⊗ V2 ⊗ I2k−1), (26)

V T2l =

l∏k=1

(I2l−k ⊗ V T

2 ⊗ I2k−1

). (27)

Taking into account the factorization of V2l and V T2l , an algorithm of the FRJT for

an input signal vector x of length 2l can be achieved as follows:

y(α) = V2l P2lΛ(α)

2l P T2l V

T2l x

=l∏

k=1

(I2l−k ⊗ V2 ⊗ I2k−1)P2lΛ(α)

2l P2l ·n∏

k=1

(I2l−k ⊗ V T

2 ⊗ I2k−1

)x, (28)

where y(α) denotes the output signal vector of length 2l depending on a given rotationparameter α in an eigenvalue-based diagonal matrix Λ

(α)

2l .

Example 1 We consider an eigen-decomposition of the Hadamard matrix H4. Ac-cording to the recursive relation expressed in Eq. (17), one obtains the FRJM H(β)

4 ofsize 4 × 4 for a given rotation parameter β , i.e.,

H(β)

4 = V4Λ(β)

4 V T4 , (29)

where Λ(β)

4 is an eigenvalue-based diagonal matrix, i.e.,

Λ(β)

4 = η−2 diag{1, e−jβ, e−2jβ, e−3jβ

}(30)

and V4 is an eigenvector-based orthogonal matrix given by

V4 =

⎛⎜⎜⎝

1 −a a2 −a

a −a2 −a 1a 1 −a −a2

a2 a 1 a

⎞⎟⎟⎠ . (31)


Actually, matrix V4 can be recursively generated from V4 = V4P4 according toEqs. (22), (24), where V4 = V2 ⊗ V2 and P4 is a permutation matrix:

P4 =

⎛⎜⎜⎝

1 0 0 00 0 0 10 1 0 00 0 1 0

⎞⎟⎟⎠ . (32)

We note that the original matrix H(β)

4 can also be calculated with a direct approachfrom Eq. (45), i.e.,

H(β)

4 =

⎛⎜⎜⎝

h11(a,β) h12(a,β) h13(a,β) h14(a,β)

h21(a,β) h22(a,β) h23(a,β) h24(a,β)

h31(a,β) h32(a,β) h33(a,β) h34(a,β)

h41(a,β) h42(a,β) h43(a,β) h44(a,β)

⎞⎟⎟⎠ , (33)

where the elements hij (a,β), ∀i, j ∈ {1,2,3,4}, are represented as

h11(a,β) = η−2(1 + a2e−jβ + a4e−2jβ + a2e−j3β),

h22(a,β) = η−2(a2 + a4e−jβ + a2e−2jβ + e−j3β),

h33(a,β) = η−2(a2 + e−jβ + a2e−2jβ + a4e−j3β),

h44(a,β) = η−2(a4 + a2e−jβ + e−2jβ + a2e−j3β),

h21(a,β) = h12(a,β) = η−2(a + a3e−jβ − a3e−2jβ − ae−j3β),

h31(a,β) = h13(a,β) = η−2(a − ae−jβ − a3e−2jβ + a3e−j3β),

h43(a,β) = h34(a,β) = η−2(a3 + ae−jβ − ae−2jβ − a3e−j3β),

h41(a,β) = h14(a,β) = η−2(a2 − a2e−jβ + a2e−2jβ − a2e−j3β),

h42(a,β) = h24(a,β) = η−2(a3 − a3e−jβ − ae−2jβ + ae−j3β),

h32(a,β) = h23(a,β) = η−2(a2 − a2e−jβ + a2e−2jβ − a2e−j3β).

Subsequently, for an input signal vector x of length 4, the algorithm for computingthe FRJT will be described as

y(β) =2∏

k=1

(I22−k ⊗ V2 ⊗ I2k−1)P4Λ(β)

4 P T4 ·

2∏k=1

(I22−k ⊗ V T

2 ⊗ I2k−1

)x. (34)

A data flow diagram of this algorithm for a given rotation parameter β is shown inFig. 1, where λ

(β)k = e−jkβ , ∀k ∈ {0,1,2,3}. It shows that the number of additions

and multiplications required for the direct computing approach can be calculated as2l(2l − 1) and 22l for l = 2, respectively. However, the total number of additions andmultiplications for the proposed FRJT should be at most l2l+1 and (l +1)2l for l = 2,respectively. It is obvious that even for a small integer l the number of arithmeticoperations required for the realization of the FRJT is several times less than that ofthe direct computation approach, as shown in Table 1. Specifically, taking β = π , thetotal number of additions and multiplications for the proposed FRJT are at most l2l+1

and l2l , respectively. It is obvious that the proposed algorithm with the fixed rotation


Fig. 1 The data flow diagram of the fast FRJT with the eigenvalue-based diagonal matrix Λ(β)4 for a given

rotation parameter β

Table 1 Computational

complexity of the FRJM J (α)

2l

with the given rotationparameter α

Rotated angle Direct approach Fast algorithm

Addition α ∈ (0,π) 2l (2l − 1) l2l+1

Multiplication α ∈ (0,π) 22l (l + 1)2l

Addition α = π 2l (2l − 1) l2l+1

Multiplication α = π 22l l2l

parameter β = π is better than the algorithm with an arbitrary rotation parameterβ ∈ (0,π).

3.2 The Generalized FRJM from the Jacket Matrix

So far we have investigated the structure of the FRJM based on an eigen-decomposi-tion of the Hadamard matrix. Now we move onto the extension of the FRJM basedon the derived eigen-decomposition of the jacket matrix.

A jacket matrix J4 is initially derived from a center weighted Hadamard matrix,i.e.,

J4 =

⎛⎜⎜⎝

a b b a

b −c c −b

b c −c −b

a −b −b a

⎞⎟⎟⎠ , (35)

where a, b, and c denote the weighted factors. We note that the matrix J4 is neatlyweighted by the core matrix

(−c c

c −c

),

which is located at the center of the matrix J4. Specifically taking a = 1, b = 1, andc = 1, the center weighted Hadamard matrix is can be achieved as follows [5]:

H4 =

⎛⎜⎜⎝

1 1 1 11 −1 1 −11 1 −1 −11 −1 −1 1

⎞⎟⎟⎠ . (36)

It is obvious the inverse matrices H−14 and J −1

4 of the above-mentioned matrices H4and J4 can be conveniently calculated from the simple element-wise inverse of the


respective matrices, i.e.,

H−14 = 1

4

⎛⎜⎜⎝

1−1 1−1 1−1 1−1

1−1 −1−1 1−1 −1−1

1−1 1−1 −1−1 −1−1

1−1 −1−1 −1−1 1−1

⎞⎟⎟⎠ , (37)

and

J −14 = 1

4

⎛⎜⎜⎝

a−1 b−1 b−1 a−1

b−1 −c−1 c−1 −b−1

b−1 c−1 −c−1 −b−1

a−1 −b−1 −b−1 a−1

⎞⎟⎟⎠ . (38)

It is necessary to note that the matrix J4 (or H4) and its inverse matrix J −14 (or H−1

4 )have some similarity in configuration. Consequently, the generalized matrix J4 iscalled a jacket matrix because it highlights the relationship between the matrix andthe conventional jacket in appearance [6].

Based on the jacket matrix J4, the eigenvalue-based matrix can be calculated as

Λ4 = diag{−2b,2b,−2c,2a} (39)

with the eigenvector-based matrix given by

Z4 =

⎛⎜⎜⎝

−1 1 0 11 1 −1 01 1 1 01 −1 0 1

⎞⎟⎟⎠ . (40)

Therefore the eigen-decomposition of the jacket matrix J4 can be represented by

J4 = Z4Λ4ZT4 . (41)

From Definition 3, the FRJM J (β)

4 of size 4 × 4 for a given rotation parameter β isdescribed as

J (β)

4 = Z4Λ(β)

4 ZT4 , (42)

where Λ(β)

4 is an eigenvalue-based matrix with the rotation parameter β , i.e.,

Λ(β)

4 = diag{2b,2be−jβ,2ce−2jβ,2ae−3jβ

}. (43)

In view of the properties of the jacket matrix, the jacket transform and its genera-tions have been applied in a wide range [6, 7, 14]. Now we present an algorithm forthe fast eigen-decomposition of the FRJM in a way similar to that of the jacket matrixin order to implement an FRJT of any size in practical applications.

A generalized FRJM of any size can be defined by analogy to the FRJM basedon the eigen-decomposition of the Hadamard matrix. According to the eigen-decomposition of the jacket matrix, the FRJM can be extended to be a fractionalmatrix of size 4n × 4n with two rotation parameters in a recursive relation described


as

J (α,β)

4l+1 = J (β)

4 ⊗H(α)

4l = (I4 ⊗H(α)

4l

)(J (β)

4 ⊗ I4l

), (44)

where H(α)

4l is the FRJM achieved from the eigen-decomposition of the Hadamard

matrix of size 4l × 4l for a given rotation parameter α, which can be decomposed ina recursive relation according to Theorem 1, i.e.,

H(α)

4l =l∏

k=1

(I4l−k ⊗ V4 ⊗ I4k−1)(P4l Λ

(α)

4l P T4l

) l∏k=1

(I4l−k ⊗ V T

4 ⊗ I4k−1

). (45)

Here Λ(α)

4l denotes an eigenvalue-based diagonal matrix.We note that an arbitrary jacket matrix can be expressed as the Kronecker product

of sparse matrices. Combining with Eqs. (42), (44), we have the eigen-decompositionof the FRJM J (α,β)

4l+1 of large size 4l+1 × 4l+1 for two rotation parameters α and β ,i.e.,

J (α,β)

4l+1 = (Z4Λ

(β)

4 ZT4

) ⊗[

l∏k=1

(I4l−k ⊗ V4 ⊗ I4k−1)

· (P4l Λ(α)

4l P T4l

) l∏k=1

(I4l−k ⊗ V T

4 ⊗ I4k−1

)], (46)

which can be rewritten as

J (α,β)

4l+1 =[(Z4 ⊗ I4 ⊗ I4l−1)

l−1∏k=1

(I4l−k ⊗ V4 ⊗ I4k )

]

· [(I4 ⊗ P4l )(Λ

(β)

4 ⊗ I4l

)(I4 ⊗ Λ

(α)

4l

)(I4 ⊗ P T

4l

)]

·[(

ZT4 ⊗ IT

4 ⊗ I4l−1

) l−1∏k=1

(I4l−k ⊗ V T

4 ⊗ I4k

)]. (47)

The FRJM J (α,β)

4l+1 of size 4l+1 × 4l+1 with two rotation parameters α and β mayplay a role in multidimensional signal processing [4], such as image compression,orthogonal coding, cryptography, and so forth.

Example 2 Based on the fractional Hadamard matrix H(α)4 of size 4 × 4 with a ro-

tation parameter α expressed in Eq. (45) and the FRJM J (β)

4 of size 4 × 4 with

another rotation parameter β in Eq. (42), we obtain the combined FRJM J (α,β)

16 ofsize 16 × 16 using the recursive relation given by

J (α,β)

16 = J (β)

4 ⊗H(α)4

= (Z4 ⊗ I4)(V4 ⊗ I4)(I4 ⊗ P4)(Λ

(β)

4 ⊗ I4)

· (I4 ⊗ Λ(α)4

)(I4 ⊗ P T

4

)(ZT

4 ⊗ I4)(

V T4 ⊗ I4

). (48)


Fig. 2 The data flow diagram of the fast FRJT for the FRJM J (α,β)16 based on H(α)

4 with eigenvalues

{λ(α)k

: k = 0,1,2,3} and H(β)4 with eigenvalues {λ(β)

k: k = 0,1,2,3}

Here Λ(β)

4 is an eigenvalue-based diagonal matrix with eigenvalues {λ(β)k : k =

0,1,2,3}, and Λ(α)4 is another eigenvalue-based diagonal matrix with eigenvalues

{λ(α)k : k = 0,1,2,3}. Taking a signal vector x of length 16 into account, we obtain

the data flow diagram described as

y(α,β) = J (α,β)

16 x, (49)

which is shown in Fig. 2. We illustrate the computational complexity of this algorithmin Table 2. The proposed algorithm on the basis of the FRJM requires 192 additionsand 160 multiplications. However, the direct computational approach needs 240 addi-tions and 256 multiplications. This shows that the proposed algorithm in terms of thecalculational complexity is obviously faster than the direct computational approach.


Table 2 Computationalcomplexity of the combined

FRJM J (α,β)16 for the proposed

algorithm and the directcalculation approach

Direct approach Fast algorithm

Additions 240 192

Multiplications 256 160

Generally, making full use of the properties of the Kronecker product [11], anFRJM of any size may undergo fast decomposition (or construction) on a basis of thefactorization.

Theorem 2 Suppose J (α)p and J (β)

q are both FRJMs for two rotation parameters α

and β with sizes p and q , respectively. Let m and n be two positive integers. Then theFRJM J (α,β)

N=pmqn of size N × N can be decomposed (or constructed) as follows:

J (α,β)N = J (α)

pm ⊗J (β)qn

=m∏

k=1

(Ipm−k ⊗ Vp ⊗ Ipk−1 ⊗ Iqn) · [(PpmΛ(α)pmP T

pm

) ⊗ Ipm

]

·m∏

k=1

(Ipm−k ⊗ V T

p ⊗ Ipk−1 ⊗ Iqn

) ·n∏

k=1

(Ipm ⊗ Iqn−k ⊗ Vq ⊗ Iqk−1)

· [Ipm ⊗ (PqnΛ

(β)qn P T

qn

)] ·n∏

k=1

(Ipm ⊗ Iqn−k ⊗ V T

q ⊗ Iqk−1

), (50)

where Vp and Vq denote the eigenvector-based matrices of Jp and Jq and Λ(α)pm

and Λ(β)qn denote the eigenvalue-based matrices with rotation parameters α and β ,

respectively.

Proof Using the eigen-decomposition of J (α)pm and J (β)

qn for their respective rotationparameters α and β , we obtain

J (α)pm =

m∏k=1

(Ipm−k ⊗ Vp ⊗ Ipk−1) · (PpmΛ(α)pmP T

pm

)

·m∏

k=1

(Ipm−k ⊗ V T

p ⊗ Ipk−1

), (51)

and

J (β)qn =

n∏k=1

(Iqn−k ⊗ Vq ⊗ Iqk−1) · (PqnΛ(β)qn P T

qn

)

·n∏

k=1

(Iqn−k ⊗ V T

q ⊗ Iqk−1

). (52)


Table 3 Computational complexity of algorithms for the FRJMs. The abbreviations DC and FA denotethe direction computational and fast algorithm approach while Add. and Mul. denote additions and multi-plications

DC FA (N = pm) FA (N = pmqn,n �= 0)

Add. (N − 1)N 2m(p − 1)N 2(mp + nq − m − n)N

Mul. N2 (2p−1m(p − 1)2 + 1)N 2Nmp−1(p − 1)2

+ Nnq−1(q − 1)2 + N

Subsequently, the combined FRJM J (α,β)N can be described as

J (α,β)pmqn = J (α)

pm ⊗J (β)qn

= (J (α)

pm · Ipm

) ⊗ (Iqm ·J (β)

qn

)

= (J (α)

pm ⊗ Iqn

)(Ipm ⊗J (β)

qn

). (53)

Making use of the properties of the Kronecker product of matrices [11], we completethe proof of this theorem. �

Theorem 2 suggests that a factorable FRJM of any size can be fast decomposed (orconstructed) using the proposed algorithm expressed in Eq. (50). This demonstratesa close relationship between the large-size FRJM and the small one.

Next, we have to evaluate the computational complexity of the proposed FRJTstep by step with the assistance of the eigen-decomposition procedure expressed inEq. (50). Without loss of generality, we consider the computational complexity ofthe FRJT based on the FRJM J (α)

N=pm using the factorized representation of matricesin Eq. (51). The input vector is x of length N . According to the algorithm for fastdecomposition of the jacket matrix [14], the calculation of the product of the initialjacket matrix JN and a vector x of length N = pm requires at most m(p − 1)N

additions and p−1(p − 1)2N multiplications, respectively. Subsequently, we obtainanother vector of length N . As mentioned in the present algorithm, the multiplicationof the permutation matrix P T

N (or PN ) by a vector does not require any arithmetic

operations. After that, the calculation of the product of diagonal Λ(α)N and the result-

ing vector requires at most N multiplications. The yielded vector denoted by e is thencomplex-valued of length N . In a similar way, the calculation of ZN e requires at mostm(p − 1)N additions and p−1(p − 1)2N multiplications, respectively. It is obviousthat the total numbers of arithmetic operations are at most 2m(p − 1)N additions and2p−1(p − 1)2N + 2N multiplications. Generally, the computational complexity ofthe algorithm for the FRJM J (α,β)

N=pmqn is detailed in Table 3. Compared to the directcomputation approach, the proposed algorithm is obviously faster.

4 Conclusion

We have exploited an algorithm for the fast decomposition or construction of anFRJM with arbitrary rotation parameters based on eigen-decomposition in a recur-sive fashion using the Kronecker product of the successive FRJMs. This algorithm


is compact with a simple mathematical formalism suitable for a parallel realization.It may enable us to adequately represent an explicit structure of the computationalprocess for hardware realizations. Compared with direct computation, the proposedalgorithm obviously decreases the computational complexity, and may be applicablefor use in image compression, signal processing, and information theory.

Acknowledgements This work was supported by the National Natural Science Foundation of China(61071096, 61379153), the bilateral cooperation of the science foundations between China and Korea(NSFC-NRF 61140391), and MEST 2012-0025-21, National Research Foundation, Korea.

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