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Journal of Engineering Technology Volume 6, Special Issue on Technology Innovations and Applications

Oct. 2017, PP. 392-404

392

A note on the singular value decomposition of inverse-orthogonal circulant

jacket matrices

K. M. Cho1

, Md. Abdul Latif Sarker1

, N. Yankov2

, Moon Ho Lee1*.

1Department of Electronics and Information Engineering, Conbuk National University, Jeonju-si, 54896 Korea.

2The Faculty of Mathematics and Informatics, Shumen University, Shumen 9712, Bulgaria.

*Corresponding author: Moon Ho Lee

Abstract. This study investigates the singular value decomposition (SVD) of inverse orthogonal circulant

Jacket (CJ) matrices, which are able to be constructed by inverse orthogonal Toeplitz matrices. These CJ

matrices are applied to a variety of fields of engineering and mathematical disciplines such as signal

processing, wireless communications, and so forth. The SVD of inverse orthogonal CJ matrices is enabled to

compute small, medium, and large size matrices with higher accuracy.

Keywords: Toeplitz and Jacket matrices, Circulant Jacket (CJ) matrices, Eigen value decomposition

(EVD), Singular value decomposition (SVD).

1. Introduction

The matrices such as Toeplitz [1], circulant [1, 2, 3], Hadamard [4], and Jacket matrices [5, 6, 7], have an

important role in numerical analysis and signal processing. The practical values of Hadamard transformation

such as center weighted Hadamard transformation represent their corresponding signals and images. The class

of Jacket matrices also contains the class of real and complex Hadamard matrices. Jacket matrices generated

by inverse orthogonal Toeplitz matrices are in fact circulant Jacket (CJ) matrices. We present a very simple

decomposition of CJ matrices which can provide efficient eigenvalue and singular-value decomposition

strategies in this paper. These eigenvalue and singular-value decomposition techniques are extensively used in

signal processing and wireless communications which are broadly used in practice thanks to their efficiently

computable inverse matrices. The Jacket matrices are a generalization of complex Hadamard matrices. A

square n × n matrix ( )ijM m is called a Jacket matrix [5] if its inverse satisfies-1

,

,

]1

[ i j

j i

Mnm

, i.e., the inverse

matrix can be obtained by taking element-wise inverse and transpose, up to a negligible constant factor.

Equivalently, these matrices satisfy the following relations:

,

,

1 ,

, , 1, , ,n

i k

i j

k j k

mn i j n

m

(1)

where ,i j is the Kronecker delta - a function of two variables, usually integers,

0, if .

1, if ij

i j

i j

(2)


Oct. 2017, PP. 389-402

393

If C is nonzero constant, then the definition of Jacket matrix can be rewritten as follows. A square m × m

matrix

0,0 0,1 0, 1

1,0 1,1 1, 1

1,0 11 1, 1

m

m

m

m m m m

j j j

j j jJ

j j j

is called a Jacket matrix if its normalized element-wise inverse and transpose

0,0 0,1 0, 1

1,0 1,1 1, 1

1,0 11 1, 1

1/ 1/ 1/

1/ 1/ 1/1

1/ 1/ 1/

T

m

m

m

m m m m

j j j

j j jJ

C

j j j

satisfies .m m mJ J I

A Jacket matrix in which the modulus of each entry is unity, is called a complex Hadamard matrix [4]. It is

easy to see that if K is a jacket matrix, then for every permutation matrices 1 2,P P , and for every invertible

diagonal matrix 1 2,D D , the matrix

1 1 2 2H PD KD P is a Jacket matrix as well. Jacket matrices related in this

fashion are called equivalent. Finding all jacket matrices up to equivalent ones turns out to be a challenging

problem, and has been solved only up to orders 5n [8].

The property of Jacket matrices is, that for any two different rows ,1 ,, ,i i na a and ,1 ,, ,j j na a , it is

necessary to have the Jacket condition

,

1 ,

0.n

i s

s j s

a

a

(3)

Requirement of Eq. (1) instead of usual inner product is given as bad algebraic properties, about Jacket

matrices - for example, multiplication of two Jacket matrices in general is not a Jacket matrix. However,

Jacket matrices have some interesting combinatorial properties. For example, if we multiply some rows or

columns by any non-zero element, then the resulting matrix remains a Jacket matrix. This type of equivalence

operation splits the space in large classes of matrices. In an orthogonal case, we can multiply only by 1 . We

conclude this section with some examples of Jacket matrices. Examples of Jacket matrices are the following

matrices of order 2n and 3, respectively:

2

1 1,

1 1A

2

3

2

1 1 1

1

1

A

,

where 2 1 0 , i.e., 1 3

,2 2

so alternative notation is

3

1 1 1

1 3 1 31 .

2 2 2 2

1 3 1 31

2 2 2 2

A

Numerous examples can be found in [6]. Some of the Jacket matrices can have parameters. For example, for

every nonzero a, the following matrix


Oct. 2017, PP. 389-402

394

4

1 1 1 1

1 1

1 1

1 1 1 1

a aA

a a

,

is not only a Jacket matrix but also a symmetric matrix. Also, it is easy to check that the Vandermonde matrix

of n-th roots of unity1, , nx x ,

1 2

2 2 2

1 2

1 1 1

1 2

1 1 1

n

n

n n n

n

x x x

W x x x

x x x

,

is a Jacket matrix, too [7].

This paper is organized as follows. First, we investigate the system model in Section 2. In Section 3, we show

the relation between the circulant and inverse orthogonal circulant Jacket (CJ) matrices. In Section 4, we

calculate the eigenvector and eigenvalues of inverse CJ matrices. In Section 5, we compute the SVD of

inverse CJ matrices, which can be applied to wireless MIMO communications shown in Section 6. Finally, the

conclusions are drawn in Section 7.

2. System Model

A Toeplitz matrix is an n × n matrix such that ,{ ; , 0,1, , 1},n k jT t k j n where

,k j k jt t for some elements

1 1 0 1 1, , , , , ,n nt t t t t . In other words, the Toeplitz matrixes have diagonal-constant elements and have

the form

0 1 2 1

1 0 1

2 1

1 2

1 0 1

1 2 1 0

.

n

n

n

t t t t

t t t

t tT

t t

t t t

t t t t

(4)

Such matrices have been found in many applications. For example, suppose that 0 1 1( , , , )T

nx x x x is a column

vector denoting an “input” and that kt is zero for 0k . Then the vector,

y x znT (5)

with entries 1

0

k

k k i i

i

y t x

represents the output of the discrete time causal time-invariant filter h with

“impulse response” .kt Equivalently, this is a matrix and vector formulation of a discrete-time convolution of

a discrete time input with a discrete time filter.

As another example, suppose that { }nX is a discrete time random process with mean function given by the

expectations ( )k km E X and covariance function given by the expectations ( , ) [( )( )].X k k j jK k j E X m X m

The signal processing theory such as prediction, estimation, detection, classification, regression, and


Oct. 2017, PP. 389-402

395

communications, and information theory are most thoroughly developed under the assumption that the mean

is constant and that the covariance is Toeplitz, i.e., ( , ) ( ),X XK k j K k j in which case the process is said to

be weakly stationary. (The terms “covariance stationary” and “second order stationary” also are used when the

covariance is assumed to be Toeplitz.) In this case, the n × n covariance matrices

[ ( , ); , 0,1, , 1]n XK K k j k j n are Toeplitz matrices. Many theories of weakly stationary processes involve

applications of Toeplitz matrices. Toeplitz matrices also give a hand with solutions to differential and integral

equations, spline functions, problems, and methods in physics, mathematics, statistics, and signal processing.

3. Circulant and Inverse Orthogonal Circulant Jacket Matrices

A common special case of Toeplitz matrices – which will result in significant simplification and play a

fundamental role in developing more general results when every row of the matrix is a right cyclic shift of the

row above it so that ( )k n k k nt t t for 1,2, , 1.k n In this case, the resulting matrix becomes,

0 1 2 1

1 0 1 2

1 0

2 1

1 2 1 0

.

n

n

n

n n

n n

c c c c

c c c c

C c c

c c

c c c c

(6)

A matrix of this form is called a circulant matrix and usually is denoted by 0 1 1circ( , , , ).nc c c

Circulant

matrices have been made good use of in various engineering areas such as the discrete Fourier transform

(DFT) and cyclic codes for error correction.

In [9], there is an example of inverse-orthogonal Toeplitz or Toeplitz Jacket (TJ) matrices of order 4

24

2 2

2

( , , ) .

bca b c

a

aba b c

cTJ a b c

a aba b

c c

a b a aba

c cc

(7)

We take advantage of the diagonal matrices that can be used to replace the matrix with the circulant Jacket

matrix. One interesting thing about the aforementioned matrix has three independent parameters which are

one more than those of circulant matrices with the same size. This extra parameter, however, is giving rise due

to the following non-trivial symmetry of Toeplitz matrices. We use the notation 1 2, ,..., nDiag a a a to refer to the

n × n diagonal matrix with diagonal entries ai, i=1,2,…,n. Therefore, a square n × n inverse-orthogonal

Toeplitz matrix is called a CJ matrix where ai is an arbitrary complex number and b is a nonzero complex

number. Then, we arrive at Lemma 1 as follows:

Lemma 1. Let nT be a Toeplitz matrix of order n, and let a be an arbitrary complex number while let b be

a nonzero complex number. Then, the following matrix is a Toeplitz matrix as well:

' 1 2 1 2 11, , , . . . , 1, , , . . . , n n

n nT aDiag b b b T Diag b b b . (8)


Oct. 2017, PP. 389-402

396

Proof: It is easy to see that the (i, j)-th entry of 'nT leads to

j iijt ab

in [9].

Now we show a structural result w.r.t. Toeplitz-Jacket matrices

Proposition 1. Suppose that T is a Toeplitz Jacket matrix. Then, with the notations from Eq. (4), we have

1

1

, 1,2,... 1ll n l

n l

tt t l n

t

(9)

Proof: Note that the Eq. (9) holds trivially for 1l . We begin the proof by showing that it holds for 2l .

We assume that 3n .

Considering the Eq. (3) within the first two rows of nT , i.e., for , 2,1i j , we reach

0 21

0 1 1

0n

n

t tt

t t t

(10)

Next, the Eq. (3) for the pair of rows , 3,2i j results in

0 32 1

1 0 1 2

0n

n

t tt t

t t t t

Adding 2

1

n

n

t

t

to both sides and by means of Eq. (10), it is easy to prove that the Eq. (9) holds for 2l . In

addition, when 1l , we get to

1 1 , 1,2,... 2l n l

l n l

t tl n

t t

(11)

Now, by using mathematical induction, we prove that the Eq. (11) holds for all 1 2l n . We can assume

that 4n because it was already proven that the Eq. (11) holds for 3l n . Then, we consider the Eq. (3)

for the pair of rows 3, 2l l :

12

10

0

nl k

l kk

t

t

,

and rewrite it using the Eq. (11) as

12 2

1 10

0

nl n k

l n kk

t t

t t

.

Adding 2 1/n l n lt t to both sides and by means of Eq. (10), it is easy to prove that the Eq. (11) holds for

1.l l We complete the proof by using the Eq. (10) for the consecutive terms lt and 1lt , hence

1 11 1

n l l n l ll l n l n l

n l n l n l n l

t t t tt t t t

t t t t

.

Now, we move to Theorem 1 about a Toeplitz-Jacket n

TJ and a circulant-Jacket n

CJ matrix.

Theorem 1. Every Toeplitz-Jacket matrix is equivalent to a circulant-Jacket matrix.

Proof Theorem 1. Let n

TJ be a Toeplitz-Jacket matrix as in the Eq. (4) and assume1 1/n

nx t t , where the

operator n denotes the principal n th root. By Lemma 1, the matrix


Oct. 2017, PP. 389-402

397

1 1 1 11 1n n

n nCJ diag x x TJ diag x x (12)

is a Toeplitz matrix equivalent to n

TJ . This n

CJ is called a circulant-Jacket matrix. To see this, it is

enough to show that 1, 2,1j n jc c for every 2,...,j n . It is clear that

11, 1

jj jc t x

. On the other hand, by

using Proposition 1, we have,

1 1 112,1 1 1 1

1

n j n j jn j n j j j

n

tc t x t x t x

t

.

Using the Eq. (7) in the Eq. (12), if a=b=c=1, and n=4, then the [CJ]4 is given by

0 1

41 0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

C CCJ

C C

. (13)

Proposition 2. Let 0 1

1 0n

C CCJ

C C

be a 2×2 block matrix of order n=2p. If [C0]p and [C1]p are p×p

Jacket matrices, then [CJ]n is a block circulant Jacket matrix if and only if

0 1 1 0 0RT RT

nC C C C , where RT is the reciprocal transpose. (14)

Proof. Since 0C and 1C are Jacket matrices, we have 0 0

RT

pC C p I and 1 1

RT

pC C p I . Note that [CJ]n is a

block circulant Jacket matrix if and only if RT

nn nCJ CJ nI . Then, [CJ]n is a Jacket matrix if and only if

0 1 1 00 1 0 1

1 0 1 0 0 1 1 0

2

2

RT RTRTpRT

n n nRT RT

p

p I C C C CC C C CCJ CJ n I

C C C C C C C C p I

(15)

Hence, [CJ]n is a Jacket matrix if and only if

0 1 1 0 0RT RT

nC C C C .

By using Proposition 2, we may construct many block circulant matrices.

One example of block circulant jacket matrices: Let

0

1 1

1 1C

, 1 1 1

a a

C

a a

(16)

Since 0 0 22RTC C I and 1 1 2

2RTC C I , 0C and 1C are Jacket matrices of order 2. Moreover,

0 1 1 0 2

1

1 1 1 101 1

1 1 1 1 1

RT RT

a aaa

C C C C

a a aa

. (17)

And we reach

0 1

1 0

1

1 1

1 1 1 11 11 1

1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 11 1

n

a

a a

C C a aCJ

C C a a

a a

, (18)


Oct. 2017, PP. 389-402

398

Therefore, [CJ]n is a block circulant inverse orthogonal Jacket matrix.

Circulant submatrices of size 2×2, are such that each submatrix has the property that all rows are cyclic shifts

of the first row and all columns are cyclic shifts of the first column. We shall refer to these submatrices as

circulant blocks. Also, we define user’s equivalence classes as the sets of user groups whose corresponding

columns of Hadamard matrices form circulant blocks. A circulant matrix is fully specified by one vector

which appears as its first column. Therefore, these matrices are important because they are diagonalized by

the DFT, and hence their linear equations can be quickly solved using the unitary fast Fourier transform.

4. Eigenvector and eigenvalue decomposition (EVD)

Because by Theorem 1, every Toeplitz Jacket matrix is equivalent to circulant matrices [9], it is natural to

look into details how the eigenvalues, eigenvectors, and eigenvalue decomposition (EVD) of a circulant

matrix can be obtained. Also note that in contrast to the usual symmetric matrix S (that have the propertyTS S ), the Toeplitz matrices and circulant matrices in particular are persymmetric [3]. This means that a real

matrixn nB ,

11 12 1

21 22 2

1 2

n

n

n n nn

b b b

b b bB

b b b

is persymmetric if it is symmetric about its northeast-southwest diagonal, i.e., for all 1 ,i j n , , 1, 1.i j n j n ib b

This is equivalent to requiring ,TB EB E where T is the transpose and E is the n×n exchange matrix

0 0 0 1

0 0 1 0

.

0 1 0 0

1 0 0 0

E

Lemma 2. If T is a Toeplitz (circulant) matrix, then the following statements are satisfied:

a) T is a persymmetric matrix;

b) If T is nonsingular, then its inverse 1T is also persymmetric.

In contrast to diagonal matrices whose eigenvalues are their diagonal elements [10], the numerical

determination of the eigenvalues of persymmetric matrices requires tedious calculation. Their

eigendecomposition depends on whether their dimension is odd or even.

We will delve into the EVD of circulant matrices following [1]. Let be the universal circulant matrix given

by

0 1 0 0

0 0 1 0

circ(0,1,0, ,0) .

0 0 0 1

1 0 0 0

(19)

It is easy to prove that a matrix C is circulant if and only if CΠ=ΠC


Oct. 2017, PP. 389-402

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Definition 1. Let n be a fixed integer ≥1 and 1i is the imaginary unit. Let 2

2 2cos sin .

i

ne in n

Denote

2 1( ) diag(1, , , , ).n

n (20)

Theorem 2. F F , where * is the conjugate transpose and F is the normalized matrix of the unitary

discrete Fourier Transform 0,0 0,1 0,( 1)

1,0 1,1 1,( 1)

2 i/

( 1),0 ( 1),1 ( 1),( 1)

1 1( ) .

n

n n n

n

jk n n n n

n n n n

n n n

F en n

(21)

Using the universal circulant matrix Π and Theorem 2, we have

Theorem 3. If C is circulant, it is diagonalized by F.

More precisely, ,C F F where 1diag( (1), ( ), , ( )),np p p for 1

0 1 1 .( ) n

np x c c x c x

If 2 ij

nj e

are the n-th roots of unity, the eigenvalues of C are 2 1

0 1 2 1 , 0,.... 1,n

j n j n j j j nc c c c

and the eigenvectors 1 , 0,1,..(1 ) . 1n T

j j jv j n .

As a consequence of the explicit formula for the eigenvalues above, the determinant of circulant matrix can be

computed as: 1

2 1

0 1 2 1

0

det( ) ( ).n

n

n j n j j

j

C c c c c

(22)

5. The Singular-value Decomposition

Using the Eq. (7) [9, Theorem 1.1], we can get to the diagonal matrices that can be used to replace the matrix

with circulant Jacket matrices. We have

44

1 0 0 0 1 0 0 0

0 0 0 0 0 0

( , , )0 0 0 0 0 0

0 0 0 0 0 0

c a

a c

CJ T a b cc a

a c

c c a a

a a c c

ab aba a

c c

ab aba a

c c

ab aba a

c c

ab aba a

c c

. (23)


Oct. 2017, PP. 389-402

400

The Eq. (23) is enabled to have Jacket matrices in the pattern of 0 1

1 0

C C

C C

where

0

a

C

a

ab

c

ab

c

and 1

a

C

a

ab

c

ab

c

.

The characteristic polynomial of C is 3 2 2 2 2

2 2 2 3 4

2 22 2 2 2 2

16 16 4det( ) 4 4

4 4( 4 4 ) ( 2 ) ,

a b a b abC xI x a x x ax x

c c c

ab abx x a a x x a

c c

so the eigenvalues are 2 2

2 ,2 , ,i ab i ab

a ac c

and the eigenvectors are

{(0,1,0,1),(1,0,1,0),( , 1, ,1),( , 1, ,1)}.i i i i

Since we do not have any simple spectrum of the matrix C, we proceed with the singular value decomposition

(SVD):

4

1 1 1 10 0 0 0

2 0 0 02 2 2 20 2 0 01 1 1 1

0 0 0 022 2 2 2

0 0 01 1 1 1

0 0 0 02 2 2 22

0 0 01 1 1 1

0 0 0 02 2 2 2

.

T

a

a

abCJ

c

ab

c

(24)

Another example of the Toeplitz Jacket 6×6 matrix

6

1 1 11

3 9 27 81 243

1 1 13 1

3 9 27 81

1 19 3 1

3 9 27

127 9 3 1

3 9

181 27 9 3 1

3

243 81 27 9 3 1

w w

ww

ww

TJ

ww

w w

w w

, (25)

can be replaced with the circulant Jacket matrix using x=3i as


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401

1 2

0 1

1

5 2 5

6 6

0

diag(1, , , , ) diag(1, , , , )

circ(1, , , ,1, )

1 1

1 1

1 1

1 1

1 1

1 1

,

CJ x x x TJ x x x

i i i

i i i

i i i

i i i

i i i

i i i

i i i

C C

C C

(26)

where 0

1

1

1 1

i

C i i

i

, 1

1

1

i i

C i

i i

, and 2

31 3

2 2

i

e i

, respectively.

If n=8, a=1, and b=0, the circulant vector x(a,b)=x(1,0) is constructed as follows

2 2

22 21,0 exp exp exp

2 8 2 8i

ai ix bi i

n

i i i ,

Therefore, this vector looks like as

7 7

8 8 8 82 2 21,0 exp 1 ,exp 2 ,...,exp 8 , , 1, , ,18 8 8

i i i i

x e e e e

i i ii, i, .

Therefore, an instance of 8×8 CJ matrix [CJ]8(a,b) is given by

7 78 8 8 8

7 78 8 8 8

7 78 8 8 8

7 78 8 8 8

7 78 8 8 8

7 78 8 8 8

7 78 8 8 8

78

18 88 8

1, ,

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1

i i

i i

i i

i i

i i

i i

i i

i

i i i i

i i i i

i i i i

i i i i

i i i i

i i i i

i i i i

i

CJ a b D TJ a b Db

e e e e

e e e e

e e e e

e e e e

e e e e

e e e e

e e e e

e e

78 8 8

0 1

1 0

1ii i i

C C

C C

e e

, (27)

where

78 8

78 8

8 8

8 8

0

1

1

1

1

i

i

i

i

i i

i i

i i

i i

e e

e eC

e e

e e

,

78 8

78 8

7 78 8

7 78 8

1

1

1

1

1

i

i

i

i

i i

i i

i i

i i

e e

e eC

e e

e e

, the diagonal matrix 2 3 4 5 6 71, , ,a , , ,D diag a a a a a ,

the inverse of this diagonal matrix 1 2 3 4 5 6 71,1/ ,1/ ,1/ a ,1/ ,1/ ,1/D diag a a a a a , and Toeplitz-Jacket

matrix


Oct. 2017, PP. 389-402

402

7 78 8 8 8

7 78 8 8 8

7 788 8 8

2

7 788 8 8

3 2

78 8

4 3 2

2 3 4 5 6 7

2 3 4 5 6

2 3 4 5

2 3 4

8,

i i i i

i i i i

ii i i

ii i i

i i

ba

be baa

ib be baa a

be ib be baa a a

be iab a be a b a be ia b a be a b

be iab a be a b a be ia b a be

be iab a be a b a be ia b

be iab a be a b a be

TJ a bbe

78 8

778 8

8 85 4 3 2

7 78 8 8

86 5 4 3 2

7 78 8 8

87 6 5 4 3 2

2 3

2

i i

i ii i

i i ii

i i ii

b be ib be baa a a a

be b be ib be baa a a a a

ib be b be ib be baa a a a a a

iab a be a b

be iab a be

be iab

be

.

We also note that from Eq. (27), the eigenvalue of 8×8 CJ matrices is 2 2 whose multiplicity is 8 as that of

8×8 DFT matrices.

6. Wireless MIMO Applications

We consider,

1

1 1 1

1

2 2 2

1

2 2 2

1

, ,H I T

I J Σ J

I J I Σ I J

JΣJ

n n n

n

n n n

a b c

(28)

where J is the n×n Jacket matrix that satisfies JJ-1

=I,

1 1

2

2 2 2

0

0

J

J=

Jn n

,

its inverse Jacket matrix

1 1

1

2

1

1

2 2 2

0

0

J

J =

Jn n

,

and Σ is the diagonal matrix given by

1 1

1

2

2 2 2

, ,

0

0

Σ J T J

Σ

Σ

n

n n

a b c

.

The well-known capacity formula is given by

2log det / ,H

n n nC bps Hz I H H (29)


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403

where γ = ρ / N0 is the signal-to-noise ratio (SNR), ρ is the energy of the transmitted signals, and N0 is the

power spectral density of the additive noise 1

n

i iz

. The SVD of H H

H

n nis given by

,H H H H H H H

n n H H JΣJ JΣ J JΣΣ J QΛQ (30)

where QQ IH and Λ

n n is the diagonal matrix with its elements given as

2

min

min

, 1,...,

0, 1,...

i

i

if i n

if i n n

. (31)

However, the system capacity can be written as

min

2

1

log 1 /r n

i

i

C bps Hzn

, (32)

where r = min(nmin, n) = nmin denotes the rank of Hn and

i is the eigenvalue of the matrix H HH

n n.

In Figure 1, we compare the inverse orthogonal Jacket channel against the conventional channel [11-12].

Using MATLAB software, we perform the Monte Carlo simulation at γ = 20[dBs] and generate by 1000

channel realizations in the simulations. We observe that in Figure 1, the channel capacity over the inverse

orthogonal channel is slightly increased at high SNR regime compared to Toeplitz and Jacket channels,

which can be a big help for meeting exponentially growing demand for higher capacity communication

systems.

Figure 1. Comparison of MIMO channel capacity employing Eq. (7) in Eq. (32).

7. Conclusions

To sum up, we have studied the CJ matrices that have a similar structure of a special kind of Toeplitz such as

traditional circulant matrices in this paper. The greatest advantages of these CJ matrices are that they facilitate

very fast and instantaneous computation of inverse CJ matrices stemming from the fact that inverse CJ

matrices are diagnosable by fast Fourier transform matrices and that they acquaint with the singular value

decomposition technique. There are many applications of CJ matrices such as the Kronecker MIMO channel

[6] and a block circulant Jacket in [11].

Acknowledgement

This work was supported by MEST 2015R1A2A1A05000977, NRF, Korea and thanks to Ferenc Szöllősi, he

has derived Eq. (27).


Oct. 2017, PP. 389-402

404

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