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Communications and Network, 2013, 5, 20-24
doi:10.4236/cn.2013.52B004 Published Online May 2013 (http://www.scirp.org/journal/cn)
Jacket Matrix Based on Modular (3, 5, 6) Lattice
Triangular Expansion
Wei Duan1, Haiya ng Yu2, Wenjun Yu3, Moon Ho Lee4
1Division of Electronics and Information Engineering, Chonbuk National University, Baekje-Daero, Republic of Korea
2Division of Electronics and Information Engineering, Chonbuk National University, Deokjin-Gu, Republic of Korea
3Division of Electronics and Information Engineering, Chonbuk National University, Jeonju-Si, Republic of Korea
4Division of Electronics and Information Engineering, Chonbuk National University, Jeollabuk-Do, Republic of Korea
Email: [email protected], yuha[email protected], [email protected], [email protected]
Received 2013
ABSTRACT
A Lattice triangular expansion matrix is presented based on the classical Hadamard matrices, which is defined over the
fields of finite characteristic. Also, the modular Lattice and Pentagon expansion matrices are structured from triangular
matrix, each of the expansion matrices are modular the sides of the shape p. The issue for the existence (neces-
sary conditions) of odd and even order matrices of that kind is addressed. The modular Lattice code is highly efficient
since it requires only additions, multiplications by constant modulo p. The modular 6 Lattice triangular expanded con-
stellation is even possible efficiency to gain advantage from the channel selection and maximum likelihood (ML) de-
coding in the interference Lattice alignment (IA) system.
77
Keywords: Element-Wise Inverse; Modulo Jacket Matrix; the Sides of Shape; Lattice Alignment; ML Decoding
1. Introduction
The generalized reverse jacket transforms (GRJT) as
multi-phase or multilevel generalizations of the WHT
and the even-length DFT were introduced in [1]. With
the rapid technological development, many different and
generalized forms of signal processing transforms with
independent parameters have been proposed. It has been
discovered that the new proposed transforms with many
parameters have been widely used in various signal
processing, CDMA, cooperative relay MIMO system
analysis. However, it can be proven that matrices having
the abovementioned properties with entries from the field
of complex numbers do exist only for even orders [2]. So,
it seems the problem of the existence of similar trans-
forms on distinct odd dimension spaces sounds natural.
By that motivation, in this Letter, we consider a family
of matrices (under the name jacket modulo prime matri-
ces) over the fields of finite characteristic, the properties
of which resemble very closely th ose of the conventio nal
Ha damard matrices.
For basic definitions and notions the reader is referred
to [3]. The primary generalized reverse jacket transform
, defined in [4], is a permuted version of the
DFT, so called mixed-radix representation
of integers from the set , which retains the
first n rows and columns unchanged, and reverses the
last n rows and columns of the corresponding DFT ma-
trix (a Vandermonde matrix based on a primitive 2nth
root of unity on the complex circle).
(GRJT s
2lenn
For the two-user interference channel, one of the best
known achievable regions is that introduced by Han and
Kobayashi [5]. This achievable region can be naturally
generalized to more than two-users. However, a “good”
choice for the auxiliary random variables and their joint
distribution in the generalization of the Han & Kobaya-
shi coding sche me is not known. In [6], it is shown th at a
layered lattice coding scheme can result in an improved
set of achievable rates than an i.i.d. Gaussian Han &
Kobayashi region. The layered lattice coding 1It is
known from [7] that i.i.d. Gaussian is in fact a reasonably
good choice for the two user Gaussian interference
channel. It is therefore somewhat surprising that this is
not true for the K > 2 user case sch eme in [6] attempts to
separate the signal and interference signals into non-
interfering levels. Alth ough the scheme in [6] achieves a
higher DoF (and a better set of rates at any SNR) than
i.i.d. Gaussian Han & Kobayashi -style coding, it does
not achieve the same DoF as obtained using the schemes
in [8-10]. In order to obtain a better achievable region
than in [6], we allow the signal and interference lattices
to interact with one another in the case of channels with
integer channel gains in [12], and determine algebraic
mechanisms of separating signal and interference. Al-
though the sche me in [12] achieves a strictly b etter set of
rates than in [6], it still falls short, in terms of degrees of
freedom, than that achieved in [8,9].
)
gth {0,1,...,2 1}n
Copyright © 2013 SciRes. CN
W. DUAN ET AL. 21
2. Center Weighted Hadamard Matrix
In this section, we introduce some definitions and nota-
tions. First, we recall the center weighted Hadamard ma-
trix of orde r 4 i n [12]
4
111 1
11
[CWH] 11
1111











where
is a nonzero complex parameter. The inverse
of this basic matrix can be easily obtained by element-
wise inverse matrix as follows:
1
4
111 1
11
11
[CWH] 11
11
11 11














Definition 2.1: A matrix ,
[]( )
N
Nik
J
j
of order N
whose entries are complex is called a Jacket matrix, if
the element in the entry of its inverse matrix is
equal to the product of
(, )ik
1N and the inverse of the ele-
ment in the entry of
(,)ki []
N
N
J. In other words, if
0,00,10, 1
1,01,11, 1
1,01,11,1
[]
N
N
NN
NN NN
jj j
jj j
J
jj j
 





and its inverse
0,00,10, 1
11,01,11, 1
1,01,11, 1
11 1
11 1
[]
11 1
N
N
NN
NN NN
jj j
jj j
J
jj j
 


then is called a Jacket matrix.
From the definition of Jacket matrices, it is easy to see
that any Hadamard matrices of order are Jacket matrices.
In addition, the center weighted Hadamard (CWH) is
also a Jacket matrix.
We can find that Jacket matrices have reciprocal or-
thogonality and reciprocal relation. The basic Jacket ma-
trix of orde r 3 i s de fi ned as
2
32
11 1
[] 1
1
J





where
is the third primitive root of unity. The in-
verse of 3
J
is
1
32
2
11 1
111
[] 1
311
1
J









which satisfies
1
33
[][] []
3
J
JI
where n
I
is th e identity matrix of order n. From (10), it
is easy to see that the inverse of 1
3
[]
J
can be easily
obtained from the forward matrix 3
J
by taking the in-
verse of each entry 3
J
of and then transposing the re-
sulting matrix. Hence th e Jacket transform has following
two advantages:
1) Element-wise inverse orthogonality.
2) The entries of the forward and the inverse trans-
forms have a reciprocal relationship.
3. Jacket Matrix over Finite Characteristic
Fields
Without loss of generality we may focus on the fields
, where p is a prime and define the notion of the
jacket modulo prime matrix over them.
()GF p
Definition 3.1: A jacket modulo prime ()
J
MP matrix
J of order n over is an non-singular ma-
trix of ()GF pnn
1
s
that field such that
Tn
J
JnI (1)
where n
I
is the identity matrix of order n.
As usual, the notation T
M
is used for the transpose
matrix of a given matrix M. We shall use also the nota-
tion ()
J
MPp (GF for the set of jacket modulo prime matri-
ces over .
)p
Example 1: Triangular matrix (Figure 1) 77
Let n
J
, where 4npk
and be a
square matrix of order n consisting of with the following
description. Its first row and column consist entirely of
1,2,...,k
1
s
; its last row and column consist o f 1
s
with excep-
tion of the corner entries, and all other entries are equal
to1with the exception of those on the main d iagonal. For
instance, 7(3Jp ,1)k
looks as:
7
1111111
1-11111-1
1 1-1 111-1
=11 1-11 1-1
1111-11-1
11111-1-1
1-1-1-1-1-11
J











Copyright © 2013 SciRes. CN
W. DUAN ET AL.
22
also
-1
7
1111111
1-11111-1
1 1-1 111-1
1
=11 1-11 1-1
71111-11-1
11111-1-1
1-1-1-1-1-11
J











The inner product of a pair of rows equals either to
, i.e. in the following matrix equa-
tion holds:
31 3pk  (3)GF
77 7
7
T
J
JI (2)
Clearly, 7
1
77
T
J
JJ
, where 7
T
J
is the transpose
matrix of 7
J
. So, 7
J
is an orthogonal matrix over the
filed . We stress once again that in this ex-
ample the operations are taken modulo 3. Thus,
(3)pGF
n
J
is a
J
MP matrix over . (3)GF P
Example 2: Extended Lattice Triangular 10 10
ma-
trix (Figure 2)
Similarly, by the same way as shown in the example 1,
the matrix can be ex pressed as follow
10 10
10
1111111111
111111111 1
11 11111111
111 1111111
1111 111111
,
11111 11111
111111 1111
1111111 111
11111111 11
1111111111
L
























Figure 1. Triangular and circular internally tangent.
Figure 2. Lattice and circular internally tangent.
The inverse of 10
J
can be easily calculate as
1
10
1111111111
111111111 1
11 1111111 1
111 111111 1
11 1 111 1 1 11
1
11 1 1111 1 11
10
111111 111 1
1111111 11 1
11111111 11
1111111111
L





















Clearly, that
1
10 10 10
T
LLL
 (3)
1
10 1010
10(in 6)LL IGF
1T
(4)
By the Definition 2.1, are also Jacket
matrices over . 10 1010
,,LLL
(6)GF
Example 3: Extended Pentagon Triangular 99
ma-
trix
Also, the Pentagon Triangular matrix can be
structured as 99
9
p
9
111111111
11111111 1
11 111111 1
111 11111 1
11 1 111 1 11
11 1 1 111 11
111111 11 1
1111111 11
111111111
p



















Clearly, that
1
99 9
T
PPP
 (5)
1
99 9
9(in5PP IGF
1T
) (6)
By the Definition 2.1, are also Jacket ma-
trices over 99 9
,,PP P
(5)GF
Over these 3 examples, the modular (5,6) Jacket ma-
trix Jn is constructed based on the triangular (7 7)
matrix, where n = p + 4, 5,6p
is the number of sides for
the shape (that’s meaning pentagon, lattice). Note that
this scheme is highly efficient since it requires on ly addi-
tions, multiplications by constant modulo p, and it is even
possible to gain advantage from interference alignment.
4. Lattice Alignment Application
In this section, we consider 3-pairs interference system,
Copyright © 2013 SciRes. CN
W. DUAN ET AL. 23
where each transmitter Ti and receiver Ri equipped with
one antenna, respectively. The channel coefficients ,ij
H
define links from transmitter i to the receiver j, where
. ,1,2,3ij
4.1. Channel Selection with Lattice Constellation
Motivated by the advantage of having a structured inter-
ference, we propose an approximate lattice alignment
scheme in which the precoders are designed to best align
the receiving lattices. However, we accept the fact that
lattice alignment may not be perfect (due to infeasible
configurations and imperfect CSI effects) and try to
model and minimize the effect of the residual lattice
alignment errors. The lattice alignment error is given by
e=| |
H
ajijjL
uhv a (7)
where
L
a
, }a
is the lattice coordinate as shown in the Fig-
ures 3 and 4. As a result, the design parameters
ii L
in (7) are chosen to minimize the effects of
the lattice alignment errors.
{,uv
The error is smaller, the channel state information
(CSI) is better. The optimal is .The conditional
error probability given e=0
a
j
u can be upper bounded as
follows:
22
(| ){}
{|||||||| }
H
jijj
H
jjijjL
H
jijj L
uhv
Pe uPuhva
Puhv a



where is the maximized distance in the cons tellation ,
on the other ha nd is the length of side.
Figure 3. Pentagon and circular internally tangent.
Figure 4. Lattice alignment constellation with imperfect CSI.
4.2. Lattice Alignment in 3-pairs Interference
Chanel
In the conventional works, the perfect IA requirements
for all kK
are summarized as
,
UHV0,
H
jijj
(8)
,
rank(UHV)=.
H
iiiii
d (9)
Eq. (8) guarantees that all the interfering signals at
destination lK
are aligned in a subspace of ki
Nd
dimensions and can be zero-forced by
j
Z
. Eq. (9) guar-
antees that destination kK
is able to decode all
j
d
intended data streams successfully. When both equations
(8) and (9) are satisfied, the interference alignment is
feasible for the given DoF.
We will work with a many-to-one Gaussian interfer-
ence channel with 3 users, where interference is only
present at receiver 1. The desired symbols of receiver
1, 2,3k
can be estimated as
interference signa
desired signal
,,
y=u+u+u n
l
k
HH
iiiii jijji
ij
xhx


hH
i
where ,ij
is the nnh
channel matrix from transmitter
j to receiver i,
j
x
is the transmitted symbols and ni is
the additive white Gaussian noise with variance 2
.
At receiver 1, there is interference from users 2 and 3.
By suitably choosing and in such a way that
2
v3
v
12213 3.hv hv
We can perform lattice align ment of the int erfering sig-
nals from users 2 and 3 at the first receiver’s as follow:
12213 313 3
()(LhvhvLhv)
where is the lattice generated by the matrix
()
n
LJ n
J
.
Then the desired sign al belongs to the lattice1111
L
hva ,
while the sum of the interfering signals
12 223
[(x x)Lh v]
is aligned in the lattice 2122
L
hv a,
where
L
a is the Lattice alignment coordinate which
will be introduced in the next section. Then the received
signal at the receiver 1 can be rewritten as:
112
yxxn
1

where 11111
x
hvs
and 221223
[( )]
x
Lh vxx belong to
the Lattice constellation coordinate.
After successfully channel selection, we wish to de-
code the desired signal k
i
x
at stage-II as illustrated in
Figure 5. The desired signal is detected given by
1
||
kHkHH
iiii j
yuyuxux

2
(10)
Compared with the alignment error and ML decoding
algorithm in first and second stage, both of them are the
Euclidean distance. Let’s focus on:
Copyright © 2013 SciRes. CN
W. DUAN ET AL.
Copyright © 2013 SciRes. CN
24
Figure 5. 3-piars interference channel.
Figure 6. ML decoding based on Lattice expansion triangular.
,,
|(
||
k
kHkH H
iiiiiiii jijjj
ij
Hk
iii ajLj
yuyuhvxuhvx
uyx exax
 

)|
(11)
where
L
a is given as the coordinate in each constella-
tion, ,i
H
iiiii
x
uhvx
. We can find that the ML decoding
mapping constellation should include the lattice constel-
lation.
Also, in the Figures 4 and 6, the area of the lattice
expansion triangular is greater than the lattice. It’s also
satisfied the formula in (7).
5. Conclusions
Clearly, the Lattice triangular expansion matrix, which
can be given for an arbitrary field of finite characteristic,
is presented based on the conventional real Hadamard
matrices. In this paper, we have also addressed a neces-
sary condition for the existence and presented a con-
struction of odd and even order JMP matrices. The
modular Lattice and Pentagon expansion matrices are
structured by the triangular 77
matrix, and modular
the sides of the shape p.The modular Lattice is highly
efficient since it requires only additions, multiplications
by constant modulo p. The modular 6 Lattice triangular
expanded constellation is even possible to gain advan-
tage from the channel selection and maximum likelihood
(ML) decoding in the interference alignment (IA) sys-
tem.
6. Acknowledgements
his work was supported by World Class University,
R32-2012-000-20014-0, Basic Science Research Pro-
gram 2010-0020942, MEST 2012-002521 and NRF
China-Korea International Corsearch (D00066, I00026).
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