Extracted Text
IJAM_48_2_11.pdf
The Quaternion Domain Fourier Transform and its
Application in Mathematical Statistics
Mawardi Bahri, Amir Kamal Amir, Resnawati, and Chrisandi Lande
AbstractRecently a generalization of the quaternion
Fourier transform over quaternion domains so-called the
quaternion domain Fourier transform (QDFT) has been in-
troduced, including its properties such as shift, modulation,
convolution theorem and uncertainty principle. In the present
paper we explore more properties of the QDFT such as the
correlation and product theorems and propose its application
in probability theory and mathematical statistics.
Index Termsquaternion domain Fourier transform, quater-
nion random variable
I. INTRODUCTION
It is well known that in signal and image processing, the
classical Fourier transform is a very important tool (see,
e.g., [6], [16]). The quaternion Fourier transform (QFT) (for
e.g., [1], [2], [3], [4], [5], [7]) is also very useful tool
for signal processing for quaternion signals with domain
R
2
. The quaternion domain Fourier transform (QDFT) is a
generalization of the QFT over the quaternion domain. The
rst work concerning the denition of the QDFT and its
relation to the denition of the QFT was done by Hitzer
[15]. The QDFT also can regarded as an extension of the
classical Fourier transform (FT) using quaternion algebra. It
transforms quaternion valued signals dened over a quater-
nion domain from a quaternion position space to a quaternion
frequency space. A number of useful properties of the QDFT
have been found including shift, modulation, convolution,
correlation, differentiation, energy conservation, uncertainty
principle and so on. It is well known that the classical
Fourier transform plays crucial roles in probability theory
and mathematical statistics. It is related to the characteristic
function of any real-valued random variable to compute the
distribution function. Therefore, in the present paper, we
rst investigate some important properties of the QDFT such
as derivative, convolution, correlation and product theorems.
We then establish the relationship between the quaternion
characteristic function and the QDFT. We nally apply this
relation to derive the properties of quaternion probability
function and quaternion moments in the framework of the
quaternion algebra of mathematical statistics.
The remainder of this paper is organized as follows.
In Section II we briey review the basic knowledge of
Manuscript received January 11, 2018; revised April 17, 2018. This work
was supported by Research Collaboration Between Indonesian Universities
(PEKERTI) with contract number: 622.48/UN28.2/PL/2017.
Mawardi Bahri is with the Department of Mathematics, Hasanuddin
University, Makassar 90245, Indonesia (corresponding author: e-mail:
mawardibahri@gmail.com)
Amir Kamal Amir is with the Department of Mathematics, Hasanuddin
University, Makassar, Indonesia. e- mail: amirkamalamir@yahoo.com
Resnawati is with the Department of Mathematics, Tadulako University,
Palu 94148, Indonesia. e- mail: r35n4w4t1@yahoo.com
Chrisandi Lande is with Politeknik Ilmu Pelayaran Makassar, Makassar
90165, Indonesia. e-mail: sandhylande@gmail.com
quaternion and derivative operators used in the next section.
In Section III we derive some useful properties of the QDFT
such as the convolution, correlation and product theorems.
In Section IV we discuss the application of the QDFT in
probability theory and mathematical statistics. Finally, in
Section V we give conclusion.
II. QUATERNIONS
We rst review the basic concepts and denition of quater-
nions. The quaternions, a generalization of complex numbers,
are an associative but noncommutative overR. The set of
quaternions is denoted byH. Every element ofHcan be
written in the following form
H=fq=qa+iqb+jqc+kqd;qa; qb; qc; qd2Rg;(1)
which obeys the following multiplication rules:
ij= ji=k;jk= kj=i;ki= ik=j;
i
2
=j
2
=k
2
=ijk= 1: (2)
For a quaternionq=qa+iqb+jqc+kqd2H,qais called
thescalarpart ofqdenoted bySc(q)andiqb+jqc+kqd
is called thevector(orpure) part ofq. The vector part of
qis conventionally denoted byq. Letp,q2Handp,q
be their vector parts, respectively. From (2) we obtain the
quaternionic multiplicationqpas
qp=qapa qp+qap+paq+qp; (3)
where
qp=qbpb+qcpc+qdpd (4)
qp=i(qcpd qdpc) +j(qdpb qbpd) +k(qbpc qcpd):
(5)
Analogously as in the complex case, the quaternion con-
jugate ofqis dened by
q=qa iqb jqc kq3; qa; qb; qc; qd2R:(6)
It is an anti-involution, i.e.
qp= pq: (7)
Notice that conjugate switches the order of multiplication.From (6) we obtain the norm or modulus ofq2Hdened
as
jqj=
p
qq=
q
q
2
a+q
2
b
+q
2
c+q
2
d
: (8)
It is routine to check that
jqpj=jqjjpjandjq+pj jq j+jpj; 8p; q2H:(9)
Using the conjugate (6) and the modulus ofq, we get the
inverse of a non-zero quaternionq2Has
q
1
=
q
jqj
2
; (10)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
which shows thatHis a normed division algebra. When
jqj= 1; qis a unit quaternion. A quaternionqwithqa= 0
is called a pure quaternion and its square is negative sum of
three squares:
q
2
= (q
2
a+q
2
b+q
2
c) = 1: (11)
According to (4) we can get a scalar part of two quaternions
p; qas
pq=Sc(pq) =
1
2
(pq+qp) =paqa+pbqb+pcqc+pdqd:
(12)
A quaternion numberqmay be dened as a complex
number with complex and imaginary parts.
q=z1+jz2; z1=qa+iqb; z2=qc+iqd:(13)
Equation (13) is known as the Cayley-Dickson form.
We dene derivative operators as
~
@=@xa
+@xb
i+@xc
j+@xd
k
@=@xa
@xb
i @xc
j @xd
k; (14)
where@xa=@=@xaand so on.
Using the orthogonal planes split ofq2Hwith respect
to the pure quaternion2H;
2
= 1we dene
q=
1
2
(qq); q =qa+q;
q+=q+q= (q+q) (15)
where rotation operatorR= (i+)i; =RjR
1
and=
RkR
1
;
2
=
2
= 1; qa; q; q; q2R.
Similar to the complex case, we may dene an inner
product for two functionsf; g:H !Has
(f; g)
L
2
(H;H)=
Z
H
f(x)g(x)d
4
x; (16)
wherex2H; d
4
x=dxadxbdxcdxd2R. Notice that
every quaternion domain functionfmapsH !H, its
decomposition will take the form
f(x) =fa(x) +fb(x)i+fc(x)j+fd(x)j (17)
=fa(xa; xb; xc; xd) +fb(xa; xb; xc; xd)i
+fc(xa; xb; xc; xd)j+fd(xa; xb; xc; xd)k; x; y2H;
(18)
where four coefcient functionsfa; fb; fcandfdare in turn
real valued quaternion domain function.
In particular, forf=g, we also may dene theL
2
(H;H)-
norm as
kfk=
Z
H
jf(x)j
2
d
4
x
1=2
: (19)
III. THEQUATERNIONDOMAINFOURIERTRANSFORM
AND ITSUSEFULPROPERTIES
In this section we introduce the quaternion domain Fourier
transform (QDFT) and its basic properties, which is taken
from [15]. We then make some observations about some
further properties related the QDFT such as the correlation
and product theorems, which will be very useful later.
Denition 1.The quaternion domain Fourier transform of
the quaternion functionf2L
1
(H;H)is given by the integral
Fff)g(!) =
Z
H
e
!x
f(x)d
4
x; (20)
wherex; !2Hand some constant2H.
Notice that constant pure quaternioncan be chosen
for each problem. For example, if we take=jand the
quaternion functionfis decomposed as in (13), then the
QDFT takes the form
Fjffg(!) =
Z
H
e
j!x
f(x)d
4
x: (21)
Expandingfin (21) into real and imaginary parts with
respect toiand using the Euler formula for the quaternion
Fourier kernel we obtain
Fjffg(!)
=
Z
H
(cos(! x) +jsin(!x)(f0(x) +f1(x)i)d
4
x
=
Z
H
f0(x) cos(!x)d
4
x+j
Z
H
f0(x) sin(!x)d
4
x
+
Z
H
f1(x) cos(!x)id
4
x+j
Z
H
f1(x) sin(!x)id
4
x:
(22)
Equation (22) clearly shows how the QDFT separates quater-
nion signal into the odd and even parts of real and imaginary
parts in four different components in the QDFT domain.
Denition 2.Iff2L
1
(H;H)and its QDFTFffg 2
L
1
(H;H), then the inverse transform of the QDFT is given
by the integral
F
1
[Fffg](x) =f(x) =
1
2
Z
H
e
!x
Fffg(!)d
4
!:
(23)
Like the polynomial Fourier transform [19], the convolu-
tion of two quaternion functionsf; g2L
1
(H;H)is dened
by
(fg)(x) =
Z
H
f(x y)g(y)d
4
y: (24)
The following theorem provide the convolution theorem
which describes how the QDFT behaves under the quaternion
convolution.
Theorem 1.Supposef2L
1
(H;H)andg2L
1
(H;H)are
integral functions. Then we have
Fffgg(!)
=Fff g(!)Ffgg(!) +Fff+g(!)Ffgg( !):(25)
Moreover,
(fg)(x) =F
1
Fff g(!)Ffgg(!)+
+Fff+g(!)Ffgg( !)
(x): (26)
Proof: LetFffgandFfggdenote the QDFT off
andg, respectively. It follows from(20)and(24)that
Fff ? gg(!) =
Z
H
e
!x
(f ? g)(x)d
4
x
=
Z
H
Z
H
e
!x
f(x y)g(y)d
4
y
d
4
x
=
Z
H
e
!x
f(x y)
Z
H
g(y)d
4
y
d
4
x:
(27)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
After the change of variablesz=x y, the above expression
becomes
Fff ? gg(!)
=
Z
H
Z
H
e
!(y+z)
f(z)g(y)d
4
y d
4
z
=
Z
H
Z
H
e
!z
e
!y
f(z)g(y)d
4
z d
4
y
=
Z
H
Z
H
e
!z
e
!y
(f (z) +f+(z))g(y)d
4
z d
4
y
=
Z
H
Z
H
e
!z
(f (z)e
!y
+f+(z)e
!y
)g(y)d
4
z d
4
y
=
Z
H
Z
H
e
!z
f (z)e
!y
g(y)d
4
z d
4
y
+
Z
H
Z
H
e
!z
f+(z)e
!y
g(y)d
4
z d
4
y
=
Z
H
e
!z
f (z)d
4
z
Z
H
e
!y
g(y)d
4
y
+
Z
H
e
!z
f+(z)d
4
z
Z
H
e
!y
g(y)d
4
y
=Fff g(!)Ffgg(!) +Fff+g(!)Ffgg( !):(28)
In view of(28)and inversion formula for the QDFT(23), the
relation(26)holds. This completes the proof of the theorem.
Denition 3.The correlation for the QDFT of two quater-
nion functionsf; g2L
1
(H;H)is given by
(fg)(x) =
Z
H
f(x+y)g(y)d
4
y: (29)
We derive the following correlation theorem for the QDFT
using the relationship between the quaternion convolution
and quaternion correlation (compare to [17]).
Theorem 2.Supposef2L
1
(H;H)andg2L
1
(H;H)are
integral functions. Thus
Fffgg(!)
=Fff g(!)Ffgg( !) +Fff+g(!)Ffgg(!):(30)
Proof:A simple computation yields
(fg)(x)
=
Z
H
f(x+y)g(y)d
4
y
=
Z
H
f(x u)g( u)d
4
u
=
Z
H
f(x u)h(u)d
4
u
= (fh)(x)
(26)
=F
1
h
Fff g(!)Ffhg(! )
+Fff+g(!)Ffhg( ! )
i
(x): (31)
It is easily seen that
Ffhg(! ) =
Z
H
e
!u
g( u)d
4
u=Ffgg( !);(32)
and
Ffhg( ! ) =
Z
H
e
!u
g( u)d
4
u=Ffgg(!):(33)
Due to (32) and (33), equation (31) can be expressed as
(fg)(x)
=F
1
[Fff g(!)Ffgg( !) +Fff+g(!)Ffgg(!)] (x):
Or, equivalently,
Fffgg(!)
=Fff g(!)Ffgg( !) +Fff+g(!)Ffgg(!);
which was to be proved.
Theorem 3.Supposef2L
1
(R;H)such thatf(x)is con-
tinuousn-times differentiable, then forlimx!1f(x) = 0
the following holds
Ff
~
@
n
fg(!) =!
n
( )
n
Fffg(!); n2N: (34)
Proof:Consider rst the casen= 1. Indeed, we have
Ff
~
@fg(!)
=
Z
H
e
!x
(@xa
+@xb
i+@xc
j+@xd
k)f(x)d
4
x
=
Z
H
e
!x
@xaf(x)d
4
x+
Z
H
e
!x
i@xaf(x)d
4
x
+
Z
H
e
!x
j@xa
f(x)d
4
x+
Z
H
e
!x
k@xa
f(x)d
4
x
=
Z
H
(@xa
e
!x
)f(x)d
4
x
Z
H
(@xb
e
!x
)f(x)d
4
xi
Z
H
(@xb
e
!x
)f(x)d
4
xj
Z
H
(@xb
e
!x
)f(x)d
4
xk
=
Z
H
(!a+!bi+!cj+!dk)( ) e
!x
f(x)d
4
x
=!( )
Z
H
e
tx
f(x)d
4
x:
Using mathematical induction we can nish the proof of the
theorem.
The following theorem describes the relationship between
the product of two quaternion functions and its QDFT.
Theorem 4.Letf; g2L
2
(H;H). Then the QDFT of product
of two quaternion functionsfandgis given by
Fffgg(!)
=
1
(2)
2
(Fqff g Ffgg)(!) + (F f
f+g Fqfgg)(!)
:
(35)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
Proof:Applying the QDFT denition we obtain
Fffgg(!)
=
Z
H
e
!x
f(x)g(x)d
4
x
=
Z
H
e
!x
f(x)
1
(2)
2
Z
H
e
ux
Ffgg(u)d
4
u
d
4
x
=
Z
H
e
!x
f(x)
1
(2)
2
Z
H
e
ux
Ffgg(u)d
4
u
d
4
x
=
1
(2)
2
Z
H
e
!x
(f (x) +f+(x))
Z
H
e
ux
Ffgg(u)d
4
u
d
4
x
=
1
(2)
2
Z
H
Z
H
e
ux
e
!x
f (x)F fgg(u)d
4
u d
4
x
+
1
(2)
2
Z
H
Z
H
e
ux
e
!x
f+(x)F fgg(u)d
4
u d
4
x
=
1
(2)
4
Z
H
Z
H
Z
H
e
ux
e
!x
e
vx
Fff g(v)Ffgg(u)
d
4
u d
4
v d
4
x
+
1
(2)
4
Z
H
Z
H
Z
H
e
ux
e
!x
e
vx
Fff+g(v)Ffgg(u)
d
4
u d
4
v d
4
x
=
1
(2)
4
Z
H
Z
H
"
Z
H
e
(! u v )x
d
4
x
Fff g(v)Ffgg(u)
#
d
4
u d
4
v
+
1
(2)
4
Z
H
Z
H
#
Z
H
e
(!+u v )x
d
4
x
Fff+g(v)Ffgg(u)jbigg]d
4
u d
4
v
=
1
(2)
2
Z
H
Z
H
(w u v)Fff g(v)Ffgg(u)d
4
u d
4
v
+
1
(2)
2
Z
H
Z
H
(!+u v)Fff+g(v)Ffgg(u)d
4
u d
4
v
=
1
(2)
2
Z
H
Fff g(! u)Ffgg(u)d
4
u
+
1
(2)
2
Z
H
Fff+g(!+u)Ffgg(u)d
4
u: (36)
This gives the required result.
As an immediate consequence of the above theorem, we
get the following corollary.
Corollary 1.Letf; g2L
2
(H;H). Assume that the QDFT
ofgis a real-valued function, then Theorem 4 will reduce to
Fqffgg(!) =
1
(2)
2
(Fqffg Fqfgg)(!): (37)
IV. APPLICATION OF THEQUATERNIONDOMAIN
FOURIERTRANSFORM INMATHEMATICALSTATISTICS
Following [8], [9] we dene the probability density func-
tion of a quaternion random variable, which is algebraically
similar to real probability density function of the correspond-
ing associated probability quantity [6].
Denition 4.LetX=Xa+Xbi+Xcj+Xdkbe a quaternion
random variable. A quaternion functionfX(x) =fXa
(x) +
fXb
(x)i+fXc(x)j+fXd
(x)kof the quaternion variable
x=xa+xbi+xcj+xdkis called the quaternion probability
density function (qpdf) ofXif
Z
H
fXi(x)d
4
x= 1; andfXi(x)08x2H; i=a; b; c; d:
We also dene the quaternion cumulative distribution func-
tion (compare to [18])
fX(x) =
~
@FX(x);
where the probabilityP ris related toFXgiven by
FX(x) =P r(Xaxa; Xbxb; Xcxc; Xdxd):
Denition 5(Expected value). LetXbe a quaternion-valued
random variable with quaternion density functionf(x). The
expected valuem=E[X]is dened by
m=E[X]
=
Z
H
xfX(x)d
4
x
=
Z
H
x
fXa
(x) +fXb
(x)i+fXc
(x)j+fXd
(x)k
d
4
x
=
Z
H
xfXa(x)d
4
x+
Z
H
xfXb
(x)id
4
x
+
Z
H
xfXc
(x)jd
4
x+
Z
H
xfXd
(x)kd
4
x
=E[Xa] +E[Xb]i+E[Xc]j+E[Xd]k; (38)
provided the integral exists. The expected value ofX2His
usually called the mean.
Two quaternion random variablesX=Xa+Xbi+Xcj+
XdkandY=Ya+Ybi+Ycj+Ydkare independent if
(Xa; Ya);(Xb; Yb);(Xc; Yd)and(Xd; Yd)are independent.
Using (2) we can obtain the product of two quaternion
random variablesXandY. If the quaternion variablesX
andYare independent, thenE[XY] =E[X]E[Y].
Corollary 2.IfXandYare quaternion random variables,
then
1)E[ X+ Y] = E[X] + E[Y]for any constants
; 2H;
2)jE[X]j E[jXj]for any the quaternion probability
density functionfX(x)2R; fX(x)>08x2H.
Denition 6.IfXis a quaternion random variable with the
quaternion density functionfX(x), then the characteristic
function X(t)of the random variableXor the distribution
functionF(x)is dened by formula
X(t) =E[e
tX
]
(12)
=E[e
1
2
(t
X+Xt)
]
=
Z
H
e
tx
fX(x)d
4
x: (39)
This shows that the characteristic function X(t)can be
regarded as the quaternion domain Fourier transform of the
density functionfX(x). Applying (12) we can rewrite (39)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
mentioned above in the form
X(t)
=
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXa(x) +fXb
(x)i+fXc(x)j+fXd
(x)k
d
4
x:
=
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXa
(x)d
4
x
+
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXb
(x)d
4
xi
+
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXc
(x)d
4
xj
+
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXd
(x)d
4
xk
= Xa
(t) + Xb
i+ Xc
j+ Xd
k: (40)
Some basic properties of the characteristic function are listed
in the following corollary.
Corollary 3.LetXbe a quaternion random variable with
quaternion density functionfX(x).
1)If quaternion random variablesXandYare indepen-
dent, then
X+Y(t) = X(t) + Y(t); (41)
2) + X(t) =e
t
X(t ).
Proof:For the rst assertion, simple computations gives
X+Y(t)
=E[e
t(X +Y)
]
= Xa+Ya
(t) + Xb+Yb
(t)i+ Xc+Yc
(t)j+ Xd+Yd
(t)k
=
Xa(t) + Xb
(t)i+ Xc(t)j+ Xd
(t)k
+
Ya
(t) + Yb
(t)i+ Yc
(t)j+ Yd
(t)k
= X(t) + Y(t): (42)
For the second, In fact, we have
+ X(t)
=E[e
t( + X )
]
=e
t
E[e
t X
]
=e
t
aXa
(t) + bXb
(t)i+ cXc
(t)j+ dXd
(t)k
=e
t
X(t ): (43)
This is the desired result.
The following theorem is an extension of the Riemann-
Lebesgue lemma to the quaternion density function.
Theorem 5(Riemann-Lebesgue lemma of density function).
For a quaternion density functionfX2L
1
(H;H)the
quaternion characteristic function X(t)satises
lim
jtj!1
j X(t)j= 0; (44)
Proof:Because of
e
tx
= e
t(x+
t
jtj
2
)
; (45)
we have
X(t) =
Z
H
e
tx
fX(x)d
4
x=
Z
H
e
t(x+
t
jtj
2
)
fX(x)d
4
x:
By substitution, we have
X(t)
=
1
2
Z
H
e
tx
fX(x)d
4
x
Z
H
e
t(x+
t
jtj
2
)
fX(x)d
4
x
=
1
2
Z
H
e
ty
f(y)d
4
y
Z
H
e
ty
f(y
t
jtj
2
)d
4
y
=
1
2
Z
H
e
ty
f(y) f(y
t
jtj
2
)
d
4
y:
Hence,
lim
jtj!1
j X(t)j
1
2
lim
jtj!1
Z
H
je
ty
j
f(y) f(y
t
jtj
2
)
d
4
y
=
1
2
lim
jtj!1
Z
H
f(y) f(y
t
jtj
2
)
d
4
y= 0:
The proof is complete.
The following theorem describes an important property of
the characteristic function.
Theorem 6(Continuity). If the quaternion density function
f2L
1
(H;H), then the characteristic function X(t)of a
quaternion-valued random variableXis continuous function
onH.
Proof:It follow directly from the QDFT denition (20)
that
j X(t+h) X(t)j
=
E
h
e
(t+h)X
i
E
e
tX
=
Z
H
e
tx
e
hx
fX(x)d
4
x
Z
H
e
tx
fX(x)d
4
x
=
Z
H
e
tx
(e
hx
1)fX(x)d
4
x
Z
H
j(e
hx
1)jjf X(x)jd
4
x: (46)
We know from the triangle inequality for quaternions that
je
hx
1j je
hx
j+ 1 = 2:
Therefore,
j X(t+h) X(t)j 2
Z
H
jfX(x)jd
4
x:(47)
Applying the Lebesgue dominated convergence theorem to
(46) gives
lim
h!0
j X(t+h) X(t)j= 0: (48)
This shows the characteristic function X(t)is continuous
onH.
Theorem 7(Parseval identity). If the quaternion character-
istic functions X(t)and X(t)of the random variableX
are dened by
X(t) =
Z
H
e
tx
fX(x)d
4
x; X(t) =
Z
H
e
tx
gX(x)d
4
x;
(49)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
wherefX(x)andgX(x)are quaternion density functions
with respect to X(t)and X(t), respectively. Then we have
Z
H
gX(t)e
ty
(t)d
4
t
=
Z
H
X (x y)fX(x)d
4
x+
Z
H
X+(y x)fX(x)d
4
x:
(50)
Proof:Applying the characteristic function (39) we
obtain
e
ty
X(t) =
Z
H
e
t(x y )
fX(x)d
4
x: (51)
Multiplying both sides of the above identity bygX(t)and
then integrating with respect tod
4
twe immediately get
Z
H
gX(t)e
ty
X(t)d
4
t
=
Z
H
Z
H
gX(t)e
t(x y)
fX(x)d
4
x d
4
t
=
Z
H
Z
H
(gX (t) +gX+(t))e
t(x y )
d
4
t
fX(x)d
4
x
=
Z
H
Z
H
e
t(x y )
gX (t)d
4
t
fX(x)d
4
x
+
Z
H
Z
H
e
t( x y)
gX+(t)d
4
t
fX(x)d
4
x
=
Z
H
X (x y)fX(x)d
4
x+
Z
H
X+(y x)fX(x)d
4
x:
(52)
This is the desired result.
The next, we rst observe that
Ff
~
@FXg(t) =Fffg(t) = X(t); (53)
whereFX(x)is quaternion distribution function of random
variableX. Furthermore, application of (34) we easily get
(t) =t( )F fFXg(t); (54)
and thus
FfFXg(t) =t
1
X(t): (55)
As easy consequence of (55), we obtain the following
corollary.
Corollary 4.LetXbe a quaternion random variable. If the
composition of two quaternion distribution functionsFX(x)
andGX(x)is given by
H(x) =FX(x)GX(x) =
Z
H
F(x y)
~
@G(y)d
4
y(56)
the following holds
'(t) = X (t) X(t)) + X+(t) X( t)); (57)
where X(t)and X(t)are the characteristic functions of
the distributions functionsFX(x)andGX(x), respectively.
Proof:Simple computation shows that
t
1
'(t)
=F
Z
H
F(x y)
~
@G(y)d
4
y
(t)
(55)
=FfFX g(t)F f
~
@Gg(t) + FfFX+g(t)F f
~
@Gg( t)
=t
1
X (t) X(t)) +t
1
X+(t) X( t)):
The proof is complete.
From (38) we introduce thenthmoment of a quaternion-
valued random variableXdened by
mn=E[X
n
] =
Z
H
x
n
fX(x)d
4
x; n= 1;2;3; : : : ;(58)
provided the integral exists. It is obvious that forn= 1in
(58) we obtain the rst momentm1(simplym), which is
called the expectation ofX. This gives the following result.
Theorem 8.IfXis quaternion random variable, then
there existsn-th continuous derivatives for the quaternion
characteristic function X(t)which is given formula
~
@
k
X(t) =
k
Z
H
x
k
e
tx
fX(x)d
4
x: (59)
Moreover,
mk=E[X
k
] = ( )
k~
@
k
X(0); k = 1;2;3; : : : ; n:
(60)
Proof:The proof of this theorem is quite similar to the
proof of Theorem 3.
Denition 7.LetXbe a any quaternion random variable.
The variance ofXis dened by
2
= var(X)
=E
h
(X E[X])(X E[X])
i
=
E[X
2
a] E[Xa]
2
+
E[X
2
b] E[Xb]
2
+
E[X
2
c]
E[Xc]
2
+
E[X
2
d] E[Xd]
2
= var(Xa) + var(Xb) + var(Xc) + var(Xd): (61)
Next, we can obtain the variance
2
of a quaternion
random variable in terms of the characteristic function as
2
=
Z
H
(x m)(x m)fX(x)d
4
x
= ( )
2~
@
2
X(0) f( )
~
@ X(0)g
2
=f
~
@ X(0)g
2
~
@
2
X(0):
Example 1.Find the moments of the normal distribution
dened by the density function (compare to [6])
f(x) =
1
p
2
e
(x m) (x m)
2
2
: (62)
It follows from (39) that
(t) =
Z
H
1
p
2
e
jx mj
2
2
2
e
tx
d
4
x:
Making the change of variablex m=ywe obtain
(t) =
Z
H
1
p
2
e
jyj
2
2
2
e
t(m+y )
d
4
y
=
e
tm
p
2
Z
H
e
jyj
2
2
2
e
ty
d
4
y
=
e
tm
p
2
p
2
2
e
2
jtj
2
2
=e
tm
2
jtj
2
2:
Therefore,
~
@ (t) = (m t
2
)e
tm
2
jtj
2
2: (63)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
Combining (60) and (63) yields
m1= ( )
~
@ (0)
= ( )(m) = m
m2=m
2
+
2
m3=m(m
2
+ 3
2
): (64)
This means that the variance of the normal distribution is
2
=m2 m
2
1: (65)
V. CONCLUSION
In this paper, we derived more properties of the QDFT
such as the convolution, correlation and product theorems.
We presented the probability density function of a quaternion
random variable in the framework of quaternion algebra. We
studied the application of the QDFT in probability theory
and mathematical statistics.
REFERENCES
[1] M. Bahri, E. Hitzer, A. Hayashi, and R. Ashino, An Uncertainty Prin-
ciple for Quaternion Fourier Transform,Computer and Mathematics
& Applications, vol. 56, no.9, pp. 24112417, 2008.
[2] M. Bahri, R. Ashino and R. Vaillancourt, Convolution Theorems for
Quaternion Fourier Transform: Properties and Applications,Abstract
and Applied Analysis, vol. 2013, Article ID 162769, 10 pages.
[3] M. Bahri, and R. Ashino, Continuous Quaternion Fourier and Wavelet
Transforms,International Journal of Wavelets, Multiresolution and
Information Processing, vol. 12, no. 4, 21 pages, 2014.
[4] M. Bahri, Quaternion algebra-valued wavelet transform,Applied
Mathematical Sciences, vol.5, no. 71, 3531-3540, 2011.
[5] M. Bahri, Zulfajar, and R. Ashino, Convolution and Correlation
Theorem for Linear Canonical Transform and Properties,Information,
vol.17, no. 6B, pp. 25092521, 2014.
[6] L. Debnath and F. A. Shah,Wavelet Transforms and Their Applications,
Birkh¨auser, USA, 2015.
[7] T. A. Ell, Quaternion Fourier Transform: re-Tooling Image and Signal
Processing Analysis. In: E. Hitzer, S.J. Sangwine (eds.): Quaternion
and Clifford Fourier Transforms and Wavelets. Trends in Mathematics
(TIM), vol.27, pp. 3-14, Birkha¨user, Basel (2013).
[8] M. T. Loots,The Development of the Quaternion Normal Distribution,
Master Thesis, University of Pretoria, Pretoria, 2010.
[9] M. T. Loots, A. Bekker, M. Arashi and J.J. Rou, On the Real Repre-
sentation of Quaternion Random Variables,Statistics A: A Journal of
Theoretical and Applied Statistics, vol. 47, no. 6, pp. 12241240, 2013.
[10] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and
Generalizations,Advances in Applied Clifford Algebras, vol. 17, no.
13, pp. 497517, 2007.
[11] E. Hitzer, Directional Uncertainty Principle for Quaternion Fourier
Transform,Advances in Applied Clifford Algebras, vol. 20, no.2, pp.
271284, 2010.
[12] E. Hitzer, General Two-Sided Quaternion Fourier Transform, Con-
volution and Mustard Convolution,Advances in Applied Clifford
Algebras, vol. 27, no. 1, pp.381395, 2017.
[13] E. Hitzer and S. J. Sangwine, The Orthogonal 2D Planes Split of
Quaternions and Steerable Quaternion Fourier Transformations, In: E.
Hitzer, S.J. Sangwine (eds.) Quaternion and Clifford Fourier Transforms
and Wavelets,Trends in Mathematics. Birkh ¨auser, pp. 15-40, 2013.
[14] E. Hitzer, Quaternionic Wiener-Khinchine Theorems and Spectral
Representation of Convolution With Steerable Two-Sided Quaternion
Fourier Transform,Advances in Applied Clifford Algebras, vol. 27,
no. 2, pp. 13131328, 2017.
[15] E. Hitzer, The Quaternion Domain Fourier Transform and Its Prop-
erties,Advances in Applied Clifford Algebras, vol. 26, no. 3, pp. 969
984, 2016.
[16] S. Mallat,A Wavelet Tour of Signal Processing, Academic Press, 1999.
[17] M. Bahri, A. Lawi, N. Aris, M. Saleh AF and M. Nur, Relation-
ships Between Convolution and Correlation for Fourier Transform and
Quaternion Fourier Transform,Int. Journal of Math. Analysis, vol.7,
no. 43, pp. 21012109, 2013.
[18] S. C. Olhede, On Probability Density Function for Complex Vari-
ables,IEEE Transaction on Information Theory, vol. 52, no. 3, pp.
12121217, 2006.
[19] D.Urynbassarova, B. Z.Li, and Z.C. Zhang, A Convolution Theorem
for the Polynomial Fourier Transform,IAENG International Journal
of Applied Mathematics, vol. 47, no. 4, pp. 381-387, 2017.
Mawardi Bahrireceived his B.S. degree in mathematics from Hasanuddin
University, Makassar, Indonesia, in 1997 and his M.S. degree in mathematics
from the Bandung Institute of Technology, Bandung, Indonesia, in 2001.
He received Ph.D. degree in 2007 from the University of Fukui, Japan.
Presently, he is a professor with the Department of Mathematics, Hasanuddin
University. His research interests are Clifford algebra and wavelet and
Fourier transforms in quaternion algebra.
Amir Kamal Amirreceived his B.S degree in mathematics from Hasanud-
din University, Makassar, Indonesia, in 1990 and his M.Sc. degree in
industrial mathematics from Kaiserslautern University, Germany, in 1998.
He received Ph.D. degree from the Bandung Institute of Technology,
Bandung, Indonesia, in 2011. He is a professor with the Department of
Mathematics, Hasanuddin University. His research interests are ring theory
specially skew polynomial ring. In recent time, he is also working in skew
polynomial ring over quaternion.
Resnawatireceived her B.S. degree in mathematics from from Hasanuddin
University, Makassar, Indonesia, in 2005 and her M.S. degree in mathemat-
ics from Hasanuddin University, Makassar, Indonesia, in 2013. Her research
interests are in the areas of the quaternion Fourier transform, quaternion
linear canonical transform, and time-frequency signal processing.
Chrisandi Landereceived his B.S. degree in mathematics from Hasanuddin
University, Makassar, Indonesia, in 2008 and his M.S. degree in mathematics
from Hasanuddin University, Makassar, Indonesia, in 2017. His research
interests are in the areas of the linear canonical transform, Stockwell
transform, and time-frequency signal processing.IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
Application in Mathematical Statistics
Mawardi Bahri, Amir Kamal Amir, Resnawati, and Chrisandi Lande
AbstractRecently a generalization of the quaternion
Fourier transform over quaternion domains so-called the
quaternion domain Fourier transform (QDFT) has been in-
troduced, including its properties such as shift, modulation,
convolution theorem and uncertainty principle. In the present
paper we explore more properties of the QDFT such as the
correlation and product theorems and propose its application
in probability theory and mathematical statistics.
Index Termsquaternion domain Fourier transform, quater-
nion random variable
I. INTRODUCTION
It is well known that in signal and image processing, the
classical Fourier transform is a very important tool (see,
e.g., [6], [16]). The quaternion Fourier transform (QFT) (for
e.g., [1], [2], [3], [4], [5], [7]) is also very useful tool
for signal processing for quaternion signals with domain
R
2
. The quaternion domain Fourier transform (QDFT) is a
generalization of the QFT over the quaternion domain. The
rst work concerning the denition of the QDFT and its
relation to the denition of the QFT was done by Hitzer
[15]. The QDFT also can regarded as an extension of the
classical Fourier transform (FT) using quaternion algebra. It
transforms quaternion valued signals dened over a quater-
nion domain from a quaternion position space to a quaternion
frequency space. A number of useful properties of the QDFT
have been found including shift, modulation, convolution,
correlation, differentiation, energy conservation, uncertainty
principle and so on. It is well known that the classical
Fourier transform plays crucial roles in probability theory
and mathematical statistics. It is related to the characteristic
function of any real-valued random variable to compute the
distribution function. Therefore, in the present paper, we
rst investigate some important properties of the QDFT such
as derivative, convolution, correlation and product theorems.
We then establish the relationship between the quaternion
characteristic function and the QDFT. We nally apply this
relation to derive the properties of quaternion probability
function and quaternion moments in the framework of the
quaternion algebra of mathematical statistics.
The remainder of this paper is organized as follows.
In Section II we briey review the basic knowledge of
Manuscript received January 11, 2018; revised April 17, 2018. This work
was supported by Research Collaboration Between Indonesian Universities
(PEKERTI) with contract number: 622.48/UN28.2/PL/2017.
Mawardi Bahri is with the Department of Mathematics, Hasanuddin
University, Makassar 90245, Indonesia (corresponding author: e-mail:
mawardibahri@gmail.com)
Amir Kamal Amir is with the Department of Mathematics, Hasanuddin
University, Makassar, Indonesia. e- mail: amirkamalamir@yahoo.com
Resnawati is with the Department of Mathematics, Tadulako University,
Palu 94148, Indonesia. e- mail: r35n4w4t1@yahoo.com
Chrisandi Lande is with Politeknik Ilmu Pelayaran Makassar, Makassar
90165, Indonesia. e-mail: sandhylande@gmail.com
quaternion and derivative operators used in the next section.
In Section III we derive some useful properties of the QDFT
such as the convolution, correlation and product theorems.
In Section IV we discuss the application of the QDFT in
probability theory and mathematical statistics. Finally, in
Section V we give conclusion.
II. QUATERNIONS
We rst review the basic concepts and denition of quater-
nions. The quaternions, a generalization of complex numbers,
are an associative but noncommutative overR. The set of
quaternions is denoted byH. Every element ofHcan be
written in the following form
H=fq=qa+iqb+jqc+kqd;qa; qb; qc; qd2Rg;(1)
which obeys the following multiplication rules:
ij= ji=k;jk= kj=i;ki= ik=j;
i
2
=j
2
=k
2
=ijk= 1: (2)
For a quaternionq=qa+iqb+jqc+kqd2H,qais called
thescalarpart ofqdenoted bySc(q)andiqb+jqc+kqd
is called thevector(orpure) part ofq. The vector part of
qis conventionally denoted byq. Letp,q2Handp,q
be their vector parts, respectively. From (2) we obtain the
quaternionic multiplicationqpas
qp=qapa qp+qap+paq+qp; (3)
where
qp=qbpb+qcpc+qdpd (4)
qp=i(qcpd qdpc) +j(qdpb qbpd) +k(qbpc qcpd):
(5)
Analogously as in the complex case, the quaternion con-
jugate ofqis dened by
q=qa iqb jqc kq3; qa; qb; qc; qd2R:(6)
It is an anti-involution, i.e.
qp= pq: (7)
Notice that conjugate switches the order of multiplication.From (6) we obtain the norm or modulus ofq2Hdened
as
jqj=
p
qq=
q
q
2
a+q
2
b
+q
2
c+q
2
d
: (8)
It is routine to check that
jqpj=jqjjpjandjq+pj jq j+jpj; 8p; q2H:(9)
Using the conjugate (6) and the modulus ofq, we get the
inverse of a non-zero quaternionq2Has
q
1
=
q
jqj
2
; (10)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
which shows thatHis a normed division algebra. When
jqj= 1; qis a unit quaternion. A quaternionqwithqa= 0
is called a pure quaternion and its square is negative sum of
three squares:
q
2
= (q
2
a+q
2
b+q
2
c) = 1: (11)
According to (4) we can get a scalar part of two quaternions
p; qas
pq=Sc(pq) =
1
2
(pq+qp) =paqa+pbqb+pcqc+pdqd:
(12)
A quaternion numberqmay be dened as a complex
number with complex and imaginary parts.
q=z1+jz2; z1=qa+iqb; z2=qc+iqd:(13)
Equation (13) is known as the Cayley-Dickson form.
We dene derivative operators as
~
@=@xa
+@xb
i+@xc
j+@xd
k
@=@xa
@xb
i @xc
j @xd
k; (14)
where@xa=@=@xaand so on.
Using the orthogonal planes split ofq2Hwith respect
to the pure quaternion2H;
2
= 1we dene
q=
1
2
(qq); q =qa+q;
q+=q+q= (q+q) (15)
where rotation operatorR= (i+)i; =RjR
1
and=
RkR
1
;
2
=
2
= 1; qa; q; q; q2R.
Similar to the complex case, we may dene an inner
product for two functionsf; g:H !Has
(f; g)
L
2
(H;H)=
Z
H
f(x)g(x)d
4
x; (16)
wherex2H; d
4
x=dxadxbdxcdxd2R. Notice that
every quaternion domain functionfmapsH !H, its
decomposition will take the form
f(x) =fa(x) +fb(x)i+fc(x)j+fd(x)j (17)
=fa(xa; xb; xc; xd) +fb(xa; xb; xc; xd)i
+fc(xa; xb; xc; xd)j+fd(xa; xb; xc; xd)k; x; y2H;
(18)
where four coefcient functionsfa; fb; fcandfdare in turn
real valued quaternion domain function.
In particular, forf=g, we also may dene theL
2
(H;H)-
norm as
kfk=
Z
H
jf(x)j
2
d
4
x
1=2
: (19)
III. THEQUATERNIONDOMAINFOURIERTRANSFORM
AND ITSUSEFULPROPERTIES
In this section we introduce the quaternion domain Fourier
transform (QDFT) and its basic properties, which is taken
from [15]. We then make some observations about some
further properties related the QDFT such as the correlation
and product theorems, which will be very useful later.
Denition 1.The quaternion domain Fourier transform of
the quaternion functionf2L
1
(H;H)is given by the integral
Fff)g(!) =
Z
H
e
!x
f(x)d
4
x; (20)
wherex; !2Hand some constant2H.
Notice that constant pure quaternioncan be chosen
for each problem. For example, if we take=jand the
quaternion functionfis decomposed as in (13), then the
QDFT takes the form
Fjffg(!) =
Z
H
e
j!x
f(x)d
4
x: (21)
Expandingfin (21) into real and imaginary parts with
respect toiand using the Euler formula for the quaternion
Fourier kernel we obtain
Fjffg(!)
=
Z
H
(cos(! x) +jsin(!x)(f0(x) +f1(x)i)d
4
x
=
Z
H
f0(x) cos(!x)d
4
x+j
Z
H
f0(x) sin(!x)d
4
x
+
Z
H
f1(x) cos(!x)id
4
x+j
Z
H
f1(x) sin(!x)id
4
x:
(22)
Equation (22) clearly shows how the QDFT separates quater-
nion signal into the odd and even parts of real and imaginary
parts in four different components in the QDFT domain.
Denition 2.Iff2L
1
(H;H)and its QDFTFffg 2
L
1
(H;H), then the inverse transform of the QDFT is given
by the integral
F
1
[Fffg](x) =f(x) =
1
2
Z
H
e
!x
Fffg(!)d
4
!:
(23)
Like the polynomial Fourier transform [19], the convolu-
tion of two quaternion functionsf; g2L
1
(H;H)is dened
by
(fg)(x) =
Z
H
f(x y)g(y)d
4
y: (24)
The following theorem provide the convolution theorem
which describes how the QDFT behaves under the quaternion
convolution.
Theorem 1.Supposef2L
1
(H;H)andg2L
1
(H;H)are
integral functions. Then we have
Fffgg(!)
=Fff g(!)Ffgg(!) +Fff+g(!)Ffgg( !):(25)
Moreover,
(fg)(x) =F
1
Fff g(!)Ffgg(!)+
+Fff+g(!)Ffgg( !)
(x): (26)
Proof: LetFffgandFfggdenote the QDFT off
andg, respectively. It follows from(20)and(24)that
Fff ? gg(!) =
Z
H
e
!x
(f ? g)(x)d
4
x
=
Z
H
Z
H
e
!x
f(x y)g(y)d
4
y
d
4
x
=
Z
H
e
!x
f(x y)
Z
H
g(y)d
4
y
d
4
x:
(27)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
After the change of variablesz=x y, the above expression
becomes
Fff ? gg(!)
=
Z
H
Z
H
e
!(y+z)
f(z)g(y)d
4
y d
4
z
=
Z
H
Z
H
e
!z
e
!y
f(z)g(y)d
4
z d
4
y
=
Z
H
Z
H
e
!z
e
!y
(f (z) +f+(z))g(y)d
4
z d
4
y
=
Z
H
Z
H
e
!z
(f (z)e
!y
+f+(z)e
!y
)g(y)d
4
z d
4
y
=
Z
H
Z
H
e
!z
f (z)e
!y
g(y)d
4
z d
4
y
+
Z
H
Z
H
e
!z
f+(z)e
!y
g(y)d
4
z d
4
y
=
Z
H
e
!z
f (z)d
4
z
Z
H
e
!y
g(y)d
4
y
+
Z
H
e
!z
f+(z)d
4
z
Z
H
e
!y
g(y)d
4
y
=Fff g(!)Ffgg(!) +Fff+g(!)Ffgg( !):(28)
In view of(28)and inversion formula for the QDFT(23), the
relation(26)holds. This completes the proof of the theorem.
Denition 3.The correlation for the QDFT of two quater-
nion functionsf; g2L
1
(H;H)is given by
(fg)(x) =
Z
H
f(x+y)g(y)d
4
y: (29)
We derive the following correlation theorem for the QDFT
using the relationship between the quaternion convolution
and quaternion correlation (compare to [17]).
Theorem 2.Supposef2L
1
(H;H)andg2L
1
(H;H)are
integral functions. Thus
Fffgg(!)
=Fff g(!)Ffgg( !) +Fff+g(!)Ffgg(!):(30)
Proof:A simple computation yields
(fg)(x)
=
Z
H
f(x+y)g(y)d
4
y
=
Z
H
f(x u)g( u)d
4
u
=
Z
H
f(x u)h(u)d
4
u
= (fh)(x)
(26)
=F
1
h
Fff g(!)Ffhg(! )
+Fff+g(!)Ffhg( ! )
i
(x): (31)
It is easily seen that
Ffhg(! ) =
Z
H
e
!u
g( u)d
4
u=Ffgg( !);(32)
and
Ffhg( ! ) =
Z
H
e
!u
g( u)d
4
u=Ffgg(!):(33)
Due to (32) and (33), equation (31) can be expressed as
(fg)(x)
=F
1
[Fff g(!)Ffgg( !) +Fff+g(!)Ffgg(!)] (x):
Or, equivalently,
Fffgg(!)
=Fff g(!)Ffgg( !) +Fff+g(!)Ffgg(!);
which was to be proved.
Theorem 3.Supposef2L
1
(R;H)such thatf(x)is con-
tinuousn-times differentiable, then forlimx!1f(x) = 0
the following holds
Ff
~
@
n
fg(!) =!
n
( )
n
Fffg(!); n2N: (34)
Proof:Consider rst the casen= 1. Indeed, we have
Ff
~
@fg(!)
=
Z
H
e
!x
(@xa
+@xb
i+@xc
j+@xd
k)f(x)d
4
x
=
Z
H
e
!x
@xaf(x)d
4
x+
Z
H
e
!x
i@xaf(x)d
4
x
+
Z
H
e
!x
j@xa
f(x)d
4
x+
Z
H
e
!x
k@xa
f(x)d
4
x
=
Z
H
(@xa
e
!x
)f(x)d
4
x
Z
H
(@xb
e
!x
)f(x)d
4
xi
Z
H
(@xb
e
!x
)f(x)d
4
xj
Z
H
(@xb
e
!x
)f(x)d
4
xk
=
Z
H
(!a+!bi+!cj+!dk)( ) e
!x
f(x)d
4
x
=!( )
Z
H
e
tx
f(x)d
4
x:
Using mathematical induction we can nish the proof of the
theorem.
The following theorem describes the relationship between
the product of two quaternion functions and its QDFT.
Theorem 4.Letf; g2L
2
(H;H). Then the QDFT of product
of two quaternion functionsfandgis given by
Fffgg(!)
=
1
(2)
2
(Fqff g Ffgg)(!) + (F f
f+g Fqfgg)(!)
:
(35)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
Proof:Applying the QDFT denition we obtain
Fffgg(!)
=
Z
H
e
!x
f(x)g(x)d
4
x
=
Z
H
e
!x
f(x)
1
(2)
2
Z
H
e
ux
Ffgg(u)d
4
u
d
4
x
=
Z
H
e
!x
f(x)
1
(2)
2
Z
H
e
ux
Ffgg(u)d
4
u
d
4
x
=
1
(2)
2
Z
H
e
!x
(f (x) +f+(x))
Z
H
e
ux
Ffgg(u)d
4
u
d
4
x
=
1
(2)
2
Z
H
Z
H
e
ux
e
!x
f (x)F fgg(u)d
4
u d
4
x
+
1
(2)
2
Z
H
Z
H
e
ux
e
!x
f+(x)F fgg(u)d
4
u d
4
x
=
1
(2)
4
Z
H
Z
H
Z
H
e
ux
e
!x
e
vx
Fff g(v)Ffgg(u)
d
4
u d
4
v d
4
x
+
1
(2)
4
Z
H
Z
H
Z
H
e
ux
e
!x
e
vx
Fff+g(v)Ffgg(u)
d
4
u d
4
v d
4
x
=
1
(2)
4
Z
H
Z
H
"
Z
H
e
(! u v )x
d
4
x
Fff g(v)Ffgg(u)
#
d
4
u d
4
v
+
1
(2)
4
Z
H
Z
H
#
Z
H
e
(!+u v )x
d
4
x
Fff+g(v)Ffgg(u)jbigg]d
4
u d
4
v
=
1
(2)
2
Z
H
Z
H
(w u v)Fff g(v)Ffgg(u)d
4
u d
4
v
+
1
(2)
2
Z
H
Z
H
(!+u v)Fff+g(v)Ffgg(u)d
4
u d
4
v
=
1
(2)
2
Z
H
Fff g(! u)Ffgg(u)d
4
u
+
1
(2)
2
Z
H
Fff+g(!+u)Ffgg(u)d
4
u: (36)
This gives the required result.
As an immediate consequence of the above theorem, we
get the following corollary.
Corollary 1.Letf; g2L
2
(H;H). Assume that the QDFT
ofgis a real-valued function, then Theorem 4 will reduce to
Fqffgg(!) =
1
(2)
2
(Fqffg Fqfgg)(!): (37)
IV. APPLICATION OF THEQUATERNIONDOMAIN
FOURIERTRANSFORM INMATHEMATICALSTATISTICS
Following [8], [9] we dene the probability density func-
tion of a quaternion random variable, which is algebraically
similar to real probability density function of the correspond-
ing associated probability quantity [6].
Denition 4.LetX=Xa+Xbi+Xcj+Xdkbe a quaternion
random variable. A quaternion functionfX(x) =fXa
(x) +
fXb
(x)i+fXc(x)j+fXd
(x)kof the quaternion variable
x=xa+xbi+xcj+xdkis called the quaternion probability
density function (qpdf) ofXif
Z
H
fXi(x)d
4
x= 1; andfXi(x)08x2H; i=a; b; c; d:
We also dene the quaternion cumulative distribution func-
tion (compare to [18])
fX(x) =
~
@FX(x);
where the probabilityP ris related toFXgiven by
FX(x) =P r(Xaxa; Xbxb; Xcxc; Xdxd):
Denition 5(Expected value). LetXbe a quaternion-valued
random variable with quaternion density functionf(x). The
expected valuem=E[X]is dened by
m=E[X]
=
Z
H
xfX(x)d
4
x
=
Z
H
x
fXa
(x) +fXb
(x)i+fXc
(x)j+fXd
(x)k
d
4
x
=
Z
H
xfXa(x)d
4
x+
Z
H
xfXb
(x)id
4
x
+
Z
H
xfXc
(x)jd
4
x+
Z
H
xfXd
(x)kd
4
x
=E[Xa] +E[Xb]i+E[Xc]j+E[Xd]k; (38)
provided the integral exists. The expected value ofX2His
usually called the mean.
Two quaternion random variablesX=Xa+Xbi+Xcj+
XdkandY=Ya+Ybi+Ycj+Ydkare independent if
(Xa; Ya);(Xb; Yb);(Xc; Yd)and(Xd; Yd)are independent.
Using (2) we can obtain the product of two quaternion
random variablesXandY. If the quaternion variablesX
andYare independent, thenE[XY] =E[X]E[Y].
Corollary 2.IfXandYare quaternion random variables,
then
1)E[ X+ Y] = E[X] + E[Y]for any constants
; 2H;
2)jE[X]j E[jXj]for any the quaternion probability
density functionfX(x)2R; fX(x)>08x2H.
Denition 6.IfXis a quaternion random variable with the
quaternion density functionfX(x), then the characteristic
function X(t)of the random variableXor the distribution
functionF(x)is dened by formula
X(t) =E[e
tX
]
(12)
=E[e
1
2
(t
X+Xt)
]
=
Z
H
e
tx
fX(x)d
4
x: (39)
This shows that the characteristic function X(t)can be
regarded as the quaternion domain Fourier transform of the
density functionfX(x). Applying (12) we can rewrite (39)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
mentioned above in the form
X(t)
=
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXa(x) +fXb
(x)i+fXc(x)j+fXd
(x)k
d
4
x:
=
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXa
(x)d
4
x
+
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXb
(x)d
4
xi
+
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXc
(x)d
4
xj
+
Z
H
e
(taxa+tbxb+tcxc+tdxd)
fXd
(x)d
4
xk
= Xa
(t) + Xb
i+ Xc
j+ Xd
k: (40)
Some basic properties of the characteristic function are listed
in the following corollary.
Corollary 3.LetXbe a quaternion random variable with
quaternion density functionfX(x).
1)If quaternion random variablesXandYare indepen-
dent, then
X+Y(t) = X(t) + Y(t); (41)
2) + X(t) =e
t
X(t ).
Proof:For the rst assertion, simple computations gives
X+Y(t)
=E[e
t(X +Y)
]
= Xa+Ya
(t) + Xb+Yb
(t)i+ Xc+Yc
(t)j+ Xd+Yd
(t)k
=
Xa(t) + Xb
(t)i+ Xc(t)j+ Xd
(t)k
+
Ya
(t) + Yb
(t)i+ Yc
(t)j+ Yd
(t)k
= X(t) + Y(t): (42)
For the second, In fact, we have
+ X(t)
=E[e
t( + X )
]
=e
t
E[e
t X
]
=e
t
aXa
(t) + bXb
(t)i+ cXc
(t)j+ dXd
(t)k
=e
t
X(t ): (43)
This is the desired result.
The following theorem is an extension of the Riemann-
Lebesgue lemma to the quaternion density function.
Theorem 5(Riemann-Lebesgue lemma of density function).
For a quaternion density functionfX2L
1
(H;H)the
quaternion characteristic function X(t)satises
lim
jtj!1
j X(t)j= 0; (44)
Proof:Because of
e
tx
= e
t(x+
t
jtj
2
)
; (45)
we have
X(t) =
Z
H
e
tx
fX(x)d
4
x=
Z
H
e
t(x+
t
jtj
2
)
fX(x)d
4
x:
By substitution, we have
X(t)
=
1
2
Z
H
e
tx
fX(x)d
4
x
Z
H
e
t(x+
t
jtj
2
)
fX(x)d
4
x
=
1
2
Z
H
e
ty
f(y)d
4
y
Z
H
e
ty
f(y
t
jtj
2
)d
4
y
=
1
2
Z
H
e
ty
f(y) f(y
t
jtj
2
)
d
4
y:
Hence,
lim
jtj!1
j X(t)j
1
2
lim
jtj!1
Z
H
je
ty
j
f(y) f(y
t
jtj
2
)
d
4
y
=
1
2
lim
jtj!1
Z
H
f(y) f(y
t
jtj
2
)
d
4
y= 0:
The proof is complete.
The following theorem describes an important property of
the characteristic function.
Theorem 6(Continuity). If the quaternion density function
f2L
1
(H;H), then the characteristic function X(t)of a
quaternion-valued random variableXis continuous function
onH.
Proof:It follow directly from the QDFT denition (20)
that
j X(t+h) X(t)j
=
E
h
e
(t+h)X
i
E
e
tX
=
Z
H
e
tx
e
hx
fX(x)d
4
x
Z
H
e
tx
fX(x)d
4
x
=
Z
H
e
tx
(e
hx
1)fX(x)d
4
x
Z
H
j(e
hx
1)jjf X(x)jd
4
x: (46)
We know from the triangle inequality for quaternions that
je
hx
1j je
hx
j+ 1 = 2:
Therefore,
j X(t+h) X(t)j 2
Z
H
jfX(x)jd
4
x:(47)
Applying the Lebesgue dominated convergence theorem to
(46) gives
lim
h!0
j X(t+h) X(t)j= 0: (48)
This shows the characteristic function X(t)is continuous
onH.
Theorem 7(Parseval identity). If the quaternion character-
istic functions X(t)and X(t)of the random variableX
are dened by
X(t) =
Z
H
e
tx
fX(x)d
4
x; X(t) =
Z
H
e
tx
gX(x)d
4
x;
(49)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
wherefX(x)andgX(x)are quaternion density functions
with respect to X(t)and X(t), respectively. Then we have
Z
H
gX(t)e
ty
(t)d
4
t
=
Z
H
X (x y)fX(x)d
4
x+
Z
H
X+(y x)fX(x)d
4
x:
(50)
Proof:Applying the characteristic function (39) we
obtain
e
ty
X(t) =
Z
H
e
t(x y )
fX(x)d
4
x: (51)
Multiplying both sides of the above identity bygX(t)and
then integrating with respect tod
4
twe immediately get
Z
H
gX(t)e
ty
X(t)d
4
t
=
Z
H
Z
H
gX(t)e
t(x y)
fX(x)d
4
x d
4
t
=
Z
H
Z
H
(gX (t) +gX+(t))e
t(x y )
d
4
t
fX(x)d
4
x
=
Z
H
Z
H
e
t(x y )
gX (t)d
4
t
fX(x)d
4
x
+
Z
H
Z
H
e
t( x y)
gX+(t)d
4
t
fX(x)d
4
x
=
Z
H
X (x y)fX(x)d
4
x+
Z
H
X+(y x)fX(x)d
4
x:
(52)
This is the desired result.
The next, we rst observe that
Ff
~
@FXg(t) =Fffg(t) = X(t); (53)
whereFX(x)is quaternion distribution function of random
variableX. Furthermore, application of (34) we easily get
(t) =t( )F fFXg(t); (54)
and thus
FfFXg(t) =t
1
X(t): (55)
As easy consequence of (55), we obtain the following
corollary.
Corollary 4.LetXbe a quaternion random variable. If the
composition of two quaternion distribution functionsFX(x)
andGX(x)is given by
H(x) =FX(x)GX(x) =
Z
H
F(x y)
~
@G(y)d
4
y(56)
the following holds
'(t) = X (t) X(t)) + X+(t) X( t)); (57)
where X(t)and X(t)are the characteristic functions of
the distributions functionsFX(x)andGX(x), respectively.
Proof:Simple computation shows that
t
1
'(t)
=F
Z
H
F(x y)
~
@G(y)d
4
y
(t)
(55)
=FfFX g(t)F f
~
@Gg(t) + FfFX+g(t)F f
~
@Gg( t)
=t
1
X (t) X(t)) +t
1
X+(t) X( t)):
The proof is complete.
From (38) we introduce thenthmoment of a quaternion-
valued random variableXdened by
mn=E[X
n
] =
Z
H
x
n
fX(x)d
4
x; n= 1;2;3; : : : ;(58)
provided the integral exists. It is obvious that forn= 1in
(58) we obtain the rst momentm1(simplym), which is
called the expectation ofX. This gives the following result.
Theorem 8.IfXis quaternion random variable, then
there existsn-th continuous derivatives for the quaternion
characteristic function X(t)which is given formula
~
@
k
X(t) =
k
Z
H
x
k
e
tx
fX(x)d
4
x: (59)
Moreover,
mk=E[X
k
] = ( )
k~
@
k
X(0); k = 1;2;3; : : : ; n:
(60)
Proof:The proof of this theorem is quite similar to the
proof of Theorem 3.
Denition 7.LetXbe a any quaternion random variable.
The variance ofXis dened by
2
= var(X)
=E
h
(X E[X])(X E[X])
i
=
E[X
2
a] E[Xa]
2
+
E[X
2
b] E[Xb]
2
+
E[X
2
c]
E[Xc]
2
+
E[X
2
d] E[Xd]
2
= var(Xa) + var(Xb) + var(Xc) + var(Xd): (61)
Next, we can obtain the variance
2
of a quaternion
random variable in terms of the characteristic function as
2
=
Z
H
(x m)(x m)fX(x)d
4
x
= ( )
2~
@
2
X(0) f( )
~
@ X(0)g
2
=f
~
@ X(0)g
2
~
@
2
X(0):
Example 1.Find the moments of the normal distribution
dened by the density function (compare to [6])
f(x) =
1
p
2
e
(x m) (x m)
2
2
: (62)
It follows from (39) that
(t) =
Z
H
1
p
2
e
jx mj
2
2
2
e
tx
d
4
x:
Making the change of variablex m=ywe obtain
(t) =
Z
H
1
p
2
e
jyj
2
2
2
e
t(m+y )
d
4
y
=
e
tm
p
2
Z
H
e
jyj
2
2
2
e
ty
d
4
y
=
e
tm
p
2
p
2
2
e
2
jtj
2
2
=e
tm
2
jtj
2
2:
Therefore,
~
@ (t) = (m t
2
)e
tm
2
jtj
2
2: (63)IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________
Combining (60) and (63) yields
m1= ( )
~
@ (0)
= ( )(m) = m
m2=m
2
+
2
m3=m(m
2
+ 3
2
): (64)
This means that the variance of the normal distribution is
2
=m2 m
2
1: (65)
V. CONCLUSION
In this paper, we derived more properties of the QDFT
such as the convolution, correlation and product theorems.
We presented the probability density function of a quaternion
random variable in the framework of quaternion algebra. We
studied the application of the QDFT in probability theory
and mathematical statistics.
REFERENCES
[1] M. Bahri, E. Hitzer, A. Hayashi, and R. Ashino, An Uncertainty Prin-
ciple for Quaternion Fourier Transform,Computer and Mathematics
& Applications, vol. 56, no.9, pp. 24112417, 2008.
[2] M. Bahri, R. Ashino and R. Vaillancourt, Convolution Theorems for
Quaternion Fourier Transform: Properties and Applications,Abstract
and Applied Analysis, vol. 2013, Article ID 162769, 10 pages.
[3] M. Bahri, and R. Ashino, Continuous Quaternion Fourier and Wavelet
Transforms,International Journal of Wavelets, Multiresolution and
Information Processing, vol. 12, no. 4, 21 pages, 2014.
[4] M. Bahri, Quaternion algebra-valued wavelet transform,Applied
Mathematical Sciences, vol.5, no. 71, 3531-3540, 2011.
[5] M. Bahri, Zulfajar, and R. Ashino, Convolution and Correlation
Theorem for Linear Canonical Transform and Properties,Information,
vol.17, no. 6B, pp. 25092521, 2014.
[6] L. Debnath and F. A. Shah,Wavelet Transforms and Their Applications,
Birkh¨auser, USA, 2015.
[7] T. A. Ell, Quaternion Fourier Transform: re-Tooling Image and Signal
Processing Analysis. In: E. Hitzer, S.J. Sangwine (eds.): Quaternion
and Clifford Fourier Transforms and Wavelets. Trends in Mathematics
(TIM), vol.27, pp. 3-14, Birkha¨user, Basel (2013).
[8] M. T. Loots,The Development of the Quaternion Normal Distribution,
Master Thesis, University of Pretoria, Pretoria, 2010.
[9] M. T. Loots, A. Bekker, M. Arashi and J.J. Rou, On the Real Repre-
sentation of Quaternion Random Variables,Statistics A: A Journal of
Theoretical and Applied Statistics, vol. 47, no. 6, pp. 12241240, 2013.
[10] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and
Generalizations,Advances in Applied Clifford Algebras, vol. 17, no.
13, pp. 497517, 2007.
[11] E. Hitzer, Directional Uncertainty Principle for Quaternion Fourier
Transform,Advances in Applied Clifford Algebras, vol. 20, no.2, pp.
271284, 2010.
[12] E. Hitzer, General Two-Sided Quaternion Fourier Transform, Con-
volution and Mustard Convolution,Advances in Applied Clifford
Algebras, vol. 27, no. 1, pp.381395, 2017.
[13] E. Hitzer and S. J. Sangwine, The Orthogonal 2D Planes Split of
Quaternions and Steerable Quaternion Fourier Transformations, In: E.
Hitzer, S.J. Sangwine (eds.) Quaternion and Clifford Fourier Transforms
and Wavelets,Trends in Mathematics. Birkh ¨auser, pp. 15-40, 2013.
[14] E. Hitzer, Quaternionic Wiener-Khinchine Theorems and Spectral
Representation of Convolution With Steerable Two-Sided Quaternion
Fourier Transform,Advances in Applied Clifford Algebras, vol. 27,
no. 2, pp. 13131328, 2017.
[15] E. Hitzer, The Quaternion Domain Fourier Transform and Its Prop-
erties,Advances in Applied Clifford Algebras, vol. 26, no. 3, pp. 969
984, 2016.
[16] S. Mallat,A Wavelet Tour of Signal Processing, Academic Press, 1999.
[17] M. Bahri, A. Lawi, N. Aris, M. Saleh AF and M. Nur, Relation-
ships Between Convolution and Correlation for Fourier Transform and
Quaternion Fourier Transform,Int. Journal of Math. Analysis, vol.7,
no. 43, pp. 21012109, 2013.
[18] S. C. Olhede, On Probability Density Function for Complex Vari-
ables,IEEE Transaction on Information Theory, vol. 52, no. 3, pp.
12121217, 2006.
[19] D.Urynbassarova, B. Z.Li, and Z.C. Zhang, A Convolution Theorem
for the Polynomial Fourier Transform,IAENG International Journal
of Applied Mathematics, vol. 47, no. 4, pp. 381-387, 2017.
Mawardi Bahrireceived his B.S. degree in mathematics from Hasanuddin
University, Makassar, Indonesia, in 1997 and his M.S. degree in mathematics
from the Bandung Institute of Technology, Bandung, Indonesia, in 2001.
He received Ph.D. degree in 2007 from the University of Fukui, Japan.
Presently, he is a professor with the Department of Mathematics, Hasanuddin
University. His research interests are Clifford algebra and wavelet and
Fourier transforms in quaternion algebra.
Amir Kamal Amirreceived his B.S degree in mathematics from Hasanud-
din University, Makassar, Indonesia, in 1990 and his M.Sc. degree in
industrial mathematics from Kaiserslautern University, Germany, in 1998.
He received Ph.D. degree from the Bandung Institute of Technology,
Bandung, Indonesia, in 2011. He is a professor with the Department of
Mathematics, Hasanuddin University. His research interests are ring theory
specially skew polynomial ring. In recent time, he is also working in skew
polynomial ring over quaternion.
Resnawatireceived her B.S. degree in mathematics from from Hasanuddin
University, Makassar, Indonesia, in 2005 and her M.S. degree in mathemat-
ics from Hasanuddin University, Makassar, Indonesia, in 2013. Her research
interests are in the areas of the quaternion Fourier transform, quaternion
linear canonical transform, and time-frequency signal processing.
Chrisandi Landereceived his B.S. degree in mathematics from Hasanuddin
University, Makassar, Indonesia, in 2008 and his M.S. degree in mathematics
from Hasanuddin University, Makassar, Indonesia, in 2017. His research
interests are in the areas of the linear canonical transform, Stockwell
transform, and time-frequency signal processing.IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_11 (Advance online publication: 28 May 2018)
______________________________________________________________________________________