Journal of Applied Mathematics and Physics, 2013, 1, 36-38
http://dx.doi.org/10.4236/jamp.2013.14007 Published Online October 2013 (http://www.scirp.org/journal/jamp)
Copyright © 2013 SciRes. JAMP
Spectrum of Signals on the Quaternion
Fourier Transform Domain
Guicheng Yang1, Yingxiong Fu2*
1Business College, Hubei University of Technology, Wuhan, China
2Faculty of Mathematics and Computer Science, Hubei University, Wuhan, China
Email: *834082116@qq.com, fyx@hubu.edu.cn
Received August 2013
ABSTRACT
The quatern ion Fourier transform plays a vital role in the representation of two-dimensional signals. This paper charac-
terizes spectrum of quaternion-valued signals on the quaternion Fourier transform domain by the partial derivative.
Keywords: Spectrum; Quaternion Fourier Transform; Partial Derivative
1. Introduction
The quaternion Fourier transform (QFT) is a nontrivial
generalization of the real and complex Fourier transform
to quaternion cases. The four QFT components separate
four cases of symmetry in real signals instead of only
two in the complex FT. The QFT plays a vital role in the
representation of signals and transforms a quaternion 2D
signal into a quaternion-valued frequency domain signal.
Many efforts had been devoted to some important prop-
erties and appli c a t ions of the QFT [1-7].
In the last few years, there has been a great interest to
the study of the spectrum of signals, i.e. the support of
the transform of these signals relatively to certain
integral transforms [8-15].
Motivated by the treatment of the QFT in quaternion
algebra, in this paper we will characterize the quater-
nion-valued signals whose QFT has compact support.
The main difficulty lies in the fact that the quaternion
algebra is non-commutative, so one cannot directly ex-
tend the results for the Fourier transform to those for the
QFT.
This paper is organized as follows: Section 2 is de-
voted to reviewing some necessary results about the qua-
ternion algebra. In Section 3, based on the definition and
some properties of the QFT, we get a result to describe
the spectrum for the QFT.
2. Preliminaries
The quaternion algebra
is an extension of the alge-
bra of complex numbers to a four dimensional real alg e-
bra. It is given by
{ }
0 1230123
|,,,, ,q qiqjqkqqqqq=++ +∈  
where the elements
,,i jk
obey Hamilton’s multiplica-
tion rules
222
,,
1.
ijjikjkkjikiikj
ij kijk
=−==−= =−=
====−
The conjugate of a quaternion
q
is obtained by
changing the sign of the pure quaternion part, i.e.,
00123
.q qq qiqjqkq
= −= −−−
The modulus
q
of a
quaternion
q
is defined by
2222
01 23
.qqq qqqq== +++
Using the conjugate and
the modulus of a quaternion, we can define the inverse of
{ }
\0q
by
which shows that
is a
normed division algebra. Moreover, for arbitrary
,ab
the following i de ntity hol ds
.aba b=
(2. 1)
We introduce the space
( )
22
;L 
as the left mod-
ule of all quaternion-valued functions
2
:f 
with
finite norm
()
2
1
22
2
()ffx dx=

(2.2)
where
212
dxdx dx=
represents the usual Lebesgue
measure in
2
. Moreover, denote the space
( )
12
;L 
the left module of all quaternion-valued functions
2
:f 
satisfying
22
()fxdx<∞

.
3. Main Results
Note that
( )( )
12 22
;;LL  
is dense in
*Corresponding a uthor.
G.-C. YANG, Y.-X. FU
Copyright © 2013 SciRes. JAMP
37
( )
22
;.L 
Hence, standard density arguments allow us
to extend the definition of the QFT of
( )
12
;fL 
in a unique way to the whole of
( )
22
;.L 
We give
the following definition of the QFT as an operator from
( )
22
;L 
into
( )
22
;L 
[3].
Definition 1
The tw o-sided QFT of
( )
22
;fL 
is the function
q
Ff
define d by
( )( )( )
112 2
2
2
2
1
2
ix jx
q
Ffef xedx
ωω
ωπ
−−
=
(3.1)
with arbitrary frequency
12
(, )
ω ωω
=
.
The QFT can be inverted by
( )( )( )
112 2
2
2
2
1
2
ix jx
q
f xeFfed
ωω
ωω
π
−−
=
with
212
d dd
ω ωω
=
.
In what follows, we review some properties of the
QFT, such as the Parseval theorem and the partial deriva-
tive. For more details, we refer to [3].
Lemma 2
For
( )
22
;fL 
we have
12 ,
q
f Ff
π
=
where the norm
is defined by Equat i on (2.2).
Lemma 3
If
( )
( )
22 0
12
; ,,
mn
mn
f xLmn
xx
+
∈∈
∂∂  
and
( )
22
;fL 
. Then we have
()()()
12
12
{} ,
mn m mnn
qq
mn
Ff xiFfj
xx
ω ωωω
+
=
∂∂
where the QFT
q
Ff
is defined by Equation (3.1).
Given a multi-index
( )
2
12
,,
α αα
+
= ∈
we write as
usual
12
,
ααα
= +
12
12
12
Dxx
αα
α
αα
∂∂
=∂∂
for the partial de-
rivative.
Moreover, we denote by
supp
q
Ff
the support of
q
Ff
describing the smallest close set in
2
outside
which
q
Ff
vanishes almost everywhere. The following
theorem describes the spectrum of signals for the QFT,
i.e. the compactness of the support of
q
Ff
by means of
the norm of its partial derivative on
2
.
Theorem 4
Let
( )
22
;fL 
. Then the QFT
( )
q
Ff
ω
is
compactly supported in
[ ]
2
,
σσ
if and only if partial
derivatives
( )
22
;,Df L
α
 
( )
12
22
12
;
q
Ff L
αα
ωω
 
for all
2
α
+
and
1
lim ,Df
α
α
α
σ
→∞
=
where
( )
{ }
2
sup,1, 2:0,.
kq
k Ff
σ ωωω
= =≠∈ 
PROOF. Firstly, we prove the necessity. Suppose that
( )
supp
q
Ff
ω
=
[ ]
2
,
σσ
. The compactness of the
support of
q
Ff
and
( )
22
;fL 
imply that
12
12q
Ff
αα
ωω
belongs to
( )( )
12 22
;;LL  
, thus
partial derivatives
Df
α
exist and belong to
( )
22
;L 
for all
2.
α
+
Moreover, by Lemma 3 we
have
{ }
11 22
12
()
qq
FDfiFfj
αα αα
α
ωωω
=
.
Applying Lemma 2, it follows
( )( )
11 22
2
2
22
2
2
1,
2
q
D fiFfjd
αα αα
α
ω ωωω
π
=
that is,
[ ]
12
2
22
22 2
12
2,
1()
(2 )q
D fFfd
αα
α
σσ
ωωω ω
π
=
(3.2)
based on Equation (2.1) and
12
1.ij
αα
= =
Thus, we
obtain
( )( )
[ ]
( )
12
2
22
22 2
12
2,
22
22
2
1
2
1,
2
q
q
D fFfd
Ff f
αα
α
σσ
αα
ωωω ω
π
σσ
π
=
≤=
which leads to
11
Df C
αα
α
σ
with the constant
Cf=
independent of .
α
Then, we have
1
limsup Df
α
α
α
σ
→∞
due to
1
lim 1C
α
α
→∞ =
for all
0C< <∞
. On the other
hand, using Equation (3.2) again, for
(0, 2),
εσ
it
holds
( )()
[ ]
() ( )( )
[ ]
12
2
2
22
22 2
12
22,
2
22
22,
1
2
1
2,
2
q
q
D fFfd
Ff d
αα
α
σεσε
α
σεσε
ωωωω
π
σεω ω
π
−−
−−
≥−
which leads to
1
liminf2 .Df
α
α
α
σε
→∞ ≥−
The arbitrariness of
ε
implies
1
liminf Df
α
α
α
σ
→∞
.
Therefore, we can conclude that
1
lim Df
α
α
α
σ
→∞
=
.
Secondly, we prove the sufficiency. Suppose that par-
tial derivatives
( )
( )
22
;,Dfx L
α
 
( )
12
22
12
;
q
Ff L
αα
ωω
 
for all
2
α
+
and
G.-C. YANG, Y.-X. FU
Copyright © 2013 SciRes. JAMP
38
1
lim Dfd
α
α
α
→∞
= <∞
. (3.3)
We need to prove that
sup{,1,2:( )0},
kq
k Ff
σω ω
= =≠
Otherwise,
() 0
q
Ff
ω
holds for almost everywhere
2
ω
and thus, associated with Equation (3.2) we ob-
tain that for arbitrary
M
it hol ds
( )( )
( )( )
12
2
12
22
22 2
12
2
22
22 2
12
2
1
2
1,
2
q
q
E
D fFfd
F fdCM
αα
α
α
αω
ωωω ω
π
ωωω ω
π
=
≥≥
where
{ }
2
:,1, 2
k
E Mk
ωω
=∈ ≥=
and
C
is some
positive constant independent of
α
, that is to say,
11
.Df MC
αα
α
(3.4)
The above inequality (3.4) implies
1
lim Df
α
α
α
→∞
= ∞
,
which contradicts the assumption (3.3). Thus, we have
sup{,1,2:( )0},
kq
k Ff
σω ω
==≠ <∞
which means
( )
q
Ff
ω
is compactly supported in
[ ]
2
,
σσ
. Finally, the same technique as the part of the
proof for the necessity yields that
d
σ
=
. Thus, the
proof is complete.
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