Spectrum of Signals on the Quaternion Fourier Transform Domain

doi:10.4236/jamp.2013.14007

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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)

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: *[email protected], [email protected]

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

ijjikjkkjikiikj

ij kijk

=−==−= =−=

====−

The conjugate of a quaternion

is obtained by

changing the sign of the pure quaternion part, i.e.,

00123

.q qq qiqjqkq

−= −= −−−

The modulus

of a

quaternion

is defined by

2222

01 23

.qqq qqqq== +++

Using the conjugate and

the modulus of a quaternion, we can define the inverse of

{ }

\0q∈

−=

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

( )

;L 

as the left mod-

ule of all quaternion-valued functions

:f→ 

with

finite norm

()

()ffx dx=

∫



(2.2)

where

212

dxdx dx=

represents the usual Lebesgue

measure in



. Moreover, denote the space

( )

;L 

the left module of all quaternion-valued functions

:f→ 

satisfying

()fxdx<∞

∫



3. Main Results

Note that

( )( )

12 22

;;LL  

is dense in

*Corresponding a uthor.

G.-C. YANG, Y.-X. FU

( )

;.L 

Hence, standard density arguments allow us

to extend the definition of the QFT of

( )

;fL∈ 

in a unique way to the whole of

( )

;.L 

We give

the following definition of the QFT as an operator from

( )

;L 

into

( )

;L 

[3].

Definition 1

The tw o-sided QFT of

( )

;fL∈ 

is the function

define d by

( )( )( )

112 2

ix jx

Ffef xedx

ωω

ωπ

−−

=∫

(3.1)

with arbitrary frequency

(, )

ω ωω

The QFT can be inverted by

( )( )( )

112 2

ix jx

f xeFfed

ωω

−−

∫

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

( )

;fL∈ 

we have

12 ,

f Ff

where the norm

⋅

is defined by Equat i on (2.2).

Lemma 3

( )

22 0

; ,,

f xLmn

∂∈∈

∂∂  

and

( )

;fL∈ 

. Then we have

()()()

{} ,

mn m mnn

Ff xiFfj

ω ωωω

∂=

∂∂

where the QFT

is defined by Equation (3.1).

Given a multi-index

( )

α αα

= ∈

we write as

usual

ααα

= +

Dxx

αα

∂∂

=∂∂

for the partial de-

rivative.

Moreover, we denote by

supp

the support of

describing the smallest close set in



outside

which

vanishes almost everywhere. The following

theorem describes the spectrum of signals for the QFT,

i.e. the compactness of the support of

by means of

the norm of its partial derivative on



Theorem 4

Let

( )

;fL∈ 

. Then the QFT

( )

compactly supported in

[ ]

σσ

−

if and only if partial

derivatives

( )

;,Df L

∈ 

( )

;

Ff L

αα

ωω

∈ 

for all

∈

and

lim ,Df

→∞

where

( )

{ }

sup,1, 2:0,.

k Ff

σ ωωω

= =≠∈ 

PROOF. Firstly, we prove the necessity. Suppose that

( )

supp

[ ]

σσ

−

. The compactness of the

support of

and

( )

;fL∈ 

imply that

12q

αα

ωω

belongs to

( )( )

12 22

;;LL  

, thus

partial derivatives

exist and belong to

( )

;L 

for all

∈

Moreover, by Lemma 3 we

have

{ }

11 22

()

FDfiFfj

αα αα

ωωω

Applying Lemma 2, it follows

( )( )

11 22

D fiFfjd

αα αα

ω ωωω

=∫



that is,

[ ]

22 2

1()

(2 )q

D fFfd

αα

σσ

ωωω ω

−

=∫

(3.2)

based on Equation (2.1) and

1.ij

αα

= =

Thus, we

obtain

( )( )

[ ]

( )

22 2

D fFfd

Ff f

αα

σσ

αα

ωωω ω

σσ

−

≤=

∫

which leads to

Df C

αα

≤

with the constant

Cf=

independent of .

Then, we have

limsup Df

→∞

≤

due to

lim 1C

→∞ =

for all

0C< <∞

. On the other

hand, using Equation (3.2) again, for

(0, 2),

εσ

∈

holds

( )()

[ ]

() ( )( )

[ ]

22 2

22,

D fFfd

Ff d

αα

σεσε

ωωωω

σεω ω

−−

≥

≥−

∫

which leads to

liminf2 .Df

σε

→∞ ≥−

The arbitrariness of

implies

liminf Df

→∞

≥

Therefore, we can conclude that

lim Df

→∞

Secondly, we prove the sufficiency. Suppose that par-

tial derivatives

( )

;,Dfx L

∈ 

( )

;

Ff L

αα

ωω

∈ 

for all

∈

and

G.-C. YANG, Y.-X. FU

lim Dfd

→∞

= <∞

. (3.3)

We need to prove that

sup{,1,2:( )0},

k Ff

σω ω

= =≠

Otherwise,

() 0

≠

holds for almost everywhere

∈

and thus, associated with Equation (3.2) we ob-

tain that for arbitrary

it hol ds

( )( )

22 2

D fFfd

F fdCM

αα

αω

ωωω ω

≥≥

∫



where

{ }

:,1, 2

E Mk

ωω

=∈ ≥=

and

is some

positive constant independent of

, that is to say,

.Df MC

αα

≥

(3.4)

The above inequality (3.4) implies

lim Df

→∞

= ∞

which contradicts the assumption (3.3). Thus, we have

sup{,1,2:( )0},

k Ff

σω ω

==≠ <∞

which means

( )

is compactly supported in

[ ]

σσ

−

. Finally, the same technique as the part of the

proof for the necessity yields that

. Thus, the

proof is complete.

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