CONVOLUTION THEOREMS FOR CLIFFORD FOURIER TRANSFORM AND PROPERTIES | Bahri | Journal of the Indonesian Mathematical Society 143 496 1 PB
J. Indones. Math. Soc.
Vol. 20, No. 2 (2014), pp. 125–140.
CONVOLUTION THEOREMS FOR CLIFFORD
FOURIER TRANSFORM AND PROPERTIES
´mi Vaillancourt3
Mawardi Bahri1 , Ryuichi Ashino2 , and Re
1
3
Department of Mathematics, Hasanuddin University
Tamalanrea Makassar 90245,Indonesia
[email protected]
2
Division of Mathematical Sciences, Osaka Kyoiku University
Osaka 582-8582, Japan
[email protected]
Department of Mathematics and Statistics, University of Ottawa
Ottawa, Ontario, Canada K1N 6N5
[email protected]
Abstract. The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis. We establish main properties of convolution
theorems for the Clifford Fourier transform. Some properties of these generalized
convolutions are extensions of the corresponding convolution theorems of the classical Fourier transform.
Key words: Clifford convolution, Clifford algebra, Clifford Fourier transform.
Abstrak. Tujuan tulisan ini adalah untuk membangun sifat-sifat penting dari teorema konvolusi untuk transformasi Fourier Clifford (TFC). Karena perkalian Clifford adalah tidak komutatif, maka diperoleh sifat-sifat konvolusi dari TFC ini adalah
perluasan dari konvolusi dari transformasi Fourier.
Kata kunci: Konvolusi Clifford, aljabar Clifford, transformasi Fourier Clifford.
1. Introduction
Recently, several attempts have been made to generalize the classical Fourier transform in the framework of Clifford algebra, so-called the Clifford Fourier transform
(CFT). It was first introduced from the mathematical aspect by Brackx et al.
[5, 6]. The CFT was recently used in signal processing [9, 14] and in other fields of
mathematics and applications. Many generalized transforms, such as the Clifford
2000 Mathematics Subject Classification:15A66, 42B10.
Received: 21-09-2013, revised: 22-04-2014, accepted: 21-05-2014.
125
M. Bahri et al.
126
wavelet transform, fractional Clifford Fourier transform, and Clifford windowed
Fourier transform (see, for example, [2, 7, 8, 10, 12, 15]) are closely related to the
CFT. One of the most fundamental and important properties of the CFT is the
convolution theorem.
Convolution is a mathematical operation with several applications in pure and
applied mathematics such as numerical analysis, numerical linear algebra and the
design and implementation of finite impulse response filters in signal processing. In
[3, 13], authors generalized convolution to the quaternion Fourier transform (QFT).
They found that the QFT of the real-valued signals are very similar to the classical
ones. In this paper, we establish convolution theorems for the CFT. Here we adopt
the definition of the CFT suggested by several authors [1, 4, 9]. Because the Clifford
multiplication is not commutative, we find important properties of the relationship
between the convolution theorems and the CFT. We finally establish the inverse
CFT of the product of two CFTs, which is very useful in solving partial differential
equations in the Clifford algebra.
This paper is organized as follows. In section 2, we provide some basic knowledge of real Clifford algebra used in the paper. Subsequently, in section 3, we define
the CFT and discuss its important properties, which are used to construct the properties of generalized convolution. Next in section 4, we introduce convolution on
Clifford algebra Cln,0 and derive its useful properties. Finally, in section 5, we
investigate the important properties of the CFT of convolution of Clifford-valued
functions and derive the inverse CFT of the product of two CFTs.
2. Preliminaries
We shall be working with real Clifford algebras. Let {e1 , e2 , e3 , · · · , en } be
an orthonormal vector basis of the n-dimensional Euclidean vector space Rn . The
real Clifford algebra over Rn , denoted by Cln,0 , has the graded 2n -dimensional
basis
{1, e1 , e2 , · · · , en , e12 , e31 , e23 , · · · , in = e1 e2 · · · en }.
(1)
Obviously, for n = 2(mod 4), the pseudoscalar in = e1 e2 · · · en anti-commutes with
each basis of the Clifford algebra while i2n = −1. The noncommutative multiplication of the basis vectors satisfies the rules ei ej + ej ei = 2δij (δij denotes the Dirac
distribution whose support is {i, j}).
An element of the Clifford algebra is called a multivector and has the following
form
∑
e A fA ,
f=
A
where fA ∈ R, A ⊂ {1, 2, · · · , n}. For convenience, we introduce ⟨f ⟩k =
to denote the k-vector part of f (k = 0, 1, 2, · · · , n). Then
f=
k=n
∑
⟨f ⟩k = ⟨f ⟩ + ⟨f ⟩1 + ⟨f ⟩2 + · · · + ⟨f ⟩n ,
k=0
where ⟨. . .⟩ = ⟨. . .⟩0 .
∑
A
f A eA
Convolution Theorems for Clifford Fourier Transform
127
A multivector f ∈ Cln,0 , n = 2 (mod 4) can be decomposed as a sum of its
even grade part, feven , and its odd grade part, fodd . Thus, we have
f = feven ⊕ fodd ,
(2)
where
n
,
2
feven = ⟨f ⟩ + ⟨f ⟩2 + · · · + ⟨f ⟩r ,
r = 2s, s ∈ N, s ≤
fodd = ⟨f ⟩1 + ⟨f ⟩3 + · · · + ⟨f ⟩r ,
r = 2s + 1, s ∈ N, s <
n
.
2
The reverse f˜ of a multivector f is an anti-automorphism given by
fe =
and hence
k=n
∑
(−1)k(k−1)/2 ⟨f ⟩k ,
(3)
k=0
ffg = g˜f˜ for arbitrary f, g ∈ Cln,0 .
Decomposition (2) gives the following important proposition (see [4]).
(4)
Proposition 2.1. Given a multivector f ∈ Cln,0 with n = 2(mod 4). For λ ∈ R
we have
f ein λ = e−in λ fodd + ein λ feven ,
in λ ]
feven .
f¯ein λ = e−in λ fg
odd + e
(5)
(6)
The Clifford product of two vectors splits up into a scalar part (the inner
product) and a so-called bivector part (the wedge product):
xy = x · y + x ∧ y,
where
x·y =
n
∑
xi yi
and
x∧y =
n
n
∑
∑
ei ej (xi yj − xj yi ).
i=1 j=i+1
i=1
Observe that the square of a vector x is scalar-valued and x2 = x · x + x ∧ x = |x|2 .
We introduce a first order vector differential operator by
∂x =
m
∑
∂xi e i .
i=1
This operator is the so-called Dirac operator, which may be looked upon as the
square root of the Laplacian operator in Rn : △n = ∂x2 .
Let us consider L2 (Rn ; Cln,0 ) as a left module. For f , g ∈ L2 (Rn ; Cln,0 ), an
inner product is defined by
∫
g dn x
f (x)g(x)
(f, g)L2 (Rn ;Cln,0 ) =
Rn
∫
∑
f
e A eB
fA (x)gB (x) dn x.
(7)
=
A,B
Rn
M. Bahri et al.
128
In particular, if f = g, then the scalar part of the above inner product gives the
L2 -norm
∫ ∑
fA2 (x) dn x.
(8)
∥f ∥2L2 (Rn ;Cln,0 ) =
Rn A
Hereinafter, if not otherwise stated, n is assumed to be n = 2 (mod 4).
3. Clifford Fourier Transform (CFT)
3.1. Fundamental Operators. Before we define the CFT, we need to introduce
some notation, which will be used in the next section. For f ∈ L2 (Rn ; Cln,0 ), we
define the translation and modulation as follows:
τa f (x) = f (x − a),
Mω0 f (x) = ein ω0 ·x f (x),
(9)
and their composition, which is called the time-frequency shift,
Mω0 τa f (x) = ein ω0 ·x f (x − a),
a, ω 0 ∈ Rn .
(10)
Just as in the classical case, we obtain the canonical commutation relations
τa Mω0 f = e−in ω0 ·a Mω0 τa f.
(11)
The following lemma describes the behavior of translation, modulation, and
time-frequency shift in the Clifford algebra Cln,0 .
Lemma 3.1. If a, ω 0 ∈ Rn and f, g ∈ L2 (Rn ; Cln,0 ), then we have the following:
(i) (f, τa g)L2 (Rn ;Cln,0 ) = (τ−a f, g)L2 (Rn ;Cln,0 ) .
(ii) For g ∈ L2 (Rn ; Cln,0 ), n = 3 (mod 4), we obtain
(f, Mω0 g)L2 (Rn ;Cln,0 ) = (Mω0 fodd + M−ω0 feven , g)L2 (Rn ;Cln,0 ) .
(iii) For f ∈ L2 (Rn ; Cln,0 ), n = 3 (mod 4), we get
(Mω0 f, g)L2 (Rn ;Cln,0 ) = (f, Mω0 godd + M−ω0 geven )L2 (Rn ;Cln,0 ) .
(iv) Under the assumption stated in (iii), we obtain
(Mω0 τa f, g)L2 (Rn ;Cln,0 ) = (f, τ−a Mω0 godd + τ−a M−ω0 geven )L2 (Rn ;Cln,0 ) .
Proof. Proof of (i). It follows from (7) that
∫
^
f (x)g(x
− a) dn x
(f, τa g)L2 (Rn ;Cln,0 ) =
n
∫R
g dn x
f (x + a)g(x)
=
Rn
∫
g dn x.
τ−a f (x)g(x)
=
Rn
Proof of (ii). By equations (7) and (9), we easily obtain
∫
f (x){ein ω0 ·x g(x)}∼ dn x
(f, Mω0 g)L2 (Rn ;Cln,0 ) =
Rn
Convolution Theorems for Clifford Fourier Transform
=
=
=
=
=
∫
f (x){g(x) ein ω0 ·x }∼ dn x
∫
Rn
∫
Rn
∫
∫
129
Rn
Rn
Rn
g dn x
f (x) e−in ω0 ·x g(x)
g dn x
(fodd (x) + feven (x)) e−in ω0 ·x g(x)
g dn x
(ein ω0 ·x fodd (x) + e−in ω0 ·x feven (x))g(x)
g dn x.
(Mω0 fodd (x) + M−ω0 feven (x))g(x)
(12)
Here, in the second equality of (12), we have used the assumption that ensures to
interchange the position.
Proof of (iii). The proof is similar to (ii) and is left to the reader.
Proof of (iv). By simple computations, we get
(Mω0 τa f, g)L2 (Rn ;Cln,0 )
∫
g dn x
ein ω0 ·x f (x − a)g(x)
=
Rn
∫
in ω 0 ·(a+y
f (y)(ein ω0 ·(y+a) gg
)]
geven (y + a)) dn y
=
odd (y + a) + e
n
∫R
f (y){godd (y + a)e−in ω0 ·(y+a }∼ dn y
=
Rn
∫
f (y){geven (y + a)e−in ω0 ·(y+a) }∼ dn y
+
Rn
∫
f (y){ein ω0 ·(y+a) godd (y + a)}∼ dn y
=
n
R
∫
f (y){e−in ω0 ·(y+a) geven (y + a)}∼ dn y
+
Rn
∫
f (y)({τ−a Mω0 godd (y)}∼ + {τ−a M−ω0 geven (y)}∼ ) dn y.
(13)
=
Rn
Here, in the second and third equalities of (13), we have used the assumption and
properties of the decomposition of multivector g.
3.2. Definition of CFT. The Cln,0 Clifford Fourier transform (CFT) is a generalization of the FT in Clifford algebra obtained by replacing the FT kernel with
the Clifford Fourier kernel. For detailed discussions of the properties of the CFT
and their proofs, see, e.g., [4, 9].
Definition 3.2. The CFT of f ∈ L2 (Rn ; Cln,0 ) is the function F{f } ∈ L2 (Rn ; Cln,0 )
given by
∫
f (x) e−in ω·x dn x,
(14)
F{f }(ω) = fˆ(ω) =
with x, ω ∈ Rn .
Rn
M. Bahri et al.
130
Decomposing the multivector f into feven and fodd , equation (14) can be
rewritten as
∫
∫
e−in ω·x feven (x) dn x.
(15)
ein ω·x fodd (x) dn x +
F{f }(ω) =
Rn
Rn
The Clifford exponential e−in ω·x is often called the Clifford Fourier kernel.
For dimension n = 3 (mod 4), this kernel commutes with all elements of the Clifford algebra Cln,0 , but for n = 2 (mod 4) it does not. Notice that the different
commutation rules of the pseudoscalar in play a crucial rule in establishing the
properties of the convolution theorems of the CFT.
In the following, we collect the fundamental properties of the CFT.
Lemma 3.3. If f ∈ L2 (Rn ; Cln,0 ), then the following results hold:
F{τa f }(ω) = F{f }(ω) e−in ω·a
= ein ω·a F{fodd }(ω) + e−in ω·a F{feven }(ω).
F{Mω0 f }(ω) = F{fodd }(ω 0 − ω) + F{feven }(ω + ω 0 ).
F{Mω0 τa f }(ω) = e−in (ω0 −ω)·a F{fodd }(ω 0 − ω)
+ e−in (ω0 +ω)·a F{feven }(ω + ω 0 ).
Theorem 3.4. Suppose that F{f } ∈ L2 (Rn ; Cln,0 ). Then F{f } is invertible and
its inverse is calculated by the formula
∫
1
F{f }(ω) ein ω·x dn ω.
(16)
F −1 [F{f }(ω)](x) = f (x) =
(2π)n Rn
Proof. Substituting (14) into (16) yields
∫ ∫
1
F −1 [F{f }ω](x) =
f (y) e−in ω·y dn y ein ω·x dn ω
(2π)n Rn Rn
∫
∫
1
f (y)
=
ein (x−y)·ω dn ω dn y
(2π)n Rn
Rn
∫
f (y) δ(x − y) dn y
=
Rn
= f (x).
Equation (16) is called the Clifford Fourier integral theorem. It describes how to
get from the transform F{f } back to the original function f .
It is straightforward to see that the inverse CFT and the CFT share the same
properties. One may check the properties of the inverse CFT analogous to those in
Lemma 3.3. For an example,
F −1 [F{τa f }(ω)](x)
∫
1
=
F{τa f }(ω) ein ω·x dn ω
(2π)n Rn
Convolution Theorems for Clifford Fourier Transform
131
∫
( in ω·a
)
1
e
F{fodd }(ω) + e−in ω·a F{feven }(ω) ein ω·x dn ω
n
(2π) Rn
∫
1
=
ein ω·a F{fodd }(ω) ein ω·x dn ω
(2π)n Rn
∫
1
e−in ω·a F{feven }(ω) ein ω·x dn ω
+
(2π)n Rn
=
= ein ω·a F −1 [F{fodd }(ω)](x) + e−in ω·a F −1 [F{feven }(ω)](x).
4. Clifford Convolution And Its Properties
In this section, we introduce the Clifford convolution and establish its important properties. Ebling and Scheuermann [9] distinguish between right and left
Clifford convolutions due to the non-commutative property of the Clifford multiplication. Here, we only consider one kind of Clifford convolution. Let us first define
the convolution of two Clifford-valued functions.
Definition 4.1. The Clifford convolution f ⋆ g of f and g belong to L2 (Rn ; Cln,0 )
is defined by
∫
f (y)g(x − y) dn y
(f ⋆ g)(x) =
Rn
∫ ∑
eA eB fA (y)gB (x − y) dn y.
(17)
=
Rn A,B
Since, in general, the basis vectors eA eB ̸= eB eA , the Clifford convolution
is not commutative, i.e., (f ⋆ g) ̸= (g ⋆ f ). It is clear that the Clifford convolution
of f and g is a binary operation, which combines shifting, geometric product and
integration.
If we perform the change of variables z = x − y and relabel z back to y, then
equation (17) can be written as
∫
f (x − y)g(y) dn y.
(18)
(f ⋆ g)(x) =
Rn
Lemma 4.2 (Linearity). Let f , g, h ∈ L2 (Rn ; Cln,0 ) and α, β ∈ Cln,0 . Then, we
have
(αf + βg) ⋆ h = α(f ⋆ h) + β(g ⋆ h).
h ⋆ (f α + gβ) = (h ⋆ f )α + (h ⋆ g)β.
(19)
Lemma 4.3 (Shifting). Let f ∈ L2 (Rn ; Cln,0 ). Then we have
τa (f ⋆ g)(x) = (f ⋆ τa g)(x) = (τa f ⋆ g)(x),
τa (g ⋆ f )(x) = (g ⋆ τa f )(x) = (τa g ⋆ f )(x),
(20)
n
a∈R .
(21)
132
M. Bahri et al.
Proof. We only prove (20), the proof of (21) being similar. A direct calculation
yields
∫
f (y)g(x − y − a) dn y
τa (f ⋆ g)(x) =
Rn
∫
f (y)τa g(x − y) dn y
=
Rn
= (f ⋆ τa g)(x).
On the other hand, by the change of variables, z = x − y − a, we easily get
∫
f (x − z − a)g(z) dn z
τa (f ⋆ g)(x) =
Rn
∫
τa f (x − z)g(z) dn z
=
Rn
= (τa f ⋆ g)(x).
This completes the proof.
Equations (20) and (21) tell us that the Clifford convolutions commute with
translations.
Remark 4.4. Due to the noncommutativity of the Clifford convolution, it is easy
to see that (f ⋆ τa g)(x) ̸= (τa g ⋆ f ) and (τa f ⋆ g)(x) ̸= (g ⋆ τa f ) in general.
Lemma 4.5 (Reversion). Let f , g ∈ L2 (Rn ; Cln,0 ). Then, we have
^
(f
⋆ g)(x) = (˜
g ⋆ f˜)(x).
(22)
Proof. A straightforward computation gives
∫
^
{f (y)g(x − y)}∼ dn y
(f
⋆ g)(x) =
Rn
∫
(4)
g˜(x − y)f˜(y) dn y
=
Rn
∫
g˜(z)f˜(x − z) dn z
=
Rn
= (˜
g ⋆ f˜)(x),
which was to be proved.
5. Main Results
In this section, we investigate some important properties of the CFT of convolution of two Clifford-valued functions. We find that most of these properties
are extensions of the classical case. The following theorem gives the relationship
between the reversion of Clifford convolution and its CFT.
Convolution Theorems for Clifford Fourier Transform
133
Theorem 5.1. Let f, g ∈ L2 (Rn ; Cln,0 ). Denote by godd (resp. geven ), the odd
(resp. even) grade part of g. Then
^
^
^
F{f]
⋆ g}(ω) = (F{g
odd }(−ω) + F{geven }(ω)) F{fodd }(ω)
^
^
^
+ (F{g
odd }(ω) + F{geven }(−ω)) F{feven }(−ω).
(23)
Proof. An application of the CFT definition combined with the Clifford convolution
property of Lemma 4.5 gives
F{f]
⋆ g}(ω) = F{˜
g ⋆ f˜}(ω)
∫
(14)
(˜
g ⋆ f˜) e−in ω·x dn x
=
Rn
∫ ∫
(17)
^
g˜(y) f (x
− y) dn y e−in ω·x dn x
=
Rn Rn
]
[∫
∫
^
f (x
− y)e−in ω·x dn x dn y
g˜(y)
=
n
Rn
]
[∫R
∫
=
fg
(z)e−in ω·(y+z) dn z dn y,
g˜(y)
Rn
Rn
where the last equality follows from the change of variables z = x− y. By splitting
f into its even grade and odd grade parts, the above identity may be rewritten as
F{f]
⋆ g}(ω)
[∫
∫
g˜(y)
ein ω·y {ein ω·z fodd (z)}∼ dn z
=
Rn
Rn
]
∫
e−in ω·y {ein ω·z feven (z)}∼ dn z dn y
+
Rn
[∫
∫
=
ein ω·y {fodd (z)e−in ω·z }∼ dn z
g˜(y)
Rn
Rn
]
∫
e−in ω·y {feven (z)ein ω·z }∼ dn z dn y
+
n
∫
∫ R
in ω·y n
^
^
g˜(y) e−in ω·y dn y F{feven
}(−ω).
g˜(y) e
d y F{fodd }(ω) +
=
Rn
Rn
Again we decompose the multivector g into its even grade and odd grade parts to
get
F{f]
⋆ g}(ω)
∫
in ω·y n
(gg
]
d y F{f^
=
odd (y) + g
even (y)) e
odd }(ω)
Rn
∫
−in ω·y n
^
(gg
]
d y F{feven
}(−ω)
+
odd (y) + g
even (y)) e
Rn
]
[∫
∫
−in ω·y ∼ n
in ω·y ∼ n
{geven (y)e
} d y F{f^
=
{godd (y)e
} d y+
odd }(ω)
Rn
Rn
M. Bahri et al.
134
+
[∫
{godd (y)e−in ω·y }∼ dn y +
Rn
∫
Rn
]
^
}(−ω)
{geven (y)ein ω·y }∼ dn y F{feven
^
^
^
= (F{g
odd }(−ω) + F{geven }(ω)) F{fodd }(ω)
^
^
^
+ (F{g
odd }(ω) + F{geven }(−ω)) F{feven }(−ω).
This concludes the proof.
As an immediate consequence of Theorem 5.1, we get the following corollaries.
Corollary 5.2. Let g be as above. For f ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4),
equation (23) reduces to
^
^
^
F{f]
⋆ g}(ω) = (F{g
odd }(ω) + F{geven }(−ω)) F{f }(−ω).
2
(24)
n
Corollary 5.3. If f , g ∈ L (R ; Cln,0 ) with n = 3 (mod 4), then equation (23)
becomes
^ F{f^
^
^
F{f]
⋆ g}(ω) = F{g}(ω)
(25)
odd }(ω) + F{g}(−ω) F{feven }(−ω).
Proof. An alternative proof of Corollary 5.2 uses [4, (4.59) of Theorem 4.33], i.e.,
F{f]
⋆ g}(ω)
= F{˜
g }(ω) F{f˜}(ω)
]
] [∫
[∫
−in ω·x n
−in ω·y n
˜
f (x) e
d x
g˜(y) e
d y
=
n
Rn
[∫R
]
∫
in ω·y ∼ n
−in ω·y ∼ n
=
{geven (y) e
} d y
{godd (y) e
} d y+
Rn
Rn
[∫
]
×
{g(x) ein ω·x }∼ dn x
Rn
^
^
= (F{g
odd }(ω) + F{geven }(−ω)).
It is worth noting here that a similar argument cannot be applied to prove Corollary
5.3. Its proof follows directly from Theorem 5.1.
We next establish the shift property of the convolution theorem of the CFT.
The proof of this property uses the shift property of the CFT and the decomposition
of a multivector f .
Theorem 5.4. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{τa f ⋆ g}(ω) = F{f ⋆ τa g}(ω) = F{f }(−ω) ein ω·a F{godd }(ω)
+ F{f }(ω) e−in ω·a F{geven }(ω).
Proof. By the definition of CFT, we easily get
F{τa f ⋆ g}(ω)
∫ ∫
(14)
f (y − a) g(x − y) dn y e−in ω·x dn x
=
Rn
Rn
(26)
Convolution Theorems for Clifford Fourier Transform
135
]
g(x − y) e−in ω·x dn x dn y
n
Rn
[∫R
]
∫
f (y − a)
g(z)e−in ω·(y+z) dn z dn y
=
n
Rn
]
[∫R
∫
∫
e−in ω·y geven (z) e−in ω·z dn z dn y
ein ω·y godd (z) +
f (y − a)
=
n
Rn
Rn
∫R
f (y − a) ein ω·y dn yF{godd }(ω)
=
Rn
∫
f (y − a) e−in ω·y dn yF{geven }(ω)
+
=
∫
f (y − a)
[∫
Rn
= F{f }(−ω) ein ω·a F{godd }(ω) + F{f }(ω) e−in ω·a F{geven }(ω).
For the third equality, we have used the substitution of variable z = x − y. For
the last equality, we have used the shift property of the CFT. This completes the
proof of (26).
We have the following simple corollary to Theorem 5.4.
Corollary 5.5. When g ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4), Theorem 5.4 takes
the form
F{τa f ⋆ g}(ω) = F{f }(ω) e−in ω·a F{g}(ω).
(27)
Or, equivalently,
F{τa f ⋆ g}(ω) = (ein ω·a F{fodd }(ω) + e−in ω·a F{feven }(ω))F{g}(ω).
(28)
Proof. Using a similar argument as the proof of equation (24), we immediately get
F{τa f ⋆ g}(ω)
= F{τa f }(ω) F{g}(ω)
= F{f }(ω) e−in ω·a F{g}(ω)
= (ein ω·a F{fodd }(ω) + e−in ω·a F{feven }(ω))F{g}(ω),
where, in the last line of (29), we have used the first equation in Lemma 3.3.
(29)
Now we establish the modulation property of the convolution theorem of the
CFT.
Theorem 5.6. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{f ⋆ Mω0 g}(ω) = F{f }(−ω)F{godd }(ω + ω 0 )
+ F{f }(ω)F{geven }(ω − ω 0 ),
(30)
and
F{Mω0 f ⋆ g}(ω) = (F{fodd }(ω 0 − ω) + F{feven }(−ω − ω 0 ))F{godd }(ω)
+ (F{fodd }(ω + ω 0 ) + F{feven }(ω − ω 0 ))F{geven }(ω).
(31)
M. Bahri et al.
136
Proof. We only sketch the proof of (30), the other being similar. A direct computation gives
F{f ⋆ Mω0 g}(ω)
∫ ∫
(14)
f (y) ein ω0 ·(x−y) g(x − y) dn y e−in ω·x dn x
=
Rn Rn
[∫
]
∫
in ω 0 ·(x−y)
−in ω·x n
f (y)
e
g(x − y) e
d x dn y
=
Rn
Rn
]
[∫
∫
in ω 0 ·z
−in ω·(y+z) n
e
g(z) e
d z dn y
f (y)
=
Rn
Rn
[∫
]
∫
in ω 0 ·z
−in ω·z −in ω·y n
f (y)
e
g(z) e
e
d z dn y
=
Rn
Rn
[∫
∫
ein ω·y godd (z) e−in (ω+ω0 )·z dn z
f (y)
=
Rn
Rn
]
∫
−in ω·y
−in (ω−ω 0 )·z n
e
geven (z) e
d z dn y
+
Rn
∫
∫
e−in ω·y dn yF{geven }(ω − ω 0 )
f (y) ein ω·y dn yF{godd }(ω + ω 0 ) +
=
Rn
Rn
= F{f }(−ω)F{godd }(ω + ω 0 ) + F{f }(ω)F{geven }(ω − ω 0 ),
which was to be proved.
Remark 5.7. Note that if g ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4), then Theorem
5.6 has the form
F{f ⋆ Mω0 g) = F{f }(ω)F{g}(ω − ω 0 ),
which is of the same form as the modulation property of the convolution of the FT
(see [11]).
We further establish the time-frequency shift of the convolution theorem of
the CFT.
Theorem 5.8. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{Mω0 τa f ⋆ g}(ω) = (ein ω·a F{fodd }(ω 0 − ω)
+ e−in ω·a F{feven }(−ω − ω 0 ))F{godd }(ω) + (ein ω·a F{fodd }(ω + ω 0 )
= +e−in ω·a F{feven }(ω − ω 0 ))F{godd }(ω).
Proof. Applying equations (10) and (14), we immediately obtain
F{Mω0 τa f ⋆ g}(ω)
∫
(Mω0 τa f ⋆ g) e−in ω·x dn x
=
Rn
∫ ∫
ein ω0 ·y f (y − a) g(x − y) dn y e−in ω·x dn x
=
Rn
Rn
Convolution Theorems for Clifford Fourier Transform
137
]
g(x − y) e−in ω·x dn x dn y
n
Rn
[∫R
]
∫
ein ω0 ·y f (y − a)
g(z) e−in ω·(y+z) dn z dn y
=
n
Rn
]
[∫R
∫
g(z) e−in ω·y e−in ω·z dn z dn y.
ein ω0 ·y f (y − a)
=
=
∫
ein ω0 ·y f (y − a)
[∫
Rn
Rn
We decompose f and g into fodd + feven and godd + geven , respectively. Then, we
obtain
F{Mω0 τa f ⋆ g}(ω)
[∫
∫
ein ω·y godd (z) e−in ω·z dn z
ein ω0 ·y f (y − a)
=
Rn
Rn
]
∫
e−in ω·y geven (z) e−in ω·z dn z dn y
+
n
[∫ R
fodd (y − a) e−in (ω0 −ω)·y dn y
=
Rn
]
∫
feven (y − a) e−in (−ω−ω0 )·y dn y F{godd }(ω)
+
n
[∫R
fodd (y − a) e−in (ω+ω0 )·y dn y
+
Rn
]
∫
feven (y − a) e−in (ω−ω0 )·y dn y F{geven }(ω).
+
Rn
which was to be proved.
Corollary 5.9. If g ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4). Then, Theorem 5.8
reduces to
F{Mω0 τa f ⋆ g}(ω) = (ein (ω0 −ω)·a F{fodd }(ω + ω 0 )
+ e−in (ω−ω0 )·a F{feven }(ω − ω 0 ))F{g}(ω).
2
(32)
n
If f ∈ L (R ; Cln,0 ) with n = 3 (mod 4), then
F{Mω0 τa f ⋆ g}(ω) = (ein (ω0 +ω)·a F{f }(−ω − ω 0 )F{godd }(ω)
+ e−in (ω−ω0 )·a F{f }(ω − ω 0 ))F{geven }(ω).
(33)
Theorem 5.10. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{f }(−ω − ω 0 ) ein (ω−ω0 )·a F{godd }(ω)
+ F{f }(ω − ω 0 ) e−in (ω+ω0 )·a F{geven }(ω).
Proof. By the definition of the CFT (14) and Clifford convolution (17), we have
F{f ⋆ τa Mω0 g}(ω)
M. Bahri et al.
138
∫ ∫
f (y) ein ω0 ·(y−a) g(x − y − a) dn y e−in ω·x dn x
=
Rn Rn
[∫
]
∫
g(x − y − a) e−in ω·x dn x dn y
f (y) ein ω0 ·(y−a)
=
n
Rn
]
[∫R
∫
−in ω·(y+z) n
in ω 0 ·(y−a)
g(z − a) e
d z dn y
f (y) e
=
Rn
Rn
[∫
]
∫
−in ω·y −in ω·z n
in ω 0 ·(y−a)
g(z − a) e
e
d z dn y
f (y) e
=
Rn
Rn
[∫
∫
in ω 0 ·(y−a)
ein ω·y godd (z − a) e−in ω·z dn z
f (y) e
=
Rn
Rn
]
∫
−in ω·y
−in ω·z n
e
geven (z − a) e
d z dn y
+
Rn
∫
f (y) e−in (−ω−ω0 )·y dn y ein (ω−ω0 )·a F{godd }(ω)
=
Rn
∫
f (y) e−in (ω−ω0 )·y dn ye−in (ω+ω0 )·a F{geven }(ω)
+
(14)
Rn
= F{f }(−ω − ω 0 ) ein (ω−ω0 )·a F{godd }(ω)
+ F{f }(ω − ω 0 ) e−in (ω+ω0 )·a F{geven }(ω),
which was to be proved.
Corollary 5.11. It is straightforward to check that for g ∈ L2 (Rn ; Cln,0 ) with
n = 3 (mod 4), Theorem 5.8 reduces to
F{f ⋆ τa Mω0 g}(ω) = F{f }(ω − ω 0 ) e−in (ω+ω0 )·a F{g}(ω).
By applying the inverse CFT we get the following result, which is very important for solving partial differential equations in Clifford algebra.
Theorem 5.12. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F −1 [F{f }(ω)F{g}(ω)](x) = (f ⋆ godd )(−x) + (f ⋆ geven )(x).
Or, equivalently,
F{f }(ω)F{g}(ω) = F{(f ⋆ godd )(−·)}(ω) + F{f ⋆ geven }(ω).
(34)
Convolution Theorems for Clifford Fourier Transform
139
Proof. According to the inverse CFT, the left hand-side of (34) leads to
F −1 [F{f }(ω)F{g}(ω)](x)
∫ ∫
1
(16)
=
F{f }(ω) g(y) e−in ω·y dn y ein ω·x dn ω
(2π)n Rn Rn
∫ ∫
1
=
F{f }(ω) (godd (y) + geven (y)) ein ω·(x−y) dn ω dn y
(2π)n Rn Rn
∫ ∫
1
F{f }(ω) ein ω·(y−x) dn ω godd (y) dn y
=
(2π)n Rn Rn
∫ ∫
1
+
F{f }(ω) ein ω·(x−y) dn ω geven (y) dn y
(2π)n Rn Rn
∫
∫
n
f (x − y) geven (y) dn y
f (y − x) godd (y) d y +
=
Rn
Rn
(18)
= (f ⋆ godd )(−x) + (f ⋆ geven )(x),
which finishes the proof.
The following corollary is a special case of Theorem 5.12.
Corollary 5.13. Let f ∈ L2 (Rn ; Cln,0 ). If we assume that g ∈ L2 (Rn ; Cln,0 ) for
n = 3 (mod 4), then
F{f }(ω)F{g}(ω) = F{f ⋆ g}(ω).
(35)
Remark 5.14. According to (2), formula (35) also holds if only if g ∈ L2 (Rn ; Cln,0 )
is an even grade multivector.
Acknowledgements. The authors would like to thank the anonymous reviewer for
valuable comments and suggestions which improved the clarity and quality of this
paper. The first author is partially supported by Hibah Penelitian Kompetisi Internal tahun 2013 (No. 110/UN4-.42/LK.26/SP-UH/2013) from the Hasanuddin University, Indonesia. The second author is partially supported by JSPS.KAKENHI
(C)25400202 of Japan and the third author is partially supported by NSERC of
Canada.
References
[1] Mawardi, B. and Hitzer, E., Clifford Fourier transformation and uncertainty principle for the
Clifford geometric algebra Cl3,0 , Adv. Appl. Clifford Algebr., 16(1) (2006), 41–61.
[2] Mawardi, B., Adji, S. and Zhao, J., Clifford algebra-valued wavelet transform on multivector
fields, Adv. Appl. Clifford Algebr., 21(1) (2011), 13–30.
[3] Hitzer, E., Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl.
Clifford Algebr., 17(3) (2007), 497–517.
[4] Hitzer, E. and Mawardi, B., Clifford Fourier transform on multivector fields and uncertainty
principle for dimensions n = 2(mod 4) and n = 3(mod 4), Adv. Appl. Clifford Algebr., 18(3-4)
(2008), 715–736.
140
M. Bahri et al.
[5] Brackx, F., Delanghe, R. and Sommen, F., Clifford Analysis, Research Notes in Mathematics,
76. Pitman, Boston, MA, 1982.
[6] Sommen, F., A product and an exponential function in hypercomplex function theory, Applicable Anal., 12(1) (1981), 13–26.
[7] De Bie, H., De Schepper, N. and Sommen, F., The class of Clifford-Fourier transforms, J.
Fourier Anal. Appl., 17(6) (2011), 1198–1231.
[8] De Bie, H. and De Schepper, N., The fractional Clifford-Fourier transforms, Complex Anal.
Oper. Theory, 6(5) (2012), 1047–1067.
[9] Ebling, J. and Scheuermann, G., Clifford Fourier transform on vector fields, IEEE Transactions on Visualization and Computer Graphics, 11(4) (2005), 469–479.
[10] Bahri, M., Clifford windowed Fourier transform applied to linear time-varying systems, Appl.
Math. Sci. (Ruse), 6(58) (2012), 2857–2864.
[11] Bracewell, R. N., The Fourier Transform and its Applications, McGraw-Hill International
Editions, Singapore, 2000.
[12] Bujack, R., Scheuermann, G. and Hitzer, E., A general geometric Fourier transform convolution theorem, Adv. Appl. Clifford Algebr., 23(1) (2013), 15–38.
[13] Sangwine, S. J., Color images edge detector based on quaternion convolution, Electronics
Letters, 34(10) (1998), 969–971.
[14] Batard, T., Berthier, M. and Saint-Jean, C., Clifford-Fourier transform for color image processing, In: Geometric Algebra Computing in Engineering and Computer Science, pp. 135–
162, Springer London, 2010.
[15] Fu, Y., K¨
ahler, U. and Cerejeiras, P., The Balian-Low theorem for the windowed quaternionic
Fourier transform, Adv. Appl. Clifford Algebr., 22(4) (2012), 1025–1040.
Vol. 20, No. 2 (2014), pp. 125–140.
CONVOLUTION THEOREMS FOR CLIFFORD
FOURIER TRANSFORM AND PROPERTIES
´mi Vaillancourt3
Mawardi Bahri1 , Ryuichi Ashino2 , and Re
1
3
Department of Mathematics, Hasanuddin University
Tamalanrea Makassar 90245,Indonesia
[email protected]
2
Division of Mathematical Sciences, Osaka Kyoiku University
Osaka 582-8582, Japan
[email protected]
Department of Mathematics and Statistics, University of Ottawa
Ottawa, Ontario, Canada K1N 6N5
[email protected]
Abstract. The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis. We establish main properties of convolution
theorems for the Clifford Fourier transform. Some properties of these generalized
convolutions are extensions of the corresponding convolution theorems of the classical Fourier transform.
Key words: Clifford convolution, Clifford algebra, Clifford Fourier transform.
Abstrak. Tujuan tulisan ini adalah untuk membangun sifat-sifat penting dari teorema konvolusi untuk transformasi Fourier Clifford (TFC). Karena perkalian Clifford adalah tidak komutatif, maka diperoleh sifat-sifat konvolusi dari TFC ini adalah
perluasan dari konvolusi dari transformasi Fourier.
Kata kunci: Konvolusi Clifford, aljabar Clifford, transformasi Fourier Clifford.
1. Introduction
Recently, several attempts have been made to generalize the classical Fourier transform in the framework of Clifford algebra, so-called the Clifford Fourier transform
(CFT). It was first introduced from the mathematical aspect by Brackx et al.
[5, 6]. The CFT was recently used in signal processing [9, 14] and in other fields of
mathematics and applications. Many generalized transforms, such as the Clifford
2000 Mathematics Subject Classification:15A66, 42B10.
Received: 21-09-2013, revised: 22-04-2014, accepted: 21-05-2014.
125
M. Bahri et al.
126
wavelet transform, fractional Clifford Fourier transform, and Clifford windowed
Fourier transform (see, for example, [2, 7, 8, 10, 12, 15]) are closely related to the
CFT. One of the most fundamental and important properties of the CFT is the
convolution theorem.
Convolution is a mathematical operation with several applications in pure and
applied mathematics such as numerical analysis, numerical linear algebra and the
design and implementation of finite impulse response filters in signal processing. In
[3, 13], authors generalized convolution to the quaternion Fourier transform (QFT).
They found that the QFT of the real-valued signals are very similar to the classical
ones. In this paper, we establish convolution theorems for the CFT. Here we adopt
the definition of the CFT suggested by several authors [1, 4, 9]. Because the Clifford
multiplication is not commutative, we find important properties of the relationship
between the convolution theorems and the CFT. We finally establish the inverse
CFT of the product of two CFTs, which is very useful in solving partial differential
equations in the Clifford algebra.
This paper is organized as follows. In section 2, we provide some basic knowledge of real Clifford algebra used in the paper. Subsequently, in section 3, we define
the CFT and discuss its important properties, which are used to construct the properties of generalized convolution. Next in section 4, we introduce convolution on
Clifford algebra Cln,0 and derive its useful properties. Finally, in section 5, we
investigate the important properties of the CFT of convolution of Clifford-valued
functions and derive the inverse CFT of the product of two CFTs.
2. Preliminaries
We shall be working with real Clifford algebras. Let {e1 , e2 , e3 , · · · , en } be
an orthonormal vector basis of the n-dimensional Euclidean vector space Rn . The
real Clifford algebra over Rn , denoted by Cln,0 , has the graded 2n -dimensional
basis
{1, e1 , e2 , · · · , en , e12 , e31 , e23 , · · · , in = e1 e2 · · · en }.
(1)
Obviously, for n = 2(mod 4), the pseudoscalar in = e1 e2 · · · en anti-commutes with
each basis of the Clifford algebra while i2n = −1. The noncommutative multiplication of the basis vectors satisfies the rules ei ej + ej ei = 2δij (δij denotes the Dirac
distribution whose support is {i, j}).
An element of the Clifford algebra is called a multivector and has the following
form
∑
e A fA ,
f=
A
where fA ∈ R, A ⊂ {1, 2, · · · , n}. For convenience, we introduce ⟨f ⟩k =
to denote the k-vector part of f (k = 0, 1, 2, · · · , n). Then
f=
k=n
∑
⟨f ⟩k = ⟨f ⟩ + ⟨f ⟩1 + ⟨f ⟩2 + · · · + ⟨f ⟩n ,
k=0
where ⟨. . .⟩ = ⟨. . .⟩0 .
∑
A
f A eA
Convolution Theorems for Clifford Fourier Transform
127
A multivector f ∈ Cln,0 , n = 2 (mod 4) can be decomposed as a sum of its
even grade part, feven , and its odd grade part, fodd . Thus, we have
f = feven ⊕ fodd ,
(2)
where
n
,
2
feven = ⟨f ⟩ + ⟨f ⟩2 + · · · + ⟨f ⟩r ,
r = 2s, s ∈ N, s ≤
fodd = ⟨f ⟩1 + ⟨f ⟩3 + · · · + ⟨f ⟩r ,
r = 2s + 1, s ∈ N, s <
n
.
2
The reverse f˜ of a multivector f is an anti-automorphism given by
fe =
and hence
k=n
∑
(−1)k(k−1)/2 ⟨f ⟩k ,
(3)
k=0
ffg = g˜f˜ for arbitrary f, g ∈ Cln,0 .
Decomposition (2) gives the following important proposition (see [4]).
(4)
Proposition 2.1. Given a multivector f ∈ Cln,0 with n = 2(mod 4). For λ ∈ R
we have
f ein λ = e−in λ fodd + ein λ feven ,
in λ ]
feven .
f¯ein λ = e−in λ fg
odd + e
(5)
(6)
The Clifford product of two vectors splits up into a scalar part (the inner
product) and a so-called bivector part (the wedge product):
xy = x · y + x ∧ y,
where
x·y =
n
∑
xi yi
and
x∧y =
n
n
∑
∑
ei ej (xi yj − xj yi ).
i=1 j=i+1
i=1
Observe that the square of a vector x is scalar-valued and x2 = x · x + x ∧ x = |x|2 .
We introduce a first order vector differential operator by
∂x =
m
∑
∂xi e i .
i=1
This operator is the so-called Dirac operator, which may be looked upon as the
square root of the Laplacian operator in Rn : △n = ∂x2 .
Let us consider L2 (Rn ; Cln,0 ) as a left module. For f , g ∈ L2 (Rn ; Cln,0 ), an
inner product is defined by
∫
g dn x
f (x)g(x)
(f, g)L2 (Rn ;Cln,0 ) =
Rn
∫
∑
f
e A eB
fA (x)gB (x) dn x.
(7)
=
A,B
Rn
M. Bahri et al.
128
In particular, if f = g, then the scalar part of the above inner product gives the
L2 -norm
∫ ∑
fA2 (x) dn x.
(8)
∥f ∥2L2 (Rn ;Cln,0 ) =
Rn A
Hereinafter, if not otherwise stated, n is assumed to be n = 2 (mod 4).
3. Clifford Fourier Transform (CFT)
3.1. Fundamental Operators. Before we define the CFT, we need to introduce
some notation, which will be used in the next section. For f ∈ L2 (Rn ; Cln,0 ), we
define the translation and modulation as follows:
τa f (x) = f (x − a),
Mω0 f (x) = ein ω0 ·x f (x),
(9)
and their composition, which is called the time-frequency shift,
Mω0 τa f (x) = ein ω0 ·x f (x − a),
a, ω 0 ∈ Rn .
(10)
Just as in the classical case, we obtain the canonical commutation relations
τa Mω0 f = e−in ω0 ·a Mω0 τa f.
(11)
The following lemma describes the behavior of translation, modulation, and
time-frequency shift in the Clifford algebra Cln,0 .
Lemma 3.1. If a, ω 0 ∈ Rn and f, g ∈ L2 (Rn ; Cln,0 ), then we have the following:
(i) (f, τa g)L2 (Rn ;Cln,0 ) = (τ−a f, g)L2 (Rn ;Cln,0 ) .
(ii) For g ∈ L2 (Rn ; Cln,0 ), n = 3 (mod 4), we obtain
(f, Mω0 g)L2 (Rn ;Cln,0 ) = (Mω0 fodd + M−ω0 feven , g)L2 (Rn ;Cln,0 ) .
(iii) For f ∈ L2 (Rn ; Cln,0 ), n = 3 (mod 4), we get
(Mω0 f, g)L2 (Rn ;Cln,0 ) = (f, Mω0 godd + M−ω0 geven )L2 (Rn ;Cln,0 ) .
(iv) Under the assumption stated in (iii), we obtain
(Mω0 τa f, g)L2 (Rn ;Cln,0 ) = (f, τ−a Mω0 godd + τ−a M−ω0 geven )L2 (Rn ;Cln,0 ) .
Proof. Proof of (i). It follows from (7) that
∫
^
f (x)g(x
− a) dn x
(f, τa g)L2 (Rn ;Cln,0 ) =
n
∫R
g dn x
f (x + a)g(x)
=
Rn
∫
g dn x.
τ−a f (x)g(x)
=
Rn
Proof of (ii). By equations (7) and (9), we easily obtain
∫
f (x){ein ω0 ·x g(x)}∼ dn x
(f, Mω0 g)L2 (Rn ;Cln,0 ) =
Rn
Convolution Theorems for Clifford Fourier Transform
=
=
=
=
=
∫
f (x){g(x) ein ω0 ·x }∼ dn x
∫
Rn
∫
Rn
∫
∫
129
Rn
Rn
Rn
g dn x
f (x) e−in ω0 ·x g(x)
g dn x
(fodd (x) + feven (x)) e−in ω0 ·x g(x)
g dn x
(ein ω0 ·x fodd (x) + e−in ω0 ·x feven (x))g(x)
g dn x.
(Mω0 fodd (x) + M−ω0 feven (x))g(x)
(12)
Here, in the second equality of (12), we have used the assumption that ensures to
interchange the position.
Proof of (iii). The proof is similar to (ii) and is left to the reader.
Proof of (iv). By simple computations, we get
(Mω0 τa f, g)L2 (Rn ;Cln,0 )
∫
g dn x
ein ω0 ·x f (x − a)g(x)
=
Rn
∫
in ω 0 ·(a+y
f (y)(ein ω0 ·(y+a) gg
)]
geven (y + a)) dn y
=
odd (y + a) + e
n
∫R
f (y){godd (y + a)e−in ω0 ·(y+a }∼ dn y
=
Rn
∫
f (y){geven (y + a)e−in ω0 ·(y+a) }∼ dn y
+
Rn
∫
f (y){ein ω0 ·(y+a) godd (y + a)}∼ dn y
=
n
R
∫
f (y){e−in ω0 ·(y+a) geven (y + a)}∼ dn y
+
Rn
∫
f (y)({τ−a Mω0 godd (y)}∼ + {τ−a M−ω0 geven (y)}∼ ) dn y.
(13)
=
Rn
Here, in the second and third equalities of (13), we have used the assumption and
properties of the decomposition of multivector g.
3.2. Definition of CFT. The Cln,0 Clifford Fourier transform (CFT) is a generalization of the FT in Clifford algebra obtained by replacing the FT kernel with
the Clifford Fourier kernel. For detailed discussions of the properties of the CFT
and their proofs, see, e.g., [4, 9].
Definition 3.2. The CFT of f ∈ L2 (Rn ; Cln,0 ) is the function F{f } ∈ L2 (Rn ; Cln,0 )
given by
∫
f (x) e−in ω·x dn x,
(14)
F{f }(ω) = fˆ(ω) =
with x, ω ∈ Rn .
Rn
M. Bahri et al.
130
Decomposing the multivector f into feven and fodd , equation (14) can be
rewritten as
∫
∫
e−in ω·x feven (x) dn x.
(15)
ein ω·x fodd (x) dn x +
F{f }(ω) =
Rn
Rn
The Clifford exponential e−in ω·x is often called the Clifford Fourier kernel.
For dimension n = 3 (mod 4), this kernel commutes with all elements of the Clifford algebra Cln,0 , but for n = 2 (mod 4) it does not. Notice that the different
commutation rules of the pseudoscalar in play a crucial rule in establishing the
properties of the convolution theorems of the CFT.
In the following, we collect the fundamental properties of the CFT.
Lemma 3.3. If f ∈ L2 (Rn ; Cln,0 ), then the following results hold:
F{τa f }(ω) = F{f }(ω) e−in ω·a
= ein ω·a F{fodd }(ω) + e−in ω·a F{feven }(ω).
F{Mω0 f }(ω) = F{fodd }(ω 0 − ω) + F{feven }(ω + ω 0 ).
F{Mω0 τa f }(ω) = e−in (ω0 −ω)·a F{fodd }(ω 0 − ω)
+ e−in (ω0 +ω)·a F{feven }(ω + ω 0 ).
Theorem 3.4. Suppose that F{f } ∈ L2 (Rn ; Cln,0 ). Then F{f } is invertible and
its inverse is calculated by the formula
∫
1
F{f }(ω) ein ω·x dn ω.
(16)
F −1 [F{f }(ω)](x) = f (x) =
(2π)n Rn
Proof. Substituting (14) into (16) yields
∫ ∫
1
F −1 [F{f }ω](x) =
f (y) e−in ω·y dn y ein ω·x dn ω
(2π)n Rn Rn
∫
∫
1
f (y)
=
ein (x−y)·ω dn ω dn y
(2π)n Rn
Rn
∫
f (y) δ(x − y) dn y
=
Rn
= f (x).
Equation (16) is called the Clifford Fourier integral theorem. It describes how to
get from the transform F{f } back to the original function f .
It is straightforward to see that the inverse CFT and the CFT share the same
properties. One may check the properties of the inverse CFT analogous to those in
Lemma 3.3. For an example,
F −1 [F{τa f }(ω)](x)
∫
1
=
F{τa f }(ω) ein ω·x dn ω
(2π)n Rn
Convolution Theorems for Clifford Fourier Transform
131
∫
( in ω·a
)
1
e
F{fodd }(ω) + e−in ω·a F{feven }(ω) ein ω·x dn ω
n
(2π) Rn
∫
1
=
ein ω·a F{fodd }(ω) ein ω·x dn ω
(2π)n Rn
∫
1
e−in ω·a F{feven }(ω) ein ω·x dn ω
+
(2π)n Rn
=
= ein ω·a F −1 [F{fodd }(ω)](x) + e−in ω·a F −1 [F{feven }(ω)](x).
4. Clifford Convolution And Its Properties
In this section, we introduce the Clifford convolution and establish its important properties. Ebling and Scheuermann [9] distinguish between right and left
Clifford convolutions due to the non-commutative property of the Clifford multiplication. Here, we only consider one kind of Clifford convolution. Let us first define
the convolution of two Clifford-valued functions.
Definition 4.1. The Clifford convolution f ⋆ g of f and g belong to L2 (Rn ; Cln,0 )
is defined by
∫
f (y)g(x − y) dn y
(f ⋆ g)(x) =
Rn
∫ ∑
eA eB fA (y)gB (x − y) dn y.
(17)
=
Rn A,B
Since, in general, the basis vectors eA eB ̸= eB eA , the Clifford convolution
is not commutative, i.e., (f ⋆ g) ̸= (g ⋆ f ). It is clear that the Clifford convolution
of f and g is a binary operation, which combines shifting, geometric product and
integration.
If we perform the change of variables z = x − y and relabel z back to y, then
equation (17) can be written as
∫
f (x − y)g(y) dn y.
(18)
(f ⋆ g)(x) =
Rn
Lemma 4.2 (Linearity). Let f , g, h ∈ L2 (Rn ; Cln,0 ) and α, β ∈ Cln,0 . Then, we
have
(αf + βg) ⋆ h = α(f ⋆ h) + β(g ⋆ h).
h ⋆ (f α + gβ) = (h ⋆ f )α + (h ⋆ g)β.
(19)
Lemma 4.3 (Shifting). Let f ∈ L2 (Rn ; Cln,0 ). Then we have
τa (f ⋆ g)(x) = (f ⋆ τa g)(x) = (τa f ⋆ g)(x),
τa (g ⋆ f )(x) = (g ⋆ τa f )(x) = (τa g ⋆ f )(x),
(20)
n
a∈R .
(21)
132
M. Bahri et al.
Proof. We only prove (20), the proof of (21) being similar. A direct calculation
yields
∫
f (y)g(x − y − a) dn y
τa (f ⋆ g)(x) =
Rn
∫
f (y)τa g(x − y) dn y
=
Rn
= (f ⋆ τa g)(x).
On the other hand, by the change of variables, z = x − y − a, we easily get
∫
f (x − z − a)g(z) dn z
τa (f ⋆ g)(x) =
Rn
∫
τa f (x − z)g(z) dn z
=
Rn
= (τa f ⋆ g)(x).
This completes the proof.
Equations (20) and (21) tell us that the Clifford convolutions commute with
translations.
Remark 4.4. Due to the noncommutativity of the Clifford convolution, it is easy
to see that (f ⋆ τa g)(x) ̸= (τa g ⋆ f ) and (τa f ⋆ g)(x) ̸= (g ⋆ τa f ) in general.
Lemma 4.5 (Reversion). Let f , g ∈ L2 (Rn ; Cln,0 ). Then, we have
^
(f
⋆ g)(x) = (˜
g ⋆ f˜)(x).
(22)
Proof. A straightforward computation gives
∫
^
{f (y)g(x − y)}∼ dn y
(f
⋆ g)(x) =
Rn
∫
(4)
g˜(x − y)f˜(y) dn y
=
Rn
∫
g˜(z)f˜(x − z) dn z
=
Rn
= (˜
g ⋆ f˜)(x),
which was to be proved.
5. Main Results
In this section, we investigate some important properties of the CFT of convolution of two Clifford-valued functions. We find that most of these properties
are extensions of the classical case. The following theorem gives the relationship
between the reversion of Clifford convolution and its CFT.
Convolution Theorems for Clifford Fourier Transform
133
Theorem 5.1. Let f, g ∈ L2 (Rn ; Cln,0 ). Denote by godd (resp. geven ), the odd
(resp. even) grade part of g. Then
^
^
^
F{f]
⋆ g}(ω) = (F{g
odd }(−ω) + F{geven }(ω)) F{fodd }(ω)
^
^
^
+ (F{g
odd }(ω) + F{geven }(−ω)) F{feven }(−ω).
(23)
Proof. An application of the CFT definition combined with the Clifford convolution
property of Lemma 4.5 gives
F{f]
⋆ g}(ω) = F{˜
g ⋆ f˜}(ω)
∫
(14)
(˜
g ⋆ f˜) e−in ω·x dn x
=
Rn
∫ ∫
(17)
^
g˜(y) f (x
− y) dn y e−in ω·x dn x
=
Rn Rn
]
[∫
∫
^
f (x
− y)e−in ω·x dn x dn y
g˜(y)
=
n
Rn
]
[∫R
∫
=
fg
(z)e−in ω·(y+z) dn z dn y,
g˜(y)
Rn
Rn
where the last equality follows from the change of variables z = x− y. By splitting
f into its even grade and odd grade parts, the above identity may be rewritten as
F{f]
⋆ g}(ω)
[∫
∫
g˜(y)
ein ω·y {ein ω·z fodd (z)}∼ dn z
=
Rn
Rn
]
∫
e−in ω·y {ein ω·z feven (z)}∼ dn z dn y
+
Rn
[∫
∫
=
ein ω·y {fodd (z)e−in ω·z }∼ dn z
g˜(y)
Rn
Rn
]
∫
e−in ω·y {feven (z)ein ω·z }∼ dn z dn y
+
n
∫
∫ R
in ω·y n
^
^
g˜(y) e−in ω·y dn y F{feven
}(−ω).
g˜(y) e
d y F{fodd }(ω) +
=
Rn
Rn
Again we decompose the multivector g into its even grade and odd grade parts to
get
F{f]
⋆ g}(ω)
∫
in ω·y n
(gg
]
d y F{f^
=
odd (y) + g
even (y)) e
odd }(ω)
Rn
∫
−in ω·y n
^
(gg
]
d y F{feven
}(−ω)
+
odd (y) + g
even (y)) e
Rn
]
[∫
∫
−in ω·y ∼ n
in ω·y ∼ n
{geven (y)e
} d y F{f^
=
{godd (y)e
} d y+
odd }(ω)
Rn
Rn
M. Bahri et al.
134
+
[∫
{godd (y)e−in ω·y }∼ dn y +
Rn
∫
Rn
]
^
}(−ω)
{geven (y)ein ω·y }∼ dn y F{feven
^
^
^
= (F{g
odd }(−ω) + F{geven }(ω)) F{fodd }(ω)
^
^
^
+ (F{g
odd }(ω) + F{geven }(−ω)) F{feven }(−ω).
This concludes the proof.
As an immediate consequence of Theorem 5.1, we get the following corollaries.
Corollary 5.2. Let g be as above. For f ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4),
equation (23) reduces to
^
^
^
F{f]
⋆ g}(ω) = (F{g
odd }(ω) + F{geven }(−ω)) F{f }(−ω).
2
(24)
n
Corollary 5.3. If f , g ∈ L (R ; Cln,0 ) with n = 3 (mod 4), then equation (23)
becomes
^ F{f^
^
^
F{f]
⋆ g}(ω) = F{g}(ω)
(25)
odd }(ω) + F{g}(−ω) F{feven }(−ω).
Proof. An alternative proof of Corollary 5.2 uses [4, (4.59) of Theorem 4.33], i.e.,
F{f]
⋆ g}(ω)
= F{˜
g }(ω) F{f˜}(ω)
]
] [∫
[∫
−in ω·x n
−in ω·y n
˜
f (x) e
d x
g˜(y) e
d y
=
n
Rn
[∫R
]
∫
in ω·y ∼ n
−in ω·y ∼ n
=
{geven (y) e
} d y
{godd (y) e
} d y+
Rn
Rn
[∫
]
×
{g(x) ein ω·x }∼ dn x
Rn
^
^
= (F{g
odd }(ω) + F{geven }(−ω)).
It is worth noting here that a similar argument cannot be applied to prove Corollary
5.3. Its proof follows directly from Theorem 5.1.
We next establish the shift property of the convolution theorem of the CFT.
The proof of this property uses the shift property of the CFT and the decomposition
of a multivector f .
Theorem 5.4. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{τa f ⋆ g}(ω) = F{f ⋆ τa g}(ω) = F{f }(−ω) ein ω·a F{godd }(ω)
+ F{f }(ω) e−in ω·a F{geven }(ω).
Proof. By the definition of CFT, we easily get
F{τa f ⋆ g}(ω)
∫ ∫
(14)
f (y − a) g(x − y) dn y e−in ω·x dn x
=
Rn
Rn
(26)
Convolution Theorems for Clifford Fourier Transform
135
]
g(x − y) e−in ω·x dn x dn y
n
Rn
[∫R
]
∫
f (y − a)
g(z)e−in ω·(y+z) dn z dn y
=
n
Rn
]
[∫R
∫
∫
e−in ω·y geven (z) e−in ω·z dn z dn y
ein ω·y godd (z) +
f (y − a)
=
n
Rn
Rn
∫R
f (y − a) ein ω·y dn yF{godd }(ω)
=
Rn
∫
f (y − a) e−in ω·y dn yF{geven }(ω)
+
=
∫
f (y − a)
[∫
Rn
= F{f }(−ω) ein ω·a F{godd }(ω) + F{f }(ω) e−in ω·a F{geven }(ω).
For the third equality, we have used the substitution of variable z = x − y. For
the last equality, we have used the shift property of the CFT. This completes the
proof of (26).
We have the following simple corollary to Theorem 5.4.
Corollary 5.5. When g ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4), Theorem 5.4 takes
the form
F{τa f ⋆ g}(ω) = F{f }(ω) e−in ω·a F{g}(ω).
(27)
Or, equivalently,
F{τa f ⋆ g}(ω) = (ein ω·a F{fodd }(ω) + e−in ω·a F{feven }(ω))F{g}(ω).
(28)
Proof. Using a similar argument as the proof of equation (24), we immediately get
F{τa f ⋆ g}(ω)
= F{τa f }(ω) F{g}(ω)
= F{f }(ω) e−in ω·a F{g}(ω)
= (ein ω·a F{fodd }(ω) + e−in ω·a F{feven }(ω))F{g}(ω),
where, in the last line of (29), we have used the first equation in Lemma 3.3.
(29)
Now we establish the modulation property of the convolution theorem of the
CFT.
Theorem 5.6. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{f ⋆ Mω0 g}(ω) = F{f }(−ω)F{godd }(ω + ω 0 )
+ F{f }(ω)F{geven }(ω − ω 0 ),
(30)
and
F{Mω0 f ⋆ g}(ω) = (F{fodd }(ω 0 − ω) + F{feven }(−ω − ω 0 ))F{godd }(ω)
+ (F{fodd }(ω + ω 0 ) + F{feven }(ω − ω 0 ))F{geven }(ω).
(31)
M. Bahri et al.
136
Proof. We only sketch the proof of (30), the other being similar. A direct computation gives
F{f ⋆ Mω0 g}(ω)
∫ ∫
(14)
f (y) ein ω0 ·(x−y) g(x − y) dn y e−in ω·x dn x
=
Rn Rn
[∫
]
∫
in ω 0 ·(x−y)
−in ω·x n
f (y)
e
g(x − y) e
d x dn y
=
Rn
Rn
]
[∫
∫
in ω 0 ·z
−in ω·(y+z) n
e
g(z) e
d z dn y
f (y)
=
Rn
Rn
[∫
]
∫
in ω 0 ·z
−in ω·z −in ω·y n
f (y)
e
g(z) e
e
d z dn y
=
Rn
Rn
[∫
∫
ein ω·y godd (z) e−in (ω+ω0 )·z dn z
f (y)
=
Rn
Rn
]
∫
−in ω·y
−in (ω−ω 0 )·z n
e
geven (z) e
d z dn y
+
Rn
∫
∫
e−in ω·y dn yF{geven }(ω − ω 0 )
f (y) ein ω·y dn yF{godd }(ω + ω 0 ) +
=
Rn
Rn
= F{f }(−ω)F{godd }(ω + ω 0 ) + F{f }(ω)F{geven }(ω − ω 0 ),
which was to be proved.
Remark 5.7. Note that if g ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4), then Theorem
5.6 has the form
F{f ⋆ Mω0 g) = F{f }(ω)F{g}(ω − ω 0 ),
which is of the same form as the modulation property of the convolution of the FT
(see [11]).
We further establish the time-frequency shift of the convolution theorem of
the CFT.
Theorem 5.8. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{Mω0 τa f ⋆ g}(ω) = (ein ω·a F{fodd }(ω 0 − ω)
+ e−in ω·a F{feven }(−ω − ω 0 ))F{godd }(ω) + (ein ω·a F{fodd }(ω + ω 0 )
= +e−in ω·a F{feven }(ω − ω 0 ))F{godd }(ω).
Proof. Applying equations (10) and (14), we immediately obtain
F{Mω0 τa f ⋆ g}(ω)
∫
(Mω0 τa f ⋆ g) e−in ω·x dn x
=
Rn
∫ ∫
ein ω0 ·y f (y − a) g(x − y) dn y e−in ω·x dn x
=
Rn
Rn
Convolution Theorems for Clifford Fourier Transform
137
]
g(x − y) e−in ω·x dn x dn y
n
Rn
[∫R
]
∫
ein ω0 ·y f (y − a)
g(z) e−in ω·(y+z) dn z dn y
=
n
Rn
]
[∫R
∫
g(z) e−in ω·y e−in ω·z dn z dn y.
ein ω0 ·y f (y − a)
=
=
∫
ein ω0 ·y f (y − a)
[∫
Rn
Rn
We decompose f and g into fodd + feven and godd + geven , respectively. Then, we
obtain
F{Mω0 τa f ⋆ g}(ω)
[∫
∫
ein ω·y godd (z) e−in ω·z dn z
ein ω0 ·y f (y − a)
=
Rn
Rn
]
∫
e−in ω·y geven (z) e−in ω·z dn z dn y
+
n
[∫ R
fodd (y − a) e−in (ω0 −ω)·y dn y
=
Rn
]
∫
feven (y − a) e−in (−ω−ω0 )·y dn y F{godd }(ω)
+
n
[∫R
fodd (y − a) e−in (ω+ω0 )·y dn y
+
Rn
]
∫
feven (y − a) e−in (ω−ω0 )·y dn y F{geven }(ω).
+
Rn
which was to be proved.
Corollary 5.9. If g ∈ L2 (Rn ; Cln,0 ) with n = 3 (mod 4). Then, Theorem 5.8
reduces to
F{Mω0 τa f ⋆ g}(ω) = (ein (ω0 −ω)·a F{fodd }(ω + ω 0 )
+ e−in (ω−ω0 )·a F{feven }(ω − ω 0 ))F{g}(ω).
2
(32)
n
If f ∈ L (R ; Cln,0 ) with n = 3 (mod 4), then
F{Mω0 τa f ⋆ g}(ω) = (ein (ω0 +ω)·a F{f }(−ω − ω 0 )F{godd }(ω)
+ e−in (ω−ω0 )·a F{f }(ω − ω 0 ))F{geven }(ω).
(33)
Theorem 5.10. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F{f }(−ω − ω 0 ) ein (ω−ω0 )·a F{godd }(ω)
+ F{f }(ω − ω 0 ) e−in (ω+ω0 )·a F{geven }(ω).
Proof. By the definition of the CFT (14) and Clifford convolution (17), we have
F{f ⋆ τa Mω0 g}(ω)
M. Bahri et al.
138
∫ ∫
f (y) ein ω0 ·(y−a) g(x − y − a) dn y e−in ω·x dn x
=
Rn Rn
[∫
]
∫
g(x − y − a) e−in ω·x dn x dn y
f (y) ein ω0 ·(y−a)
=
n
Rn
]
[∫R
∫
−in ω·(y+z) n
in ω 0 ·(y−a)
g(z − a) e
d z dn y
f (y) e
=
Rn
Rn
[∫
]
∫
−in ω·y −in ω·z n
in ω 0 ·(y−a)
g(z − a) e
e
d z dn y
f (y) e
=
Rn
Rn
[∫
∫
in ω 0 ·(y−a)
ein ω·y godd (z − a) e−in ω·z dn z
f (y) e
=
Rn
Rn
]
∫
−in ω·y
−in ω·z n
e
geven (z − a) e
d z dn y
+
Rn
∫
f (y) e−in (−ω−ω0 )·y dn y ein (ω−ω0 )·a F{godd }(ω)
=
Rn
∫
f (y) e−in (ω−ω0 )·y dn ye−in (ω+ω0 )·a F{geven }(ω)
+
(14)
Rn
= F{f }(−ω − ω 0 ) ein (ω−ω0 )·a F{godd }(ω)
+ F{f }(ω − ω 0 ) e−in (ω+ω0 )·a F{geven }(ω),
which was to be proved.
Corollary 5.11. It is straightforward to check that for g ∈ L2 (Rn ; Cln,0 ) with
n = 3 (mod 4), Theorem 5.8 reduces to
F{f ⋆ τa Mω0 g}(ω) = F{f }(ω − ω 0 ) e−in (ω+ω0 )·a F{g}(ω).
By applying the inverse CFT we get the following result, which is very important for solving partial differential equations in Clifford algebra.
Theorem 5.12. Let f , g ∈ L2 (Rn ; Cln,0 ). Then
F −1 [F{f }(ω)F{g}(ω)](x) = (f ⋆ godd )(−x) + (f ⋆ geven )(x).
Or, equivalently,
F{f }(ω)F{g}(ω) = F{(f ⋆ godd )(−·)}(ω) + F{f ⋆ geven }(ω).
(34)
Convolution Theorems for Clifford Fourier Transform
139
Proof. According to the inverse CFT, the left hand-side of (34) leads to
F −1 [F{f }(ω)F{g}(ω)](x)
∫ ∫
1
(16)
=
F{f }(ω) g(y) e−in ω·y dn y ein ω·x dn ω
(2π)n Rn Rn
∫ ∫
1
=
F{f }(ω) (godd (y) + geven (y)) ein ω·(x−y) dn ω dn y
(2π)n Rn Rn
∫ ∫
1
F{f }(ω) ein ω·(y−x) dn ω godd (y) dn y
=
(2π)n Rn Rn
∫ ∫
1
+
F{f }(ω) ein ω·(x−y) dn ω geven (y) dn y
(2π)n Rn Rn
∫
∫
n
f (x − y) geven (y) dn y
f (y − x) godd (y) d y +
=
Rn
Rn
(18)
= (f ⋆ godd )(−x) + (f ⋆ geven )(x),
which finishes the proof.
The following corollary is a special case of Theorem 5.12.
Corollary 5.13. Let f ∈ L2 (Rn ; Cln,0 ). If we assume that g ∈ L2 (Rn ; Cln,0 ) for
n = 3 (mod 4), then
F{f }(ω)F{g}(ω) = F{f ⋆ g}(ω).
(35)
Remark 5.14. According to (2), formula (35) also holds if only if g ∈ L2 (Rn ; Cln,0 )
is an even grade multivector.
Acknowledgements. The authors would like to thank the anonymous reviewer for
valuable comments and suggestions which improved the clarity and quality of this
paper. The first author is partially supported by Hibah Penelitian Kompetisi Internal tahun 2013 (No. 110/UN4-.42/LK.26/SP-UH/2013) from the Hasanuddin University, Indonesia. The second author is partially supported by JSPS.KAKENHI
(C)25400202 of Japan and the third author is partially supported by NSERC of
Canada.
References
[1] Mawardi, B. and Hitzer, E., Clifford Fourier transformation and uncertainty principle for the
Clifford geometric algebra Cl3,0 , Adv. Appl. Clifford Algebr., 16(1) (2006), 41–61.
[2] Mawardi, B., Adji, S. and Zhao, J., Clifford algebra-valued wavelet transform on multivector
fields, Adv. Appl. Clifford Algebr., 21(1) (2011), 13–30.
[3] Hitzer, E., Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl.
Clifford Algebr., 17(3) (2007), 497–517.
[4] Hitzer, E. and Mawardi, B., Clifford Fourier transform on multivector fields and uncertainty
principle for dimensions n = 2(mod 4) and n = 3(mod 4), Adv. Appl. Clifford Algebr., 18(3-4)
(2008), 715–736.
140
M. Bahri et al.
[5] Brackx, F., Delanghe, R. and Sommen, F., Clifford Analysis, Research Notes in Mathematics,
76. Pitman, Boston, MA, 1982.
[6] Sommen, F., A product and an exponential function in hypercomplex function theory, Applicable Anal., 12(1) (1981), 13–26.
[7] De Bie, H., De Schepper, N. and Sommen, F., The class of Clifford-Fourier transforms, J.
Fourier Anal. Appl., 17(6) (2011), 1198–1231.
[8] De Bie, H. and De Schepper, N., The fractional Clifford-Fourier transforms, Complex Anal.
Oper. Theory, 6(5) (2012), 1047–1067.
[9] Ebling, J. and Scheuermann, G., Clifford Fourier transform on vector fields, IEEE Transactions on Visualization and Computer Graphics, 11(4) (2005), 469–479.
[10] Bahri, M., Clifford windowed Fourier transform applied to linear time-varying systems, Appl.
Math. Sci. (Ruse), 6(58) (2012), 2857–2864.
[11] Bracewell, R. N., The Fourier Transform and its Applications, McGraw-Hill International
Editions, Singapore, 2000.
[12] Bujack, R., Scheuermann, G. and Hitzer, E., A general geometric Fourier transform convolution theorem, Adv. Appl. Clifford Algebr., 23(1) (2013), 15–38.
[13] Sangwine, S. J., Color images edge detector based on quaternion convolution, Electronics
Letters, 34(10) (1998), 969–971.
[14] Batard, T., Berthier, M. and Saint-Jean, C., Clifford-Fourier transform for color image processing, In: Geometric Algebra Computing in Engineering and Computer Science, pp. 135–
162, Springer London, 2010.
[15] Fu, Y., K¨
ahler, U. and Cerejeiras, P., The Balian-Low theorem for the windowed quaternionic
Fourier transform, Adv. Appl. Clifford Algebr., 22(4) (2012), 1025–1040.