DEVELOPMENT OF CONVOLUTION THEOREM IN FRFT DOMAIN

Ashutosh Kumar Singh Rajiv Saxena

Department of Electronics & ommunication Department of Electronics & ommunication
Jaypee Institute of Engineering & echnology Jaypee Institute of Engineering Technology
Raghogarh, Guna-473226, India Raghogarh, Guna-473226, India
E-mail: ashutoshsingh79@yahoo.com E-mail: rajiv.saxena@jiet.ac.in

Abstract— The fractional Fourier transform Reflection operator when α becomes π and
(FRFT), which is a generalization of the Fourier Inverse Fourier operator when α becomes 3π/2.
transform (FT), has emerged as a very efficient Many properties of FRFT are recently derived
mathematical tool for signals which are having and published including convolution theorem by
time-dependent frequency component. FRFT has Luis B. Almeida [3], Ahmed I. Zayed [4] and
an advantage over other transforms being used Deyun Wei et all [5].
in the application areas like: signal processing In 1997, Almeida [3] gave a definition by
and optics. Many properties of this transform are considering two functions, x, y Є L1(R). Their
already known, but an extension of convolution convolution is given by
theorem of Fourier transform is still not having a ∞

widely accepted closed form expression. In the w(t ) = ( x ∗ y)(t ) = ∫ x(τ ) y(t − τ ) dτ (2)
literature of recent past different authors have −∞
tried to formulate convolution theorem for Then FRFT of the convolved signal was defined
FRFT, but none have received acclamation as,
because their definition do not generalize very − j(
u2
) Tan α
nicely the classical result for the FT. A new Wα (u ) = Sec α e 2

convolution theorem for FRFT is proposed in ∞ v2 (3)

j( ) Tan α
this article which is supposed to be a better
realizable proposition. ∫ X α (v) y[(u − v) Sec α ] e
−∞
2
dv
INTRODUCTION This is actually, the FRFT of a convolution
The fractional Fourier transform (FRFT) was
which can be obtained by taking the FRFT of
introduced way back in 1920’s, but remained
one of the signals, multiplying by a chirp,
largely unknown until the work of Victor
convolving with a scaled version of the other
Namias [1] in 1980. After that it has been
signal, and multiplying again by a chirp and by a
analyzed by many authors and due to their
scale factor. But this definition of convolution
contribution now it is used in many areas of
theorem does not converge to the classical
communication systems, signal processing and
definition of convolution theorem (FT) by
optics. FRFT is a powerful tool for analyzing
putting value of α as π/2, as FRFT at angle π/2
time-varying signals. With the advent of FRFT
becomes FT.
and related properties [2], it has been observed
After one year, Zayed [4] has documented a
that the properties and application of FT are
different method to calculate FRFT of the
special cases of those of FRFT. The FRFT with
convolution of two signals. A different
angle parameter α of a signal f(t) is defined as,
∞ j
convolution process was first defined by him as
1− jcotα {(t2+u2)cotα − 2t u cscα}

2π −∫∞
j
Fα (u) = f (t) e2
dt , if α ≠ kπ 1 − j cot α − 2 x 2 Cot α
h( x) = ( f ∗ g )( x) = e
= f (u) , if α =2kπ
2π
∞ j 2 j
τ Cot α ( x − y ) 2 Cot α
= f (−u) , if α =(2k +1)π
(1) ∫ f ( y) e
−∞
2
g ( x − y) e 2
dy (4)

The expression of FRFT is defined for time- And FRFT of above convolved signal is
frequency plane. And the kernel of FRFT is formulated as,
converted into Identity operator when α becomes
zero, Fourier operator when α becomes π/2,

978-1-4244-7138-6/10/$26.00 ©2010 IEEE

 j ∞
− u 2 Cot α
H α (u ) = e 2
Fα (u ) Gα (u ) (5) h(t ) = ∫ f (τ ) g (t − τ ) m(t ,τ ) dτ
−∞
(9)

Although this definition given by Zayed [4]

converges to classical definition for convolution By using a weighting function m(t,τ) as
theorem of FT, but during the process of
j τ ( τ − t ) cot α
evaluation of convolution, the signals are m (t ,τ ) = e (10)
multiplied three times by the different chirp
signals. From realization point of view this Theorem: Let h(t) is defined as weighted
would impose difficulty because in convolution of two functions f(t) and g(t) and
communication systems it is nearly impossible Fα(u), Gα(u) and Hα(u) are defined as FRFT of
to generate a chirp signal accurately. f(t), g(t) and h(t) respectively. Then
After Zayed [4], it took almost ten years to
propose a different definition of convolution j
theorem in FRFT domain by Deyun Wei [5] in 2π − u2 cotα
Hα (u) = e2 Fα (u) Gα (u) (11)
2009. Prior to defining convolution theorem, 1 − j Cotα
author defines a τ-generalized translation of
signal f(t) denoted by f(tθτ) where, Proof: From the definition of FRFT, we have
∞
f (tθτ ) = ∫ Fα (u) Kα (u,τ ) K (u, t ) du
∗
(6) ∞ j
1− j cotα {(t 2 +u2 ) Cotα − 2t u Cscα}
−∞
Where K and K* represent FRFT and IFRFT
Hα (u) =
2π ∫
−∞
h(t) e2 dt

kernel respectively. Based on this generalized

⎛ 1− j cotα ⎞ (12)
function convolution process is defined as = ⎜⎜ ⎟
⎟
∞
⎝ 2 π ⎠
( f D g )( t ) = ∫ f (τ ) g ( t θτ ) d τ (7) ∞∞ j 2 2
{(t +u ) Cotα − 2t u Cscα + 2τ (τ −t ) Cotα}
−∞
And its FRFT is calculated as,
∫ ∫ f (τ ) g(t −τ ) e
−∞−∞
2
dτ dt

FRFT (( f D g )(t ) ) = X α (u ) Yα (u ) (8) By putting the change of variable, t – τ = x, we

obtain
Where Fα(u) and Gα(u) are FRFT of f(t) and g(t)
respectively. ⎛ 1− j cotα ⎞ ∞ ∞
Hα (u) =⎜⎜ ⎟ ∫∫ f (τ) g(x)
But in this definition the time-domain 2π ⎟
convolution contains a generalized quantity of a ⎝ ⎠ −∞−∞
j
signal which is function of FRFT of that signal {((τ+x)2+u2)Cotα − 2(τ+x)uCscα−2τ xCotα}
itself, in this relation the defined convolution e2 dτ dx
(13)
process is function of not only time variable but ⎛ 1− j cotα ⎞ ∞⎧∞ {(τ +u )Cotα − 2τ uCscα} ⎫
j 2 2

⎟∫ ∫
also of the transform variable. This generates =⎜⎜ ⎟ ⎨ f (τ) e2 dτ⎬
error. ⎝ 2π ⎠ −∞⎩−∞ ⎭
In this article, a new expression for the FRFT of j 2
{x cotα − 2xucscα}
convolution integral is proposed to overcome the g(x)e2 dx
difficulties and shortcomings of the previous
reported definitions.
j
PROPOSED CONVOLUTION THEOREM 1 − j cot α 2 u 2 cot α
Multiply and divide by e ,
Let us introduce a new definition. 2π
Definition: For any two functions f(t) and g(t), we get
we define the weighted convolution operation as
Let us introduce a new definition.
 2π
j
− u2 Cotα 1− j Cotα CONCLUSION
Zα (u) = e2 We have introduced new expression for the
1− j Cotα 2π FRFT of convolution integral. This can be
⎧ ∞ ⎧⎪ 1− j Cotα ∞ j 2 2
{(τ +u ) Cotα − 2τ u Cscα} ⎫⎪ ⎫ considered as convolution theorem for FRFT
⎪ ∫⎨
2π −∫∞
f (τ ) e 2
d τ ⎬ ⎪⎪ and enhances the support for FRFT to be
⎪ −∞⎪ ⎪⎭
⎨ ⎩ ⎬ considered as integral transform. In the proposed
⎪ j 2 2
{(x +u ) Cotα − 2 x u Cscα } ⎪ relationship, the shortcomings and difficulties of
⎪⎩ g(x) e2 dm⎪⎭ the work reported by Almeida [3], Zayed [4] and
j Wei [5] has been removed.
2π − u2 Cotα
= e2 Fα (u) Gα (u)
1− j Cotα REFERENCES
(13) [1] V. Namias, “The fractional Fourier transform
So we have a convolution theorem transform and its application to quantum mechanics,” J.
pair as: Inst. Math. Appl. vol. 25, pp 241-265, 1980.
[2] Rajiv Saxena, Kulbeer Singh, “Fractional
∞ Fourier transform: a novel tool for signal
∫ f (τ ) g(t −τ ) e processing,” J. Indian Inst. Sci., Jan-Feb, 2005,
j τ (τ −t ) Cotα
dτ
−∞ 85, 11-26.
j [3] Luis B. Almeida, “Product and convolution
2π − u 2 cotα
theorems for the fractional Fourier transform,”
⇔ e 2 Fα (u) Gα (u)
1− j Cotα IEEE Signal Processing Letters, vol. 4, no. 1,
(14) Jan 1997.
[4] Ahmed I. Zayed, “A convolution and product
theorem for the fractional Fourier transform,”
IEEE Signal Processing Letters, vol. 5, no. 4,
Special Case: By putting α = π/2 in the April 1998.
transform pair, we get classical convolution [5] Deyun Wei, Qiwen Ran, Yuanmin Li, Jing
theorem for Fourier transform. Ma, and Liying Tan, “A convolution and
∞ product theorem for the linear canonical
∫ f (τ) g(t −τ) dτ
−∞
⇔ 2π Fπ / 2 (u) Gπ / 2 (u) (15) transform,” IEEE Signal Processing Letters, vol.
16, no. 10, Oct 2009.
Where Fπ / 2 (u ) and Gπ / 2 (u ) are Fourier
transform of f(t) and g(t) respectively.