CHAPTER
CONVOLUTION & PRODUCT
THEOREMS FOR FRFT
3
M
ANY properties of FRFT were derived, developed or established earlier as described in
previous chapter. This includes multiplication property, differentiation property, shifting
property, modulation property, and few more. Since, the convolution theorem of the transform plays an
important role in digital signal processing, so it is extensively investigated always for the refinement to a
well accepted closed-form expression.
3.1
INTRODUCTION
In Fourier transform, the convolution theorem states that the Fourier transform of a convolution
of two signals is the point wise product of respective Fourier transforms (FT) of both the signals. In other
words, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other
domain (e.g., frequency domain). The usefulness of convolution theorem can be best explained by its
application in filtering. Since filtering can be performed both way i.e., time domain filtering and
frequency domain filtering. Simultaneously, if the computational complexity is a basis parameter then it
can be shown that under different input conditions one type of filtering has advantage over other [30] and
vice-versa.
One of the inferences of convolution is the central limit theorem. The Central Limit Theorem is
an important tool in probability theory because it mathematically explains why the Gaussian probability
distribution is observed so commonly in nature. For example: the amplitude of thermal noise in electronic
circuits follows a Gaussian distribution; the cross-sectional intensity of a laser beam is Gaussian; even the
pattern of holes around a dart board bull's eye is Gaussian. In its simplest form, the Central Limit
Theorem states that a Gaussian distribution results when the observed variable is the sum of many random
processes, each with finite mean and variance. Even if the component processes do not have a Gaussian
distribution, the sum of them will behave as Gaussian distribution. The Central Limit Theorem has an
interesting implication for convolution. If a pulse-like signal is convolved with itself many times, a
Gaussian is produced. Other applications of convolution are –

In electrical engineering, the convolution of input signal with the impulse response gives the
output of a linear time-invariant system (LTI). At any given moment, the output is an
[45]
accumulated effect of all the prior values of the input function, with the most recent values
typically having the most influence. Convolution amplifies or attenuates each frequency
component of the input independent to the other components.

In statistics, as noted above, a weighted moving average is a convolution.

In probability theory, the probability distribution of the sum of two independent random variables
is the convolution of their individual distributions.

In optics, many kinds of "blur" are described by convolutions. A shadow (e.g., the shadow on the
table when you hold your hand between the table and a light source) is the convolution of the
shape of the light source that is casting the shadow and the object whose shadow is being cast. An
out-of-focus photograph is the convolution of the sharp image with the shape of the iris
diaphragm.

Similarly, in digital image processing, convolution filtering plays an important role in many
important algorithms in edge detection and related processes.

In linear acoustics, an echo is the convolution of the original sound with a function representing
the various objects that are reflecting it.

In artificial reverberation (digital signal processing, pro-audio), convolution is used to map the
impulse response of a real room on a digital audio.

In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta
pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.

In physics, wherever there is a linear system with a "superposition principle", a convolution
operation makes an appearance.

In computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the
convolution operation to lower the range of length scales necessary in computation thereby
reducing computational cost.
For non-stationary signals and noise, the time and frequency domain filtering both fails because
the signal and noise may have their respective Wigner distribution overlapping to each other in time and
frequency domains. In this case, fractional Fourier transform based filtering can provide a better solution,
where for a rotated domain in time-frequency plane corresponding to an optimum value of angle
[46]
parameter, the Wigner distribution of signal and noise may be separated. The filtering of the signal from
the noise can be performed by designing a filter in FRFT domain [121] with this optimum angle
parameter value.
3.2
CONVOLUTION THEOREM FOR FT
Classically, the convolution theorem of the FT for the signals
( ) and ( ) with associated
Fourier transforms, ( ) and ( ), respectively is given by( )⊗ ( ) =∫
( ) ( − )
↔ √2
( )
( )
(3.2.1)
Where, ⊗ denotes the linear convolution operation.
This theorem states that convolution of two signals in time domain results in simple
multiplication of their Fourier transforms in frequency domain.
3.2.1 Properties
The convolution theorem defined for any integral transform has to satisfy a set of properties in
order to establish its utility in various application areas. The set of properties needed is- Commutative
property, Associative property, and Distributive property.
3.2.1.1 Commutative property
In mathematics, an operation is commutative if changing the order of the operands does not
change the end result. Records of the implicit use of the commutative property go back to ancient times.
The Egyptians used the commutative property of multiplication to simplify computing products. Formal
uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians
began to work on a theory of functions. The first recorded use of the term commutative was in a memoir
by Francois Servois in 1814, which used the word commutative when describing functions that have what
is now called the commutative property. The word is a combination of the French word commuter
meaning "to substitute or switch" and the suffix ative meaning "tending to" so the word literally means
[47]
"tending to substitute or switch." Mathematically, the commutative property is defined for the two
functions &
as –
⋆
=
⋆
(3.2.2)
Where, ′ ⋆ ′, represents a mathematical operation which is commutative in nature. For example,
convolution is a commutating operation. In words, the order in which two signals are convolved makes no
difference; the results are identical which facilitate, in any linear system, the input signal and the system's
impulse response can be exchanged without changing the output signal.
3.2.1.2
Associative property
The associative property is closely related to the commutative property. The associative property
of an expression containing two or more occurrences of the same operator states that the order in which
operations are performed does not affect the final result, as long as the order of terms is not changed. In
contrast, the commutative property states that the order of the terms does not affect the final result. The
associative property is used in system theory to describe how cascaded systems behave, for example, two
or more systems are said to be in a cascade if the output of one system is used as the input for the next
system. From the associative property, the order of the systems can be rearranged without changing the
overall response of the cascade. Further, any number of cascaded systems can be replaced with a single
system. The impulse response of the replacement system is found by convolving the impulse responses of
all of the original systems. Mathematically, the commutative property is defined for the two functions ,
&
as –
( ∘ )∘ =
∘( ∘ )
(3.2.3)
Where, ′ ∘ ′, represents a mathematical operation which is associative in nature.
3.2.1.3
Distributive property
In equation form, the distributive property can be written as –
⋄( + )=
⋄
+ ⋄
[48]
(3.2.4)
Where, ′ ⋄ ′, represents a mathematical operation which is distributive in nature. The distributive
property describes the operation of parallel systems with added outputs. Here, two or more systems can
share the same input, x[n], and have their outputs added to produce a resultant output. The distributive
property allows this combination of systems to be replaced with a single system, having an impulse
response equal to the sum of the impulse responses of the original systems.
3.3
EXISTING DEFINITIONS OF CONVOLUTION THEOREM FOR FRFT
In the available literature, many definitions of convolution theorem for FRFT are being proposed
such as by Almeida [77], Zayed [6], Deng Bing et al [28] and Deyun Wei et al [35]. In this section a brief
review of these definitions are presented.
In 1997, Almeida [77] gave a definition for convolution theorem by considering two signals, ( )
and ( ) as-
∫
( ) ( − )
↔ |sec |
( ) [( − ) sec ]
∫
(3.3.1)
The FRFT of a convolution can be obtained by multiplying a chirp to the FRFT of one of the
signals and convolving it with a scaled version of the other signal, subsequently multiplying again by
another chirp and a scale factor, as evident from (3.3.1).
Similarly, the product theorem was given by( ) ( )↔
|
|
√
( )
∫
[( − ) csc ]
(3.3.2)
Where, Y(u) is Fourier transform of y(t).
Later on, in year 1998, Zayed [6] has considered the approach of multiplying chirp to the signals
before convolving them. Subsequently, the FRFT of the obtained convolution is derived which is entirely
in the FRFT domain. The convolution theorem was given as -
∫
( )
( − )
(
)
Along with the product theorem given by Zayed [6] was-
[49]
↔
( )
( )
(3.3.3)
( ) ( )
↔
( )
∫
(
( − )
)
(3.3.4)
Then for a bigger set, i.e., linear canonical transform (LCT) a convolution theorem was proposed
by Deng et al [28] in 2006 as given below ( , , , )[
( ) ⊘ ( )] =
( , , , )[
( )]
( )]
( , , , )[
(3.3.5)
Where, operator ′ ⊘ ′ is dependent on the parameter of the LCT, and its definition is as follows ( ) ⊘ ( )=
( )
∫
(
( − )
)
(3.3.6)
Since, FRFT is a special case of the LCT. Actually given identity for LCT converts into the
equivalent definition for FRFT when the parameter set ( , , , ) of LCT is changed as =
−
( )
( )
( )
( )
(3.3.7)
Therefore, the convolution theorem for the FRFT will be obtained by using definition of FRFT,
(3.3.5), (3.3.6), and (3.3.7) asℑ[ ( ) ⊘ ( )] =
ℑ[ ( )] ℑ[ ( )]
(3.3.8)
Where,
( ) ⊘ ( )=
∫
( )
(
( − )
)
(3.3.9)
And the converted identity for FRFT (3.3.8 – 3.3.9) resembles with the definition given by Zayed [6].
Later on in year 2009, a different definition of convolution theorem in LCT domain is given by
Deyun Wei et al [35]. Prior to defining convolution theorem, author defines a τ-generalized translation of
signal ( ) denoted by (
(
) where)=
√
√
∫
( , , , )(
)
(
By considering the inverse expression for LCT, (3.3.10) may be written as -
[50]
)
(3.3.10)
(
Based on this
) ==
(
√
(3.3.11)
, , )
,
− generalized translation of the signal ( ) the convolution theorem was given as –
∫
( ) (
)
↔
( , , , )[
( )]
( , , , )[
( )]
(3.3.12)
Again by using definition of FRFT, (3.3.11) and (3.3.12), the convolution theorem in FRFT
domain given by Deyun Wei et al [35] is written by defining the
− generalized translation of the signal
( ) in FRFT domain by(
)=∫
( ) K (u, τ) K ∗ (u, t) du
(3.3.13)
Where K and K ∗ represent FRFT and IFRFT kernel respectively. By utilizing this generalized
function, the convolution theorem for FRFT was defined as∫
( ) (
)
↔
( )
( )
(3.3.14)
As per the available literature studied, these convolution theorems were established and projected without
any comparative study with the earlier existing such relationships. In the study, an effort is made to define
some performance metrics for evaluating these identities. These indices are included in the next section.
3.4
PERFORMANCE METRICES FOR EVALUATION OF CONVOLUTION
THEOREMS
Some parameter indices have been proposed and defined in this study to make a comparison of
the various available convolution theorems. These are – Fourier transform (FT) convertibility, variable
dependability, computational error and simulation approach.
3.4.1 FT Convertibility
Since, fractional Fourier transform is defined as a generalization of Fourier transform in timefrequency plane, or in other words, the FRFT converts into FT at α (angle parameter value) becomes π / 2.
Hence, any property defined for FRFT should be convertible into its equivalent version of the classical
property of FT for the angle parameter value, α = π / 2. This aspect of FRFT and FT is considered as a
parameter index and named as ‘FT Convertibility’.
[51]
3.4.2 Variable Dependability
By analyzing the convolution theorem for Fourier transform, it can clearly be noticed that, the
convolution integral is dependent on ‘time variable’ only and its Fourier transformed version is function
of ‘frequency’ variable only. For example, when the time domain convolution of rectangular window
function, which depends on ‘time’ variable only, is Fourier transformed, it is having a closed form
expression of ‘sinc’ function, dependent upon frequency as variable only. This inherent quality of a
convolution theorem of FT is now considered as a performance index for the convolution theorem for
FRFT, which can be summarized as, the convolution defined in one domain and its transformed
counterpart in transformed domain should have mathematical expressions in terms of respective domain
variables only. And this parameter is named as ‘variable dependability’.
3.4.3 Computational Error
Chirp is a signal in which the frequency is changing with time, and practically, it is a signal
having finite bandwidth along with finite time duration. Usually, a signal having finite duration and
infinite bandwidth or infinite duration and finite bandwidth are easy to generate. But in the case of chirp
signals, where, both bandwidth and duration is finite, the practical generation of chirp signal always
posses some error. Due to this inherent error in practical generation it is considered as another
performance matrix, i.e., the method which has minimum number of chirp multiplications will be a better
definition for convolution theorem for FRFT.
3.4.4 Simulation Comparison
When a unity rectangular window function is convolved linearly with itself, gives a triangular (or
Bartlett) window function of duration double to it, in the time domain. Due to this reason, the convolution
theorems are compared by evaluating the convolution of rectangular function in the fractional Fourier
transformed domain with the FRFT of triangular (or Bartlett) window function. The convolution method
simulation result which is more near to the FRFT of triangular function is considered as a better approach
for convolution theorem for FRFT.
3.5
DEFICIENCIES IN EXISTING METHODS
During the examination of these propositions the following observations are made –
[52]
a) The definition given by Almeida [77] has failed to convert into its Fourier transform (FT)
equivalent, when α = π/2.
b) The definitions given by Zayed [6] and Deng et al [28] have required three chirp
multiplications to evaluate the defined convolution integral.
c) In the definition given by Deyun et al [35], the generalized convolution operation defined
in time domain is not only dependent on time variable but it also depends on transform
domain variable ‘u’ in which it has to be transformed.
This necessitates that a new convolution theorem should be developed in a manner  Which is capable to convert the convolution identity into its FT equivalent, at α = π/2,
 In which the convolution integral should be dependent on time variable only and its
transform will depend on transform domain variable, and
 Where, the computation of convolution identity requires less number of chirp
multiplications.
3.6
PROPOSED CONVOLUTION & PRODUCT THEOREM
The unavailability of a well defined and structured convolution and product theorems, as
explained in the previous section, has motivated to propose another method to define these identities for
FRFT which is a modified version of the identities proposed by Zayed [6]. These modified identities not
only satisfy variable dependability and FT conversion (for α = π/2) but also satisfy all the properties that
these classical identities in FT domain satisfies.
3.6.1 Proposed Convolution Theorem
The circular convolution has been defined separately for Discrete Fourier Transform (DFT) in
place of linear convolution, whereas the circular and linear convolutions are related and one can be
determined if other is known. Similarly, a weighted convolution is proposed for FRFT, which can be
made linear with α = π/2.
Definition: For any two functions ( ) and ( ), the weighted convolution operation is defined as (here
⊖ symbol is used to represent proposed modified convolution operation).
[53]
( ) = ( ⊖ y)( ) = ∫
∞
∞
( ) ( − )
(
)
(3.6.1)
Theorem: Let ( ) is weighted convolution of two functions ( ) and ( ), and
( ),
( ) and
( )
are FRFT at an angle α of functions ( ), ( ) and ( )respectively. Then( )=∫
∞
∞
( ) ( − )
(
)
( )=
↔
( ) ( ) (3.6.2)
Proof: Considering the LHS of this identity
( ) = ( ⊖ y)( ) = ∫
∞
∞
( ) ( − )
(
)
(3.6.3)
Taking its FRFT
( )=
∫
∞
∞
( )
(3.6.4)
Putting the expression of ( ) from (3.6.3) into (3.6.4)
( )=
∫
∞
∞
∞
∞
( ) ( − )
(
)
(3.6.5)
∞
∞
∞∫ ∞
( ) ( − )
(
)
(3.6.6)
∫
This can be rearranged as –
( )=
∫
Substituting, − =
⟹
( )=
∫
= +
∞
∞
∞∫ ∞
⟹
=
(
( ) ( )
)
(
)
(3.6.7)
After some manipulation
( )=
∫
∞
∞
∞∫ ∞
( ) ( )
After multiplying and dividing by
[54]
(
)
(3.6.8)
( )=
2
∞
1 − cot
2
1 − cot
( )
∞
∫
∞
∞
( )
(3.6.9)
This can be written as –
( )=
( ) ( )
(3.6.10)
Finally, convolution theorem transform pair for FRFT is given as-
∫
3.6.1.1
∞
∞
( ) ( − )
(
)
( ) ( )
↔
(3.6.11)
FT convertibility
By putting α = π/2 in the transform pair, the linear convolution for Fourier transform can be
obtained as∫
( )and
∞
∞
( ) ( − )
↔ √2
⁄
( )
⁄
⁄
3.6.1.2
Properties satisfied by proposed convolution theorem
(a)
(3.6.12)
( ) are Fourier transform of ( ) and ( ) respectively.
Where,
⁄
( )
Commutative law
Lemma:
( ⊖ y)( ) ↔
( )
[55]
( )
(3.6.13)
( ⊖ x)( ) ↔
( )
( )
(3.6.14)
( ⊖ y)( ) = ( ⊖ x)( )
(3.6.15)
Proof: Considering the LHS of (3.6.14)
( ) = ( ⊖ x)( ) = ∫
∞
∞
( ) ( − )
(
)
(3.6.16)
Taking its FRFT
( )=
∞
∞
( )
(3.6.17)
( ) from (3.6.16) into (3.6.17)
Putting the expression of
( )=
∫
∫
∞
∞
∞
∞
( ) ( − )
(
)
(3.6.18)
∞
∞
∞∫ ∞
( ) ( − )
(
)
(3.6.19)
∫
This can be rearranged as –
( )=
∫
Substituting, − =
⟹
( )=
∫
= +
∞
∞
∞∫ ∞
⟹
=
(
( ) ( )
)
(
)
(3.6.20)
After some manipulation
( )=
∫
∞
∞
∞∫ ∞
(
( ) ( )
After multiplying and dividing by
[56]
)
(3.6.21)
( )=
2
∞
1 − cot
2
1 − cot
∫
( )
∞
∞
∞
( )
(3.6.22)
( ) ( )
(3.6.23)
This can be written as –
( )=
Hence from (3.6.10) and (3.6.23), it can be conclude that,
( ⊖ y)( ) = ( ⊖ x)( )
(3.6.24)
This proves that the proposed convolution theorem for FRFT satisfies the commutative law.
(b)
Associative law
Lemma:
{( ⊖ y) ⊖ }( ) ↔
( )
( )
( )
(3.6.25)
{ ⊖ ( ⊖ f)}( ) ↔
( )
( )
( )
(3.6.26)
{( ⊖ y) ⊖ }( ) = { ⊖ ( ⊖ f)}( )
(3.6.27)
Proof: Considering the LHS of (3.6.25)
( ) = {( ⊖ y) ⊖ }( ) = { ⊖ }( )
(3.6.28)
Where,
( ) = ( ⊖ y)( ) = ∫
∞
∞
( ) ( − )
[57]
(
)
(3.6.29)
Now,
( ) = {( ⊖ y) ⊖ }( ) = { ⊖ }( ) = ∫
∞
∞
(
( ) ( − )
)
(3.6.30)
Substituting the expression of ( ) from (3.6.29) into (3.6.30)
( ) = { ⊖ }( ) = ∫
∞
∞
∞∫ ∞
(
( ) ( − )
)
(
( − )
)
(3.6.31)
Taking FRFT of (3.6.31)
( )=
Substituting,
∫
∞
∞
( )
(3.6.32)
( ) from (3.6.31) into (3.6.32)
∞
1 − cot
2
( )=
∞
∞
(
( ) ( − ) ( − )
∞
∞
)
(
)
∞
(3.6.33)
Substituting −
=
⟹
( )=
=
1 − cot
2
+
⟹
∞
∞
=
∞
(
( ) ( − ) ( )
∞
∞
)
∞
(
)
(
)
(3.6.34)
After some manipulation,
( )=
( )∫
∞
∞
∞∫ ∞
Again substituting
− =
( )=
( )∫
⟹
∞
∞
∞∫ ∞
(
( ) ( − )
= +
)
⟹
(3.6.35)
=
(
( ) ( )
[58]
)
(
)
(3.6.36)
Multiplying and dividing by
( )=
∫
2
( )
1 − cot
∞
∞
∞∫ ∞
1 − cot
2
(
( ) ( )
)
(3.6.37)
After some manipulation (3.6.37) can be rearranged as –
( )=
2
( )
1 − cot
∞
1 − cot
2
( )
∞
∫
∞
∞
( )
(3.6.38)
And by applying the definition of FRFT, (3.6.38) can be written as –
( )=
Similarly, it can also be proved that the FRFT of
( )
( )
( )
(3.6.39)
( ) = { ⊖ ( ⊖ f)}( ) will be –
( )=
( )
( )
( )
(3.6.40)
Hence from (3.6.39) and (3.6.40)
{( ⊖ y) ⊖ }( ) = { ⊖ ( ⊖ f)}( )
(3.6.41)
This proves that the proposed convolution theorem for FRFT satisfies the associative law.
(c)
Distributive law
Lemma:
{ ⊖ ( + f)}( ) ↔
( ){ ( )+
{( ⊖ y) + (x ⊖ )}( ) ↔
{
[59]
( )
( )+
( )}
( ) ( )}
(3.6.42)
(3.6.43)
{ ⊖ ( + f)}( ) = { ⊖ y + x ⊖ }( )
(3.6.44)
Proof: Considering the LHS of (3.6.43)
∞
( ) = {( ⊖ y) + (x ⊖ )}( ) =
( ) ( − )
(
)
∞
+∫
∞
∞
(
( ) ( − )
)
(3.6.45)
Taking FRFT of (3.6.45)
( )=
∞
∞
( )
(3.6.46)
( ) from (3.6.45) into (3.6.46)
Putting the expression of
( )=
∫
∞
1 − cot
2
∞
∞
(
( ) ( − )
∞
)
( ) ( − )
+
∞
(
)
∞
(3.6.47)
Substituting, − =
( )=
⟹
= +
∞
1 − cot
2
⟹
=
∞
∞
( ) ( )
∞
( ) ( )
+
∞
∞
(
)
(
)
(3.6.48)
And it can be rearranged as –
( )=
∞
1 − cot
2
∞
(
( ) ( )
∞
∫
∫
)
(
)
+
∞
∞
∞
(
( ) ( )
And after some manipulation –
[60]
)
(
)
(3.6.49)
( )=
1 − cot
2
∞
∞
(
( ) ( )
∞
∫
)
+
∞
∫
∞
∞
(
( ) ( )
Multiplying and dividing by
)
(3.6.50)
and doing some step of calculations –
( )=
{
( )
( )+
( ) ( )}
(3.6.51)
( )}
(3.6.52)
Utilizing the linearity property of FRFT, (3.6.51) can be written as –
( )=
( ){ ( )+
Similarly, it can also be proved that the FRFT of
( )=
( ) = { ⊖ ( + f)}( ) will be –
( ){ ( )+
( )}
(3.6.53)
Hence from (3.6.52) and (3.6.53)
{ ⊖ ( + f)}( ) = {( ⊖ y) + (x ⊖ )}( )
(3.6.54)
This proves that the proposed convolution theorem for FRFT satisfies the distributive law.
3.6.2 Proposed Product Theorem
Definition: For any two functions ( ) and ( ), the modified product operation,
( ), is defined as
( )= ( ) ( )
Theorem: Let
(3.6.55)
( ) is weighted product of two functions ( ) and ( ), and
( ),
( ) and
( ) are
FRFT of ( ), ( ) and ( ) respectively. Then –
( )= ( ) ( )
↔
( )=
∫
[61]
( )
( − )
(
)
(3.6.56)
Proof: Considering the RHS of identity (3.5.56) as –
( )=
( )
∫
( )=
( )=
∫
∞
∞
( )
1 + cot
2
⟹
( )=
(3.6.58)
=
∞
1 + cot
2
(
=
(3.6.57)
( ) from (3.6.57) into (3.6.58) –
Putting, the expression of
−
)
( ) as –
Taking, Inverse FRFT of
Substituting,
(
( − )
+
⟹
1 + cot
2
( )
( − )
∞
)
(3.6.59)
=
1 + cot
2
∞
( )
( )
∞
(
)
(
)
(3.6.60)
After some step of manipulation –
( )=
1 + cot
2
∞
1 + cot
2
( )
( )
(
)
∞
(3.6.61)
Adding and subtracting
( )=
in (3.6.61) and after rearranging them –
∫
( )
∫
[62]
( )
(3.6.62)
Applying the definition of Inverse FRFT on (3.6.62) –
( )=
( ) ( )
(3.6.63)
Finally, convolution theorem transform pair for FRFT is given as( ) ( )
↔
∫
( )
( − )
(
)
(3.6.64)
Properties satisfied by product theorem: Same as that of convolution theorem.
3.7
COMPARATIVE ANALYSIS OF THE PROPOSED THEOREM TO EXISTING
METHODS
A comparative analysis of available definitions of convolution function on the following
parameters is presented in this section.
3.7.1 FT Convertibility
The proposed convolution theorem should be converted into classical convolution theorem for
Fourier transform with angle parameter, α = π/2, it is due to the basic property of FRFT as expression of
FRFT converts into expression of FT at angle parameter, α = π/2. In the Table-3.1, ‘Satisfying’ is entered
for the method where the relation is converted into classical convolution theorem of Fourier transform at
α = π/2 and ‘Not Satisfying’ is mentioned otherwise. In the existing definitions, the identity given by
Almeida [77] is not satisfying this parameter.
Table-3.1: FT Convertibility
Name of Methods
Almeida’s Method
Performance Index –
FT convertibility
Not Satisfying
Zayed’s Method
Satisfying
Deng et al Method
Satisfying
Deyun et al Method
Satisfying
Proposed Method
Satisfying
[63]
3.7.2 Variable Dependability
The convolution defined in one domain and its transformed counterpart in transformed domain
should have mathematical expressions in terms of respective domain variables only. This parameter is
assumed in order to assure that a quantity defined in one domain when transformed will result in an
equivalent quantity in transformed domain. In the Table-3.2, ‘Satisfying’ is included for the method,
which transform a convolution function defined in one domain variable into equivalent function of
transform domain variable and ‘Not Satisfying’ is for the method in which either convolution function is
dependent on both variable or transformed equivalent quantity is function of both variable.
Table-3.2: Variable Dependability
Name of Methods
Almeida’s Method
Performance Index –
Variable Dependability
Satisfying
Zayed’s Method
Satisfying
Deng et al Method
Satisfying
Deyun et al Method
Not Satisfying
Proposed Method
Satisfying
3.7.3 Computational Error
A simple block diagram implementation of the mathematical relations given by various authors
for establishing the convolution theorem is used for the comparison purpose based on the computational
error. The relationships (both the LHS and RHS) given by Deyun et al [35; 2009], Deng et al [28; 2006],
Zayed [6; 1998], Almeida [77; 1997] and proposed one are shown with the help of their block diagram
realizations in Figure-3.1 to Figure-3.5 respectively. The nomenclature of various building blocks is given
in Figure-3.6.
[64]
 t 2  u 2   2tu
x(t)
 dt
y(t)
 dt
t2
F1(α)
 u 2   2tu
F1(α)
 1
z(t)
(a) RHS
  2  u 2  2 u
y(t)
F1(α)
t2
 dt
  2
 d
 du
z(t)
F2(α)
F1(α)
 2 u  2t u
(b) LHS
Figure-3.1: Block diagram representation of Deyun et al [35] method.
 t 2  u 2   2tu
x(t)
 dt
y(t)
 dt
t2
F1(α)
 u 2   2tu
 u 2
 1
F1(α)
z(t)
(a) RHS
[65]
 2
 t 2
F1(α)
 d
x(τ)
y(τ)
z(t)
De(t)
& In
 2
(b) LHS
Figure-3.2: Block diagram representation of Deng et al [28] method.
 t 2  u 2   2tu
x(t)
 dt
y(t)
 dt
t
2
u
2
 u 2
F1(α)
  2tu
 1
F1(α)
z(t)
(a) RHS
 t 2
 2
F1(α)
 d
x(τ)
y(τ)
z(t)
De(t)
& In
 2
(b) LHS
Figure-3.3: Block diagram representation of Zayed [6] method.
[66]
 t 2  v2
  2tv
x(t)
F1(α)
 v2
 dt
VC
t_v Secα
t _ v Sec α
y(t)
De(u)
& In
& In
z(t)
 dv
 u 2
 1
F4(α)
(a) RHS
y(τ)
De(t)
& In
& In
 d
z(t)
x(τ)
(b) LHS
Figure-3.4: Block diagram representation of Almeida [77] method
 t 2  u 2   2tu
F1(α)
 u 2
F5(α)
x(t)
 dt
y(t)
 dt
t2
 u 2   2tu
 1
F1(α)
z(t)
(a) RHS
[67]
 2
 2t
x(τ)
 d
De(t)
& In
z(t)
y(τ)
(b) LHS
Figure-3.5: Block diagram representation of proposed method.
t
2
→
Chirp Function ( e
j(
t2
) Cot ( )
2
)
F1(α)
→
 dt
→ Integrator (with respect to‘t’)
F2(α)
→
De(t)
& In
→ Delay (with respect to‘t’) & Inversion
F3(α)
→
VC
t_v Sec α
→ Variable Conversion (‘t’ to ‘v Secα’)
 2tu
1 jCot ( )
2
1  jCot ( )
2
Csc( )
2
→
Chirp Function ( e  j 2 tu Csc ( ) )
 1
→
F5(α)
→
F4(α)
→
IFRFT
2
1  jCot ( )
Sec ( )
Figure-3.6: Blocks used in the block diagram representation
Based on the above block diagram realization, the numbers of chirp multiplications are calculated
for each of the methods and resulting analysis is shown in the Table-3.3 (LHS represents the defined
convolution process by different methods and RHS represents their transforms).
Table-3.3: Computational complexity of all the methods for Convolution Theorem
Parameter
Hardware Complexity
No. of Chirp functions
Deyun et al
[35]
LHS RHS
7
6
Deng et al
[28]
LHS RHS
3
7
[68]
Zayed[6]
LHS RHS
3
7
Almeida[77]
Proposed
LHS RHS LHS RHS
5
2
7
3.7.4 Simulation Based Comparison
The mathematical relations describing the convolution operation in fractional domain by Zayed
[5] and Deyun et al [35] is compared with the proposed identity for convolution by simulating both the
expressions on the platform of Wolfram Mathematica® software (version-7.0) on a system having
configuration Pentium-4, Intel(R) CPU–1.8 GHz processor having 1GB RAM.
A rectangular window function ′ ( )′ of unity amplitude and unit duration (0 ≤ t ≤ 1.0) is
convolved linearly with itself ( ⊖ r)( ), gives a triangular (or Bartlett) window function of duration
double to it, in the time domain as shown in Figure-7. As, the methods given by Zayed [6] and Deng et al
[28] are having similar closed form expressions, therefore, the definition given by Zayed is only
simulated for the analysis purpose. The source codes for the convolution theorem given by Zayed [6],
Deyun et al [35] and of the proposed one are tested by choosing the two functions as ′ ( )′ in respective
convolution integral. The results of the FRFT determined, at an angle α = π/4, by the Zayed’s method,
Deyun’s method and also by the proposed convolution identity are shown in the Figure-8 and for angle α
= π/3 are shown in Figure-9. Simultaneously, the FRFT of the triangular window function is also
evaluated for the same angle to make a comparison. It has been shown, that the FRFT for α = π/4 and for
α = π/3 of the triangular window function resembles maximally, i.e. the real (Re), imaginary (Im) and
absolute (Abs) components to the FRFT of the convolved output with the proposed theorem. It is also
visible from the Figure-8(a) and Figure-8(b) in case of angle α = π/4 and in the Figure-9(a) and Figure9(b) in case of angle α = π/3 that the Real and Imaginary components are more oscillatory in the case of
Deyun’s method than the similar components determined by the proposed method.
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1
1
2
1
3
(a)
1
2
3
(b)
Figure-3.7: (a) Rectangular function, ‘ r t  ’, and (b) Convolved signal ‘ rr t  ’ i.e.,
Bartlet window.
[69]
Bartlet
Bartlet
Proposed
Proposed
Deyun's
Deyun's
Zayed's Deng's
Zayed's Deng's
(a)
(b)
Bartlet
Proposed
Deyun's
Zayed's Deng's
(c)
Figure-3.8: FRFT of rr t  for α = π/4 by Zayed, Deyun and Proposed method along with FRFT
of Bartlett window (a) Real component of, (b) Imaginary component of, and (c) Absolute
component of FRFT.
Bartlet
Bartlet
Proposed
Proposed
Deyun's
Deyun's
Zayed's Deng's
Zayed 's Deng's
(a)
(b)
[70]
Bartlet
Proposed
Deyun's
Zayed's Deng's
(c)
Figure-3.9: FRFT of rr t  for α = π/3 by Zayed, Deyun and Proposed method along with FRFT
of Bartlett window (a) Real component of, (b) Imaginary component of, and (c) Absolute
component of FRFT.
3.8 SUMMARY
A modified expression for the convolution integral for FRFT has been introduced after successful
derivation. This can be treated as convolution theorem for FRFT and enhances the support for FRFT for
its consideration as an integral transform. The proposed definition satisfies all the properties of
classical convolution theorem for Fourier transform, i.e. the commutative property, associative property
and distributive property. As shown in the Table-3.2, the definition given by Deyun et al [35] has failed to
satisfy the variable dependability condition, while all other obeys this aspect of classical convolution
theorem. Similarly, the definition given by Almeida [77] is not converted to the classical convolution
theorem with α = π/2, while others satisfy this parameter, as seen from Table-3.1. So analyzing all the five
method of defining the convolution theorem on the FT convertibility and variable dependability
parameters, it can be clearly noticed that the definition given by Zayed [6] and proposed one are only
methods (Since, the method given by Deng [28] is same as given by Zayed [6]) having satisfied both the
parameters, variable dependability and equivalent Fourier transform conversion.
[71]
As can be seen from the simulation results of Figure-3.8 and Figure-3.9, the proposed weighted
convolution theorem is giving results better than the convolution theorem given by Zayed [6] and Deyun
et al [35]. The results determined by the proposed theorem are closer in shape and of matching values to
the FRFT of a triangular window function. The results determined by the convolution expression of
Deyun et al [35] have more oscillations in both the Real and Imaginary components, as it is visible from
the Figure-3.8 and Figure-3.9 for different value of angle α. These oscillations are significant and present
due to the chirp signal included in the calculation of convolution integral by Deyun et al [35], which also
contains variable of the transformed domain. In the context of computational error, the comparison has
been established in terms of the number of chirp multiplications performed in realizing the different
convolution theorems. Finally, comparing the definition given by Zayed [6] and proposed one on the
basis of computational error, it is found that number of chirp multiplications is lesser in the case of
proposed method. Therefore, proposed modified definition of convolution theorem is found to be a
better proposition to other four definitions given by Almeida [77], Zayed [6], Deng et al [28] and Deyun
et al [35]. This type of convolution is termed as weighted convolution.
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[72]