[13]

CHAPTER 2 LITERATURE

SURVEY

HE literature review summarizes, interprets, and evaluates existing "literature" (or

published material) in order to establish current knowledge of a subject. The purpose for

doing so relates to ongoing research to develop that knowledge. The literature review may resolve a

controversy, establish the need for additional research, and define a topic of inquiry.

2.1 LCT AND ITS TYPES

The linear canonical transformation (LCT) is a family of integral transforms that generalizes

many classical transforms. It has four parameters and one constraint, so it is a 3-dimensional family, and

can be visualized as the action of the special linear group on the time–frequency plane [73, 84, 126, and

127]. It was first introduced in 1970s [128, 144]. Many operations, such as the Fresnel transform [71],

Fourier transform (FT), fractional Fourier transform (FRFT) [76, 151], and scaling operations are the

special cases of the LCT.

The linear canonical transform of a signal 푓(푡) with parameter set (푎, 푏, 푐,푑) is defined [134] as -

퐿( , , , )[푓(푡)] = 퐹( , , , )(푢) = 푒

∫ 푒 ∞ 푒

푓(푡) 푑푡 ∀ 푏 ≠ 0

√푑 푒 푓(푑 푢) ∀ 푏 = 0 (2.1.1)

Where, 퐿( , , , )[ ] is the LCT operator, 퐹( , , , )(푢) is the LCT of signal 푓(푡) and the four parameters

(푎, 푏, 푐,푑) associated with LCT should satisfy the following relation-

푎 푑 − 푏 푐 = 1 (2.1.2)

For the different values of parameter set (푎, 푏, 푐,푑), LCT transforms into special transform technique

which has different application areas. For –

푎 푏푐 푑 = 1 휆푧

0 1 (2.1.3)

Where, 휆 is the wavelength and 푧 is the distance, the LCT converts into Fresnel transform. The

Fresnel transform is the operation that describes the monochromic light propagating through the free

space [71]. The 2-Dimensional expression of Fresnel transform is given as –

T

[14]

퐹( , , , )(푢) =

∫ ∫ 푒 ( ) ( ) 푓(푥,푦) 푑푥 푑푦∞∞ (2.1.4)

Where, 푓(푥,푦) is the distribution of the monochromatic light, 휆 is its wavelength, and 푧 is the

propagation distance.

Similarly, fractional Fourier transform (FRFT) is a type of LCT when the parameter set

(푎, 푏, 푐,푑) of LCT is changing as -

푎 푏푐 푑 = 푐표푠(훼) 푠푖푛(훼)

− 푠푖푛(훼) 푐표푠(훼) (2.1.5)

Where, ‘α’ being the angle parameter related with FRFT, defined in time-frequency plane. Also,

LCT converts into Fourier transform (FT), when –

푎 푏푐 푑 = 0 1

− 1 0 (2.1.6)

Analyzing (2.1.5) and (2.1.6), it is evident that FRFT converts into FT when 훼 = (휋 2⁄ ).

Therefore, FRFT may be considered as a generalization of Fourier transform (FT), which was initially

one of the most frequently used tool in signal processing [120].

2.2 MATHEMATICAL PROPERTIES OF FRFT

The definition of FRFT (1.3.13 – 1.3.15), shows that for angle ‘α’ (not a multiple of π) the

computation of FRFT corresponds to following steps [76]:

Multiply by a chirp

Fourier transform with its argument scaled by ‘cosec(α)’

Multiply with another chirp

Multiply again by a complex amplitude factor

The utility of any transform technique is based on the set of mathematical properties possessed by

it because these properties of transform are actually responsible for the application area of the respective

transform technique. For any transform, the closed form expression of its inverse should exist in order to

satisfy unitary property which is a necessary condition for a transform. Similarly, the modulation,

shifting, integration and differentiation properties are facilitating the transform to be used in various

[15]

application areas. The convolution property for a transform is required at most which fits the transform in

filtering and amplification areas. Subsequently, if the transform is having the correlation theorem then it

may be used in radar communication, missile tracking and navigation purpose. Hence, a set of properties

is needed for any transform to have some realizable applications. This aspect of the FRFT is going to be

covered in following sections.

2.2.1 Properties Satisfied by the Kernel of FRFT

The kernel of FRFT ‘K (t, u)’ satisfies the following properties [10, 76] –

K (t, u) = K (u, t) (2.2.1)

K (t, u) = K∗ (t, u) (2.2.2)

K (−t, u) = K (t,−u) (2.2.3)

∫ K (t, u) K (u, z) du = K (t, z) (2.2.4)

∫ K (t, u) K∗ (t, v) dt = δ(u− v) (2.2.5)

Where the asterisk ‘∗’ represent the complex conjugate. The first three properties represented by

(2.2.1 – 2.2.3) can be easily derived from the definition of FRFT. The derivation of property (2.2.4) is

based on the definition of kernel of the FRFT and property (2.2.5) can be derived by considering the LHS

of identity (2.2.5)-

∫ K (t, u) K∗ (t, v) dt (2.2.6)

Using (2.2.1), K (t, u) is changed as K (u, t), and (2.2.6) converts into -

∫ K (u, t) K∗ (t, v) dt (2.2.7)

Now using (2.2.2), the conjugate of kernel is replaced and (2.2.7) becomes-

∫ K (u, t) K (t, v) dt (2.2.8)

And finally using (2.2.4) and definition of FRFT for α = 0, (2.2.8) can be written as

[16]

∫ K (u, t) K (t, v) dt = K ( )(u, v) = δ(u− v) (2.2.9)

As mentioned above, in comparison of FRFT, STFT and WD as a signal processing tool in time-

frequency plane, FRFT comes out as the winner based on its advantage over STFT in terms of

mathematical properties and clarity and over WD based on the cross-term interference, which is totally

removed in the FRFT. Motivated by this superiority of FRFT for analyzing the non-stationary signals,

next section is dedicated to the development of FRFT as a tool in time-frequency plane.

2.2.2 FRFT in Time-Frequency Plane

The expression of FRFT is defined for entire time-frequency plane. In the definition, the value of

angle parameter ‘α’, governs the rotation of the signal to be transformed in time-frequency plane. Given

that ‘F’ is Fourier transform operator, ‘I’ is the identity operator and ‘Fα’ is fractional Fourier transform

operator, the FRFT posses the following important properties in time-frequency plane [76, 121]-

Identity operator (Zero rotation): F0 = I (2.2.10)

Fourier operator (π/2 rotation): Fπ/2 = F (2.2.11)

Reflection operator (π rotation): Fπ = -I (2.2.12)

Inverse Fourier operator (3π/2 rotation): F3π/2 = F-1 (2.2.13)

Identity operator (2 π rotation): F0 = I (2.2.14)

Angle additivity property: Fα F β = Fα+β (2.2.15)

Inverse FRFT: Fα-1 = F- α (2.2.16)

Property (2.2.10), (2.2.12) and (2.2.14) holds by definition of FRFT (1.3.13 – 1.3.15). Property

(2.2.11) is a consequence of the fact that, for α = π/2, the kernel of FRFT ‘K (t, u)’ coincides with the

kernel of Fourier transform and property (2.2.13) is a consequence of the fact that for α = - (π/2),

‘K (t, u)’ coincides with the kernel of inverse Fourier transform. Property (2.2.15) can be easily derived

by using property (2.2.4) as-

Let the FRFT of a signal x(t) is defined as-

푦(푢) = 푋 (푢) = ∫ x(t) K (t, u) dt (2.2.17)

Then FRFT of 푦(푢) will be given as-

[17]

푌 (푧) = ∫ 퐾 (푢, 푧) 푦(푢) 푑푢 =∫ 퐾 (푢, 푧) ∫ x(t) K (t, u) dt 푑푢 (2.2.18)

By changing the order of integration and rearranging the terms-

푌 (푧) = ∫ x(t) ∫ K (t, u) 퐾 (푢, 푧) 푑푢 dt (2.2.19)

And finally using the property (2.2.4) of the kernel of FRFT-

푌 (푧) = ∫ x(t) K (t, z) 푑푡 = X (z) (2.2.20)

This proof shows that two successive FRFT of a signal by angles α and β will be same as a single

FRFT of the same signal by angle α + β.

Similarly, the last property (2.2.16) can be derived by using the fact that FRFT of a signal by

angle ‘2π + α’ is same as the FRFT of the signal by angle ‘α’. By taking the FRFT of 푋 (푢) with an

angle ‘2π – α’ which will equal to the FRFT of 푋 (푢) with an angle ‘α’ and is given as-

∫ 푋 (푢) K (u, t) du =∫ 푋 (푢) K (u, t) du (2.2.21)

Using (2.2.2) -

∫ 푋 (푢) K (u, t) du = ∫ 푋 (푢) K∗ (u, t) du = x(t) (2.2.22)

Hence,

x(t) = ∫ 푋 (푢) K (u, t) du (2.2.23)

Therefore, the inverse FRFT of a transformed signal with angle ‘α’ can be calculated as the FRFT

of the transformed signal with angle ‘-α’.

Thus, FRFT may be thought as a method to express any signal on the basis by a set of

functions K (u, t), with ‘u’ acting as a parameter for spanning the set of basis functions [76]. The basis-

functions are chirp functions, orthonormal according to (2.2.5). For different value of ‘u’ the basis-

functions are only differ by a time-shift and phase factor that depends on u as -

K (t, u) = 푒 K (t − u sec α, 0) (2.2.24)

[18]

For any transform being interpretable as a rotation in time-frequency plane, it has to satisfy the

properties given by (2.2.10 – 2.2.16). Here, the properties like rotation in time-frequency plane, inverse

and angle additivity process are very well defined expressions thus making the FRFT an appropriate tool

[76]. The reason for being - if the FRFT is multiplied by 푒 with integer n, the properties of the

transform, i.e., given in (2.2.10 – 2.2.16) would still hold. In literature, the existence of any other

transform method which obeys this set of properties with such simple relationship with time-frequency

distribution has not been reported.

2.2.3 Properties of the FRFT

Various properties that a transform should uphold in order to make its application in various areas

are described in Table-2.1, such as: Translation, Modulation, Mixed product, Differentiation, Integration,

Multiplication with time variable, Division by time variable, Similarity / Parity, Scaling in time domain,

Linearity and Convolution.

The properties listed in Table-2.1 for FRFT are the extensions of the respective properties of FT.

Their proofs, except scaling property can be found in [2, 76, and 151]. Similarity/Parity property tells us

that if 푥(푡) is even, 푋 (푢) will also be even. Similarly, if 푥(푡) is odd, 푋 (푢) will also be odd. The proof

of scaling property is obtained by considering two signals 푥(푡) and 푦(푡), which are related as -

푦(푡) = 푥(푐푡) (2.2.25)

By taking the FRFT of 푦(푡) as -

푌 (푢) = ∫ 푥(푐푡) 푒 푑푡∞ (2.2.26)

Substituting, t = c t, (2.2.26) becomes -

푌 (푢) = | | ∫ 푥(푡) 푒 – 푑푡∞ (2.2.27)

After some manipulation (2.2.27) can be rearranged as -

푌 (푢) = | | ∫ 푥(푡) 푒 – 푑푡∞ (2.2.28)

[19]

Table 2.1- Properties of Fractional Fourier Transform

S.N. Properties Signal FRFT with angle α

1

Linearity

푎 푥(푡)

+ 푏 푦(푡)

푎푋 (푢) + 푏푌 (푢)

2

Scaling in time

domain

푥(푐푡) 1− 푗 cot훼푐 − 푗 cot 훼

푒 푋푢 sin훽푐 sin훼

,

for 훽 = tan (푐 tan훼)

3

Translation

푥(푡 − 휏) 푋 (푢 − 휏 csc훼) 푒 –

4

Modulation 푥(푡)푒 푋 (푢 − 푣 sin훼) 푒

5

Mixed product 푡푑 푥(푡)푑푡

− sin훼 (sin훼 − 푗푢 cos훼)푋 (푢) + 푢 cos 2훼 푋 (푢)

+ 푗sin 2훼

2푋 (푢)

6

Differentiation

푥 (푡)

푋 (푢) cos훼 + 푗 푢 푋 (푢) sin훼

7

Integration 푥(푡) 푑푡 sec훼 푒 푋 (푧) 푒 푑푧, ∀ 훼

≠ (푛 + 1)휋2

8 Multiplication

with ‘t’

푡 푥(푡)

푢 푋 (푢) cos훼 + 푗 푋 (푢) sin훼

9

Division by ‘t’

푥(푡)푡

−푗 sec훼 푒 푥(푧) 푒 푑푧, ∀ 훼

≠ 푛휋

10 Similarity /

Parity

푥(−푡)

푋 (−푢)

11 Parseval’s

Theorem 푥(푡)푦∗(푡) 푋 (푢) 푌∗(푢) 푑푢

[20]

Now by defining, 훽 = cot = tan (푐 tan훼), and rearranging terms,

푌 (푢) = ∫ 푥(푡) 푒 – 푑푡∞ (2.2.29)

After some step of manipulation (2.2.29) can be written as-

푌 (푢) =

푒 푋 (2.2.30)

The scaling property shows the effect of a change of units of the independent variable ‘푡’ in time

domain on the transformed quantity. In the classical Fourier transform, the effect of such a change is only

a corresponding change in units of the transform ‘frequency’ variable and a scaling of the amplitude of

the transformed quantity. Similarly, for the FRFT, the effects are a scaling of the transform ‘u’ variable

by, ′sin훽/푐 sin훼′, a complex amplitude scaling, a product by chirp, and most importantly, a change in

the angle at which transform is computed from 훼 to 훽 = cot (cot 훼 푐⁄ ) = tan (푐 tan훼). This effect

can be explained as - contracting the time axis in time-frequency plane has an analogous effect on

expansion of the frequency axis by the same factor. [76]. If both effects are analyzed simultaneously, as

FRFT works in time-frequency plane, then it will become clearer that the original angle at which the

transform has to be calculated will now rotate to a new angle. Rotation of domain ‘angle’ will be

counterclockwise if time axis is contracting and clockwise if time axis is expanding.

The Parseval’s theorem for FRFT [76] can easily be proved by expressing 푥(푡) as the inverse

fractional Fourier transform of 푋 (푢) as -

∫ 푥(푡)푦∗(푡)푑푡∞ = ∫ ∫ 푋 (푢) 푒 푑푢∞∞ 푦∗(푡)푑푡 (2.2.31)

After interchanging the order of integration and rearranging the terms -

∫ 푥(푡)푦∗(푡)푑푡∞ = ∫ 푋 (푢) ∫ 푦∗(푡) 푒 푑푡∞∞ 푑푢 (2.2.32)

Using (2.2.2) and the definition of FRFT, (2.2.32) can be written as -

∫ 푥(푡)푦∗(푡)푑푡∞ = ∫ 푋 (푢) 푌∗(푢) 푑푢∞ (2.2.33)

[21]

If both functions are assumed as same, then as a consequence of this equality is the energy-

preserving property of FRFT.

∫ |푥(푡)| 푑푡∞ = ∫ |푋 (푢)| 푑푢∞ (2.2.34)

Due to the energy-preserving property of the Fourier transform, the squared magnitude of the

Fourier transform of a signal |푥(휔)| is called as the energy spectrum of the signal and is interpreted as

the distribution of the signal’s energy among the different frequencies 푒 . Similarly, from the energy-

preserving property (2.2.34) of FRFT, |푋 (푢)| , the fractional energy spectrum of the signal 푥(푡) with

angle α can be interpreted as the distribution of the signal energy among the different chirps K α(t, u).

The sampling theorem of FRFT plays an important role in communication systems and signal

processing. Sampling is a process of converting a continuous-time band limited signal to obtain its

discrete-time version. Sampling theorem of FT states that “a bandlimited signal can be reconstructed

exactly if it is sampled by a rate at least twice the maximum frequency component in it”. For discrete

processing of non-stationary signals the FRFT domain sampling theorem is needed. In literature many

attempts were made to establish sampling theorem [7, 26, 122, 147, and 156]. Tao et al [122] presented a

theorem for FRFT which was based on the product theorem of FRFT and derived sampling theorem with

reconstruction formula, which explains how to sample a continuous-time signal to obtain its discrete-time

version for band limited signals in the fractional Fourier domain.

The FRFT of a signal 푥(푡) satisfies the following condition for 0 ≤ Ω ≤ Ω -

푋 (푢) = 0, 푤ℎ푒푛 |푢| > Ω or |푢| < Ω (2.2.35)

Then 푥(푡) is defined as a bandlimited signal in FRFT domain with bandwidth given below -

Ω = Ω −Ω (2.2.36)

For 푥(푡), the sampling theorem is defined for an appropriate value of 푁, for 1 ≤ 푁 ≤ 푖푛푡[Ω Ω⁄ ] as –

Ω α ≤ Ω ≤ Ω α (2.2.37)

Where, sampling rate Ω is given as Ω = 2π T⁄ for sampling interval ′T ′.

The uncertainty principle says that a function and its Fourier transform cannot be simultaneously

"small". The origin of such type of statements lies in quantum mechanics - for a quantum particle it is

[22]

impossible to determine its position and momentum simultaneously. In the classical time-frequency

domain, the product of time spread and frequency spread cannot be small unlimitedly. Their tradeoff

relationship is called the uncertainty principle, i.e., for one given real absolutely integratable signal 푥(푡)

and its Fourier transform 푋(푢), the product has the lowest bound and is given as –

∆푡 ∆푢 = ∫ |(푡 − 푡 ) 푥(푡)| ∞∞ 푑푡 ∫ |(푢 − 푢 ) 푋(푢)| ∞

∞ 푑푢 ≥ 14 (2.2.38)

Where,

푡 = ∫ 푡 | 푥(푡)| ∞∞ 푑푡 (2.2.39)

And

푢 = ∫ 푢 | 푋(푢)| ∞∞ 푑푢 (2.2.40)

For ∆푡 is the time spread and ∆푢 is the frequency spread.

It has been shown in FRFT domain that uncertainty principle has lower bounds than those of

traditional cases, which provides the possible selections for improved resolution analysis. Therefore, there

is a great need to extend the traditional entropic uncertainty principle to the FRFT domains as well. Many

articles are available in literature on uncertainty principle of FRFT [142, 154, and 155]. Guanlei et al

[154] had derived logarithmic, Heisenberg’s and short-time uncertainty principle associated with FRFT.

Subsequently, Guanlei et al [155] gave two entropic uncertainty principle based on fractional Fourier

transform, in which one was Shannon’s entropy uncertainty principle and the other was Renyi’s entropy

uncertainty principle. Shinde et al [142] introduced an uncertainty principle for real signals in the

fractional Fourier domain, in which a lower bound on the uncertainty product of signal representations in

two FRFT domains for real signals was obtained, and it was shown that a Gaussian signal achieves the

lower bound.

For a unit energy real-valued signal 푥(푡) and its fractional Fourier transform, 푋 (푢), the

uncertainty principle can be defined as –

∆푢 ∆푢 ≥ ∆푡 cos훼 cos훽 + ∆

+ { ( )} (2.2.41)

Where,

∆푢 = ∫ |(푢 − 푢 ) 푋 (푢)| ∞∞ 푑푢 (2.2.42)

[23]

And equality is achieved when

푥(푡) = 푒푥푝 −

(2.2.43)

Where, 휎, is an arbitrary real constant.

2.2.4 Existing Convolution & Product Theorem for FRFT

Studies reported in above sections reveals that almost all the properties of FRFT are already

derived except convolution theorem, product theorem and correlation theorem. The usefulness of

convolution theorem can be best explained by its application in filtering. Since, filtering can be performed

in both ways 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 and vice-versa [30]. . For non-stationary signals and noise,

filtering of the signal from the noise can be performed by designing a filter in FRFT domain with this

optimum angle parameter value [121]. FRFT domain convolution and product theorems were extensively

investigated previously and summarized as - definitions given by Almeida [77], Zayed [6], Deng Bing et

al [28], and Deyun Wei et al [35].

2.2.5 Existing Correlation Theorems and Related Theories

The fractional domain correlation theorem was analyzed and reported recently in many articles

[43, 103, 109, 123, 124, 125, and 161]. Firstly, Akay et al [103] had defined the fractional correlation

integral by using fractional shift operator, but an exclusive identity for evaluating the correlation was not

presented. Similarly, if the FRFT of correlation integral is determined then it results in a very complex

expression. Subsequently, Tao et al [123] had tried to formulate the correlation theorem especially for

FRFT, but in their definition the correlation integral contains only one chirp. It suggests that once FRFT

of this correlation integral will be carried out, the transformed quantity will not satisfy the variable

dependability constraint. Cottone [43] used fractional spectral moments to represents the fractional

correlation and power spectral density but the correlation theorem for FRFT was not documented

concisely. Torres [125] had documented another correlation theorem for the FRFT by using fractional

translational operator.

[24]

By definition, the FRFT is a subclass of integral transformations characterized by quadratic

complex exponential kernels. These complex exponential kernels often introduce very fast oscillations.

Hence, it is not possible to evaluate these transformations by direct numerical integration since these fast

oscillations require excessively large sampling rates. A possible approach is to decompose these integral

trans- formations into sub-operations. This results in greater time of computation, larger numerical

inaccuracies, and the need for more memory. Also with the increasing trend of analyzing the signal in

discrete form to utilize the advantages of digital systems over analog systems, the need of discretization

of FRFT, i.e. definition or algorithms to obtain discrete FRFT (DFRFT), arises rapidly. Next section is

dedicated to the analysis of existing DFRFT algorithms.

2.3 DISCRETE FRACTIONAL FOURIER TRANSFORM (DFRFT)

Many researchers have tried to come up with some method of evaluating DFRFT. On this

journey, many methods and algorithms had come into existence. But none got acclamation due to

deficiency of either not satisfying some of the prime properties that it’s continuous type posses or non-

existence of a closed form expression. In this section, all the approaches available in the literature are

included and divided into six classes on the basis of their methodology of evaluation.

2.3.1 Sampling Type DFRFT

The sampling theorem for the FRFT of band-limited and time-limited signals is follows from

Shannon’s sampling theorem [5, 7, 26, 74, 122, 147, and 156]. With this approach, the simplest way to

derive the DFRFT is sampling the continuous FRFT and computing it directly from the samples, but by

sampling the continuous FRFT directly the resultant discrete transform obtained will lose many important

properties. The most serious problem with this type DFRFT will be non-compliance of unitary and

reversibility properties. In addition, the DFRFT obtained by direct sampling of the FRFT lacks closed-

form expressions and is non-additive, which affects its application domain. In order to maintain some of

the FRFT properties, a type of DFRFT was derived as a special case of the continuous FRFT as reported

in [105]. Specifically, the input function was assumed as a periodic, equally spaced impulse train. Since

this type of DFRFT is a special case of continuous FRFT, many properties of the FRFT exist and have the

fast algorithm. However, this type of DFRFT cannot be defined for all values of α due to various imposed

constraints. At the same time, Ozaktas et al [55] had proposed two innovative approaches for obtaining

[25]

the DFRFT through the sampling of the FRFT. Both methods were based on the idea of manipulating the

expression of the FRFT by sampling appropriately.

In the first method, the DFRFT of a signal 푥(푡) can be obtained by, multiplying the signal with a

chirp function as -

푥 (푡) = 푒 푥(푡) (2.3.1)

For this chirp multiplication, the bandwidth and time-bandwidth product of 푥 (푡) can be as large

as twice of 푥(푡). Thus, the sampling of 푥 (푡) will be done at intervals of 12 ∆푥 . If the samples of

푥(푡) are spaced at 1∆푥 , then this need to be interpolate before multiply it by the samples of the chirp

function to obtain the desired samples of 푥 (푡).

The next step was to convolve the signal 푥 (푡) with another chirp function. To perform this

convolution, the chirp signal was replaced by its band-limited version because 푥 (푡) assumed as a band-

limited signal.

푥 (푡) = 퐴 ∫ 푥(휏) 푒 ( ) 푑휏 (2.3.2)

The band-limited version of chirp function can be expressed in the form of Fresnel integral.

Finally, again performing a chirp multiplication, the DFRFT of signal 푥(푡) can be obtained as -

ℑ[x(t)] = 푋 = 푒 푥 (푡) (2.3.3)

The convolution operation in the above equation can be achieved by sampling 푥(푡), and

performing the convolution using the fast Fourier transform (FFT). Overall, the procedure starts with N

samples spaced at 1∆푥 , which uniquely characterize the function 푥(푡), and returns the same for 푋 .

For x and 푋 denote column vectors with N elements containing the samples of 푥(푡) and its DFRFT

respectively. The above procedure can be given in a matrix notation as -

푋 = 퐷 Λ 퐻 Λ 퐽 푥 (2.3.4)

Here, 퐷 and 퐽 are matrices representing the decimation and interpolation operations, Λ is a

diagonal matrix that corresponds to chirp multiplication, and 퐻 corresponds to the convolution operation.

The above method had allowed obtaining the samples of the 푎 transform in terms of the samples of the

original function.

[26]

First method includes the analysis of one step in terms of Fresnel integral, which require large

number of computation time particularly in inverse DFRFT. To remove this discrepancy, a second

method was introduced by Ozaktas et al [55]. In this method one assumption was taken that the Wigner

distribution of function 푥(푡) could be zero outside a circle of diameter ∆푢 centered at the origin and by

limiting the order ‘a’ to the interval 0.5 ≤ |푎| ≤ 1.5. Simultaneously, the amount of vertical shear in

Wigner space resulting from the chirp modulation is bounded by ∆푢/2. The FRFT can be written as –

ℑ[x(t)] = 푋 (푢) = 퐴 푒 ∫ 푒 푒 푥(푡) 푑푡 (2.3.5)

Thus, 푒 푥(푡) can be written by using Shannon’s interpolation formula as –

푒 푥(푡) = ∑ 푒 ∆ 푥 푛2∆푢 푠푖푛푐 2∆푢 푡 − 푛

2∆푢 (2.3.6)

Where, 푁 = (∆푢) , putting (2.3.6) in (2.3.5) and interchanging the order of integration and summation -

푋 (푢) = 퐴 푒 푒 ∆ 푥푛

2∆푢푒 푠푖푛푐 2∆푢 푡 −

푛2∆푢

푑푡

(2.3.7)

And after some algebraic manipulation, (2.3.7) can be restructured as-

푋∆

=∆

푒 ( )( ∆⁄ ) ∑ 푒 ∆ 푒 ( )∆ 푥

∆ (2.3.8)

This summation can be interpreted as the convolution of 푒 ∆ and the chirp modulated

function 푒 ( )∆ 푥

∆. This convolution can be computed by fast Fourier transform

(FFT). And the overall methodology can be written in matrix form as-

푋 = 퐷 퐾 퐽 푥 (2.3.9)

Where, the 퐾 matrix was given as -

퐾 (푚,푛) =∆

푒 ∆ ( ∆ ) ∆ (2.3.10)

[27]

In the same article, Ozaktas et al [55] gave alternate methods to calculate the DFRFT by

introducing scaling process. All these methods resulted in the same time complexity. However, in

practice, these alternate methods are not preferable because they require coordinate scaling. Scaling is not

an advantageous process if the original data has already been sampled. This is because scaling requires

additional interpolations, etc., which will require additional computation.

However, in both the cases, the Wigner distribution of the signal was confined to a circle of

diameter ∆푢 around the origin. Thus, there might be several discrete fractional Fourier transform

matrices, some of which were more elegantly expressible than the others, which gave the same result

within the accuracy of this approximation. All these matrices would yield results that were in increasingly

better agreement as the signal energy contained in the circle was increasingly closer to the total energy.

2.3.2 Linear Combination Type DFRFT

In [22, 23, 24, and 41], the discrete fractional Fourier transform was derived by using the linear

combination of identity operation (퐹 ), discrete Fourier transform 퐹 ⁄ , time inverse operation (퐹 ) ,

and inverse discrete Fourier transform 퐹 ⁄ . In [24], the concept of DFRFT was given by linear

combination of Lagrange interpolation polynomial of degree 3. For any function 푥(푣)

푥(푣) = ∑ 휙 (푣) 푔(휆 ) (2.3.11)

Where,

휆 = 푒 , 푓표푟 0 ≤ 푘 ≤ 3 (2.3.12)

And, 휙 (푣) is the Lagrange interpolation polynomial of degree 3 taking value ‘one’ at 푣 = 휆

and ‘zero’ otherwise. It follows that -

휙 (푣) = ∑ 푢 푒 / (2.3.13)

Finally by taking the principle nth - root of eigenvalue 휆 , the fractional transform can be given as -

퐹 = ∑ 퐹 훼 (푛) (2.3.14)

Where,

[28]

훼 (푛) = ∑ 푒 ( ) / , 푓표푟 0 ≤ 푖 ≤ 3 (2.3.15)

Subsequently, in [22, 23], it had also been shown that the operator defined by (2.5.15) is unitary,

satisfies angle additivity and angle multiplicity properties and the operator is periodic with a fundamental

period of four. One analogous approach of this method was presented in [41], in which the DFRFT was

calculated by first chirping the signal in time domain then taking its Fourier transform and finally chirping

again in frequency domain. Depending on the character of the chirped signal, the embedded FT can be

one of the two known classes in discrete case- discrete-time and periodic discrete-time.

For discrete-time DFRFT, the expression for DFRFT of signal 푥(푡) was given as-

푋 (푓) = ∑ 푇 푥(푘푇) 퐾 (푓,푘푇) (2.3.16)

And its inverse transform as -

푥(푘푇) = ∫ 푋 (푓) 퐾∗(푓,푘푇) 푑푓 (2.3.17)

Where, 퐹 =

and 퐾 (푡,푢) is the kernel of FRFT. Similarly, for periodic discrete-time

signal 푥(푡) with period 푇 the expression for DFRFT was given as -

푋 (푛푓) = ∑ 푇 푥(푘푇) 퐾 (푛퐹,푘푇) (2.3.18)

And its inverse transform as -

푥(푘푇) = ∑ 퐹 푋 (푛퐹) 퐾∗(푛퐹,푘푇) (2.3.19)

Where, 푇 = 푁푇; 퐹 = 푁퐹 =

In this type of DFRFT, the transform matrix is orthogonal, and satisfying the additivity property

along with the reversibility property. However, the main problem with this method that the transform

result is not matched to the continuous FRFT. Besides, it performed very similarly to the original Fourier

transform or the identity operation but loses the important characteristic of fractionalization.

[29]

2.3.3 Eigenvector Decomposition Type DFRFT

The authors had derived another type of discrete fractional Fourier transform as reported in [25,

27, 67, 130, 131, 135, 136, 137, and 139], by searching the eigenvectors and eigenvalues of the DFT

matrix followed by computing the fractional power of the DFT matrix based on Hermite function. This

type of DFRFT worked very similarly to the continuous FRFT and fulfills the properties of orthogonality,

additivity, and reversibility. The fractional power of matrix could be calculated from its eigen-

decomposition and the power of eigenvalues. Unfortunately, there exist two types of ambiguity in

deciding the fractional power of the DFT kernel matrix.

i. Ambiguity in Deciding the Fractional Powers of Eigenvalues- The square roots of unity are ‘1’

and –‘1’ from elementary mathematics. This indicates that there exists root ambiguity in deciding

the fractional power of eigenvalues.

ii. Ambiguity in Deciding the Eigenvectors of the DFT Kernel Matrix- The DFT eigenvectors

constitute four major eigen-subspaces, therefore, the choices for the DFT eigenvectors to

construct the DFRFT kernel are multiple and not unique.

Because of the above mentioned ambiguities, several DFRFT kernel matrices were possible

which could obey the rotational properties. The idea for developing the DFRFT is to find the discrete

form of the following -

퐾 (푡,푢) = ∑ 푒 퐻 (푡) 퐻 (푢) (2.3.20)

Where, 훼 indicates the rotation angle associated with FRFT of signal, 퐻 (푡) represents the nth -

order normalized Hermite function with unit variance. The nth - order normalized Hermite function with

variance 휎 is defined as-

퐻 (푡) = ! √

ℎ 푒 (2.3.21)

Where, ℎ (. ) represents the nth - order Hermite polynomial. However, the normalized Hermite

functions with unitary variance 퐻 (. ) indicate the eigenfunction of the FRFT. With the help of Hermite

functions, the FRFT of a signal can be computed [136] as -

푋 (푢) = ∫ 푥(푡) 퐾 (푡,푢) 푑푡 =∑ 퐻 (푢)푒 ∫ 푥(푡) 퐻 (푡) 푑푡 (2.3.22)

[30]

The FRFT can be interpreted as a weighting summation of Hermite functions, as given in

(2.3.22). The weighting coefficients are obtained by multiplying the phase term 푒 and the inner

product of the input signal with the corresponding Hermite functions [136]. The development of this

DFRFT was based on the eigen-decomposition of the DFT kernel. The eigenvalues of DFT kernel ‘F’ can

be given as {1,−푗,−1, 푗} and corresponding to each eigenvalue one eigenvector can be evaluated. The

continuous FRFT has a Hermite function with unitary variance as its eigenfunction. The corresponding

eigenfunction property for the DFT can be given as -

퐹 ⁄ [푢⏞ ] = 푒 푢⏞ (2.3.23)

Where, 푢⏞ represents the eigenvector of DFT corresponding to the nth-order discrete Hermite

functions. In order to retain the eigenfunction properties, the unit variance Hermite function is sampled

with a period 푇 = 2휋 푁⁄ . In the case of continuous FRFT, the terms of the Hermite functions are

summed up from order zero to infinity. However, for the discrete case, only eigenvectors for the DFT

Hermite eigenvectors can be added. The selection of the DFT Hermite eigenvectors is usually made from

low to high orders, due to small approximation error of the low DFT Hermite eigenvectors [136]. Finally,

the transform kernel of DFRFT can be defined as -

퐹 ⁄ = 푈⏞ 퐷 ⁄ 푈⏞ =∑ 푒 푢⏞ 푢⏞ , ∀ 푁 = 4푚 + 1∑ 푒 푢⏞ 푢⏞ + 푒 푢⏞ 푢⏞ , ∀푁 = 4푚

(2.3.24)

Where, 푈⏞ = 푢⏞ 푢⏞ … …푢⏞ for N as odd and 푈⏞ = 푢⏞ 푢⏞ … …푢⏞ 푢⏞ for N is even, 푢⏞

represents the normalized eigenvector corresponding to the 푘 - order discrete Hermite function and 퐷

can be defined as follows-

For, N odd -

퐷 ⁄ =푒

푒 0

0⋱

푒 ( )

(2.3.25)

And, for N even -

퐷 ⁄ =푒

⋱ 0

0푒 ( )

푒

(2.3.26)

[31]

The convergence for the eigenvectors obtained from these matrixes is not so fast for the high-

order Hermite functions. To ensure orthogonality of the DFT Hermite eigenvectors 푢⏞ , the Gram–

Schmidt algorithm (GSA) or the orthogonal procrustes algorithm (OPA) can be used [136]. The GSA

minimizes the errors between the samples of the Hermite functions and the orthogonal DFT Hermite

eigenvectors. On the other hand, the OPA minimizes the total errors between those samples. The main

difference between the approach proposed in [136] and similar approaches proposed in [130, 131, and

135] is found in the eigenvectors obtained. The eigenvectors obtained by method reported in [136] were

discrete Mathieu function, although the Mathieu functions can converge to Hermite functions as obtained

by methods reported in [130, 131, and 135]. Given the number of points 푁 and the rotation angle 훼, the

DFT Hermite eigenvectors can be computed followed by determining the eigenvalues of DFRFT. The

computation of the DFRFT can be implemented only by a transform kernel matrix multiplication. The

complexity of computing the DFRFT is 푂(푁 ) same as in the DFT case. By adjusting the rotational

angles, the method for implementing the DFRFT of a signal 푥 can be given as -

퐹 ⁄ = 푈⏞ 퐷 ⁄ 푈⏞ 푥 = ∑ 푎 푒 푢⏞ (2.3.27)

The coefficients 푎 ′푠 are the inner products of signal and eigenvectors, and they can be computed

in advance. If rotation angle changed then only the diagonal matrix 퐷 ⁄ need to be recomputed.

Additionally, the authors [136] investigated the relationship between the FRFT and the DFRFT and were

found that for a sampling period equal to 2휋 푁⁄ , the DFRFT performs a circular rotation of the signal in

the time–frequency plane. However, the DFRFT becomes an elliptical rotation in the continuous time–

frequency plane for sampling periods different from 2휋 푁⁄ [136]. Therefore, for these elliptical

rotations an angle modification and a post-phase compensation in the DFRFT have been required to

obtain results similar to the continuous FRFT [136]. This approach has been extended to the so-called

multiple-parameter discrete fractional Fourier transform (MPDFRFT) [67, 137]. In fact, the MPDFRFT

maintains all desired properties and reduces to the DFRFT when all of its order parameters are same.

A similar approach to [136] has been proposed in [27]. However, authors in [27] believe that the

discrete time counterparts of the continuous time Hermite–Gaussians maintained the same properties

because these discrete time counterparts exhibited better approximations than the other proposed

approaches [25]. In order to resolve this issue about the approximation of Hermite–Gaussian functions, a

nearly tri-diagonal commuting matrix of the DFT and a corresponding version of the DFRFT were

proposed in [139]. Most of the eigenvectors of this proposed nearly tri-diagonal matrix result in a good

[32]

approximation of the continuous Hermite–Gaussian functions by providing a smaller approximation error

in comparison to the previous approaches.

2.3.4 Closed Form Type DFRFT

Pei et al [132] had derived a different type of DFRFT which was neat in concept and reversible

but lacks the additivity property. Due to the orientation of practical usage, two types of DFRFT were

derived which are differing in parameterization. The parameters of the first type have directly linked to

the continuous FRFT and suits to the applications of computing the continuous FRFT. On the other hand,

second type has the simpler parameters set and allows more elegant expression for the operator kernels

and more suitable for other applications of DFRFT, such as the filter design, pattern recognition and

phase retrieval. Considering the expression of continuous FRFT -

퐹 (푢) = 푒 ∫ 푒 푒 푓(푡) 푑푡 (2.3.28)

And to derive the DFRFT, sampling of 푓(푡) and 퐹 (푢) with the interval ∆푡 and ∆푢 need to be performed

as –

푦(푛) = 푓(푛 ∗ ∆푡) & 푌 (푚) = 퐹 (푚 ∗ ∆푢) (2.3.29)

Where, 푛 = −푁, … … ,푁 and 푚 = −푀, … … ,푀. By using this sampling, FRFT can be converted [132] as-

푌 (푚) = ∆푡 푒 ∆

∑ 푒 ∆ ∆ 푒 ∆

푦(푛) (2.3.30)

The above can be written as -

푌 (푚) = ∑ 퐹 (푚,푛)푦(푛) (2.3.31)

Where,

퐹 (푚,푛) = ∆푡 푒 ∆

푒 ∆ ∆ 푒 ∆

(2.3.32)

And 푌 (푚) can be reversible only if the inverse transform of 퐹 (푚,푛), ∀ 푀 ≥ 푁 would be

Hermitian (conjugate and transpose), given [132] as-

[33]

푦(푛) = ∑ 퐹∗(푚,푛) 푌 (푚) 푓표푟 푀 ≥ 푁 (2.3.33)

From (2.3.31) and (2.3.33),

푦(푛) = ∑ ∑ 퐹∗(푚,푛) 퐹 (푚,푘)푦(푘) (2.3.34)

After some manipulation -

푦(푛) = ∆ | | ∑ ∑ 푒

∆

푒 ( ) ∆ ∆ 푦(푘) (2.3.35)

The summation for 푚 in (2.3.35) should equal to 훿(푛 − 푘), then -

∆푡 ∆푢 =

(2.3.36)

Where, |푆| is some integer prime to (2 푚 + 1), in this case (2.3.32) becomes -

퐹 (푚,푛) = ∆푡 푒 ∆

푒 푒

∆

(2.3.37)

And,

∑ ∑ 퐹∗(푚,푛) 퐹 (푚,푘)푦(푘) = | | ∆푡 푦(푛) (2.3.38)

After normalization, the transform matrix [132] can be given as -

퐹 (푚,푛) = | | ( ) 푒 ∆

푒 ( ) 푒

∆

(2.3.39)

The definition of DFRFT is further divided into two parts [132]. The first part was defined for ′sin훼 >

0′ as -

푌 (푚) = 푒 ∆

∑ 푒 푒

∆

푦(푛) (2.3.40)

And second part was defined for ′sin 훼 < 0 ′, as -

푌 (푚) = 푒 ∆

∑ 푒 푒

∆

푦(푛) (2.3.41)

[34]

This definition of DFRFT possesses reversibility property and the periodicity property -

푌 (−푚) = 푌 (푚) & 푌 (푚) = 푌 (푚) (2.3.42)

The DFRFT of type-1 has a very important advantage in terms of its efficiency in calculating and

implementing the DFRFT [132]. Due to two chirp multiplications and one FFT required for the

implementation of this type of DFRFT, the total number of the multiplication operations required was

‘2P+(P/2) log2P’, for length of the output as P=2M+1.

Subsequently, another type of DFRFT, named as type-2, was introduced by modifying the

definitions obtained for discrete affine Fourier transform (DAFT) and presented as -

푌 (푀) = 푒

∑ 푒± 푒

푦(푛) ∀ 푀 ≥ 푁 (2.3.43)

For the DFRFT of type-2, the parameter ‘p’ is used to control the variation of the chirp in

frequency domain [132]. The DFRFT of type-2 also needed ‘2P + (P/2) log2P’ multiplication operations.

It does not follow the additivity property but observes the convertibility operation.

These definitions of DFRFT are better to previous ones, i.e. sampling type, linear combination

type, and eigen-function type, as it require less number of multiplications, having less complexity. Also it

has a closed form expression and having rich in properties.

2.3.5 Weighted Summation Type DFRFT

Yeh [92] had developed a new method for DFRFT computation. The idea of the developed

method was to compute the DFRFT at any angle by a weighted summation of the DFRFT’s with the

special angles. In this method DFRFT computation with odd point of length can be realized by the

weighted summation of the DFRFTs in special angles. The special angles were multiples of 2π / N for the

odd case and multiples of 2π / (N+1) for even length. Regardless of even- or odd-length cases, the

weighting coefficients were obtained from an IDFT operation. The N-point IDFT was needed for odd

length, and the (N + 1) - point was computed for even length.

For discrete signal ′푥′ having odd length ′푁′, the DFRFT of ′푥′ for rotation angle ‘훼’ can be

computed as -

푋 = ∑ 퐵 , 푋 , ∀ 훽 = 2휋 푁⁄ (2.3.44)

[35]

Where, the weighting coefficients 퐵 , was given as -

퐵 , = 퐼퐷퐹푇 푒 ∀ 푘 = 0,1, … . ,푁 − 1 = ∑ 푒 푒 ( ⁄ ) (2.3.45)

The weighted summation, 퐵 , has a closed form solution -

퐵 , = ( )( )

( ) , ∀ 훼 ≠ 푘훽 훿(푛 − 푘), ∀ 훼 = 푘훽

(2.3.46)

Similarly, for discrete signal ′푥′ having even length N, the DFRFT of ′푥′ for rotation angle ‘훼’ can

be computed by the following expression -

푋 = ∑ 퐵 , 푋 , ∀ 훽 = 2휋 (푁 + 1)⁄ (2.3.47)

The weighting coefficients, 퐵 , can be computed as-

퐵 , = 퐼퐷퐹푇 푒 ∀ 푘 = 0,1, … . ,푁 = ∑ 푒 푒 ( ( )⁄ ) (2.3.48)

Similar to the odd length, the weighted summation, 퐵 , for even case has also a closed form solution -

퐵 , = ( )

( ) ∀ 훼 ≠ 푘훽 훿(푛 − 푘) ∀ 훼 = 푘훽

(2.3.49)

The eigenvector decomposition methods require N2 storages to store the transform kernel. If the

computing angle is changed, the transform kernel is also changed and needs to be recomputed. In this

algorithm, the DFRFT of the specified angle was stored. The DFRFT at any angle can be obtained by

weighted summation of these specified DFRFTs. The weighted computation in this algorithm still takes

푂(푁 ) multiplications; therefore, its computation load is still 푂(푁 ). In [92], two implementation

methods for the DFRFT computation algorithm were introduced. One was called the parallel method, and

the other cascade method. Similar to the parallel method, the special angle and numbers of terms of

cascade forms are also different for the even and odd cases. Both the implementation methods can have

the advantages over each other for different application. The parallel method was suitable for the signal,

whose DFRFT’s with special angles were already known. Hence, the computation of the DFRFT will

become only a linear combination of the DFRFT’s in special angles. And, the cascade method means that

the computation of the DFRFT can be realized from the DFRFT with only one specified angle. If the

[36]

DFRFT with the specified angle can be computed efficiently, the computation of the DFRFT will become

efficient. Therefore, it has been found suitable for VLSI implementation [92].

2.3.6 Random Type DFRFT

Pei et al [138] had introduced a modified method to calculate the DFRFT based on the random

DFT eigenvectors and eigenvalues. In this method a new commuting matrix with random DFT

eigenvectors was constructed. Then a random discrete fractional Fourier transform (RDFRFT) kernel

matrix with random DFT eigenvectors and eigenvalues was developed. The magnitude and phase of the

transformed output are found to be random in RDFRFT. This can be considered as a special feature of

RDFRFT. Previously, the DFT was randomized to define the discrete random Fourier transform (DRFT)

in [159] by taking random powers of eigenvalues of the DFT matrix. However, the eigenvectors of the

DRFT were not random and computed using the same method as proposed in [135]. Although, the phase of

the DRFT was random its magnitude was not random, as shown in [159]. The eigen-decomposition of DFT

matrix ‘F’ was given as -

퐹 = ∑ 휆 푒 푒 (2.3.50)

The DFRFT with one order parameter ‘a’ was defined by [137] -

퐹 = ∑ 휆 푒 푒 (2.3.51)

The definition of RDFRFT was based on the definition of multiple-parameter DFRFT

(MPDFRFT) [137], and given as -

퐹⏞ = ∑ 휆 푒 푒 (2.3.52)

Where, the 1 × 푁 parameter vector 푎⏞ was defined as -

푎⏞ = [푎 ,푎 , … … . . ,푎 ] (2.3.53)

This order parameter of the MPDFRFT can be taken as independent random numbers to define an

MPDFRFT with the following random eigenvalues [138], 휆 ,푘 = 0, 1, … . ,푁 − 1, and random

eigenvalues were given as-

[37]

휆 =

⎩⎪⎨

⎪⎧ 푒 , 푓표푟 휆 = 1

푒 ⁄ , 푓표푟 휆 = −푗푒 , 푓표푟 휆 = −1

푒 ⁄ , 푓표푟 휆 = 푗

(2.3.54)

The generation of RDFRFT is given in the following section. In this generation of RDFRFT, the

DFT-commuting matrix with orthonormal random DFT eigenvectors is used.

For a 푁 × 푁 real random Matrix ‘A’ being ‘K’ symmetric and generated by 퐴 = 퐾 퐴 퐾, where K

represents circular reversal matrix, given by -

퐾 = 퐹 = 퐹 = 1 00 퐽 (2.3.55)

Where, 퐽 being the (푁 − 1) × (푁 − 1) reversal matrix, having nonzero entries ‘ones’ placed

on the anti-diagonal locations. Taking the K-symmetric part ‘E’ of the random matrix ‘A’ as 퐵 =

{(퐴 + 퐾 퐴 퐾) 2⁄ }. With the help of this matrix ‘a random matrix ‘C’ was generated as 퐶 = (퐵 + 퐵 ) 2⁄ ,

where ‘T’ denotes the matrix transpose operation, and finally a DFT commuting matrix ‘D’ will be

formed [138] as 퐷 = 퐶 + 퐹 퐶 퐹 . Subsequently, with the random DFT-commuting matrix ‘D’ and the

MPDFRFT, the random-DFRFT (RDFRFT) with the 1 × 푁 parameter vector 푎⏞ = [푎 ,푎 , … … . . ,푎 ]

of a signal ‘x’ can be determined [138] as-

푋 ⏞, = 퐹⏞ ∗ 푥 = ∑ 휆 푟 푟 ∗ 푥 (2.3.56)

Where, 푟 represents the orthonormal random DFT eigenvectors computed from ‘D’ and 휆

represents the DFT eigenvalue corresponding to 푟 , given by (2.3.54).

The RDFRFT kernel matrix 퐹⏞ satisfied various properties of the FRFT like - additivity, FT

convertibility and angle additivity. The computational complexity of the -point RDFRFT was 푂(푁 )

same as that of the MPDFRFT, but RDFRFT has an advantage over MPDFRFT in terms of one more free

parameter associated with it as parameter vector 푎⏞. The magnitude and phase responses of the

transformed output, with the RDFRFT, are random because the eigenvectors and eigenvalues of the

RDFRFT matrix are random [138] as shown in (2.3.56). This aspect establishes its advantageous role in

security areas like - cryptography, encryption and watermarking.

[38]

Although, the existence of one well accepted closed-form definition of DFRFT is not available, in

spite there are many definitions with their pros and cons are available. A comparison of all the proposed

classes of DFRFT based on some properties and complexity is shown in Table-2.2. In Table-2.2, YES is

included against the method which posses given property \ parameter otherwise NO is mentioned.

Table-2.2: Comparison of Different Types of DFRFT Algorithms

Properties /Parameters

Sampling Type

Linear Combination

Type

Eigenvector Decomposition

Type

Closed Form Type

Weighted Summation

Type

Random Type

Reversibility NO YES YES YES YES YES Additivity NO YES YES NO YES YES Periodicity YES YES YES YES YES YES Symmetry YES NO YES YES YES YES Angle Additivity

NO NO YES YES YES YES

Closed form expression

YES YES YES YES YES YES

Similarity with FRFT

YES NO YES YES NO YES

Complexity O(N log2N) O(N log2N) O(N2) O(N log2N) O(N2) O(N2) Constraints Many Many Many Fewer Fewer Very less

From this comparative study, it can be recapitulated that RDFRFT provides a better solution for

computation of the DFRFT. The computation of RDFRFT requires more time than sampling type, linear

combination type and closed form type, however it satisfies more properties than any other method with

very less number of imposed constraints. Previously, IEEE has conferred to use the Eigenvectors

decomposition-type DFRFT, as the most appropriate definition of DFRFT. In the year 2006, Pei et al

[137] had established the superiority of their method (MPDFRFT) over the earlier reported methods.

Later on in the year 2009, the MPDFRFT was again modified by Pei et al [138] as RDFRFT. It was also

established that RDFRFT has one more degree of freedom, while satisfying the same set of properties as

satisfied by MPDFRFT with equal computational complexity. This makes the RDFRFT as a better choice

in security applications.

2.4 ELECTROMAGNETIC PROPAGATION

A signal changes its form from guided wave to unguided wave through the antenna in wireless

communication. Till date, the majority of reported work analyzed the nature of the signal propagating

[39]

through the channel in time and frequency domains only, whereas the travelling electromagnetic (EM)

wave which is actually propagating through the channel, have both time and spatial dependencies. Very

little work has been reported in which this aspect of EM wave is being analyzed, but the time dependency

of frequency and spatial dependency of propagation constant associated with the EM wave have not been

included, which will affect the performance of the channel in practical conditions.

2.4.1 Dual-Domain Transform Based on FT

The EM wave has been analyzed with the help dual-domain transform (DDT-FT) [36], by

transforming the travelling wave expression twice, first from time domain to frequency domain and

second in spatial coordinate domain to propagation constant domain, by considering transform method as

Fourier transform in both cases. Although, the application of FRFT was investigated in the area of

electromagnetic wave propagation [52, 53, and 110] but, the time dependency of frequency components

and spatial variable dependency of propagation constant has not been considered.

2.5 RADAR SIGNAL PROCESSING

Cross-ambiguity function is used to estimate the delay and Doppler parameters for a particular

target. The Fourier transform ambiguity function is normally used in pulse-Doppler radar for detecting

targets despite the phenomenon known as Doppler smearing which limits the performance of this method

[14]. Many techniques have been introduced for the analysis of linear frequency modulated (LFM) signals

for delay and Doppler estimation of target [15, 47, 148]. Subsequently, the ambiguity function is defined

for both narrowband and wideband, where the discrimination between narrowband and wideband is done

on the basis of comparison of the received signal bandwidth to its center frequency [18, 76]. The FRFT

has been used extensively in the analysis of LFM signals for mainly Doppler estimation of moving targets

[45, 111, and 115]. If only the delay and Doppler estimation is calculated by all these methods then the

computation is more complex as either it requires the computation of cross-ambiguity function of

transmitted and received echo or by superimposing some other technique with FRFT.

2.6 APPLICATION OF FRFT IN 1-D SIGNAL PROCESSING

The FRFT has found numerous applications in signal processing and image processing. Signal

processing areas includes - filtering, de-noising, interference suppression, radar signal processing,

[40]

electromagnetic wave propagation, orthogonal frequency division multiplexing systems and wireless

communication systems. The utility of FRFT can be best explained in the processing of non-stationary

signals. Therefore, filtering and beamforming application of FRFT is considered here.

2.6.1 Filtering

To pass the signal or attenuating the noise, filtering is a useful signal processing technique. The

application areas of filtering are - signal restoration, signal reconstruction and signal synthesis. These

processes are different in nature, but all requires filtering of an incoming signal to produce the desired

signal. Both time domain filtering and frequency domain filtering are utilized and superiority of one

above other depends on the computational complexity required under different input conditions [30]. 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 based filtering can provide a better solution, where for a rotated

domain corresponding to an optimum value of angle parameter in time-frequency plane, the Wigner

distribution of signal and noise may be separated and filtering of the signal from the noise can be

performed by designing a filter in the FRFT domain corresponding to this optimum angle. Numerous

works has been carried in past on the filtering in the FRFT domain [49, 50, 78, 86, 90, 91, and 145]. An

adaptive filtering scheme in fractional Fourier domain was introduced in [82], which had shown that

fractional Fourier domain adaptive filtering scheme provides less error in the case of linear frequency

modulated (LFM) signals.

2.6.2 Beam-forming

Beamforming is a signal processing technique used to control the directionality of the reception

or transmission of a signal on an array of sensors [19]. Beamforming is widely used for signal

enhancement, interference suppression, capturing signals of interest and direction of arrival [48, 141]. The

filtering weights at the sensors are chosen to achieve a certain goal. In literature, a number of beam-

forming algorithms are accessible like - linearly constrained minimum variance (LCMV) beamformer

[106], Capon beamformer [63, 157], and maximum signal to noise ratio (SNR) beamformer [119]. These

methods are widely applied in the fields of communication [62], radar or sonar target detection and

classification [60], and biomedical imaging [149], etc. In array signal processing, due to the need of

[41]

analyzing chirp signals, a FT beamformer should be replaced by FRFT based beamformer. Yetik et al

[61] had documented a minimum mean square error beamformer based on FRFT for additive white

Gaussian noise channel, and established that the FRFT domain beamformer gave better results than

spatial domain and frequency domain beamformers for all the three cases - stationary source, source

moving with constant velocity and accelerated source.. Subsequently, this work was explored for more

practical case i.e., for Rayleigh faded channel by Khanna et al [118], where the superiority of FRFT based

beamformer was shown over spatial and frequency domain beam-formers. The optimum FRFT order for

beamforming was obtained in [61, 116, 117, and 118], by comparing the respective covariance matrices

of the delayed signals. While in [93], the optimum order was decided by maximizing the second-order

central FRFT moment. This makes the desired chirp signal substantially concentrated whereas the noise

rejected considerably.

2.7 APPLICATION OF FRFT IN 2-D IMAGE PROCESSING

Similarly, FRFT has an extensive application in the image processing areas, which includes-

watermarking, compression and encryption. The angle parameter associated with FRFT empowers it to

provide one more key, making FRFT as a better tool in security areas.

2.7.1 Encryption

Encryption is the process of transforming information (plaintext) using an algorithm (cipher) to

produce a unreadable data to anyone except those possessing special knowledge, usually referred to as a

key. The result of the process is encrypted information. Numerous works has been carried out in the

literature for the encryption of image using fractional Fourier transform (FRFT). Alok et al [11] had

proposed a technique in which the Jigsaw transform was utilized with FRFT for encryption. Phase

retrieval method and phase masking were employed with FRFT to encrypt the image [20, 70].

Subsequently, in [37, 96], the authors had proposed a method for twin color images. For triple image

encryption a method was documented by Liu et al [160]. Motivated by development of the MPDFRFT, it

had been tried by researchers in encrypting the image [69, 72, and 102]. FRFT was used along with

wavelet transform by Chen et al [79] for encryption. Here, first the dual fractional Fourier-wavelet

domain was calculated by discrete fractional Fourier transform and wavelet decomposition then different

random phases was used in different wavelet sub-bands for encryption. Similarly, the FRFT with radial

Hilbert transform (RHT) was used by Joshi et al [95]. In this method, the RHT and image subtraction

[42]

techniques were first used to segregate image into two parts and each part was encrypted independently

using double random phase encoding in the FRFT domain.

2.7.2 Watermarking

Watermarking is the process of embedding information into a digital signal in a way that it

becomes difficult to remove. It is a process by which multimedia data is protected [16, 44, 58, 59, and 89]

against illegal recording, distribution and copyright protection. A watermark needs a transformation

domain for embedding it into digital media. This domain can be spatial domain [39, 101] as well as

frequency domain [68, 88, and 153]. In frequency domain techniques, watermark is embedding in the

spread of frequencies. Therefore, the signal energy distributed among signal frequencies (thus the

watermark) becomes undetectable. Djurovic et al. [57] introduced image watermarking in the fractional

Fourier domain, which is a combination of the spatial and frequency domains. The FRFT domain

watermarking maintains its advantages over spatial and frequency domain watermarking because it offers

two more degrees of freedom associated with the FRFT angles. Many other approaches of watermarking

in FRFT domain had been presented in literature [1, 40, 46, 87, 99, 104, 143, and 150]. A normal

distributed random sequence was used to modify the transform coefficients in the middle range of

magnitude of source image in [1]. Naveen [99] had used an encryption technique for modifying the

watermark image in FRFT domain before embedding it in the host image. A bio-inspired algorithm e.g.

ant colony optimization had also been utilized for watermarking in [46], whereas [87] included Fourier

coefficients for moment-based image analysis and the fractional Fourier transformation for watermark

embedding.

2.8 FINDINGS OF THE REVIEW

In this chapter, various time-frequency distribution tools, like – ambiguity function, Wigner

distribution, short-time Fourier transform and fractional Fourier transform (FRFT), has been analyzed and

it has been found that FRFT serves as a better tool as it has no cross-term. The mathematical expression

of FRFT and its importance in time-frequency plane is discussed extensively. Subsequently, various

properties of FRFT are presented along with their proof. In exploration of the properties of FRFT it has

been concluded that the definitions of convolution theorem, product theorem and correlation theorems for

the FRFT need to be examined more rigorously. Then, the discrete version of the FRFT (DFRFT) is

studied and it is observed that there are many methods of evaluating the DFRFT are available in the

[43]

literature. A comparison of all the methods has been performed by dividing them into six categories. And

it has been established that random type DFRFT (RDFRFT) is a better option for evaluating DFRFT.

RDFRFT is based on the eigenvalue decomposition method but it is different to other methods in terms of

choice of eigenvectors and eigenvalues, as in this method a random set of DFT eigenvectors and

eigenvalues is used. Due to its random nature this algorithm is advantageous in the security applications,

e.g. encryption watermarking, cryptography, steganography etc.

Also various application areas of the FRFT and its related properties are investigated. In this

journey, it is found that FRFT has a sound application in electromagnetic (EM) wave propagation. In

wireless communication the EM wave has to undergo a medium which has different portions having

different permittivity and permeability values in turn making the propagation constant associated with the

EM wave as distance (spatial coordinate) dependent also due to random motion of transmitter and/or

receiver a frequency becomes time dependent.

Second application area of the FRFT is investigated in the field of radar communication. The

motivation behind this application is straight forward, as in radar communication; the signal being

analyzed to extract information is actually a chirp signal. And FRFT has great strength in localizing the

chirp signals in time-frequency plane.

In the same manner, noise corrupted non-stationary signals can be filtered precisely using DFRFT

and FRFT based beamformer can be utilized for directing signal from an antenna array. In both the

application, FRFT found an advantage because of the same reason that it localizes chirp signal accurately

in time-frequency plane.

And finally, the utility of FRFT is searched in the area if image processing particularly,

encryption and watermarking. As FRFT offers one more degree of freedom, in terms of angle in time-

frequency plane at which the transform is calculated, so it can be used in security application because this

one degree of freedom serves as an extra key.

2.9 STATEMENT OF THE PROBLEM

The detailed studies on FRFT and its comparability with other integral transforms confirm that

the FRFT is not a completely or perfectly defined integral transform. For the completion of the domain of

FRFT and making it compatible with other integral transforms, efforts are made in defining the various

non-existent relationships as well as modifying the existent identities in a corrective manner. Such newly

[44]

introduced identities can be very well examined and tested with some new and existing applications to

confirm their perfectness and correctness in the domain of FRFT and integral transforms.

Therefore, following problems are undertaken in this course of study –

i. A convolution and product theorem for FRFT need to be defined - which converts into its FT

counterpart for angle 훼 = 휋 2⁄ , both side of the identity is dependent on corresponding variables

only and number of chirp multiplication is less as it introduces error in computation. Also this

theorem should satisfy various properties like; commutativity, associativity and distributivity that

is being satisfied by its FT counterpart.

ii. A cross - and auto - correlation theorem for FRFT is required because this theorem has extensive

application in power spectrum estimation. Then the defined identity for FRFT should satisfies the

basic properties that the correlation theorem of FT is satisfying and these required properties are –

commutativity, associativity, distributivity and evenness.

iii. A mathematical tool is needed which should be capable of analyzing the electromagnetic wave

propagation in a medium affecting the wave in making time dependent frequency components

and spatial dependent propagation constant.

iv. A method is required to estimate the range and Doppler shift (or velocity) associated with the

echo signal in radar communication which is easy and having less complexity.

v. Due to the advantageous role of FRFT in signal processing areas like – filtering and

beamforming, a FRFT domain filtering and beamforming need to be developed.

vi. FRFT offers one more degree of freedom in terms of angular parameter associated with it. For

this reason the role of FRFT in image security areas like- encryption and watermarking need to

investigate as FRFT provides one extra key.

The above mentioned problems are dealt with detailed analysis, appropriate applications and

simulation studies in the following Chapters.

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