CONVOLUTION & PRODUCT THEOREMS FOR FRFT CHAPTER INTRODUCTION [45]

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1 CHAPTER 3 CONVOLUTION & PRODUCT THEOREMS FOR FRFT 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]

2 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]

3 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 [11] with this optimum angle parameter value. 3. CONVOLUTION THEOREM FOR FT Classically, the convolution theorem of the FT for the signals x(t) and y(t) with associated Fourier transforms, X(ω) and Y(ω), respectively is given by- x(t) y(t) = x(τ) y(t τ) dτ π X(ω) Y(ω) (3..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 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 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 18 th and early 19 th 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]

4 "tending to substitute or switch." Mathematically, the commutative property is defined for the two functions x&y as x y = y x (3..) 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 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 x, y & z as (x y) z = x (y z) (3..3) Where,, represents a mathematical operation which is associative in nature Distributive property In equation form, the distributive property can be written as x (y + z) = x y + x z (3..4) [48]

5 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 [8] 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, x(t) and y(t) as- x(τ) y(t τ) dτ sec α e X (v) y[(u v) sec α]e dv (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- x(t) y(t) e X (v) Y[(u v) csc α]e dv (3.3.) 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 - e x(τ)e y(t τ)e () dτ e X (u) Y (u) (3.3.3) Along with the product theorem given by Zayed [6] was- [49]

6 x(t) y(t)e e X (v)e Y (u v)e () dv (3.3.4) Then for a bigger set, i.e., linear canonical transform (LCT) a convolution theorem was proposed by Deng et al [8] in 006 as given below - L (,,,) [x(t) y(t)] = e L (,,,) [x(t)] L (,,,) [y(t)] (3.3.5) Where, operator is dependent on the parameter of the LCT, and its definition is as follows - x(t) y(t) = e x(τ) e y(t τ) e () dτ (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 (a, b, c, d) of LCT is changed as - a b cos(α) sin(α) = (3.3.7) c d sin(α) cos(α) 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) as- I[x(t) y(t)] = e I[x(t)] I[y(t)] (3.3.8) Where, x(t) y(t) = e x(τ)e y(t τ)e () dτ (3.3.9) And the converted identity for FRFT ( ) resembles with the definition given by Zayed [6]. Later on in year 009, 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 y(t) denoted by y(tθ ) where- y(tθ ) = e () Y (,,,)(u) e du (3.3.10) By considering the inverse expression for LCT, (3.3.10) may be written as - [50]

7 y(tθ ) == e Y (,,,) (3.3.11) Based on this τ generalized translation of the signal y(t) the convolution theorem was given as x(τ) y(tθ ) dτ L (,,,) [x(t)] L (,,,) [y(t)] (3.3.1) Again by using definition of FRFT, (3.3.11) and (3.3.1), the convolution theorem in FRFT domain given by Deyun Wei et al [35] is written by defining the τ generalized translation of the signal y(t) in FRFT domain by- y(tθ ) = Y (u) 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- x(τ) y(tθ ) dτ X (u) Y (u) (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 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 /. 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 /. This aspect of FRFT and FT is considered as a parameter index and named as FT Convertibility. [51]

8 3.4. 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 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 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 [5]

9 a) The definition given by Almeida [77] has failed to convert into its Fourier transform (FT) equivalent, when α = π/. b) The definitions given by Zayed [6] and Deng et al [8] 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/, 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/) but also satisfy all the properties that these classical identities in FT domain satisfies 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/. Definition: For any two functions x(t) and y(t), the weighted convolution operation is defined as (here symbol is used to represent proposed modified convolution operation). [53]

10 z(t) = (x y)(t) = x(τ) y(t τ)e () dτ (3.6.1) Theorem: Let z(t) is weighted convolution of two functions x(t) and y(t), and X (u), Y (u) and Z (u) are FRFT at an angle α of functions x(t), y(t) and z(t)respectively. Then- z(t) = x(τ) y(t τ)e () dτ Z (u) = e X (u)y (u) (3.6.) Proof: Considering the LHS of this identity Taking its FRFT z(t) = (x y)(t) = x(τ) y(t τ)e () dτ (3.6.3) Z (u) = z(t) e dt (3.6.4) Putting the expression of z(t) from (3.6.3) into (3.6.4) Z (u) = x(τ) y(t τ)e () dτ e dt (3.6.5) This can be rearranged as Z (u) = () x(τ) y(t τ)e e dτ dt (3.6.6) Substituting, t τ = p t = τ + p dt = dp Z (u) = x(τ) y(p)e e () () dτ dp (3.6.7) After some manipulation Z (u) = x(τ) y(p) e () dτ dp (3.6.8) After multiplying and dividing by e [54]

11 π Z (u) = 1 j cot α e 1 j cot α π x(τ)e dτ y(p)e dp (3.6.9) This can be written as Z (u) = e X (u)y (u) (3.6.10) Finally, convolution theorem transform pair for FRFT is given as- x(τ) y(t τ)e () dτ e X (u)y (u) (3.6.11) FT convertibility obtained as- By putting α = π/ in the transform pair, the linear convolution for Fourier transform can be x(τ) y(t τ) dτ π X (u) Y (u) (3.6.1) Where, X (u)and Y (u) are Fourier transform of x(t) and y(t) respectively Properties satisfied by proposed convolution theorem (a) Commutative law Lemma: (x y)(t) e X (u) Y (u) (3.6.13) [55]

12 (y x)(t) e Y (u) X (u) (3.6.14) (x y)(t) = (y x)(t) (3.6.15) Proof: Considering the LHS of (3.6.14) Taking its FRFT z (t) = (y x)(t) = y(τ) x(t τ)e () dτ (3.6.16) Z (u) = z (t) e dt (3.6.17) Putting the expression of z (t) from (3.6.16) into (3.6.17) Z (u) = y(τ) x(t τ)e () dτ e dt (3.6.18) This can be rearranged as Z (u) = () y(τ) x(t τ)e e dτ dt (3.6.19) Substituting, t τ = p t = τ + p dt = dp Z (u) = y(τ) x(p)e e () () dτ dp (3.6.0) After some manipulation Z (u) = y(τ) x(p) e () dτ dp (3.6.1) After multiplying and dividing by e [56]

13 π Z (u) = 1 j cot α e 1 j cot α π y(τ)e dτ x(p)e dp (3.6.) This can be written as Z (u) = e X (u)y (u) (3.6.3) Hence from (3.6.10) and (3.6.3), it can be conclude that, (x y)(t) = (y x)(t) (3.6.4) This proves that the proposed convolution theorem for FRFT satisfies the commutative law. (b) Associative law Lemma: {(x y) f}(t) {x (y f)}(t) e X (u) Y (u) F (u) (3.6.5) e X (u) Y (u) F (u) (3.6.6) {(x y) f}(t) = {x (y f)}(t) (3.6.7) Proof: Considering the LHS of (3.6.5) z (t) = {(x y) f}(t) = {g f}(t) (3.6.8) Where, g(t) = (x y)(t) = x(τ) y(t τ)e () dτ (3.6.9) [57]

14 Now, z (t) = {(x y) f}(t) = {g f}(t) = g(β) f(t β)e () dβ (3.6.30) Substituting the expression of g(t) from (3.6.9) into (3.6.30) z (t) = {g f}(t) = x(τ) y(β τ) e () f(t β) e () dτ dβ (3.6.31) Taking FRFT of (3.6.31) Z (u) = z (t) e dt (3.6.3) Substituting, z (t) from (3.6.31) into (3.6.3) 1 j cot α Z (u) = π x(τ) y(β τ) f(t β) e () () e e dτ dβ dt (3.6.33) Substituting t β = p t = β + p dt = dp 1 j cot α Z (u) = π x(τ) y(β τ) f(p) e () e e () () dτ dβ dp (3.6.34) After some manipulation, Z (u) = F (u) () x(τ) y(β τ) e e dτ dβ (3.6.35) Again substituting β τ = r β = r + τ dβ = dr Z (u) = F (u) x(τ) y(r) e e () () dτ dr (3.6.36) [58]

15 Multiplying and dividing by e Z (u) = π 1 j cot α 1 j cot α e F (u) π After some manipulation (3.6.37) can be rearranged as x(τ) y(r) e () dτ dr (3.6.37) Z (u) = π 1 j cot α 1 j cot α e F (u) π x(τ) e dτ y(r) e dr (3.6.38) And by applying the definition of FRFT, (3.6.38) can be written as Z (u) = e X (u) Y (u) F (u) (3.6.39) Similarly, it can also be proved that the FRFT of z (t) = {x (y f)}(t) will be Z (u) = Hence from (3.6.39) and (3.6.40) e X (u) Y (u) F (u) (3.6.40) {(x y) f}(t) = {x (y f)}(t) (3.6.41) This proves that the proposed convolution theorem for FRFT satisfies the associative law. (c) Distributive law Lemma: {x (y + f)}(t) {(x y) + (x f)}(t) e X (u) {Y (u) + F (u)} (3.6.4) e {X (u) Y (u) + X (u)f (u)} (3.6.43) [59]

16 {x (y + f)}(t) = {x y + x f}(t) (3.6.44) Proof: Considering the LHS of (3.6.43) Taking FRFT of (3.6.45) z (t) = {(x y) + (x f)}(t) = x(τ) y(t τ)e () dτ + x(τ) f(t τ)e () dτ (3.6.45) Z (u) = z (t) e dt (3.6.46) Putting the expression of z (t) from (3.6.45) into (3.6.46) 1 j cot α Z (u) = π x(τ) y(t τ)e () dτ + x(τ) f(t τ)e () dτ e dt (3.6.47) Substituting, t τ = p t = τ + p dt = dp 1 j cot α Z (u) = π x(τ) y(p)e dτ + x(τ) f(p)e dτ e () () dp (3.6.48) And it can be rearranged as 1 j cot α Z (u) = π x(τ) y(p)e e () () dτ dp + And after some manipulation [60] x(τ) f(p)e e () () dτ dp (3.6.49)

17 1 j cot α Z (u) = π x(τ) y(p) e () dτ dp + x(τ) f(p) e () dτ dp (3.6.50) Multiplying and dividing by e and doing some step of calculations Z (u) = e {X (u) Y (u) + X (u)f (u)} (3.6.51) Utilizing the linearity property of FRFT, (3.6.51) can be written as Z (u) = e X (u) {Y (u) + F (u)} (3.6.5) Similarly, it can also be proved that the FRFT of z (t) = {x (y + f)}(t) will be Z (u) = Hence from (3.6.5) and (3.6.53) e X (u) {Y (u) + F (u)} (3.6.53) {x (y + f)}(t) = {(x y) + (x f)}(t) (3.6.54) This proves that the proposed convolution theorem for FRFT satisfies the distributive law Proposed Product Theorem Definition: For any two functions x(t) and y(t), the modified product operation, w(t), is defined as w(t) = x(t) y(t)e (3.6.55) Theorem: Let w(t) is weighted product of two functions x(t) and y(t), and X (u), Y (u) and W (u) are FRFT of x(t), y(t) and w(t) respectively. Then w(t) = x(t) y(t)e W (u) = X (v) Y (u v) e () dv (3.6.56) [61]

18 Proof: Considering the RHS of identity (3.5.56) as W (u) = X (v) Y (u v) e () dv (3.6.57) Taking, Inverse FRFT of W (u) as w(t) = W (u) e du (3.6.58) Putting, the expression of W (u) from (3.6.57) into (3.6.58) 1 + j cot α 1 + j cot α w(t) = π π X (v) Y (u v) e () e dv du (3.6.59) Substituting, u v = m u = m + v du = dm 1 + j cot α 1 + j cot α w(t) = π π X (v) Y (m) e e () () dv dm (3.6.60) After some step of manipulation 1 + j cot α w(t) = π 1 + j cot α π X (v) Y (m) e () dv dm (3.6.61) Adding and subtracting e in (3.6.61) and after rearranging them w(t) = e X (v) e dv Y (m) e dm (3.6.6) [6]

19 Applying the definition of Inverse FRFT on (3.6.6) w(t) = e x(t) y(t) (3.6.63) Finally, convolution theorem transform pair for FRFT is given as- x(t) y(t)e X (v) Y (u v) e () dv (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 FT Convertibility The proposed convolution theorem should be converted into classical convolution theorem for Fourier transform with angle parameter, α = π/, it is due to the basic property of FRFT as expression of FRFT converts into expression of FT at angle parameter, α = π/. 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/ 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 Performance Index FT convertibility Almeida s Method Not Satisfying Zayed s Method Satisfying Deng et al Method Satisfying Deyun et al Method Satisfying Proposed Method Satisfying [63]

20 3.7. 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., 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.: Variable Dependability Name of Methods Performance Index Variable Dependability Almeida s Method Satisfying Zayed s Method Satisfying Deng et al Method Satisfying Deyun et al Method Not Satisfying Proposed Method Satisfying 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; 009], Deng et al [8; 006], 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]

21 t u tu F1(α) x(t) y(t) t dt dt u tu F1(α) z(t) 1 (a) RHS u u F1(α) t y(t) dt d du z(t) F(α) F1(α) u t u (b) LHS Figure-3.1: Block diagram representation of Deyun et al [35] method. t u tu F1(α) u x(t) dt y(t) t u dt 1 tu F1(α) z(t) (a) RHS [65]

22 t F1(α) x(τ) d y(τ) De(t) & In (b) LHS Figure-3.: Block diagram representation of Deng et al [8] method. z(t) t u tu F1(α) u x(t) dt y(t) t u dt 1 tu F1(α) z(t) (a) RHS t F1(α) x(τ) d y(τ) De(t) & In z(t) (b) LHS Figure-3.3: Block diagram representation of Zayed [6] method. [66]

23 t v tv F1(α) v x(t) dt y(t) VC t_v Secα t _ v Sec α De(u) & In & In dv u z(t) (a) RHS 1 F4(α) y(τ) De(t) & In & In d z(t) x(τ) (b) LHS Figure-3.4: Block diagram representation of Almeida [77] method t u tu F1(α) u x(t) dt F5(α) y(t) t u dt tu F1(α) 1 z(t) (a) RHS [67]

24 t x(τ) y(τ) De(t) & In d z(t) (b) LHS Figure-3.5: Block diagram representation of proposed method. t dt De(t) & In VC t_v Sec α tu t j( ) Cot( ) Chirp Function ( e ) Integrator (with respect to t ) Delay (with respect to t ) & Inversion 1 jcot( ) 1 jcot ( ) Csc ( ) 1 Variable Conversion ( t to v Secα ) IFRFT F1(α) F(α) F3(α) F5(α) j ( ) Chirp Function ( e tu Csc ) F4(α) Sec ( ) 1 jcot ( ) 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 Deyun et al [35] Deng et al [8] Zayed[6] Almeida[77] Proposed Hardware Complexity LHS RHS LHS RHS LHS RHS LHS RHS LHS RHS No. of Chirp functions [68]

25 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 r(t) of unity amplitude and unit duration (0 t 1.0) is convolved linearly with itself (r r)(t), 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 [8] 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 r(t) 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 Figure- 9(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 (a) (b) Figure-3.7: (a) Rectangular function, r t, and (b) Convolved signal r rt Bartlet window. i.e., [69]

26 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 Figure-3.8: FRFT of r rt (c) 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]

27 Bartlet Proposed Deyun's Zayed's Deng's Figure-3.9: FRFT of r rt (c) 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., 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/, 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 [8] is same as given by Zayed [6]) having satisfied both the parameters, variable dependability and equivalent Fourier transform conversion. [71]

28 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 [8] and Deyun et al [35]. This type of convolution is termed as weighted convolution. ******************************************************* [7]