Special Affine Wavelet Transforms and the Corresponding Poisson

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The most appropriate candidates are the wavelet functions, which inherits nice localization properties along with additional characteristic features such as orthogonality, vanishing moments and self adjustability. By intertwining the ideas of SAFT and wavelet transforms, we introduce a hybrid integral transform namely, special affine wavelet transform which is capable of providing a joint time and frequency localization of non-stationary signals with more degrees of freedom. The main contributions of the article are mentioned below: To introduce a hybrid integral transform and discuss the associated constant Q- • property in the joint time-frequency domain. To study some fundamental properties of the proposed transform including Rayleigh’s • theorem, inversion formula and characterization of range. To introduce the discrete counterpart of the proposed transform and obtain the as- • sociated inversion formula. To obtain a direct relationship between the special affine Wigner distribution and the • proposed transform. To introduce a new wave packet transform associated with the special affine Fourier • transform. To obtain an analogue of the well known Poisson summation formula for the proposed • special affine wavelet transform.

The rest of the article is organized as follows. The new hybrid transform and some of its basic properties are studied in Sections 2 and 3. Section 4 is devoted to introducing the discrete counterpart of the special affine wavelet transform. Relationship between the special affine Wigner distribution and the proposed transform is presented in Section 5. In Section 6, we study the wave packet transform associated with the special affine Fourier transform. The final section is devoted to establishing the Poisson summation formula for the special affine wavelet transform.

2. Continuous Special Affine Wavelet Transform And the Associated Constant Q-Property In this Section, our aim is to formally introduce the continuous special affine wavelet transform by intertwining the advantages of the special affine Fourier and wavelet transforms. Wavelets are arguably the most appropriate mathematical entities with affine-like struc- ture of well-localized waveforms at various scales and locations; that is, they are generated by dilation and translation of a single mother function ψ L2(R). Mathematically, the wavelet ∈ family ψb,a(t) is defined as 1 t b ψa,b(t)= ψ − , a, b R, a = 0, (2.1) a a ∈ 6 | | where a is a scaling parameterp which measures the degree of compression or scale, and b is a translation parameter which determines the location of the wavelet [4]. With major M modifications of the family (2.1), we define a new family of functions ψa,b(t) by the combined action of the unitary operators , and the uni-modular matrix M = (A,B,C,D,p,q) as Da Tb 1 t b ψM (t)= ψ − M (t, a), (2.2) a,b √a a K where 1 i M (t, a)= exp − At2 + 2t(p a) 2a(Dp Bq)+ D(a2 + p2) . (2.3) K i√2πiB 2B − − − 3 Having formulated a family of analyzing functions, we are now ready to introduce the definition of the continuous special affine wavelet transform in L2(R). Definition 2.1. For any finite energy signal f L2(R), the continuous special affine wavelet transform of f with respect to the wavelet ψ ∈L2(R) is defined by ∈ ∞ M M M ψ f (a, b)= f, ψa,b = f(t) ψa,b(t) dt. (2.4) W −∞ D E Z M where ψa,b(t) is given by (2.1). It is worth noting that the proposed transform (2.4) boils down to some existing integral transforms as well as gives birth to some new time-frequency transforms as mentioned below: (i). For M = (A,B,C,D, 0, 0), the continuous special affine wavelet transform (2.4) reduces to the linear canonical wavelet transform given by [17, 18]

∞ 2 2 M 1 t b i At 2ta + Da ψ f (a, b)= f(t) ψ − exp − dt. (2.5) W √2aπiB −∞ a 2B Z ( ) (ii). For the matrix M = (cos θ, sin θ, sin θ, cos θ, 0, 0), the continuous special aﬃne wavelet transform (2.4) boils down to the fractional− wavelet transform deﬁned by [19]

∞ 2 2 M 1 t b i t + a cot θ ψ f (a, b)= f(t) ψ − exp ita csc θ dt. (2.6) W √2aπi sin θ −∞ a 2 − Z ( ) (iii). For the matrix M = (cos θ, sin θ, sin θ, cos θ,p,q), we can obtain a new transform called the special affine fractional wavelet− transform defined by ∞ M 1 t b ψ f (a, b)= f(t) ψ − W √2aπi sin θ −∞ a Z i t2 + a2 + p2 2ap cot θ exp − + it(p a) csc θ + iaq dt. (2.7) × ( 2 − ) (iv). For the matrix M = 1, iB, 0, 1,p,q , we can obtain a new transform namely the special affine Gauss-Weierstrass-wavelet transform given by ∞ 2 2 M 1 t b (t a) + 2tp + p 2a(p iBq) ψ f (a, b)= f(t)ψ − exp − − − dt. W 2aπ( B) −∞ a 2B Z − (2.8) p (v). For the matrix M = (1, B, 0, 1,p,q), we can obtain a new transform namely the special affine Fresnel-wavelet transform given by

∞ 2 2 M 1 t b i (t a) + 2tp + p 2a(p Bq) ψ f (a, b)= f(t) ψ − exp − − − dt. W √2aπiB −∞ a 2B Z ( ) (2.9) (vi). By changing the usual matrix M = A,B,C,D,p,q to M = (1, iB, 0, 1,p,q), we can obtain special aﬃne Bilateral Laplace-wavelet transform deﬁned by − ∞ 2 2 M 1 t b (t a) + 2tp + p 2a(p + iBq) ψ f (a, b)= f(t)ψ − exp − − dt. W √2aπB −∞ a − 2B Z (2.10)

For a lucid demonstration of the continuous special aﬃne wavelet transform (2.4), we shall present some illustrative examples. 4 Example 2.2 (i). Consider the constant function f(t)= K and the Morlet function ψ(t)= 2 exp iαt t . Then, the translated and scaled versions of ψ(t) are given by − 2 t b t b t2 + b2 2tb ψ − = exp iα − − . a a − 2a2 M Consequently, the family of analyzing functions ψa,b(t) is obtained as

1 t b i ψM (t)= ψ − exp − At2 + 2t(p a) 2a(Dp Bq)+ D(a2 + p2) . a,b √2πiBa a 2B − − − To compute the continuous special aﬃne wavelet transform of f(t) with respect to the uni- modular matrix M = (A,B,C,D,p,q) and the window function ψ(t), we proceed as

M f (a, b) Wψ h Ki i = exp 2a(Dp Bq)+ D(a2 + p2) √2πiBa 2B − − ∞ iAt2 + 2t(p a) t b t2 + b2 2tb exp − exp iα − 2− dt × −∞ 2B a − 2a Z K i ibα b2 = exp 2a(Dp Bq)+ D(a2 + p2) + √2πiBa 2B − − a − 2a2 ∞ 2 1 iA p a iα b exp t 2 + t − + 2 dt × −∞ − 2a − 2B B − a a Z K i ibα b2 = exp 2a(Dp Bq)+ D(a2 + p2) + √2πiBa 2B − − a − 2a2 π2Ba2 (p a)a2 iαaB + bB 2 2a2B exp − − × B iAa2 a2B × 4(b iAa2) r − − K√a i ibα b2 = exp 2a(Dp Bq)+ D(a2 + p2) + √ 2 2B − − a − 2a2 iB + a A 2 (p a)a2 iαaB + bB exp − − . ×  2a2B(B ia2A)   − 

(ii). For the Haar function  

1, 0 t< 1/2 ψ(t)= (2.11) 1, 1/≤2 t 1, − ≤ ≤ M we ﬁrst construct the family of analyzing functions ψa,b(t) in the following way:

M ψa,b(t) 1 i a exp − At2 + 2t(p a) 2a(Dp Bq)+ D(a2 + p2) , b t< + b √ 2B 2 = 2πiBa − − − ≤  1 i a  exp − At2 + 2t(p a) 2a(Dp Bq)+ D(a2 + p2) , + b t a + b.  −√2πiBa 2B − − − 2 ≤ ≤  Then, we compute the continuous special aﬃne wavelet transform of the generalized Gaussian 2 function f(t)= e−iAt /2B ,AB > 0 in the following manner: 5

M f (a, b) Wψ h 1i i = exp 2a(Dp Bq)+ D(a2 + p2) √ 2B − − 2πiBa a 2 +b i a+b i exp 2t(p a) dt exp 2t(p a) dt 2B a 2B × ( b − − 2 +b − ) Z Z 1 i B i(p a) a = exp 2a(Dp Bq)+ D(a2 + p2) exp − + b √2πiBa 2B − − i(p a)" B 2 − i(p a)b i(p a)(a + b) i(p a) a exp − exp − + exp − + b − B − B B 2 # √iB i = exp 2a(Dp Bq)+ D(a2 + p2) 2πa(a p) 2B − − − p ib(p a) ia(p a) i(p a)a exp − 2exp − 1 exp − . × B 2B − − B " # In the following, we shall derive a fundamental relationship between the continuous special affine wavelet transform (2.4) and the special affine Fourier transform (1.1). With the aid of this formula, we shall study the Q-property of the proposed transform and obtain the associated joint time-frequency resolution. Proposition 2.3. Let M f (a, b) and M f (a) be the continuous special affine wavelet Wψ O transform and the special affine Fourier transform of any finite energy signal f L2(R), respectively. Then, we have ∈ M f (a, b) Wψ ∞ i = √a exp − 2a(ξ 1)(Dp Bq)+ Da2(1 ξ2) 2B(p a) M f (ξ) −∞ 2B − − − − − O Z i M ψ(y)exp (2a2 1)Ay2 2yp + 2ay(2Ab 2p + ξ a) (aξ) M (b, ξ) dξ × O 2B − − − − K (2.12)

where M (b, ξ) is given by (2.3). K Proof. Applying the deﬁnition of special aﬃne Fourier transform, we obtain

M ψM (t) (ξ) O a,b ∞ h Mi M = ψa,b(t) (t,ξ)dt −∞ K Z ∞ 1 t b 1 i = ψ − exp At2 + 2t(p a) 2a(Dp Bq)+ D(a2 + p) −∞ √a a √2πiB 2B − − − Z 1 i exp At2 + 2t(p ξ) 2a(Dp Bq)+ D(ξ2 + p) dt × √2πiB 2B − − − ∞ 1 1 i = ψ(y) exp A(ay + b)2 + 2(ay + b)(p a) 2a(Dp Bq)+ D(a2 + p) −∞ √a 2πiB 2B − − − Z i exp A(ay + b)2 + 2(ay + b)(p ξ) 2a(Dp Bq)+ D(ξ2 + p) a dy × 2B − − − 6 ∞ 1 i = ψ(y) exp A(a2y2 + b2 + 2ay) + 2(ay + b)(p a) 2a(Dp Bq)+ D(a2 + p) −∞ 2πiB√a 2B − − − Z i exp A(a2y2 + b2 + 2ay) + 2(ay + b)(p ξ) 2a(Dp Bq)+ D(ξ2 + p) a dy × 2B − − − ∞ √a 1 i = ψ(y) exp Ab2 + 2b(p ξ) 2ξ(Dp Bq)+ D(ξ2 + p) √2πiB −∞ √2πiB 2B − − − Z i exp 2Aa2y2 + 4Aaby + 2ay(p a) + 2ay(p a)+ Ab2 + 2b(p a) × 2B − − − i exp 2a(Dp Bq)+ D(ξ2 + p) dy × 2B − − ∞ √a i = ψ(y) M (b, ξ)exp 2Aa2y2 Ay2 + Ab2 √2πiB −∞ K 2B − Z i exp Ay2 + 2y p (aξ) 2(aξ)(Dp Bq)+ D(a2ξ2 + p) × 2B − − − i exp 4Aaby + 2ay(p a) + 4ayp 2a2y + 2bp 2ba 2a(Dp Bq)+ Da2 × 2B − − − − − i exp 2yp + 2yaξ + 2aξ(Dp Bq) D(aξ)2 dy × 2B − − − i = √a M (b, ξ) exp 2bp 2ab 2a(Dp Bq)+ Da2 + 2aξ(Dp Bq) Da2ξ2 + Ab2 K 2B − − − − − ∞ 1 i ψ(y)exp 2Aa2y2 Ay2 + 4Aaby + 4ayp 2yp + 2ayξ 2a2y × √2πiB −∞ 2B − − − Z h i i exp Ay2 + 2y[p (aξ)] 2(aξ)(Dp Bq)+ D(a2ξ2 + p) dy × 2B − − − i = √a exp 2bp 2ab 2a(Dp Bq)+ Da2 + Ab2 + 2aξ(Dp Bq) Da2ξ2 2B − − − − − i M ψ(y)exp 2Aa2y2 Ay2 + 4Aaby + 4ayp 2yp + 2ayξ 2a2y (aξ) M (b, ξ) × O 2B − − − K h i i = √a exp 2a(ξ 1)(Dp Bq)+ Da2(1 ξ2) 2b(p a)+ Da2(1 ξ2)+ Ab2 2B − − − − − − i M ψ(y)exp (2a2 1)Ay2 2yp + 2ay(2Ab 2p + ξ a) (aξ) M (b, ξ). × O 2B − − − − K h i (2.13)

By virtue of (2.13), the special aﬃne wavelet transform (2.4) can be expressed as

M f (a, b) Wψ = M f , M ψM O O a,b D ∞ E M M M = f (ξ) ψa,b (ξ) dξ −∞ O O Z ∞ i = √a M (b, ξ) M f (ξ) exp − 2a(ξ 1)(Dp Bq)+ Da2(1 ξ2) 2b(p a) −∞ K O 2B − − − − − Z i M ψ(y)exp (2a2 1)Ay2 2yp + 2ay(2Ab 2p + ξ a) (aξ) × O 2B − − − − i exp − Da2(1 ξ2)+ Ab2 dξ. × 2B − 7 This completes the proof of the Proposition 2.3.

M As a consequence of Proposition 2.3, we conclude that if the analyzing functions ψa,b(t) are supported in the time-domain or the special affine Fourier domain, then the proposed transform M f (a, b) is accordingly supported in the respective domains. This implies that Wψ the special affine wavelet transform is capable of providing the simultaneous information of the time and the special affine frequency in the time-frequency domain. To be more specific, suppose that ψ(t) is the window with centre Eψ and radius ∆ψ in the time domain. Then, M the centre and radii of the time-domain window function ψa,b(t) of the proposed transform (2.4) is given by ∞ 2 ∞ 2 M t ψa,b(t) dt t ψa,b(t) dt −∞ −∞ M Z Z E ψa,b(t) = ∞ 2 = ∞ 2 = E ψa,b(t) = b + aEψ (2.14) M h i ψa,b(t) dt ψa,b(t) dt h i −∞ −∞ Z Z

and ∞ 2 1/2 ∞ 2 1/2 M t (b + aEψ) ψa,b(t) dt t b aEψ ψa,b(t) dt −∞ − −∞ − − M Z Z ∆ ψa,b(t) =  ∞  =  ∞ 2  M 2 h i ψa,b(t) dt ψa,b(t) dt  −∞   −∞   Z   Z      =∆ ψa,b(t) = a∆ ψ, (2.15) respectively. Leth H(ξ) bei the window function in the special aﬃne Fourier transform domain given by i H(ξ)= M exp (2a2 1)Ay2 2yp + 2ay(2Ab 2p + ξ a)ψ(y) (ξ) . O 2B − − − − Then, we can derive the center and radius of the special aﬃne Fourier domain window function i H(aξ)= M exp (2a2 1)Ay2 2yp + 2ay(2Ab 2p + ξ a)ψ(y) (aξ) O 2B − − − − appearing in (2.12) as ∞ (aξ) H(aξ) 2dξ −∞ Z E H (aξ) = ∞ = a EH , (2.16) H (aξ) 2dξ −∞ Z

and

∆ H (aξ) = a ∆H . (2.17) Thus, the Q-factor of the proposed transform (2.4) is given by width of the window function ∆ H (aξ) ∆ Q = = = H = constant, (2.18) centre of the window function EH (aξ) EH which is independent of the uni-modular matrix M = (A,B,C,D,p,q ) and the scaling parameter a. Therefore, the localized time and frequency characteristics of the proposed transform (2.4) are given in the time and frequency windows

b + aE a∆ , b + aE + a∆ and a E a ∆ , a E + a ∆ , (2.19) ψ − ψ ψ ψ H − H H H respectively.h Hence, the joint resolutioni of the continuouh s special aﬃne waveleti transform (2.4) in the time-frequency domain is described by a ﬂexible window ψ having a total spread 8

4∆ψ∆H and is given by

b + aE a∆ , b + aE + a∆ a E a ∆ , a E + a ∆ . (2.20) ψ − ψ ψ ψ × H − H H H h i h i

3. Basic Properties of the Continuous Special Affine Wavelet Transform In this Section, we shall study some fundamental properties of the proposed special affine wavelet transform (2.4) such as Rayleighs theorem, inversion formula and characterization of the range are discussed using the machinery of special affine Fourier transforms and operator theory. In this direction, we have the following theorem which assembles some of the basic properties of the proposed transform. Theorem 3.1. For any f, g L2(R) and α, β R, the continuous special affine wavelet transform defined by (2.4) satisfies∈ the following propertie∈ s:

(i) Linearity: M αf(t)+ βg(t) (a, b)= α M f (b, a)+ β M g (a, b). Wψ Wψ Wψ h i i iAtα (ii) Translation: M f(t α) (a, b) = exp Aα2 α(p a) M exp f(t) (b Wψ − 2B − − Wψ B − α, a). h i

′ (iii) Parity: M f( t) (a, b) = ( i) M f ( b, a), where M ′ = (A,B,C,D, p, q). Wψ − − Wψ − − − − h i ′ a b (iv) Dilation: M D f (a, b)= M f , , where M ′ = aA, aB, aC, aD, p, q . Wψ α Wψ α α h i −1 (v) Conjugation: M f (b, a)= M f (b, a). Wψ Wψ h i Proof. For the sake of brevity, we omit the proof of (i). (ii). For any α R, we have ∈

M f(t α) (a, b) Wψ − h 1 ∞i 1 t b i = f(t α) ψ − exp At2 + 2t(p a)) √2πiB −∞ − √a a 2B − Z i exp 2a(Dp Bq)+ D(a2 + p2) dt × 2B − − 1 ∞ t α (b α) i = f(t α)ψ − − − exp A(t α)2 Aα2 + 2Aαt √2πiBa −∞ − a 2B − − Z i exp 2(t α)(p a)+ α(p a) 2a(Dp Bq)+ D(a2 + p2) dt × 2B − − − − − ∞ i 2 1 iαAt M = exp Aα + α(p a) f(t α)exp ψa,b−α(t α) (t α, a)dt 2B − − √2πiBa −∞ − B − K − Z i iAtα = exp Aα2 α(p a) M exp f (b α, a). 2B − − Wψ B − (iii). To check the behaviour of the special aﬃne wavelet transform (2.4) under the reﬂection of the input signal f, we proceed as 9

M f( t) (a, b) Wψ − h∞ i 1 (t b) 1 i = f( t) ψ − exp At2 + 2t(p a) −∞ − √a a √2πiB 2B − Z i exp 2a(Dp Bq)+ D a2 + p2 dt × 2B − − ∞ 1 t ( b) 1 i = f( t) ψ − − − exp A( t)2 + 2( t) p ( a) −∞ − √a a √2πiB 2B − − − − − Z − i exp 2( a) D( p) B( q) + D ( a)2 + ( p)2 dt × 2B − − − − − − − ∞ ′ ′ i t ( b) 1 i ′ ′ = f(t ) ψ − − exp A(t )2 + 2(t ) p ( a) −∞ √ a a √2πiB 2B − − − Z − − i exp 2( a) D( p) B( q) + D ( a)2 + ( p)2 ( dt′) × 2B − − − − − − − − ′ = ( i) M f ( b, a). − Wψ − − h i (iv). For any α R, we have ∈ M D f (a, b) Wψ α ∞ 1 t 1 t b 1 i = f ψ − exp At2 + 2t(p w) −∞ √α α √a a √2πiB 2B − Z i exp 2w(Dp Bq)+ D(w2 + p2) dt × 2B − − ∞ 2 ′ √α ′ αt b 1 i ′ 2 = f t ψ − exp A αt + 2αt(p w) −∞ √a a √2πiB 2B − Z i exp 2w(Dp Bq)+ D(w2 + p2) dt × 2B − − ∞ α t′ b α ′ − a 1 i ′ ′ 2 ′ = f t ψ ′ exp ′ A t + 2t (p w) a −∞  a √2πiB 2B − r Z  i exp 2w(D′p B′q)+ D′(w2 + p2) dt′ × 2B′ − − ′ a b = M f , . Wψ α α h i (v). It is easy to compute and is therefore omitted. This completes the proof of Theorem 3.1. In the next theorem, our aim is to establish an orthogonality formula between two signals and their respective special aﬃne wavelet transforms. As a consequence of this formula, we can deduce the resolution of identity for the proposed transform (2.4). Theorem 3.2 (Rayleigh’s Theorem). Let M f (a, b) and M g (a, b) be the special Wψ Wφ aﬃne wavelet transforms of f and g belonging to L2(R), respectively. Then, we have M f (a, b), M g (a, b) = f, g ψM , φM . (3.1) Wψ Wφ a,b a,b D E D ED E 10 Proof. Since ∞ M (t, a) M (x, a) da = δ(t x) −∞ K K − Z therefore, we have

M f (a, b), M g (a, b) Wψ Wφ D ∞ ∞ E M M = ψ f (a, b) ψ g (a, b) dadb −∞ −∞ W W Z Z ∞ ∞ ∞ ∞ M M ′ M ′ M ′ ′ = f(t) ψa,b(t) (t, a) dt g(t ) φa,b(t ) (t , a) dt −∞ −∞ −∞ K −∞ K Z ∞ Z ∞ Z ∞ ∞ Z ′ M ′ M M M ′ ′ = f(t)g(t ) ψa,b(t )φa,b(t) (t, a) (t , a) dtdt dadb −∞ −∞ −∞ −∞ K K Z ∞ Z ∞Z ∞ Z M M = f(t)g(t) ψa,b(t)φa,b(t) dt dadb −∞ −∞ −∞ Z ∞ Z Z ∞ ∞ M M = f(t)g(t) ψa,b(t) φa,b(t) dadb dt −∞ −∞ −∞ Z Z Z M M = f, g ψa,b, φa,b . D ED E This completes the proof of Theorem 3.2. Remarks: Theorem 3.2 allows us to make the following remarks: (i) For f = g, the identity (3.1) boils down to Moyal’s principle

∞ ∞ 2 M 2 2 ψ f (b, a) da db = f 2 ψ 2. (3.2) −∞ −∞ W Z Z (ii) For a normalized window function ψ L2(R ), we have ∈ ∞ ∞ 2 M 2 ψ f (b, a) da db = f 2. (3.3) −∞ −∞ W Z Z (iii) For the normalized functions f, ψ L2(R), equation (3.2) gives the Radar uncertainty principle for the special affine wavelet transform∈ ∞ ∞ 2 M ψ f (b, a) da db = 1. (3.4) −∞ −∞ W Z Z After the complete formulation of the special affine wavelet transform (2.4), our next goal is to find a way out for the reconstruction of the input signal f L2(R) via the coefficients of M ∈ f, ψa,b . D E Theorem 3.3 ( Inversion Formula). If M f (a, b) is the special affine wavelet transform Wψ of an arbitrary function f L2(R), then f can be reconstructed as ∈ ∞ ∞ 1 M M M f(t)= ψ f (a, b) (t, a) φa,b(t) da db, a.e. (3.5) M M −∞ −∞ W K ψa,b, φa,b Z Z

Proof. Implication of the Moyal’s principle (3.2) yields that ∞ ∞ 1 M M M g(t)= ψ f (a, b) (t, a) φa,b(t) da db M M −∞ −∞ W K ψa,b, φa,b Z Z

11 is well defined in L2(R). Moreover, for any g, h L2(R), the Rayleigh’s Theorem 3.2 gives ∈ ∞ ∞ ∞ 1 M M g, h = ψ f (a, b) (t, a)φa,b(t) da db h(t) dt M M −∞ −∞ −∞ W K ψa,b, φa,b Z Z Z ∞ ∞ ∞ 1 M M = ψ f (a, b) h(t) (t, a)φa,b(t) da db dt M M −∞ −∞ −∞ W K ψa,b, φa,b Z Z Z ∞ ∞ ∞ 1 M M = ψ f (a, b) h(t) (t, a)φa,b(t) dt da db M M −∞ −∞ W −∞ K ψa,b, φa,b Z Z Z ∞ ∞ 1 M M = ψ f (a, b) ψ f (a, b) da db M M −∞ −∞ W W ψa,b, φa,b Z Z 1 = M f (a, b), M h (a, b) M M Wψ Wψ ψa,b, φa,b D E = f, h . D E Since h is an arbitrary element of L2(R), therefore it follows that ∞ ∞ 1 M M M f(t)= ψ f (a, b) (t, a) φa,b(t) da db, a.e. M M −∞ −∞ W K ψa,b, φa,b Z Z This completes the proof of Theorem 3.3. Remark. For the normalized functions ψ, φ L2(R) with ψ = φ, equation (3.5) reduces to a relatively compact formula ∈ ∞ ∞ M M M f(t)= ψ f (a, b) (t, a) ψa,b(t) da db, a.e. (3.6) −∞ −∞ W K Z Z In the upcoming theorem, we shall obtain a complete characterization of the range of the transform M f (a, b) defined in (2.4). The result follows as a consequence of the inversion Wψ formula (3.5) and the well known Fubini theorem. Theorem 3.4 (Characterization of Range). If h L2(R2) and ψ is a normalized wavelet. Then, h is the special affine wavelet transform of a certain∈ square integrable function if and only if it satisfies the reproduction property: ∞ ∞ M M h(c, d)= h(a, b) ψa,b, ψc,d da db. (3.7) −∞ −∞ Z Z D E Proof. Let h belongs to the range of the transform M . Then, there exists a function Wψ f L2(R), such that M f = h. In order to show that h satisfies (3.7), we proceed as ∈ Wψ M h(c, d)= ψ f (c, d) W∞ M = f(t) ψc,d(t) dt −∞ Z ∞ ∞ ∞ 1 M M M = 2 ψ f (a, b) ψa,b(t) dadb ψc,d(t) dt −∞ ψ −∞ −∞ W Z 2 Z Z ∞ ∞ ∞ M M M = ψ f (a, b) ψa,b(t) ψc,d(t)dt da db −∞ −∞ W −∞ Z Z Z ∞ ∞ M M = h(a, b) ψa,b, ψc,d da db. −∞ −∞ Z Z D E 12 Conversely, suppose that an arbitrary function h L2(R2) satisfies (3.7). Then, we show ∈ that there exist f L2(R), such that M f = h. Assume that ∈ Wψ ∞ ∞ M f(t)= h(a, b) ψa,b(t) da db. −∞ −∞ Z Z Firstly, we claim that f L2(R). To prove this, we proceed as ∞ ∈ 2 f 2 = f(t) f(t) dt −∞ Z ∞ ∞ ∞ ∞ ∞ M M = h(a, b) ψa,b(t) dadb h(a, b) ψa,b(t) dadb dt −∞ −∞ −∞ −∞ −∞ Z ∞ Z∞ Z∞ ∞ ∞Z Z M M = h(a, b)h(a, b) ψa,b(t)ψa,b(t) dt dadb dadb −∞ −∞ −∞ −∞ −∞ Z ∞ Z ∞ Z Z ∞ ∞ Z M M = h(a, b) h(a, b) ψa,b, ψc,d dadb dadb −∞ −∞ −∞ −∞ Z Z Z Z ∞ ∞ = h(a, b) h(a, b) dadb −∞ −∞ Z ∞ Z ∞ 2 = h(a, b) 2da db −∞ −∞ Z Z 2 = h 2, which establishes our claim . Moreover, as a consequence of the well-known Fubini-theorem, we have ∞ M M ψ f (c, d)= f(t) ψc,d(t) dt W −∞ Z ∞ ∞ ∞ M M = h(a, b) ψa,b(t) dadb ψc,d(t) dt −∞ −∞ −∞ Z ∞ Z∞ Z ∞ M M = h(a, b) ψa,b(t)ψc,d(t)dt da db −∞ −∞ −∞ Z ∞ Z ∞ Z M M = h(a, b) ψa,b, ψc,d da db −∞ −∞ Z Z D E = h(c, d). This evidently completes the proof of Theorem 3.4. Corollary 3.5 (Reproducing Kernel Hilbert space). For a normalized wavelet ψ L2(R), the range of the continuous special affine wavelet transform (2.4) is a reproducing∈ kernel Hilbert space in L2(R2) with kernel given by

M (a, b, c, d)= ψM , ψM . Kψ a,b c,d D E 4. Discrete Special Affine Wavelet Transform In this Section, we introduce a discrete analogue of the proposed special affine wavelet transform (2.4) and obtained the associated reconstruction formula. Our strategy is based on an appropriate discretization of the parameters a and b involved in the newly introduced family ψM (t) given by (2.2). To facilitate this, we choose a = aj and b = kb aj ,, where j, k Z a,b 0 0 0 ∈ and a0, b0 are fixed positive constants. Consequently, the desired discretized family takes the following form: − − − ψM (t)= a j/2ψ a jt kb M t, kb a j , (4.1) j,k 0 0 − 0 K 0 0 13 M −j where (t, kb0a0 ) is the discrete version of (2.3). Formally, we have the following definition of theK discrete special affine wavelet transform. Definition 4.1. For any arbitrary function f L2(R) and uni-modular matrix M = (A,B,C,D,p,q), the discrete special affine wavelet∈ transform with respect the window function ψ L2(R) is given by ∈ ∞ M M ψ (j, k)= f(t) ψj,k(t) dt, (4.2) W −∞ Z M where ψj,k(t) is given by (4.1). For the illustration of the discrete special affine wavelet transform (4.2), we have the following example. 2 Example 4.2. We shall recall the generalized Gaussian function f(t)= e−iAt /2B, AB> 0 and from Example 2.2(ii), and compute its discrete special affine wavelet transform with respect to the window function 1, 0 t 1/2 ψ(t)= (4.3) 1, 1/≤2 ≤t 1. − ≤ ≤ The dyadic dilation and integer translations of ψ(t) are given by j j 1 − 1, 2 k t < 2 k + ψ 2 jt k = ≤ 2 (4.4) − 1, 2j k + 1 t 2j(k + 1). − 2 ≤ ≤ Therefore, the discrete special affine wavelet transform of f(t) can be computed as M (j, k) Wψ ∞ 2 iAt − = exp − 2 j/2 ψ 2−j/2t k −∞ 2B − Z i exp At2 + 2t(p k2j ) 2j+1k(Dp Bq)+ D k222j + p2 dt × 2B − − − i = 2−j/2 exp 2j+1k(Dp Bq)+ D k222j + p2 2B − − j 1 j 2 (k+ 2 ) i 2 (k+1) i exp p k2j t dt + ( 1) exp p k2j t dt × j B − j 1 − B − " Z2 k Z2 (k+ 2 ) # i = 2−j/2 exp 2j+1k(Dp Bq)+ D k222j + p2 2B − − B i 1 i exp p k2j 2j k + exp p k2j 2jk × i p k2j B − 2 − B − " − i i 1 exp p k2j 2j(k + 1) + exp p k2j 2j k + − B − B − 2 # B 2−j/2 i = exp 2j+1k(Dp Bq)+ D k222j + p2 i(p k2j) 2B − − − i i i exp p k2j 2jk 2exp p k2j 1 exp p k2j 2j . × B − 2B − − − B − " # We are now focussed to establish a reconstruction formula for the discrete special affine wavelet transform defined by (4.2). Theorem 4.3. Let M (j, k) discrete special affine wavelet transform of f L2(R), defined Wψ ∈ by (4.2). Assume that the discrete system ψM ; j, k Z as defined by (4.1) satisfies the j,k ∈ 14 stability condition 2 2 M 2 E f f, ψj,k F f , (4.5) 2 ≤ 2 ≤ 2 j∈Z k∈Z D E X X and let S be the given linear operator M M S(f)= f, ψj,k ψj,k. (4.6) ∈Z ∈Z Xj Xk D E Then, f(t) can be reconstructed as

M −1 M f(t)= f, ψj,k S ψj,k . (4.7) j∈Z k∈Z X X D E Proof. By virtue of the stability condition (4.5), it can be easily veriﬁed that the operator S deﬁned by (4.6) is one-one and bounded. Then, for S(f)= h, we have h, f = S(f),f

D E M M = f, ψj,k ψj,k, f ∈Z ∈Z D Xj Xk D E E M M = f, ψj,k f, ψj,k ∈Z ∈Z Xj Xk D ED E 2 M = f, ψj,k j∈Z k∈Z D E X X F f 2. ≤ 2 Moreover, an implication of Cauchy-Schwartz inequality yields

−1 2 −1 2 E S (h) 2 = E S S(f) 2 2 = E f 2 2 f, ψM ≤ j,k j∈Z k∈Z D E X X = h, f h f ≤ 2 2 −1 = h S (h) . 2 2 which implies that E S−1(h) h . Therefore, for every f L2(R), we have 2 ≤ 2 ∈ −1 −1 M M M −1 M f = S S(f)= S f, ψj,k ψj,k = f, ψj,k S ψj,k . j∈Z k∈Z j∈Z k∈Z X X D E X X D E This completes the proof of the Theorem 4.3.

5. Relationship Between The Special Affine Wigner Distribution and The Proposed Wavelet Transform

The Wigner-Ville distribution (WVD) is defined as the Fourier transform of the instanta- τ τ neous autocorrelation function f t + 2 f t 2 and is regarded as the main distribution of all the time-frequency distributions. It is− the premier tools in time-frequency analysis with applications in diverse areas including optical and radar systems, and non- stationary signal processing in general. The vast applicability of the WVD in time-frequency analysis has prompted serious attention in recent years (See [4]). In this direction, Urynbassarova et al.[15] introduced a generalised Wigner-Ville distribution namely, the special affine Wigner 15 distribution by substituting the Fourier transform kernel with the SAFT kernel and verified that the new Wigner distribution is more effective in applications that both the linear canonical Wigner distribution as well as the classical WVD. In this Section, we shall develop a relationship between the special affine wavelet transform as defined by (2.4) and the special affine Wigner distribution. Definition 5.1. For any f L2(R), the special affine Wigner distribution with uni-modular matrix M = (A,B,C,D,p,q∈) is defined by [] ∞ τ τ W DM f (t, a)= f t + f t M (τ, a) dτ, (5.1) −∞ 2 − 2 K h i Z where M (τ, a) is given by (2.3). K In the following theorem, we derive a relationship between special affine Wingner distribution and the proposed transform (2.4). Theorem 5.2. If ψ L2(R), M f (a, b) and W DM f (t, a) are the continuous especial ∈ Wψ affine wavelet transform and special affine Wigner distribution of f L2(R) defined by (4.2) and (5.1), respectively. Then, we have ∈

M 1 1 i 2 2 W D f (t, a)= exp 2a(Du0 Dq)+ D a + u0 2√2πiB 2 2B − − ψ 2 h i ∞ ∞ M i 2 ψ exp 2Az + 2z(u0 a 2At) f(t) (b, a) dz × −∞ −∞ W B − − Z Z 2At2 M exp − f(t) ( b 2t, a) da db.. (5.2) × Wψ B − − − τ Proof. By making a change of variables as t + 2 = z in (5.1), we obtain

W DM f (t, a) h i 1 i 2 2 = exp 2a(Du0 Dq)+ D a + u0 √2πiB 2B − − ∞ i dz f(z) f(2t z) exp A(2z 2t)2 + 2(2z 2t)(p a) × −∞ − · 2B − − − 2 Z 1 i 2 2 = exp 2a(Du0 Dq)+ D a + u0 √2πiB 2B − − ∞ i f(z) f(2t z) exp 2A(z2 + t2 2zt)2 + 2(z t)(p a) dz. (5.3) × −∞ − B − − − Z Exploiting the inversion formula (3.5) and translation property of the continuous special aﬃne wavelet transform (2.4), we get

∞ ∞ 1 M M M f(2t z)= 2 ψ f(2t z) (b, a) (2t z, a) ψa,b(2t z) da db − ψ −∞ −∞ W − K − − 2 Z Z ∞ ∞ h i 1 i 2 = 2 exp − A( 2t) 2t(u0 a) ψ −∞ −∞ 2B − − − − 2 Z Z 2 M 2At M exp − f( z) (b, a) M (2t z, a) ψ (2t z) da db ×Wψ B − K − a,b − h i 16 ∞ ∞ 1 i 2 = 2 exp 2At 2t(u0 a) ψ −∞ −∞ B − − 2 Z Z 2 M 2At M exp − f(z) ( b 2t, a) M (z, a) ψ (z) da db ×Wψ B − − − K a,b ∞h ∞ i 1 i 2 = 2 exp 2At 2t(u0 a) ψ −∞ −∞ B − − 2 Z Z 2 M 2At M exp − [f(z)] ( b 2t, a) ψ (z) da db. (5.4) ×Wψ B − − − a,b h i By substituting (5.4) in (5.3), we obtain

W DM f (t, a) h 1 i i = exp 2a(Du Dq)+ D a2 + u2 √ 2B − 0 − 0 2 2πiB ∞ ∞ ∞ 2 1 i 2 M 2At f(z) 2 exp − 2At 2t(u0 a) ψ exp − f(z) ( b 2t, a) × −∞ ψ −∞ −∞ B − − W B − − − Z 2 Z Z h i i ψM (z) da db exp A(2z 2t)2 + 2(2z 2t)(p a) dz × a,b 2B − − − 1 1 i 2 2 = exp 2a(Du0 Dq)+ D a + u0 2√2πiB 2 2B − − ψ 2 ∞ ∞ ∞ i 2 M f(z) exp 2Az + 2z(u0 a 2At) ψa,b(z) dz × −∞ −∞ −∞ B − − Z Z Z 2At2 M exp − [f] ( b 2t, a) da db × Wψ B − − − h i 1 1 i 2 2 = exp 2a(Du0 Dq)+ D a + u0 2√2πiB 2 2B − − ψ 2 ∞ ∞ M i 2 ψ exp 2Az + 2z(u0 a 2At) [f] (b, a) dz × −∞ −∞ W B − − Z Z h i 2At2 M exp − [f] ( b 2t, a) da db. × Wψ B − − − h i This completes the proof of the theorem.

6. Special Affine Wave Packet Transform In pursuit of efficient representations of functions, Córdoba and Fefferman [20] introduced the notion of wave packet systems by applying dilations, modulations and translations to the Gaussian function in the study of some classes of singular integral operators. Later on, Labate et al.[21] adopted the same strategy to define any collections of functions which are obtained by applying the same operations to a finite family of functions in L2(R). In fact, Gabor and wavelet systems are special cases of wave packet systems. Recent developments in this direction can be found in [22, 23, 24, 25, 26] and the references therein. Following the idea of Córdoba and Fefferman [20], we construct a generalized family of wave packet systems associated with the special affine Fourier transform which encompasses all the existing wave packet systems on the Euclidean space R. 17 Definition 6.1. For a uni-modular matrix M = (A,B,C,D,p,q), we define a system of the form 1 t b i ψM (t)= ψ − exp − At2 + 2t(p N) 2N(Dp Bq)+ D N 2 + p2 a,b,N √ a 2B − − − 2πiaB (6.1) called special affine wave packet system in L2(R), where ψ is a fixed function in L2(R). Definition 6.2. Given a real unimodular matrix M = A,B,C,D,p,q , the special affine wave packet transform of any square integrable function f is defined by ∞ M M W P ψ f (a, b, N)= f(t) ψa,b,N (t) dt, (6.2) −∞ h i Z M where ψa,b,N (t) is given by (6.1). This definition allows us to make the following observations: (i). If we consider only the modulations and translations of a single function ψ in (6.1), then we obtain the windowed special affine Fourier transform of the form ∞ M M W P ψ f (b, N)= f(t) ψb,N (t) dt, (6.3) −∞ h i Z where ψ(t b) i ψM (t)= − exp − At2 + 2t(p N) 2N(Dp Bq)+ D(N 2 + p2) . (6.4) b,N i√2πiB 2B − − − h i (ii). Similarly, if we consider the dilations and translations of a single function ψ in (6.1), then the special affine wave packet transform (6.2) boils down to the continuous special affine wavelet transform as defined in (2.4). (iii). For the matrix M = (cos θ, sin θ, sin θ, cos θ, 0, 0), the special affine wave packet transform (6.2) reduces to the fractional wave− packet transform given in [26]. (iv). For the matrix M = (0, 1, 1, 0, 0, 0), the special affine wave packet transform (6.2) reduces to the classical wave packet− transform given in [22]. Following theorem summarises some of the fundamental properties of the special affine wave packet transform in L2(R).

Theorem 6.3.If ψ and φ are the wavelet functions and f1 and f2 are functions which belong to L2(R), then the special affine wave packet transform defined by (6.2) satisfies the following properties: (i) Linear Property:

M M M W P ψ (αf1 + βf2) (a, b, N)= α W P ψ f1 (a, b, N)+ β W P ψ f2 (a, b, N). (ii) Anti-Linear Property:

M M M W P αψ+βφ f (a, b, N)= α W P ψ f (a, b, N)+ β W P φ f (a, b, N). (iii) Time-Shift Property: iACk2 W P M f(t k) (a, b, N) = exp − + iCk(N + p)+ iAkq W P M (a, b k, N Ak). ψ − 2 ψ − − h i (iv) Phase-Shift Property: iBDα2 W P M eiαtf (a, b, N) = exp α(Dp Bq) + iNαB) W P M f (a, b, N αB). ψ − − − 2 ψ − h i 18 (v). Joint Phase-Time Shift Property:

W P M eiαtf(t k) (a, b, N) ψ − h i(ACki 2 + BDα2) = exp − + iCk(N + p Bα)+ iq(Ak + Bα)+ αD(N p) 2 − − W P M (a, b k, N Ak αB). × ψ − − − (vi). Moyal’s Principle:

M M 2 W P ψ f1 , W P ψ f2 = ψ 2 f1,f2 . D E D E

7. Poisson Summation Formula For The Special Affine Wavelet Transform One of the fundamental formulas of classical harmonic analysis is Poisson’s summation formula. This formula asserts that the sum of infinite samples in the time domain of a function f L2(R) is equivalent to the sum of infinite samples of F [f] in the frequency domain. The Poisson∈ summation formula links the signal to the samples of its spectrum by 1 k ikt f(t + kT )= F f exp . (7.1) T T T k∈Z k∈Z X X In other words, this formula expresses the fact that discretization in one domain implies periodicity in the reciprocal domain. The Poisson summation formula has been extended in various directions and has found extensive applications in various scientific fields, particularly in signal and image processing, harmonic analysis, quantum mechanics and number theory [27, 28, 29, 30, 31]. Keeping in view that windowed SAFT enjoys several pleasant properties due to extra degrees of freedom, it is worthwhile to generalize the traditional Poisson summation formula from the classical Fourier domain to the windowed special affine Fourier domain. To achieve the desired objective, we shall apply a fundamental relationship between the conventional Fourier transform and the windowed special affine Fourier transform. The following is an analogue of the Poisson summation formula for the special affine wavelet transform. Theorem 7.1. If M f (a, b) is the continuous special affine wavelet transform of any Wψ f(t) L2(R), then associated Poisson summation formula is given by ∈ 1 t + kT b i f(t + kT )ψ − exp Ak2T 2 + 2AkTt + 2pkT √a ∈Z a 2B Xk √2iπB i = exp − At2 + 2tp + Dp2 T 2B 2 iD Bk ik ikt M Bk exp + (Dp Bq)+ f b, . (7.2) × ∈Z (−2B T T − T ) W T Xk Proof. By virtue of the continuous special affine wavelet transform (2.4), we have

∞ M Bk 1 t b ψ , b = f(t) ψ − W T √2πiBa −∞ a Z 2 i 2 Bk Bk Bk 2 exp At + 2t p 2 (Dp Bq)+ D + Duo dt. × (2B − T − T − T ) 19 The above equation can also be rewritten in the following manner:

2 √ iD Bk i 2 Bk M Bk 2πiB exp exp − Dp 2 (Dp Bq) f b, (−2B T ) 2B − T − W T ∞ i t b ikt = exp At2 + 2tp f(t) ψ − exp dt −∞ 2B a − T Z i 1 t b k = F exp At2 + 2tp f(t) ψ − . " 2B √a a # T i 2 2 (At +2tp) We shall make use of the notation h(t) = e B f(t) ψa,b(t) and applying the formula (7.1) to the function h(t), we obtain 1 k ikt h (t + kT )= F [h] exp ∈Z T ∈Z T T Xk Xk √2iπB iDp2 iD Bk 2 ik ikt Bk = exp exp + (Dp Bq)+ M , b . T − 2B −2B T T − T Wψ T k∈Z ( ) X (7.3)

Furthermore, we observe that

h (t + kT ) ∈Z Xk 1 t + kT b i = f(t + kT ) ψ − exp A(t + kT )2 + 2(t + kT )p ∈Z √a a 2B Xk 1 t + kT b i = f(t + kT ) ψ − exp At2 + Ak2T 2 + 2AkTt + 2tp + 2kTp √a ∈Z a 2B Xk 1 i t + kT b iA ipkT = exp At2 + 2tp f(t + kT ) ψ − exp k2T 2 + 2kTt + . √a 2B a 2B B k∈Z X (7.4)

Using equation (7.4) in (7.3), we get the desired result 1 t + kT b i f(t + kT )ψ − exp Ak2T 2 + 2AkTt + 2pkT √a ∈Z a 2B Xk √2iπB i = exp − At2 + 2tp + Dp2 T 2B 2 iD Bk ik ikt M Bk exp + (Dp Bq)+ ψ , b . × ∈Z (−2B T T − T ) W T Xk In particular, for t = 0, we have 1 kT b i f(kT ) ψ − exp Ak2T 2 + 2pkT √a ∈Z a 2B Xk √2iπB i iD Bk 2 ik Bk = exp − Dp2 exp + (Dp Bq) M , b . T 2B −2B T T − Wψ T k∈Z ( ) X This completes the proof of Theorem 7.1. 20 References

[1] S. Abe and J.T. Sheridan, Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation, Opt. Lett. 19 (1994) 1801-1803. [2] S. Abe and J.T. Sheridan, Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach, J. Phys. 27(12) (1994) 4179-4187. [3] L.Z. Cai, Special affine Fourier transformation in frequency-domain, Optics Communic. 2000;185:271-276. [4] L. Debnath and F.A. Shah, Wavelet Transforms and Their Applications, Birkhäuser, New York, 2015. [5] L. Debnath and F.A. Shah, Lectuer Notes on Wavelet Transforms, Birkhäuser, Boston, 2017. [6] V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl. 25 (1980) 241-265. [7] L.B. Almeida, The fractional Fourier transform and time-frequency representations, IEEE Trans. Sig. Process. 42 (1994) 3084-3091. [8] D.F.V. James and G.S. Agarwal, The generalized Fresnel transform and its application to optics, Opt. Commun. 126 (1996) 207-212. [9] J.J. Healy, M.A. Kutay, Ozaktas and J.T. Sheridan, Linear Canonical Transforms, New York, Springer, 2016. [10] X. Qiang, H.Q. Zhen and Q.K. Yu, Multichannel sampling of signals band-limited in offset linear canonical transform domains, Circ. Syst. Signal Process. 32(5) (2013) 2385-2406. [11] Q. Xiang and K. Qin, Convolution, correlation, and sampling theorems for the offset linear canonical transform. Signal Image Video Process. 8(3) (2014) 433-442. [12] X. Zhi, D. Wei and W. Zhang, A generalized convolution theorem for the special affine Fourier transform and its application to filtering. Optik 127(5) (2016) 2613-2616. [13] Z.H. Zhuo, Poisson summation formulae associated with the special affine Fourier transform and offset Hilbert transform, Math. Prob. Engineer. Article ID 1354129 (2017) (5 pages). [14] S. Xu, L. Huang, Y. Chai and Y. He, Nonuniform sampling theorems for band-limited signals in the offset linear canonical transform, Circ. Syst. Signal Process. (2018) 118. [15] D. Urynbassarova, B.Z. Li and R. Tao, Convolution and correlation theorems for Wigner-Ville distribution associated with the offset linear canonicaltransform, Optik. 157 (2018) 455-466. [16] A. Bhandari, and A.I. Zayed, Shift-invariant and sampling spaces associated with the special affine Fourier transform, Appl. Comput. Harmon. Anal. 47 (2019) 30-52. [17] D. Wei and Y.M Li, Generalized wavelet transform based on the convolution operator in the linear canonical transform domain, Optik. 125(16) (2014) 4491-4496. [18] J. Wang, Y. Wang, W. Wang and S. Ren, Discrete linear canonical wavelet transform and its applications, EURASIP J. Adv. Sig. Process. 29 (2018) 1-18. [19] H. Dai, Z. Zheng and W. Wang, A new fractional wavelet transform, Commun. Nonlinear Sci. Numer. Simulat. 44 (2017) 19-36. [20] A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations. 3 (1978) 979-1005. [21] D. Labate, G. Weiss and E. Wilson, An approach to the study of wave packet systems, Contemp. Math. Wavelets, Frames and Operator Theory, 345 (2004) 215-235. [22] O. Christensen and A. Rahimi, Frame properties of wave packet systems in L2(Rd), Adv. Comput. Math. 29 (2008) 101-111. [23] Y.M. Li and D. Wei, The wave packet transform associated with the linear canonical transform, Optik, 126 (2015) 3168-3172. [24] F.A. Shah and Abdullah, Wave packet frames on local fields of positive characteristic, Appl. Math. Comput. 249 (2014) 133-141. [25] F.A. Shah and O. Ahmad, Wave packet systems on local fields, J. Geomet. Physics, 120 (2017) 5-18. [26] F.A. Shah, O. Ahmad and P. E. Jorgensen, Fractional wave packet systems in L2(R), J. Math. Physics, 59 (2018) 073509. [27] L. Auslander and Y. Meyer, A generalized Poisson summation formula, Appl. Comput. Harmon. Anal. 3(4)(1996) 372-376. [28] A.L. Duran, R. Estrada and R.P. Kanwal, Extensions of the Poisson summation formula, J. Math. Anal. Appl. 218 (1998) 581-606. 21 [29] B.Z. Li, R. Tao, T.Z. Xu and Y. Wang, The Poisson sum formulae associated with the fractional Fourier transform, Signal Process. 89 (2009) 851-856. [30] J. Castaneda, R. Heusdens and R. Hendriks, A generalized Poisson summation formula and its application to fast linear convolution, IEEE Signal Process. Lett. 18(9) (2011) 501-504. [31] J.F. Zhang and S.P. Hou, The generalization of the Poisson summation formula associated with the linear canonical transform, J. Appl. Math. Article ID 102039 (2012) (9 pages).