Convolutions and applications for the offset linear canonical transform via Hermite weights

The main purpose of this paper is to present three new convolutions for the offset linear canonical transform, with the Hermite weights, and to illustrated their potential applications. In view of this, new factorization theorems are obtained and new Young’s convolution inequalities will be introduced. Within the more applied side, the way to design filters (including multiplicative filters in the time domain) is also discussed in the last section. INTRODUCTION The offset linear canonical transform (OLCT) (see [7]) of a signal f (t) with real parameters A = (a, b, c, d, u0, ω0), (satisfying ad − bc = 1) is defined as FA(u) := OA{ f (t)}(u) := ⎪⎪⎨⎪⎪⎩ ∫ R f (t)KA(u, t)dt, b 0 √ d e j cd 2 (u−u0)2+ jω0u f (d (u − u0)), b = 0, (1) where KA(u, t) := KAe j ( d 2b u 2− b tu+ a 2b t2+ (bω0−du0) b u+ u0 b t ) , and KA = e jdu0 2b √ 2πb j . The inverse of the OLCT is given by f (t) = OA−1 {FA(u)}(t) = C ∫ R FA(u)KA−1 (u, t)du, (2) where A−1 = (d,−b,−c, a, bω0 − du0, cu0 − aω0), and C = e j 2 (cdu0−2adu0ω0+abω0). In this paper, we will always consider b 0 since the OLCT becomes a chirp multiplication operation otherwise. We recall that the Fourier transform and its inverse are defined byΨFT ( f (t) ) (u) = ∫ R f (t)e− jutdt and f (t) = 1 2π ∫ R ΨFT ( f (t) ) (u)·e jutdu, respectively. If f , h ∈ L1(R), then the classic (Fourier) convolution in the time domain is expressed as ( f ∗ h)(t) := ∫ R f (τ)h(t − τ)dτ, (3) and the factorization property as follows ΨFT {( f ∗ h)(t)}(u) = ΨFT { f (t)}(u) · ΨFT {h(t)}(u). (4) ICNPAA 2018 World Congress AIP Conf. Proc. 2046, 020014-1–020014-10; https://doi.org/10.1063/1.5081534 Published by AIP Publishing. 978-0-7354-1772-4/$30.00 020014-1 For any real number λ 0, we have ( f ∗ h) (λt) = λ f (λt) ∗ h(λt). (5) We also have the Young’s inequality (see [2]). If f ∈ Lp(R), h ∈ Lq(R), and p + q = r + 1 (with p, q, r 1). Then, the following inequality holds ‖ f ∗ h‖r C1.‖ f ‖p · ‖h‖q, for some C1 > 0. (6) We notice that when u0 = ω0 = 0, OA is the well-known linear canonical transform (LCT) (see [4]). Remind that if A = (a, b, c, d, 0, 0), |a + d| < 2, and φn(t), μn are the eigenfunctions and the eigenvalues of the OLCT (or the LCT) (see [6]), then we have μnφn(u) = OA{φn(t)}(u), (7) where φn(t) := 1 √ β 2n n! √ π e− (1+ jα) 2β2 t2 Hn( t β ), μn := e− j( 1 2+n)θ (n ∈ N), and Hn is the n-th Hermite polynomial. The constants α, β, θ can be taken from α := sgn(b) · (a − d) √ 4 − (a + d)2 , β := 2|b| √ 4 − (a + d)2 , θ := cos−1 (a + d 2 ) . (8) Throughout this paper, for convenience, we denote EA(t) := e j( a 2b t2+ u0 b t), f (t) := EA(t) f (t), EA (t) := e EA(t) (m ∈ R). The identity (1) becomes FA(u) = OA{ f (t)}(u) = KAe j ( d 2b u 2+ (bω0−du0) b u ) ∫ R f (t)e− jut b dt. (9) This paper is divided into four sections and organized as follows. In Section 2, we introduce the relationship between the Hermite functions and the OLCT, which are displayed in Theorem 1 and Theorem 2. Three new convolutions for the OLCT with the Hermite weights and their product theorems are studied in the Section 3. Some special cases of these convolutions are also obtained. In the last section, we propose some applications of these convolutions as well as new Young’s convolution inequalities and designing multiplicative filter in the time domain. HERMITE FUNCTIONS AND THE OLCT For λ 0, it is easy to realize that 1 = ad − bc = (aλ)( d λ ) − ( b λ ) (cλ). Let λ 0, and the parameters Aλ := ( aλ, b λ , cλ, d λ , 0, 0 ) satisfy ∣∣∣∣aλ + d λ ∣∣∣∣ < 2. (10) Under the condition (10), let φn(t), μ λ n be the eigenfunctions and the corresponding eigenvalues of the OLCT with parameters Aλ = ( aλ, b λ , cλ, d λ , 0, 0 ) .We then have μn · φn(u) = OAλ {φn(t)}(u). The eigenfunctions φn(t) and the eigenvalues μn corresponding parameters Aλ can be calculated as in (8). Theorem 1 Let the parameters A1 = (a, b, c, d, 0, 0), and one of the following conditions is satisfied: (i) |a + d| < 2; (ii) |a + d| 2 and 1 − ad > 0. Then, there exists a constant λ > 0 such that the following relation holds φn(u) = 1 μn · √ λ OA1 {φn ( t λ )}(u). (11)


INTRODUCTION
The offset linear canonical transform (OLCT) (see [7]) of a signal f (t) with real parameters A = (a, b, c, d, u 0 , ω 0 ), (satisfying ad − bc = 1) is defined as where K A (u, t) := K A e j d 2b u 2 − 1 b tu+ a 2b t 2 + . The inverse of the OLCT is given by where A −1 = (d, −b, −c, a, bω 0 − du 0 , cu 0 − aω 0 ), and C = e j 1 2 (cdu 2 0 −2adu 0 ω 0 +abω 2 0 ) . In this paper, we will always consider b 0 since the OLCT becomes a chirp multiplication operation otherwise. We recall that the Fourier transform and its inverse are defined by Ψ FT f (t) (u) = R f (t)e − jut dt and f (t) = 1 2π R Ψ FT f (t) (u)·e jut du, respectively. If f, h ∈ L 1 (R), then the classic (Fourier) convolution in the time domain is expressed as and the factorization property as follows For any real number λ 0, we have We also have the Young's inequality (see [2]). If f ∈ L p (R), h ∈ L q (R), and 1 p + 1 q = 1 r + 1 (with p, q, r 1). Then, the following inequality holds f * h r C 1 . f p · h q , for some C 1 > 0. (6) We notice that when u 0 = ω 0 = 0, O A is the well-known linear canonical transform (LCT) (see [4]). Remind that if A = (a, b, c, d, 0, 0), |a + d| < 2, and φ n (t), μ n are the eigenfunctions and the eigenvalues of the OLCT (or the LCT) (see [6]), then we have where φ n (t) := 1 and H n is the n-th Hermite polynomial. The constants α, β, θ can be taken from Throughout this paper, for convenience, we denote The identity (1) becomes This paper is divided into four sections and organized as follows. In Section 2, we introduce the relationship between the Hermite functions and the OLCT, which are displayed in Theorem 1 and Theorem 2. Three new convolutions for the OLCT with the Hermite weights and their product theorems are studied in the Section 3. Some special cases of these convolutions are also obtained. In the last section, we propose some applications of these convolutions as well as new Young's convolution inequalities and designing multiplicative filter in the time domain.

Theorem 2
Let A = (a, b, c, d, u 0 , ω 0 ), and one of the following conditions be fulfilled: (i) |a + d| < 2; (ii) |a + d| 2, a + d 2 and 1 − ad > 0. Then, there exists a positive constant λ, such that the following relation holds Proof. We realize that Then, the solution of this system equations is given as Thanks to equation (11), we derive The theorem is achieved.

CONVOLUTIONS FOR THE OFFSET LINEAR CANONICAL TRANSFORM WITH HERMITE WEIGHTS
In this section, the space L p (R) will be endowed with the norm · p defined by f p : Assume that the conditions of Theorem 1 and Theorem 2 are satisfied. The convolution for the OLCT of two signals f (t) and h(t) is defined by provided that the integral in (19) is well-defined. Moreover, if f, h ∈ L 1 (R), then the function defined in (19) belongs to L 1 (R), and f where C 2 is a positive constant. We have the following product theorem.

Assume that f, h ∈ L 1 (R), F A and H A denote the OLCT of the signals f (t) and h(t) with parameters A, respectively. We have
Proof.
Using the identity (9) and Theorem 2, we realize that The proof is concluded.
By using the same method as in Theorem 3, we derive the next result.

Theorem 4 Let f, h ∈ L 1 (R), F A and H A denote the OLCT of the signals f (t) and h(t) with parameters A, respectively. The transform
defines a convolution belonging L 1 (R), and turns possible the following factorization identity

The convolution for the OLCT of two signals f (t) and h(t) associated with the Hermite functions
where κ = (3 − √ 3)(bω 0 − du 0 ) (as long as the integral in (22) is well-defined). Moreover, if f, h ∈ L 1 (R) then

Let f, h ∈ L 1 (R), F A and H A denote the OLCT of the signals f (t) and h(t), with parameter A, respectively. The following factorization identity holds
Proof. Based on (9) and (16), we have Performing the change of variables τ = τ, v = v and s = t + τ + v − κ, we achieve which proves the theorem.

Corollary 1 Let f, h ∈ L 1 (R), k ∈ {1, 2}, F A 1 and H A 1 denote the LCT of the signals f (t) and h(t) with parameters A 1 , respectively. The convolution of two signals f (t), h(t) for the LCT is defined as follows
and we have

Corollary 2 Let f, h ∈ L 1 (R), F A 1 and H A 1 are the LCT of the signals f (t) and h(t) with parameters A 1 . The convolution of two signals f (t), h(t) for the LCT with the Hermite weights
and the following relation holds

Theorem 6
Suppose that p, q, r, s 1, and 1 (ii) f k ⊗ h r C 5 φ n 1 · f p · h q , for any f ∈ L p (R), h ∈ L q (R), where C 1 , C 2 are some positive constants.

Proof.
We will present the proof for the case k = 3. The cases k ∈ {1, 2} will be omitted because the proofs are analogous. Remind that φ n (t) are rapidly decreasing functions. By applying the Minkowski's integral inequality and changing variable we obtain where C 4 is a positive constant. Thus, we obtain (i). Now, we turn to the proof of (ii). Due to the formula (5) the convolution (22) can be also expressed as where G κ (t) := . Remind that f ∈ L p (R), h ∈ L q (R). By performing a change of variable, we realize that f ( √ 3t) ∈ L p (R), h( √ 3t) ∈ L q (R). Applying the Young's inequality (6) for the case Since the Hermite functions φ n (t) are rapidly decreasing functions then, applying the Young's inequality (6) for the case 1 . Moreover, we also achieve where C 5 is a positive constant. The proof is completed.

The Multiplicative Filter in the OLCT Domain
In this subsection, we will discuss an application of the new convolution to the design of multiplicative filters in the OLCT domain (see [7]). We only consider the convolution (19) when n = 0. The Hermite function φ λ 0 (t) and the value μ 0 are given by where α, β, θ can be taken from (8). We shall denote by r in (t) and r out (t) the input signal and output signal, respectively. From (20), the output signal can be expressed as Let us now denote Thus, we obtain From (3), the output signal r out (t) can be rewritten as where the convolution function (t) is given by This shows that we can achieve the multiplicative filter through the classic Fourier convolution of r int (t) and (t) in the time domain. A realization of the method is displayed in Figure 1 (see also [7]). Using the expression (2), we obtain Then, Substituting (34) into (33), gives rise to Manipulating (15), we have Hence (t) = T πK A · sin tΩ b t . In the following example, we shall use the proposed multiplicative filter to restore an observed signal r in (t) = X(t) + N(t), where X(t), N(t) denote the desired signal and the additive noise, respectively. The real part of output signal desired signal output signal