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

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Convolutions and applications for the offset

linear canonical transform via Hermite

weights

Cite as: AIP Conference Proceedings 2046, 020014 (2018); https://doi.org/10.1063/1.5081534

Published Online: 04 December 2018 L. P. Castro, L. T. Minh, and N. M. Tuan

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Convolutions and Applications for the O

ﬀset Linear

Canonical Transform Via Hermite Weights

L.P. Castro

1,a),b)

, L.T. Minh

2,c)

and N.M. Tuan

3,d)

1_{Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics,}

University of Aveiro, Aveiro, Portugal.

2_{Department of Mathematics, Ha Noi Architectural University, Km 10, Nguyen Trai Rd., Thanh Xuan Dist., Ha Noi,}

Vietnam.

3_{Department of Mathematics, College of Education, Viet Nam National University, G7 Build., 144 Xuan Thuy Rd.,}

Cau Giay Dist., Hanoi, Vietnam.

a)_{Corresponding author: [email protected]} b)_{URL: http://sweet.ua.pt/castro/}

c)_{[email protected]} d)_{[email protected]}

Abstract. 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) := ⎧⎪⎪ ⎨ ⎪⎪⎩ Rf (t)KA(u, t)dt, b 0 √ d ejcd 2(u−u0)2+ jω0u _{f (d (u}− u 0)), b = 0, (1) whereKA(u, t) := KAej _d 2bu2− 1 btu+ a 2bt2+ (b_ω0−du0) b u+u0bt , and KA= e jdu2₀ 2b √

2πb j. The inverse of the OLCT is given by

f (t)= OA−1{FA(u)}(t) = C

RFA(u)KA−1(u, t)du, (2) where A−1= (d, −b, −c, a, bω0− du0, cu0− aω0), and C = ej

1

2(cdu20−2adu0ω0+abω20). 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 byΨFTf (t)(u)=

Rf (t)e− jutdt and f (t)= 21π

RΨFTf (t)(u)·ejutdu, respectively. If f, h ∈ L1(R), then the classic (Fourier) convolution in the time domain is expressed as

( f ∗ h)(t) :=_Rf (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)

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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

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For any real numberλ 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 1_p +1_q = 1_r + 1 (with p, q, r 1). Then, the following inequality holds

 f ∗ hr C1. f p· hq, for some C1> 0. (6) We notice that when u0 = ω0 = 0, OAis 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{φnt}(u), (7) where φn(t) := 1 β 2n_n!√π e−(1+ jα)2β2 t2_H_n₍t β), μn:= e− j( 1 2+n)θ (n∈ N),

and Hnis the n-th Hermite polynomial. The constantsα, β, θ can be taken from

α := sgn(b) · (a − d) 4− (a + d)2 , β := 2|b| 4− (a + d)2, θ := cos −1a + d 2 _. (8)

Throughout this paper, for convenience, we denote EA(t) := ej( a 2bt2+u0bt), f (t) := E A(t) f (t), EmA(t) := e jmt_E A(t) (m∈ R). The identity (1) becomes

FA(u)= OA{ f (t)}(u) = KAej _d 2bu2+ (bω0−du0) b u Rf (t)e −jut bdt. ₍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 convo-lutions 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

Forλ 0, it is easy to realize that 1 = ad − bc = (aλ)d_λ−b_λ(cλ). Let λ 0, and the parameters A_λ :=

aλ,b_λ, cλ,d_λ, 0, 0satisfy

aλ +d_λ < 2. (10)

Under the condition (10), letφλn(t), μλn be the eigenfunctions and the corresponding eigenvalues of the OLCT with parameters A_λ=aλ,b_λ, cλ,d_λ, 0, 0. We then have μ_nλ· φλ_n(u)= OAλ{φλn(t)}(u). The eigenfunctions φλn(t) and the eigen-valuesμλ_ncorresponding parameters A_λcan 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 constantλ > 0 such that the following relation holds

φλ n(u)= 1 μλ n· √ λOA1{φ λ n  t λ _}(u). (11)

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Proof. If|a + d| < 2 then from relation (7) we choose λ = 1. Thus, (11) is fulﬁlled. If|a + d| 2 and 1 − ad > 0. By changing the variable t = λτ (λ > 0), we get

KA1(u, t) = KA1e jd 2bu2− 1 btu+ a 2bt2 = KA1e jd 2bu2−λbτu+ a 2bλ2τ2 = KA1e j d λ 2b_λ u2₋1 b λ τu+ aλ 2b_λτ 2 = √1 λKAλ(u, τ). It follows KAλ(u, τ) = √ λKA1(u, t). (12)

Assume that the condition (10) is satisﬁed. We then have

(aλ2_{− 2λ + d)(aλ}2_{+ 2λ + d) < 0.} ₍₁₃₎ If a= 0, then from (13) we derive |λ| > |d|₂. Thus, there exists λ > 0 such as (10) is satisﬁed. If a 0, since δ = 1−ad > 0 then we denoteλ1 < λ2 < λ3 < λ4 be four solutions of the following equation (aλ2− 2λ + d)(aλ2+ 2λ + d) = 0. Solving this equation, we receive

λi∈⎧⎪⎨⎪⎩−1 ± √ δ a , 1± √δ a ⎫⎪⎬ ⎪⎭ , i ∈ {1, 2, 3, 4}. From (13), we deduceλ ∈ (λ1, λ2)∪ (λ3, λ4), and λ4 =1+

√ δ

|a| > 0. Hence, there exists λ > 0 such that (10) is fulﬁlled. Therefore, the OLCT with parameters A_λhas the eigenfunctionsφλ_n(t) and the eigenvaluesμλ_n:

μλ n· φλn(u)= RKAλ(u, τ) · φ λ n(τ)dτ. (14)

Substituting the relation (12) into (14) results in μλ n· φλn(u)= √ λ RKA1(u, t) · φ λ n  t λ d t λ _.

That impliesφλ_n(u)= 1 μλ_n_·√_λ RKA1(u, t) · φλn t λ dt. Hence φλ_n(u)= 1 μλ_n_·√_λOA1{φλn t λ

_{}(u). The proof is completed.}

Theorem 2 Let A= (a, b, c, d, u0, ω0), 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

ejm1u· φλ n(u− m2)= ej m3− du2₀ 2b μλ n· √ λ OA ejm1t· φλ n t − m2 λ (u), (15) provided ⎧⎪⎪ ⎪⎪⎨ ⎪⎪⎪⎪⎩

m1= (a−1)(bω_b(a+d−2)0−du0)+u0(1−d)

m2= −bω0_a+d−2−du0+u0

m3= (bω0−du0+u0)

2

2b(a+d−2) .

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Proof. We realize that RKA(u, t) · e j(m1t+m3)_e− jm1u· φλ n t − m2 λ dt = KA Rej _d 2bu2− 1 btu+ a 2bt2+ (b_ω0−du0) b u+u0bt ej(m1t+m3)_e− jm1u· φλ n _t_−m 2 λ dt = KA Rej _d 2bu2− 1 btu+ a 2bt2+ b_ω0−du0 b −m1 u+u0 b+m1 t+m3 · φλ n _t_−m 2 λ dt. (17) Let d 2bu 2₋1 btu+ a 2bt 2₊bω0− du0 b − m1 u+u0 b + m1 t+ m3= d 2b(u− m2) 2₋1 b(t− m2)(u− m2)+ a 2b(t− m2) 2_{, (18)}

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then ⎧⎪⎪ ⎪⎨ ⎪⎪⎪⎩ m1−d−1b m2 =bω0−dub 0 m1+a−1b m2= −ub0 m3 = m2₂_2bd −1_b +_2ba.

Remind that a+ d 2. Then, the solution of this system equations is given as ⎧⎪⎪ ⎪⎪⎨ ⎪⎪⎪⎪⎩ m1 =(a−1)(bωb(a0−du+d−2)0)+u0(1−d) m2 = −bω0_a−du_+d−20+u0 m3 =(bω0−du0+u0) 2 2b(a+d−2) . Substituting equation (18) into equation (17), we obtain

RKA(u, t)e j(m1t+m3)_e− jm1uφλ n t − m2 λ dt= e jdu2 0 2b RKA1(u− m2, t − m2)φ λ n t − m2 λ dt= e jdu2 0 2b O_A 1 φλ n  t λ (u− m2).

Thanks to equation (11), we derive_RKA(u, t) · ej(m1t+m3)e− jm1u· φλn _t−m 2 λ dt= ejdu22b0· μλ n· √ λ · φλ n(u− m2), which implies that e− jm3_e jdu2 0 2b μλ n· √ λ · φλ n(u− m2)= RKA(u, t) · ejm1te− jm1u· φλn _t−m 2 λ dt. This means ejm1u· φλ n(u− m2)= ejm3− du2₀ 2b μλ n· √ λ OA ejm1t· φλ n t − m2 λ (u). The theorem is achieved.

Remark 1 If a+ d = 2, and bω0− (1 − a)u0 = 0, then the relation (15) holds for the following conditions

m1=(1−a)b m2−ub0, m2∈ R and m3= 0.

Example 1 Consider the case A= (−2₃,1₃, −9, 3, 1, 3), A1 = (−2₃,1₃, −9, 3, 0, 0). Since |a + d| = 7₃ > 2, then it is easily seen that

λ ∈−3 − 3 √ 3 2 , 3− 3√3 2 ∪3 √ 3− 3 2 , 3+ 3√3 2 . If we chooseλ = 3₂, then Aλ= (−1,29, − 27

2, 2, 0, 0). The Hermite functions and the values μλncan be expressed as φλ n(t)= 3 2 2n_n!√₃_π e−243(1− j √ 3) 32 t2H n ₉√_3t 4 , μλ n = e− j( 1 2+n)π3 _(n∈ N).

Therefore, the relation (11) becomesφλ_n(t)= 2 3e j(1 2+n)π3·O_A 1 φλ n 2t 3

(u). From (16), we obtain m1= 12, m2 = 3, m3= 9

2. The relation (15) gives e12 jtφλn(t− 3) = 2 3e j(1 2+n)π3 · O_A e12 jt_φλ n 2t−6 3 (u).

CONVOLUTIONS FOR THE OFFSET LINEAR CANONICAL TRANSFORM WITH

HERMITE WEIGHTS

In this section, the space Lp(R) will be endowed with the norm · pdeﬁned by f p :=

R| f (t)|pdt 1

p

, p 1. 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 ( f ⊗ h)(t) :=1 K 2 Aej m3− du2₀ 2bE_A(t)−1 μλ n· √ λ R2 f (τ)h(v) · Em1 A (t− τ − v)φλn t − τ − v − m2 λ dτdv, (19) provided that the integral in (19) is well-defined. Moreover, if f, h ∈ L1₍_{R), then the function defined in (19) belongs} to L1_{(R), and f} _{⊗ h}1

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Theorem 3 Assume that f, h ∈ L1_{(R), F}

A and HAdenote the OLCT of the signals f (t) and h(t) with parameters

A, respectively. We have OA ( f ⊗ h)(t)1 (t)= ejm1u_e− j _d bu 2₊₂(bω0−du0) b u · φλ n(u− m2)· FA(u)· HA(u). Moreover, if ejm1u_e− j _d bu2+2 (bω0−du0) b u · φλ

n(u− m2)· FA(u)· HA(u)∈ OA(L1(R)), then ( f ⊗ h)(t) = O1 A−1 ejm1u_e− j _d bu2+2 (bω0−du0) b u · φλ n(u− m2)· FA(u)· HA(u) (t). (20)

Proof. Using the identity (9) and Theorem 2, we realize that

ejm1u_e− j _d bu2+2 (b_ω0−du0) b u · φλ n(u− m2)· FA(u)· HA(u) = ejm1u_e− j _d bu2+2 (bω0−du0) b u · φλ n(u− m2)KA2ej _d bu2+ 2(bω0−du0) b u R2 f (τ)h(v)e− juτ be− juv bdτdv = ej m3−du22b0 μλ n· √ λ K3Aej _d 2bu 2₊(bω0−du0) b u Re− jut b · Em1 A (t)· φλn _t−m 2 λ dt_R2 f (τ)h(v)e− juτ b e− juv bdτdv =ej m3−du20 2b μλ_n_·√_λ K3Aej _d 2bu2+ (bω0−du0) b u R3e− j (t+τ+v)u b · f (τ)h(v) · Em1 A (t)· φλn _t_−m 2 λ dτdvdt.

By makingτ = τ, v = v and s = t + τ + v, we obtain

ejm1u_e− j _d bu2+2 (bω0−du0) b u · φλ n(u− m2)· FA(u)· HA(u) = KAe− j _d bu 2₊₂(bω0−du0) b u Re− jus b E_A(s) K2 Ae j_m3−du2_2b0_(E A(s))−1 μλ n· √ λ R2 f (τ)h(v)E m1 A (s− τ − v) · φλn _{s−τ−v−m} 2 λ dτdv ds =_RKA(u, s) K2 Ae j_m3−du2_2b0_(E A(s))−1 μλ n· √ λ R2 f (τ)h(v) · E m1 A (s− τ − v)φλn _s_{−τ−v−m} 2 λ dτdv ds =_RKA(u, s) · ( f 1 ⊗ h)(s)ds = OA ( f ⊗ h)1 (u).

The proof is concluded.

By using the same method as in Theorem 3, we derive the next result. Theorem 4 Let f, h ∈ L1₍_{R), F}

Aand HAdenote the OLCT of the signals f (t) and h(t) with parameters A,

respec-tively. The transform

( f ⊗ h)(t) :=2 K2Ae j_m3−du20 2b (EA(t))−1 μλ_n_·√_λ R2 f (τ)h(v) · E m1 A t− τ − v + 2(bω0− du0)φλn _t_{−τ−v−m} 2+2(bω0−du0) λ dτdv (21)

deﬁnes a convolution belonging L1₍_{R), and turns possible the following factorization identity} OA ( f ⊗ h)(t)2 (t)= ejm1u_e− jdbu2· φλ n(u− m2)· FA(u)· HA(u).

The convolution for the OLCT of two signals f (t) and h(t) associated with the Hermite functionsφλ_n(√u 3 − m2) scaled by the chirp ejm1u√3 , is deﬁned as

( f ⊗ h)(t) :=3 √ 3K2 Aej m3− du2₀ 2b (EA(t))−1 μλ n· √ λ R2 f (τ)h(v) · Em1 A √ 3t− τ − v + κφλn √ 3t− τ − v − m2+ κ λ dτdv, (22)

whereκ = (3 − √3)(bω0 − du0) (as long as the integral in (22) is well-deﬁned). Moreover, if f, h ∈ L1(R) then ( f ⊗ h)(t) ∈ L3 1_{(R) since f} _{⊗ h}3

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Theorem 5 Let f, h ∈ L1_{(R), F}

Aand HAdenote the OLCT of the signals f (t) and h(t), with parameter A,

respec-tively. The following factorization identity holds

OA ( f ⊗ h)(t)3 (u)= ejm1u√3 · φλ n(√u₃− m2)· FA(√u 3)· HA( u √ 3). Moreover, if ejm1u√3 · φλ n(√u₃− m2)· FA(√u₃)· HA(√u₃)∈ OA(L1(R)), then ( f ⊗ h)(t) = O3 A−1 ejm1u√3 · φλ n( u √ 3 − m2)· FA( u √ 3)· HA( u √ 3) (t). (23)

Proof. Based on (9) and (16), we have

ejm1u· φλ n(u− m2)· FA(u)· HA(u)= ejm1u· φλn(u− m2)· K_A2ej _d bu2+ 2(bω0−du0) b u R2 f (τ)h(v)e− juτ b e− juv b dτdv = ej m3−du22b0 K3 A μλ_n_·√_λ e3 j _d 2bu2+ (bω0−du0) b u Re− jut b · Em1 A (t)· φλn _t_−m 2 λ dt_R2 f (τ)h(v)e− juτ b e− juv b dτdv = ej m3−du22b0 K3 A μλ n· √ λ ej _d 2b(u √ 3)2₊(bω0−du0) b (u √ 3) R3e− jt+τ+v−(3−√3)(bω0−du0)u b f (τ)h(v)Em1 A (t)· φλn _t−m 2 λ dτdvdt.

Performing the change of variablesτ = τ, v = v and s = t + τ + v − κ, we achieve

ejm1uφλ n(u− m2)· FA(u)· HA(u)= KAej _d 2b(u √ 3)2₊(bω0−du0) b (u √ 3) Re− ju√3 b _√s 3 · EA _s √ 3 K2 Ae j_m3−du20 2b EA s √ 3 −1 μλ n· √ λ × R2 f (τ)h(v) · E m1 A s− τ − v + κ· φλ_ns−τ−v−m2+κ λ dτdv ds =_RKA u√3, _√s 3 K2 Ae j_m3−du2_2b0 EA s √ 3 −1 μλ_n_·√_λ R2 f (τ)h(v) · E m1 A s− τ − v + κφλ_ns−τ−v−m2+κ λ dτdv ds =_RKA u√3,_√s 3 · ( f ⊗ h)3 _√s 3 · d_√s 3 = OA ( f ⊗ h)3 (u√3), which proves the theorem.

Corollary 1 Let f, h ∈ L1_{(R), k ∈ {1, 2}, F}

A1and HA1denote the LCT of the signals f (t) and h(t) with parameters

A1, respectively. The convolution of two signals f (t), h(t) for the LCT is deﬁned as follows

( f ⊗ h) (t) =k K 2 A1(EA1(t))−1 μλ n· √ λ R2 f (τ)h(v) · φλn t − τ − v λ dτdv. (24) and we have OA1 ( f ⊗ h)(t)k (u)= φλ_n(u)· e− j2bdu2· F A1(u)· HA1(u). (25) Corollary 2 Let f, h ∈ L1_{(R), F}

A1 and HA1 are the LCT of the signals f (t) and h(t) with parameters A1. The

convolution of two signals f (t), h(t) for the LCT with the Hermite weights φλ_n(√u

3) is deﬁned by ( f ⊗ h) (t) =3 √ 3K2 A1(EA1(t)) −1 μλ n· √ λ R2 f (τ)h(v) · φλn √3t − τ − v_λ dτdv. (26)

and the following relation holds

OA1 ( f ⊗ h)(t)3 (u)= φλ_n(√u 3)· FA1( u √ 3)· HA1( u √ 3). (27)

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APPLICATIONS

Young’s Convolution Inequalities

Note that 1= |ejt| = |EA(t)| = |EmA(t)| = |μλn|, and | f (t)| = | f (t)|. We derive the following theorem (see [1, 3]). Theorem 6 Suppose that p, q, r, s 1, and 1_p+1_q = 1_r + 1, k ∈ {1, 2, 3}. Then,

(i)  f ⊗ hk s C4φns· f 1· h1, for any f, h ∈ L1(R).

(ii)  f ⊗ hk r C5φn1· f p· hq, for any f ∈ Lp(R), h ∈ Lq(R), where C1, C2are 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 thatφn(t) are rapidly decreasing functions. By applying the Minkowski’s integral inequality and changing variable we obtain

R(f k ⊗ h)(t)sdt 1/s =√3K2 A √ λ R R2 f (τ)h(v) · φλn _√ 3t−τ−v−m2+κ λ dτdv s dt 1/s √3K2 A √ λ R2 Rφλn _√ 3t−τ−v−m2+κ λ s ·f (τ)s·h(v)sdt 1/s dτdv =√3K2 A √ λ R2 Rφλn _√ 3t−τ−v−m2+κ λ s dt 1/s ·f (τ) · h(v)dτdv C4φns R2f (τ) · h(v)dτdv = C4φns· f 1· h1,

where C4is 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

( f ⊗ h) (t) = 3K3 A(EA(t))−1· f (√3t)∗ h(√3t)∗ G_κ(√3t), (28) whereG_κ(t) := √3KAej m3−du22b0 μλ n· √ λ · E m1 A t+ κ· φλ_nt−m2+κ λ

. Remind that f ∈ Lp₍_{R), h ∈ L}q₍_{R). By performing a change} of variable, we realize that f (√3t) ∈ Lp(R), h(√3t) ∈ Lq(R). Applying the Young’s inequality (6) for the case

1 p+ 1 q = 1 r + 1, we have f ( √

3t)∗ h(√3t)∈ Lr(R). Since the Hermite functions φn(t) are rapidly decreasing functions then, applying the Young’s inequality (6) for the case 1_r +1₁ = 1_r + 1, we getf (√3t)∗ h(√3t)∗ G_κ(√3t)∈ Lr(R). Moreover, we also achieve

 f ⊗ hk r C5φn1· f p· hq, where C5is 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 φλ₀(t) and the valueμ0are given by

φλ 0(t)= 1 β√π e−(1+ jα)2β2 t2, μ 0= e − jθ 2 ,

whereα, β, θ can be taken from (8). We shall denote by rin(t) and rout(t) the input signal and output signal, respectively. From (20), the output signal can be expressed as

rout(t)= OA−1 OA{rin(t)} (u) · ejm1ue− j( d bu2+2 (b_ω0−du0) b u)· φλ 0(u− m2)· HA(u) (t). (29)

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rin(t)

EA(t)

(rin∗ )(t)

KA(EA(t))−1

rout(t)

FIGURE 1. Method of achieving multiplicative ﬁlter in time domain.

Let us now denote

HA(u)= ejm1ue− j( d bu2+2 (bω0−du0) b u)· φλ 0(u− m2)· HA(u). (30) Then, it follows HA(u)= e− jm1uej(

d bu2+2 (bω0−du0) b u)·φλ 0(u− m2) −1_·_H A(u).

Based on diﬀerent transforms H(u), there are many ways to design a multiplicative ﬁlter. For instance, we can

choose the function h(t) such that HA(u) is constant over [−Ω, Ω], and zero or with rapid decay outside that region. Let T be a constant and

HA(u)= T, u ∈− Ω, Ω 0, u − Ω, Ω . (31) Thus, we obtain rout(t)= T · OA−1 OA{rin(t)} (u) (t).

From (3), the output signal rout(t) can be rewritten as

rout(t)= KA(EA(t))−1

rin∗

(t), (32)

where the convolution function(t) is given by

(t) = KAej m3− du2₀ 2b μλ 0· √ λ Rh(v)· E m1 A (t− v) · φλ0( t− v − m2 λ )dv. (33)

This shows that we can achieve the multiplicative ﬁlter through the classic Fourier convolution of rint(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

h(t) = CKA−1 RHA(u)e − jd 2bu2−1btu+2bat2+ (bω0−du0) b u+u0bt du = CKA−1 Re − jm1u_ej _d bu 2₊₂(bω0−du0) b u ·φλ 0(u− m2) −1_·_H A(u)e− j _d 2bu 2₋1 btu+ a 2bt 2₊(bω0−du0) b u+u0bt du = CKA−1(EA(t))−1 Re − jm1u_ej(2bdu2+ (bω0−du0) b u)·φλ 0(u− m2)−1· HA(u)e jut bdu. Then, h(t)= CKA−1 R e− jm1u_ej _d 2bu2+ (bω0−du0) b u ·φλ 0(u− m2)−1· HA(u)e jut bdu. ₍₃₄₎

(10)

Wigner distribution of desired signal Time -10 -8 -6 -4 -2 0 2 4 6 8 10 Frequency -6 -4 -2 0 2 4

6 Wigner distribution of observed signal

-10 -8 -6 -4 -2 0 2 4 6 8 10 -6 -4 -2 0 2 4 6

FIGURE 2. Wigner distributions of desired and observed signals.

Substituting (34) into (33), gives rise to

(t) = KAej m3− du2₀ 2b μλ 0· √ λ Rh(v)· E m1 A (t− v) · φλ0 t − v − m2 λ dv = KAej m3− du2₀ 2b μλ 0· √ λ R CKA−1 Re − jm1u_ej(2bdu 2₊(bω0−du0) b u)·φλ 0(u− m2)−1HA(u)e juv bdu Em1 A (t− v) · φλ0 t − v − m2 λ dv = CKA−1KAej m3− du2₀ 2b μλ 0· √ λ R Re − jm1u_ej(2bdu 2₊(bω0−du0) b u)·φλ 0(u− m2) −1_·_H A(u)e iuv bejm1(t−v)E A(t− v) · φλ₀t − v − m2 λ dvdu = CKA−1ej m3− du2₀ 2b μλ 0· √ λ R e− jm1uφλ 0(u− m2) ₋₁ · HA(u) KAej( d 2bu2+ (b_ω0−du0) b u) Re juv bejm1(t−v)E A(t− v) · φλ0 t − v − m2 λ dvdu.

By taking s= t − v, it shows that

KAej( d 2bu2+ (bω0−du0) b u) Re juv b ejm1(t−v)E A(t− v)φλ0 t − v − m2 λ dv= ejutbK Aej( d 2bu2+ (bω0−du0) b u) Re −jus b E A(s) ejm1sφλ 0  s − m2 λ ds = ejut bO_A ejm1sφλ 0  s − m2 λ _(u). (35) Manipulating (15), we have KAej( d 2bu2+ (bω0−du0) b u) R ejuvbejm1(t−v)E A(t− v) · φλ0 t − v − m2 λ dv= μλ₀· √λe− jm3− du2₀ 2b ejutbejm1u· φλ 0(u− m2). (36) Therefore,(t) = CKA−1 RHA(u)e jut

bdu. Based on the relation (31), we derive (t) = CKA−1 _Ω −ΩT e jut bdu= 2bCK A−1T · sintΩ_b t . Hence(t) = _πKT_A ·sin _t_Ω b t .

In the following example, we shall use the proposed multiplicative ﬁlter to restore an observed signal rin(t) =

(11)

Time -10 -8 -6 -4 -2 0 2 4 6 8 10 Amplitude -2 -1.5 -1 -0.5 0 0.5 1

1.5 The real part of input signal

Time -10 -8 -6 -4 -2 0 2 4 6 8 10 Amplitude -0.6 -0.4 -0.2 0 0.2 0.4

0.6 The real part of output signal

desired signal output signal

FIGURE 3. Result of multiplicative ﬁlter achieved by using convolution (19).

Example 2 We use rin(t)= e−t

2

· sin(1.5t) + ei(t+10)2

, X(t) = e−t2

· sin(1.5t), and N(t) = ei(t+10)2

. For convenience, let u0 = ω0 = 0. The Wigner distributions of X(t) and rin(t) are shown in Figure 2. Thus, we can choose (see [5])

a= −2 3, b =

1

3, Ω = 2. The transfer function reads HA(u)= 1, u ∈− 2, 2 0, u − 2, 2, and(t) = 2i

3π ·sin 6tt . The output signal can be expressed as rout(t)= 3 2πieit 2 ·rin∗

(t). The consequent result of the multiplicative ﬁlter is displayed in Figure 3.

ACKNOWLEDGMENTS

This work was supported in part by FCT–Portuguese Foundation for Science and Technology through the

Cen-ter for Research and Development in Mathematics and Applications (CIDMA) of Universidade de Aveiro, within

project UID/MAT/04106/2013, and by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED).

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