Quaternion Wigner–Ville distribution associated with the linear canonical transforms

Signal Processing 130 (2017) 129–141

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Quaternion Wigner–Ville distribution associated with the linear canonical transforms Xiang-Li Fan a, Kit Ian Kou b,n, Ming-Sheng Liu a a b

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631 Guangdong, PR China Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macao, China

art ic l e i nf o

a b s t r a c t

Article history: Received 2 March 2016 Received in revised form 17 May 2016 Accepted 20 June 2016 Available online 24 June 2016

The quaternion linear canonical transform (QLCT), a generalization of the classical 2D Fourier transform, has gained much popularity in recent years because of its applications in many areas, including color image and signal processing. There are relationship between Wigner distribution and ambiguity function. But, these relations are only suitable for complex-valued signals, and have not been investigated in quaternion linear canonical transforms. The purpose of this paper is to propose an equivalent relationship for the quaternion Wigner distribution and quaternion ambiguity function in the QLCT setting. First, we propose the 2D quaternion Wigner distribution (QLWD) and quaternion ambiguity function associated with the QLCT. Next, the relationship between these two novel concepts are derived. Moreover, the connection with the corresponding analytic signal are investigated. Examples with bandpass analytic signal illustrate the features of the proposed distributions. Finally a novel algorithm for the detection of quaternion-valued linear frequency-modulated signal is presented by using the proposed QLWD. & 2016 Elsevier B.V. All rights reserved.

MSC: 00-01 99-00 Keywords: 2D quaternion Wigner–Ville distribution Quaternion ambiguity function Quaternion linear canonical transform Linear frequency modulation

1. Intoduction The linear canonical transform (LCT) is a family of linear integral transforms, which has found broad applications in signal processing and optics [1,8,9,17]. Many operations such as the Fourier transform (FT), fractional Fourier transform, Lorentz transform and scaling operations are the special cases of the LCT. The LCT can be seen as an effective processing tool for chirp signal analysis such as the parameter estimation, sampling progress for non-bandlimited signals with non-linear Fourier atoms [28] and the LCT filtering [12,34,38]. As shown in [2,37] the LCT has been successfully used to study the generalized Wigner distribution (WD) and ambiguity function (AF). They have been shown to be very useful in non-stationary signal processing [2,37,17]. The ambiguity function (AF) also plays an important role in the non-stationary signal analysis and processing theory in classical timefrequency distribution and has applied in many fields such as sonar technology, radar signal processing and optical information processing (refer to [19]). We can use the AF to calculate the distance between the target and the radar, the speed of the target, the distance resolution and radia velocity resolution. In [5,33], the AF n

Corresponding author. E-mail addresses: [email protected] (X.-L. Fan), [email protected] (K.I. Kou), [email protected] (M.-S. Liu). http://dx.doi.org/10.1016/j.sigpro.2016.06.018 0165-1684/& 2016 Elsevier B.V. All rights reserved.

associated with the LCT are discussed. In [36], the authors discussed the properties of generalized Wigner distribution and ambiguity function by using new integral transform. A first definition of a 2D quaternion linear canonical transform (QLCT) was introduced in [21,22], which is discussed later in this paper, and the first application of a 2D quaternion linear canonical transform to multidimensional signal analysis was reported in [21,22] involving prolate spheroidal wave signals and uncertainty principles [35]. Due to the non-commutative property of multiplication of quaternions, there are mainly three various types of 2D quaternion linear canonical transform (QLCTs): Two-sided QLCTs, Left-sided QLCTs and Right-sided QLCTs (refer to [22]). The main goals of the present paper are to study the properties of the Twosided QLCTs (TQLCTs) of 2D quaternionic signals and to derive the novel concept of quaternion Wigner distribution (QLWD) and quaternion ambiguity function (QLAF). The classical Wigner distribution is a kind of important tool in the time-frequency signal analysis (refer to [7,12]). It has advantages for the detection of linear frequency-modulated (LFM) signals, which realizes energy accumulation in the time-frequency domain. In [3], the authors introduced the 2D quaternion Wigner distribution by substituting the kernel of FT with the kernel of QFT in the classical WD definition. The theory of functions with values in the Hamiltonian quaternion algebra has been developed for decades as a generalization of the holomorphic function theory in the complex plane to 3D

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X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

and 4D, and considered as a refinement of classical harmonic analysis; see e.g. [13,14,26,27]. So far, quaternionic analysis has been successfully used in a wide variety of fields from theoretical to practical problems such as gauge theories, mathematical physics, signal and image processing, navigation, computer vision, robotics as well as natural sciences and engineering [14,13,27,39]. In quaternionic analysis, the quaternion Fourier transform (QFT) [6,16,29–31] plays a vital role in the representation of (multidimensional) signals. It transforms a real (or quaternionic) 2D signal into a quaternion-valued frequency domain signal. The four components of the QFT separate four cases of symmetry into real signals instead of only two as in the complex FT. In [32] the authors applied the QFT to proceed color image analysis. The efficient implementation of the QFT was studied in [31]. In [4] the authors applied the QFT to image preprocessing and neural computing techniques for speech recognition. The QLCT, as the generalization of the QFT, was firstly studied in [22]. In this paper [22] the authors investigated the uncertainty principle of the (rightsided) QLCT within quaternionic analysis. From a signal processing point of view, this principle prescribes a lower bound on the product of the effective widths of quaternionic signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty. Recently, certain asymptotic properties of the QFT were analyzed and quaternionic counterparts of classical Bochner–Minlos theorems were derived in [10] and [11]. In [23] authors established a generalized Riemann–Lebesgue lemma for the (right-sided) QLCT, which prescribes the asymptotic behavior of the QLCT extending and refining the classical Riemann–Lebesgue lemma for the Fourier transform of 2D quaternion signals. The QLCT of a probability measure was introduced and some of its basic properties such as linearity, reconstruction formula, continuity, boundedness, and positivity were studied [25]. Different approaches of the 2D quaternion Hilbert transforms [24] are proposed recently which allow the calculation of the associated analytic signals in the QLCT domains. The corresponding analytic signals can suppress the negative frequency components in the QLCT domains. Envelope detection for color images are investigated [24] by the generalized analytic signals in the QLCT domains [24]. The paper is organized as follows. In order to make it selfcontained, Section 2 gives a brief introduction to some general definitions and basic properties of quaternion algebra, LCTs and QLCTs of 2D Quaternion-valued signals. In Section 3 we give the definition and study the properties of a 2D QLWD and QLAF associated with QLCT. The relationship between Two-sided QLWD and QLAF is presented in Section 4. The proposed Two-sided QLWD combining with analytic signal is studied in Section 5. Examples with bandpass analytic signal illustrate the features of the proposed distributions. A novel algorithm for the detection of quaternion-valued linear frequency-modulated signal is presented in Section 6. Finally, concluding remarks are drawn in Section 7.

i2 = j2 = k2 = − 1,

ij = − ji = k,

jk = − kj = i,

ki = − ik = j.

(2)

Let [q]0 and q≔i [q]1 + j [q]2 + k [q]3 denote the real scalar part and the vector part of quaternion number q = [q]0 + i [q]1 + j [q]2 + k [q]3, respectively. Then, from [16], the real scalar part has a cyclic multiplication symmetry

∀ p, q, r ∈ .

[pqr ]0 = [qrp]0 = [rpq]0 ,

(3)

The conjugate of a quaternion q is defined by q = [q]0 − i [q]1 − j [q]2 − k [q]3, and the norm of q ∈  defined as

[q]20 + [q]12 + [q]22 + [q]23 .

∣q∣≔ qq =

(4)

It is easy to verify that

pq = qp ,

∀ p, q ∈ .

∣qp∣ = ∣q∣∣p∣,

(5)

Now we introduce an inner product of quaternion functions f , g defined on 2 with values in  as follows

(f , g )≔

∫

2

f (x) g (x) d2 x,

d2 x = dxdy ,

(6)

with symmetric real scalar part

⎫ 1⎧ 〈f , g 〉≔ ⎨ (f , g ) + (g , f ) ⎬ = ⎭ 2⎩

∫

2

[f (x) g (x) ]0 d2 x.

(7)

Its associated scalar norm ∥f ∥ can be defined by both (6) and (7):

∥f ∥L2 (2; ) ≔ (f , f ) =

〈f , f 〉 =

⎛ ⎜ ⎝

∫

2

⎞1/2 |f (x)|2 d2 x⎟ . ⎠

(8)

As a consequence of the inner product (6) we obtain the quaternion Cauchy–Schwarz inequality

∫

2

f (x) g (x) d2 x ≤

for any f , g ∈

L2 (2 ;

⎛ ⎜ ⎝

1

1

∫

2

⎞2 ⎛ ∣f (x)∣2 d2 x⎟ ⎜ ⎠ ⎝

∫

2

⎞2 ∣g (x)∣2 d2 x⎟ , ⎠

(9)

).

2.2. LCTs and QLCTs of 2D quaternion-valued signals Due to the noncommutative property of multiplication of quaternions, there are various types of LCTs and QLCTs for 2D quaternion-valued signals. In the following we briefly recall the definitions of the LCT and QLCT (refer to [22,20]).

( ) ab

Definition 2.1 (LCTs of 2D -valued signals). Let A = c d ∈ 2 × 2 be a matrix parameter such that det (A) = ad − bc = 1. The Leftsided LCT and Right-sided LCT of signals f ∈ L1 (2 ; ) are defined by

3li, A (f )(u1,

⎧ 1 a 2 1 d 2 ∫ ei ( 2b x1 − b x1u1+ 2b u1 ) f (x1, x2 ) dx1, b ≠ 0; ⎪ π i 2 b ⎨ x2 )≔ ⎪ cd 2 b=0 ⎩ d e i 2 u1 f (du1, x2 ),

and 2. Preliminaries

3 rj, A (f )(x1,

2.1. The quaternion algebra

⎧ 1 j a x 2− 1 x u + d u 2 ⎪ ∫ f (x1, x2 ) j2πb e ( 2b 2 b 2 2 2b 2 ) dx2, b ≠ 0; ⎨ u2 )≔ ⎪ cd 2 b = 0, ⎩ f (x1, du2 ) d e j 2 u2 ,

The quaternion algebra was introduced by Hamilton in 1843 and is denoted by  in his honor. Every element of  is a linear combination of a real scalar and three orthogonal imaginary units (denoted by i, j and k) with real coefficients (see [22])

Moreover, there are different types of QLCTs for 2D -valued signals.

 = {q∣q≔[q]0 + i [q]1 + j [q]2 + k [q]3 , [q]i ∈ , i = 0, 1, 2, 3} ,

Definition

(1)

where the elements i, j and k obey Hamilton's multiplication rules

respectively.

Ai =

( )∈ ai bi ci di

2.2 (QLCTs

2 × 2

of

2D

-valued

signals). Let

be a matrix parameter such that det (Ai ) = 1, bi ≠ 0

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

for i = 1, 2. The Two-sided QLCT (TQLCT), Right-sided QLCT (RQLCT) and Left-sided QLCT (LQLCT) of signals f ∈ L1 (2 ; ) are defined by

3iT, j(f )(u1, u2 )≔

∫

2

K Ai 1 (x1, u1) f (x1, x2 ) K Aj 2 (x2 , u2 ) dx1dx2,

3iR, j(f )(u1, u2 )≔

∫

2

f (x1, x2 ) K Ai 1 (x1, u1) K Aj 2 (x2 , u2 ) dx1dx2

∫

K Ai 1 (x1, u1) K Aj 2 (x2 , u2 ) f (x1, x2 ) dx1dx2,

(10)

2

The TQLCTs have many available properties, for example the additivity, the reversibility, the time shift, frequency shift properties and Parseval's formula (see [22,20]).

1

∫ K Ai

3li, A−1 (3 rj, A−1 )(3Ti, j(f )(t1, t2 ) = 1

−1 (u1, t1) 1

2

1 i2πb1

e

⎛ a ⎞ d i ⎜ 1 x12 − 1 x1u1+ 1 u12 ⎟ ⎝ 2b1 b1 2b1 ⎠ ,

1 j2πb2

e

4

1 j 2ab2 (x22 − t 22 ) −j x2b− t 2 u2 2 e 2 e du2 du1d2 2πb2

⎛ a ⎞ d j ⎜ 2 x22 − 1 x 2 u2 + 2 u22 ⎟ b2 2b2 ⎠ . ⎝ 2b2

x=

(12)

Lemma 2.3. If f ∈ ), the TQLCT of f can be decomposed into the sum of two RQLCTs or LQLCTs of f.

3Ti, j(f )(u, v) = 3Ri, j(fa )(u, v) + 3Ri, j(fb )(u, v) j,

3Lj, i (fa )(u,

v) +

3Lj, −i (fb )(u,

(13)

v) j ,

(14)

Proof. It suffices to prove Eq. (14), Eq. (13) can be proved in the similar argument, so we omit it. 2

dxdy =

∫

dsdt +

2

dsdt j = dsdt +

∫

u) fa (s ,

∫ ∫

2

t ) K Ai 2 (t ,

u) K Ai 2 (t ,

〈f , g 〉 =

∫

2

a j 2 (x22 − t 22 ) 2b2 δ (x2

− t1) f (x1, x2 )

− t2 ) d2 x = f (t1, t2 ). □

⎡

∫ ⎣⎢ ∫

=

∫ ∫

2

2

2

2

⎤ K Ai −1 (u1, x1) 3iT, j (f ) K Aj −1 (u2 , x2 ) d2ug (x) ⎥ d2 x ⎦0 1 2

[3iT, j (f ) K Aj −1 (u2 , x2 ) g (x) K Ai −1 (u1, x1)]0 d2ud2 x 2

⎡ i, j ⎢ 3T (f ) 2 ⎣

∫

⎡ i, j ⎢⎣ 3T (f )

∫

=

∫

=

∫

2

Since yields

j (a + bi) = (a − bi) j, where

[f (x) g (x)]0 d2 x

=

v) fa (s , t )

Ai−1 =

(

1

K Aj −1 (u2 , 2 2 2

⎤ x2 ) g (x) K Ai −1 (u1, x1) d2 x⎥ d2u ⎦0 1

⎤ K −i−1 (u1, x1) g (x) K −j−1 (u2 , x2 ) d2 x⎥ d2u . ⎦0 A1 A2

di −bi −ci ai

)∈

2×2

K −−i 1 (u1, x1) = K Ai 1 (x1, u1), A1

for i = 1, 2, direct computation

K −j−1 (u2 , x2 ) = K Aj 2 (x2 , u2 ). A2

Hence

Remark 2.4. (a) It is sufficient to consider the theory of TQLCT. The analogue results for LQLCT or RQLCT can be deduced by their relationships (Eqs. (13) and (14)) between TQLCT. (b) Notice that the TQLCT defined in Eq. (10) can be generalized as follows:

3 εT1, ε 2 (f )(u, v) =

a i 1 (x12 − t12 ) 2b1 δ (x1

Proof. For simplicity, we denote 3Ti, j (f ) by 3 i, j (f ). From (16) and the cyclic multiplication symmetry of quaternion number, we know

v)

K Aj1 (s , u) K A−2i (t , v) fb (s , t ) dsdt j

the last step uses the fact that □ a, b ∈ .

e

Theorem 2.6 (Plancherel formula for TQLCT). If f , g ∈ S (2 , ), then we have

K Aj1 (s , u) fb (s , t ) K A−2i (t , v)

K Aj1 (s , 2

2

The above method does not work for the RQLCT and LQLCT due to the noncommutative property of multiplication of quaternions.

K Aj1 (s , u)[fa (s , t ) + fb (s , t ) j] K Ai 2 (t , v) K Aj1 (s , 2

∫

〈f , g 〉 = 〈3iT, j(f ), 3iT, j(g )〉.

where f = fa + fb j, fa ≔f0 + if1, fs ∈ , s = 0, 1.

∫

1 i 2ab1 (x12 − t12 ) −i x1b− t1 u1 1 e 1 e f (x1, x2 ) 2πb1

∫

=

(11)

L1 (R 4 ;

3Tj, i (f )(u, v) =

K Ai 1 (x1, u1) f (x1, x2 )

2

However, the following lemma shows the relationship between various types of QLCTs.

v) =

2

du2 du1

e

3Tj, i (f )(u,

∫ ∫

K Aj 2 (x2 , u2 ) d2 xK Aj −1 (u2 , t2 )

and

K Aj 2 (x2 , u2 )≔

(16)

2

Proof. From the definition of 3Ti, j (f ) , we know that

respectively. Here

K Ai 1 (x1, u1)≔

f ∈ S (2 , ) (the

Lemma 2.5 (Inversion formula of TQLCT). If Schwartz space from 2 into ), then we have

f (t1, t2 ) = 3li, A−1 (3 rj, A−1 )(3Ti, j(f ))(t1, t2 ).

and

3iL, j(f )(u1, u2 )≔

131

∫

2

K Aε11 (s , u) f (s , t ) K Bε 2 (t , v) dsdt ,

(15)

〈f , g 〉 =

∫

⎡ i, j 3 (f ) ⎣⎢ T

=

∫

[3Ti, j(f ) 3Ti, j(g )]0 d2u = 〈3Ti, j(f ), 3iT, j(g )〉. □

2

2

∫

2

⎤ K Ai 1 (x1, u1) g (x) K Aj 2 (x2 , u2 ) d2 x⎥ d2u ⎦0

Corollary 2.7 (Parseval formula for TQLCT). If f ∈ S (2), then we have

∥f ∥2

L2 (2; )

= ∥3iT, j(f )∥2

L2 (2; )

.

where ε1≔ε1, i i + ε1, j j + ε1, k k and ε2≔ε2, i i + ε2, j j + ε2, k k so that

ε1,2i + ε1,2j + ε1,2k = ε2,2 i + ε2,2 j + ε2,2 k = 1,

(i. e ., ε12 = ε22 = − 1)

ε1, i ε2, i + ε1, j ε2, j + ε1, k ε2, k = 0. Eq. (10) is the special case of (15) in which ε1 = i and ε2 = j.

3. The 2D QLWDs and QLAFs In this section, the generalization of Wigner distribution and ambiguity function associated with the TQLCT will be discussed,

132

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

which are denoted by QLWD and QLAF. Moreover, several basic properties of them are investigated.

H0 =

∫

2

K Ai (τ1, ω1) h0 KBj (τ 2, ω2 ) d2τ ,

H1 =

∫

2

K Ai (τ1, ω1) h1KBj (τ 2, ω2 ) d2τ ,

H2 =

∫

K Ai (τ1, ω1) h2 KBj (τ 2, ω2 ) d2τ ,

H3 =

∫

K Ai (τ1, ω1) h3 KBj (τ 2, ω2 ) d2τ ,

3.1. The definition of the 2D QLWDs This section leads to the 2D quaternion Wigner distribution associated with TQLCT. Due to the noncommutative property of multiplication of quaternions, there are three different types of QLWDs: Two-sided QLWD (TQLWD), Left-sided QLWD (LQLWD) and Right-sided QLWD (RQLWD).

( ) ab

( ) a b

Definition 3.1 (TQLWD). Let A = c d , B = c′ d′ ∈ 2 × 2 be a ′ ′ matrix parameter such that det (A) = det (B ) = 1. The cross Twosided quaternion Wigner distribution (TQLWD) associated with the TQLCT of two-dimensional functions f , g ∈ L2 (2 ; ) is defined by

> Af ,,gB, T (t, ω)≔

⎛ τ⎞ ⎛ τ⎞ K Ai (τ1, ω1) f ⎜ t + ⎟ g ⎜ t − ⎟ KBj (τ 2, ω2 ) d2τ , 2⎠ ⎝ 2⎠ ⎝

∫

2

(17)

(18)

in fact, the TQLWD is the TQLCT of the signal Rf , g (t, τ ) with respect to τ , that is,

d2τ

K Ai (τ1, ω1) Rf , g (t, τ ) KBj (τ 2, ω2 ) =

3Ti, j(Rf , g (t,

·))(ω).

Definition 3.2 (LQLWD). The cross Left-sided quaternion Wigner distribution (LQLWD) associated with the TQLCT of two-dimensional functions f , g ∈ L2 (2 ; ) is defined by

> Af ,,gB, L (t, ω) =

∫

2

⎛ τ⎞ ⎛ τ⎞ K Ai (τ1, ω1) KBj (τ 2, ω2 ) f ⎜ t + ⎟ g ⎜ t − ⎟ d2τ . 2⎠ ⎝ 2⎠ ⎝

Definition 3.3 (RQLWD). The cross Right-sided quaternion Wigner distribution (RQLWD) associated with the TQLCT of twodimensional functions f , g ∈ L2 (2 ; ) is defined by

> Af ,,gB, R (t, ω) =

∫

2

 If f = [f ]0 ∈ , g = [g ]0 ∈ , then the three types of QLWDs are > Af ,,gB, T (t, ω) = > Af ,,gB, R (t, ω) = > Af ,,gB, L (t, ω) = H0.

 If f = f0 + if1, g = g0 + ig1 with f0 , f1 , g0 and g1 ∈ , then the TQLWD is identical with the RQLWD. That is,

 If f = f0 + jf2, g = g0 + jg2 with f0 , f2 , g0 and g2 ∈ , then the

⎛ τ⎞ ⎛ τ⎞ Rf , g (t, τ )≔f ⎜ t + ⎟ g ⎜ t − ⎟ , 2⎠ ⎝ 2⎠ ⎝

2

respectively. Notice that H1, H2, H3, H4 ∈ , we conclude that for the arbitrary two quaternion-valued functions f , g ∈ L2 (2 , ), the three types of QLWDs are not necessarily associated. But for some special cross functions f(t) and g(t), there are relationships between various types of QLWDs.

> Af ,,gB, T (t, ω) = > Af ,,gB, R (t, ω) = H0 + iH1.

Let us define the quaternion correlation product

∫

2

identical. That is,

where f = f0 + if1 + jf2 + kf3, g = g0 + ig1 + jg2 + kg3, K Ai (τ1, ω1) and KBj (τ 2, ω2 ) are given by (11) and (12), respectively.

> Af ,,gB, T (t, ω) =

2

⎛ τ⎞ ⎛ τ⎞ f ⎜ t + ⎟ g ⎜ t − ⎟ K Ai (τ1, ω1) KBj (τ 2, ω2 ) d2τ . 2⎠ ⎝ 2⎠ ⎝

TQLWD is identical with the LQLWD.

> Af ,,gB, T (t, ω) = > Af ,,gB, L (t, ω) = H0 + H2 j.

3.3. Some properties of TQLWD In this subsection, we discuss several basic properties of the TQLWD. In the following, denote > Af ,,gB≔> Af ,,gB, T and > Af , B = > Af ,,fB . These properties play important roles in signal representation. Lemma 3.4 (Boundedness). Let f , g ∈ L2 (2 ; ). Then > Af ,,gB (t, ω) is bounded on L4 (2 ; ), that is,

∣> Af ,,gB (t, ω)∣ ≤

2 π |bb′|

∥f ∥L2 (2; ) ∥g∥L2 (2; ) .

Proof. By using the quaternion Cauchy–Schwarz inequality (9)

∫

∣> Af ,,gB (t, ω)∣2 =

2

2 ⎛ τ⎞ ⎛ τ⎞ K Ai (τ1, ω1) f ⎜ t + ⎟ g ⎜ t − ⎟ KBj (τ 2, ω2 ) d2τ 2⎠ ⎝ 2⎠ ⎝

3.2. The relationship between various types of QLWDs

⎛ ≤ ⎜⎜ ⎝

Let fg = h0 + h1i + h2 j + h3 k , then from Definition 3.1–3.3, we have

⎛ 1 = ⎜⎜ ⎝ 4π 2|bb′|

> Af ,,gB, T (t, ω) =

∫

2

K Ai (τ1, ω1)(h0 + ih1 + h2 j + ih3 j) KBj (τ 2, ω2 )

≤

d2τ = H0 + iH1 + H2 j + iH3 j, > Af ,,gB, L (t, ω) =

∫

2

K Ai (τ1, ω1) KBj (τ 2, ω2 )

=

d2τ (h0 + h1i + h2 j + h3 k) d2τ = H0 + H1i + H2 j + H3 k, > Af ,,gB, R (t, ω) =

∫

(h0 + ih1 + jh2 + kh3 ) K Ai (τ1, ω1) KBj (τ 2, ω2 ) 2

d2τ = H0 + iH1 + jH2 + kH3, where

(19)

∫

2

⎞2 ⎛ τ⎞ ⎛ τ⎞ K Ai (τ1, ω1) f ⎜ t + ⎟ g ⎜ t − ⎟ KBj (τ 2, ω2 ) d2τ ⎟⎟ 2⎠ ⎝ 2⎠ ⎝ ⎠

⎛ 1 ⎜ 4π 2|bb′| ⎜⎝ 1

∥f ∥22

L (2; )

2

∫

2

⎛ ⎜4 ′| ⎜⎝

4π 2|bb

∫

∫

2

∥g∥22

⎞2 ⎛ τ⎞ ⎛ τ⎞ f ⎜ t + ⎟ g ⎜ t − ⎟ d2τ ⎟⎟ 2⎠ ⎝ 2⎠ ⎝ ⎠

⎞⎛ ⎛ τ⎞2 f ⎜ t + ⎟ d2τ ⎟⎟ ⎜⎜ ⎝ 2⎠ ⎠⎝

∫

⎞⎛ ∣f (x)∣2 d2 x⎟⎟ ⎜⎜ 4 ⎠⎝

⎞ 4 ∣g ( y) ∣2 d2 y⎟⎟ = 2 | π bb′| ⎠

L (2; )

∫

2

2

,

where applying the change of variables x = t + the last second step. Then we have

∣> Af ,,gB (t, ω)∣ ≤

2 π |bb′|

⎞ 2 ⎛ τ⎞ g ⎜ t − ⎟ d2τ ⎟⎟ ⎝ 2⎠ ⎠

∥f ∥L2 (2; ) ∥g∥L2 (2; ) ,

τ 2

and y = t −

τ 2

in

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X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

which completes the proof.

2

⎛ ⎞ 1 ⎜t τ ⎟ ⎛⎜ t τ ⎞⎟ − f⎜ + g 2s ⎟⎟ ⎝ s 2s ⎠ 2π jb′ ⎜s ⎝ ⎠ ⎤ ⎡ 1 d′ 2 ⎞ ⎥ 2 ⎢ ⎛ a′ 2 2 exp ⎢ j ⎜ ω2 ⎟ ⎥ d x s x2 − sx2 ω2 + ⎝ 2b′ ⎠ ⎥ b′ ⎢⎣ 2b′ ⎦

(i) Shift

> TAs,fB, Ts g (t, ω) = > Af ,,gB (t − s , ω). where Tx f (t)≔f (t − x)(t = (t1, t2 )). This can get easily from the Definition of the TQLWD. (ii) Nonlinearity

2

⎛ ⎞ 1 ⎜t τ ⎟ ⎛⎜ t τ ⎞⎟ f⎜ + g − 2s ⎟⎟ ⎝ s 2s ⎠ 2π jb′ ⎜s ⎝ ⎠ ⎡ ⎛ ⎞⎤ 1 ω d′ ⎛ ω ⎞2⎟ ⎥ ⎢ ⎜ a′ exp ⎢ j ⎜ b′ x22 − b′ x2 2 + b′ ⎜ 2 ⎟ ⎟ ⎥ d2 x s 2 2 ⎝ s ⎠ ⎠ ⎦⎥ ⎢⎣ ⎝ 2 s 2 s2 s

That is, the TQLWD of quaternion signal f + g is not simply the sum of the TQLWDs of the signals f and g. (iii) Dilation

⎛ t ω⎞ A, B (t, ω) = > Af ,1g, B1⎜ , ⎟, >D sf , Dsg ⎝s s⎠

1

( ), ab cd

B=

( ), a′ b′ c′ d′

b s2

d

⎞ ⎟, ⎟ ⎠

b′ ⎞ ⎛ a 2 B1 = ⎜⎜ 2′ sd ⎟⎟, s c′ ′ ⎝ ⎠

∫

=

2

> Af +, Bg (t, ω) =

∫

2

⎡ ⎛ ⎤ ⎛ τ⎞ τ⎞ K Ai (τ1, ω1) ⎢ f ⎜ t + ⎟ + g ⎜ t + ⎟ ⎥ ⎝ 2⎠ 2 ⎠ ⎥⎦ ⎢⎣ ⎝

⎡ ⎤ ⎢ f (t − τ ) + g ⎜⎛ t − τ ⎟⎞ ⎥ K j (τ 2, ω2 ) d2τ = ⎝ ⎠ 2 2 ⎥⎦ B ⎢⎣

∫

2

K Ai (τ1, ω1)

⎤ ⎛ ⎛ τ⎞ ⎛ τ⎞ τ⎞ ⎛ τ⎞ + g ⎜ t + ⎟ f ⎜ t − ⎟ + g ⎜ t + ⎟ g ⎜ t − ⎟ ⎥ KBj (τ 2, ω2 ) 2⎠ ⎝ 2⎠ 2⎠ ⎝ 2 ⎠ ⎦⎥ ⎝ ⎝

∫

d2τ +

∫

d2τ + d2τ +

2

2

∫

2

∫

2

From the first equality to the second equality, taking the change of τ □ variable x = s . This completes the proof. The -valued signal f can be uniquely determined in terms of its cross TQLWD with a quaternion constant as follows.

⎡ ⎛ ⎞ ⎛ ⎞ ⎢ f ⎜ t + τ ⎟ f ⎛⎜ t − τ ⎞⎟ + f ⎜ t + τ ⎟ g ⎛⎜ t − τ ⎞⎟ 2⎠ ⎝ 2⎠ 2⎠ ⎝ 2⎠ ⎝ ⎢⎣ ⎝

d2τ =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ω ⎟ ⎜t ω2 ⎟ τ ⎟ ⎛⎜ t τ ⎞⎟ j ⎜ , K Ai 1 ⎜ x1, 1 ⎟ f ⎜ + g K x − 2 2s ⎟⎟ ⎝ s 2s ⎠ B1 ⎜⎜ s ⎟ ⎜s s ⎟⎟ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎜ t ω⎟ d2 x = > Af 1, g, B1⎜ , ⎟. ⎜s s⎟ ⎝ ⎠

t

Dc f (t)≔ c f ( c ), c ≠ 0. Proof. Property (i) can be easily derived by the Definition of the TQLWD. We only prove properties (ii) and (iii). Indeed,

⎡ ⎛ ⎞⎤ 1 ω ⎢ ⎜ a d ⎛ ω ⎞2⎟ ⎥ exp ⎢ i ⎜ b x12 − b x1 1 + b ⎜ 1 ⎟ ⎟ ⎥ s 2π ib 2 2 ⎝ s ⎠ ⎠ ⎥⎦ ⎢⎣ ⎝ 2 s 2 s2 s 1

∫

=

> Af +, Bg (t, ω) = > Af ,,fB (t, ω) + > Af ,,gB (t, ω) + > gA,,fB (t, ω) + > gA,,gB (t, ω).

⎛ a A1 = ⎜⎜ 2 s c ⎝

1

∫

A, B (t, ω) = >D sf , Dsg

Proposition 3.5. Let f , g ∈ L2 (2 ; ). Then the TQLWDs of f and g have the following useful properties.

A=

⎤ ⎡ 1 ⎢ ⎛ a 2 2 d 2 ⎞⎥ exp ⎢ i ⎜ ω1 ⎟ ⎥ s x1 − sx1ω1 + ⎝ 2b ⎠ ⎥ b 2π ib ⎢⎣ 2b ⎦

□

Similar to the classical WD, we get the following useful properties of the TQLWD.

where

133

j ⎛ ⎞ τ ⎛ τ⎞ K Ai (τ1, ω1) f ⎜⎜ t + ⎟⎟ f ⎜ t − ⎟ (τ 2, ω2 ) ⎝ ⎠ 2⎠ 2 B ⎝

⎛ ⎞ τ ⎛ τ⎞ K Ai (τ1, ω1) f ⎜⎜ t + ⎟⎟ g ⎜ t − ⎟ KBj (τ 2, ω2 ) ⎝ ⎠ 2 2 ⎝ ⎠ τ τ K Ai (τ1, ω1) g (t + ) f (t − ) KBj (τ 2, ω2 ) 2 2 τ ⎛ τ⎞ K Ai (τ1, ω1) g (t + ) g ⎜ t − ⎟ KBj (τ 2, ω2 ) 2 ⎝ 2⎠

d2τ = > Af ,,fB (t, ω) + > Af ,,gB (t, ω) + > gA,,fB (t, ω) + > gA,,gB (t, ω),

Theorem 3.6 (Reconstruction formula for TQLWD). If f , g ∈ S (2 ; ), and g (0) ≠ 0 , then f can be reconstructed by the inverse TQLCT of TQLWD > Af ,,gB . That is,

f (t) =

1 g (0)

∫

2

⎛t ⎞ K i −1 (ω1, τ1) > Af ,,gB ⎜ , ω⎟ K j−1 (ω2, τ 2 ) d2ω . A ⎝2 ⎠ B

Proof. By (16) and (17), we find that

⎛ τ⎞ ⎛ τ⎞ f ⎜ t + ⎟g ⎜ t − ⎟ = 2⎠ ⎝ 2⎠ ⎝

∫

2

K i −1 (ω1, τ1) > Af ,,gB (t, ω) K j−1 (ω2, τ 2 ) A

B

d2ω .

(21)

τ

Setting t = 2 , and applying the change of variable s = 2t , we get

f (s) =

1 g (0)

∫

2

⎛s ⎞ K i −1 (ω1, s1) > Af ,,gB ⎜ , ω⎟ K j−1 (ω2, s2 ) d2ω . □ A ⎝2 ⎠ B

this implies that (ii) holds. Now we prove (iii). A, B >D (t, ω) = sf , Dsg

1 s2

∫

d2τ =

2

⎛t τ ⎞ ⎛⎜ t τ ⎞⎟ j K Ai (τ1, ω1) f ⎜ + − K (τ 2, ω2 ) ⎟g 2s ⎠ ⎝ s 2s ⎠ B ⎝s

∫

2

⎛t x⎞ K Ai (sx1, ω1) f ⎜ + ⎟ 2⎠ ⎝s

⎛t x⎞ g ⎜ − ⎟ KBj (sx2, ω2 ) d2 x. ⎝s 2⎠ Applying K Ai (τ1, ω1) and KBj (τ 2, ω2 ) defined by (11) and (12), respectively, we have

We are now ready to obtain Moyal's formula. Theorem 3.7 (Moyal's formula). If fi , gi ∈ S (2) , i = 1, 2 are quaternion-valued functions. Then

〈> Af1,,Bg1 (t, ω), > Af 2, B, g 2 (t, ω)〉 = [(f1 , f2 )(g2 , g1)]0 . Proof.

134

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

we have

〈> Af ,,Bg (t , ω) , > Af , B, g (t , ω)〉 1 1

=

2 2

∫R4 [> Af1,,Bg1 (t, ω) > Af2, B, g2 (t, ω)]0 d2td2ω

⎤ τ τ ) g2 (t − ) KBj (τ 2, ω 2 ) d 2τ ⎥ d 2 td 2ω ⎦0 2 2 ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ τ τ = ∫ 6 ⎢ > Af ,,Bg (t , ω) KB−j (τ 2, ω 2 ) g2 ⎜ t − ⎟ f2 ⎜ t + ⎟ K A−i (τ1, ω1) ⎥ d 2τd 2 td 2ω 1 1 R ⎢ ⎥⎦ 2⎠ ⎝ 2⎠ ⎝ ⎣ 0 =

⎡

∫R4 ⎢⎣ > Af1,,Bg1 (t, ω) ∫R2 K Ai (τ1, ω1) f2 (t + ⎡

=

⎛

∫R6 ⎢⎢ KA−i (τ1, ω1) > Af1,,Bg1 (t, ω) KB−j (τ 2, ω 2 ) g2 ⎜⎝ t − ⎣

⎡ = ∫ 4⎢ R ⎢ ⎣

∫R2

τ ⎞ ⎜⎛ τ ⎞⎤ 2 2 2 ⎟ f2 t + ⎟ ⎥ d τd td ω 2⎠ ⎝ 2 ⎠ ⎥⎦

〈> Af ,,Bg (t , ω) , > Af , B, g (t , ω) 〉 1 1 2 2 ⎡ ⎛ τ⎞ ⎛ τ ⎞⎤ = ∫ 4 ⎢ ∫ 2 K i −1 (ω1, τ 1) > Af ,,Bg (t , ω) K j −1 (ω 2, τ 2 ) d2ωg2 ⎜ t − ⎟ f2 ⎜ t + ⎟ ⎥ d2τd2 t A R ⎢ 1 1 B 2⎠ ⎝ 2 ⎠ ⎥⎦0 ⎝ ⎣ R =

⎡

∫R4 ⎢⎢⎣ f1 (t +

⎛ τ τ τ⎞ ⎛ τ ⎞⎤ ) g1 (t − ) g2 ⎜ t − ⎟ f2 ⎜ t + ⎟ ⎥ d2τd2 t . 2 2 2⎠ ⎝ 2 ⎠ ⎥⎦0 ⎝

Using the change of variables x = t + becomes

〈> Af1,,Bg1 (t, ω), > Af 2, B, g 2 (t, ω)〉 =

=

−2πbi 1

e

−i

2π ( − b) i

( 2ab τ e

i

d 2 2 1 1 − b τ1ω1+ 2b ω1

( −a2b τ

4

∫R

[f1 (x) g1 (y) g2 (y) f2 (x) ]0 d2 xd2 y 2

f1 (x) f2 (x) d2 x

∫R

2

⎤ g1 (y) g2 (y) d2 y⎥ ⎦0

= [(f1 , f2 )(g2 , g1)]0 . □

)

d 2 1 2 1 − −b τ1ω1+ −2b ω1

∫R

⎡ =⎢ ⎣

Because

1

τ

and y = t − 2 , the equation

0

⎛ τ⎞ ⎛ τ ⎞⎤ K A−i (τ1, ω1) > Af ,,Bg (t , ω) KB−j (τ 2, ω 2 ) d 2ωg2 ⎜ t − ⎟ f2 ⎜ t + ⎟ ⎥ d 2τd 2 t . ⎝ 1 1 2⎠ 2 ⎠ ⎥⎦ ⎝ 0

K−A i (τ1, ω1) =

τ 2

) = Ki

A−1

(ω1, τ1),

Based on the above theorem, we may conclude the following important consequences.

Fig. 1. Obtaining GAS associated with TQLCT.

2

2

Fig. 2. An example of Gaussian function f (t1, t2 ) = e−π (t1 + t 2 ) and the four components of GAS z (t1, t2 ) .

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

⎛ a where A1 = ⎜ 2 ⎜sc ⎝ (iv) Modulation

(i) If g1 = g2, then

〈> Af1,,Bg1 (t, ω), > Af 2, B, g1 (t, ω)〉 = ∥g1∥22

L (2; )

〈f1 , f2 〉.

(ii) If f1 = f2, then

〈> Af1,,Bg1 (t, ω), > Af1,,Bg 2 (t, ω)〉 = ∥f1 ∥22

L (2; )

AF

〈g1, g2 〉.

A, B (A ) (A ) (t , Ms 1f Ms 2, g 2

1

135 b s2

d

ω) ,

> Af1,,Bg1 (t,

b′ ⎞ ⎛ ⎜ a′ s2 ⎟. ⎜ s2c′ d′ ⎟ ⎝ ⎠

C,C ⎞ (- f , g1 (t, 0 ⎜ b ⎟ s1 ⎝ b1 ⎠

ω) = Ms(1D1) T ⎛

(iii) If f1 = f2 and g1 = g2, then

〈> Af1,,Bg1 (t,

⎞ ⎟, B = ⎟ 1 ⎠

C , C′ ⎞ (- f , g1 (t, 1 ⎜ b ⎟ s1 ⎝ b1 ⎠

T ⎛ b′ ⎞ Ms(2D2 ) − Ms(1D1) T ⎛

ω)〉 =

∥> Af1,,Bg1 ∥22 2 L ( ;  )

=

∥f1 ∥22 2 L ( ;  )

⎜ ⎟s ⎝ b2 ⎠ 2

∥g1∥22 2 . L ( ;  )

(D ′ )

3.4. Ambiguity function associated with TQLCT

Definition 3.8. The cross Two-sided quaternion ambiguity function (TQLAF) associated with TQLCT of the functions f , g ∈ L2 (2 ; ), (denoted by AFfA, g. B (t, ω)), is defined by 2

⎛ t⎞ ⎛ t⎞ K Ai (τ1, ω1) f ⎜ τ + ⎟ g ⎜ τ − ⎟ KBj (τ 2, ω2 ) d2τ . ⎝ 2⎠ ⎝ 2⎠

(22)

The definitions of Right-sided and Left-sided QLAF can be given accordingly. They have similar relations between TQLAF, as given in Section 3.2 for various types of QLWDs. Similar to the properties of the TQLWD, the TQLAF has the following properties. Proposition 3.9. Let f ∈ L2 (2 ; ). Then ,B AFfA(− (t, ω) = AFfA(,tB) (t, − ω). t)

∫

2

d2τ

⎛ t⎞ ⎛ t⎞ K Ai (τ1, ω1) f ⎜ −τ + ⎟ g ⎜ −τ − ⎟ KBj (τ 2, ω2 ) 2⎠ ⎝ 2⎠ ⎝ =

∫

2

C , C′ ⎞ (- f , g1 (t, 3 ⎜ b ⎟ s1 ⎝ b1 ⎠

⎜ ⎟s ⎝ b2 ⎠ 2

∫

2

(D ′ )

where

A2 =

g (t) = g0 (t) + ig1 (t) + jg2 (t) + kg3 (t), and

( )

⎛ D2′ = ⎜⎜ ⎝

0 d′ b2

The following theorem shows that the original quaternion signal can be uniquely determined by the QLAF with a quaternion constant. Theorem 3.11 (Reconstruction formula for TQLAF). If f , g ∈ S (2 ; ), and g (0) ≠ 0 , the signal f can be reconstructed by the inverse TQLCT of the cross TQLAF AFfA,,gB . That is,

1 g (0)

∫

2

⎛ ⎛ t ⎞ t ⎞ K i −1 ⎜ ω1, 1 ⎟ AFfA,,gB (t, ω) K j−1 ⎜ ω2, 2 ⎟ d2ω . A ⎝ B ⎝ 2⎠ 2⎠

⎛ t⎞ ⎛ t⎞ f ⎜ τ + ⎟g ⎜ τ − ⎟ = 2⎠ ⎝ 2⎠ ⎝

∫

2

K i −1 (ω1, τ1) AFfA,,gB (t, ω) K j−1 (ω2, τ 2 ) A

− ω2 ) d2τ ′ = AFfA(,tB) (t, − ω). □

( ). a′ b′ c′ d′

(i) Shift

AFTAs,fB, Ts g (t, ω) = Ms(1D1) Tas1 AFfA,,gB (t, ω) Ta′s 2 Ms(D2 2 ), ⎛ −1 ⎞ ⎛ −1 ⎞ 0 0 where D1 = ⎜⎜ c ac ⎟⎟, D2 = ⎜⎜ c ac′ ′ ⎟⎟, ′ ⎝ 2 ⎠ ⎝ 2 ⎠ a 2 1

d 2

a′ 2

1

d′ 2

Ms(1A) f (t) Ms(B2 )≔e−i ( 2b t1 − b t1s1+ 2b s1 ) f (t) e−j ( 2b′ t 2 − b′ t 2 s 2+ 2b′ s2 ) . (ii) Nonlinearity

AFfA+, Bg (t, ω) = AFfA, B (t, ω) + AFfA,,gB (t, ω) + AFgA,,fB (t, ω) + AFgA, B (t, ω). (iii) Dilation

⎛ t ω⎞ AFDAs, Bf , Ds g (t, ω) = AF fA,1g, B1 ⎜ , ⎟, ⎝s s⎠

a1 b1 c1 d1

,

⎛ t⎞ ⎛ t⎞ K Ai (τ1′, − ω1) f ⎜ τ ′ + ⎟ g ⎜ τ ′ − ⎟ KBj (τ2′, 2⎠ ⎝ 2⎠ ⎝ ab

( ),

⎛ ⎛ ⎞ b1 ⎞ b2 0 0 D1 = ⎜⎜ d b d d ⎟⎟, D2 = ⎜⎜ d′ b′ d′ d ⎟⎟, 1 2 − − ⎝ b1 b1 + d ⎠ ⎝ b2 b2 + d′ ⎠ ⎞ b ⎛ a − a1 b ⎞ ⎛ a′ − a2 b′ ⎞ ⎛ a′ − a2 b′ ⎞ − 2 b2 b′ ⎟ b2 −b′ ⎟ d′ ⎟ ⎜ b1 b ⎟, C1 = ⎜ ⎜ , , ′ C = = C a a 1 b′ d2 ⎟ ⎜ c − 1d d ⎟ ⎜ c′ − 2 d′ d′ ⎟ ⎜ a2 d′ −c′ d′ ⎟. + ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ b b b b2 d′ ⎠ 1 2 2

a2 b2 c 2 d2

d2ω .

( )

A1 =

Proof. From (16), we have

⎛ t⎞ K Ai ( − τ1′, ω1) f ⎜ τ ′ + ⎟ 2⎠ ⎝

Proposition 3.10. Let f , g ∈ L2 (2 ; ), A = c d and B = Then the TQLAF of f , g has the following useful properties

ω)

⎜ ⎟s ⎝ b2 ⎠ 2

⎛ t⎞ g ⎜ τ ′ − ⎟ KBj ( − τ2′, ω2 ) ⎝ 2⎠ d2τ ′ =

ω)

T ⎛ b′ ⎞ Ms2 2 k,

f (t) =

Proof. From (22), we have ,B AFfA(− (t, ω) = t)

C,C ⎞ (- f , g1 (t, 2 ⎜ b ⎟ s1 ⎝ b1 ⎠

T ⎛ b′ ⎞ Ms(2D2 ) j − Ms(1D1) T ⎛

The classical ambiguity function (AF) is firstly introduced by Woodward in 1953 for mathematical analysis of sonar and radar signal. In this section, we generalize the classical AF to the quaternion algebra setting.

∫

ω)

T ⎛ b′ ⎞ Ms2 2 i − Ms(1D1) T ⎛ ⎜ ⎟s ⎝ b2 ⎠ 2

AFfA,,gB (t, ω)≔

ω)

Fig. 3. Computation of QLWVD.

B

(23)

136

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141 t

Setting τ = 2 , Eq. (23) becomes

f (t) g (0) =

∫ ∫ 2

2

> FAA, B, A

,F 1 2 A 3, A 4

(t, ω) = 4

⎛ ⎛ ⎞ ⎛ t ⎞ t ⎞ K i −1 ⎜ ω1, 1 ⎟ AFfA,,gB ⎜ t, ω⎟ K j−1 ⎜ ω2, 2 ⎟ d2ω . A ⎝ ⎝ ⎠ B ⎝ 2⎠ 2⎠

∫

KA (2ε1, ω1)

2

F A1, A2 (t + ε) F−A3 , −A4 (ε − t) KB (2ε2, ω2 ) d2ε

Hence

f (t) =

1 g (0)

∫ ∫ 2

=4

⎛ ⎛ t ⎞ t ⎞ K i −1 ⎜ ω1, 1 ⎟ AFfA,,gB (t, ω) K j−1 ⎜ ω2, 2 ⎟ d2ω . □ A ⎝ B 2⎠ 2⎠ ⎝

2

1

∫

2

2π ib

⎡ ⎤ d ⎢ ⎥ i ⎢ 4a ε12− 2 ω1ε1+ 4 ω12 ⎥ b 2b 2b ⎦F e⎣

1

F−A3 , −A4 (ε − t)

l l

= 4AF FAA, B, A

4. The relationship between TQLWD and TQLAF

,F 1 2 −A3, −A4

The TQLWD and TQLAF usually have more flexibility and advantages than the WD and AF for non-stationary signal processing [18]. In [36], the authors discuss the relationship between the classical WD and the AF. In this section, we show that the TQLWD is the 2D TQLCT of TQLAF, and two parameter matrices corresponding to the kernel function of 2D TQLCT are the inverse matrices and vice versa. The following theorem shows that the relationship between TQLWD and TQLAF.

F A1, A2 (t)≔3 iA,1j , A2 (f )(t),

F A3, A4 (t)≔3 iA, 3j , A4 (f )(t),

Theorem 4.1. If f ∈ L2 (2 ; ), then we have (i)

> FAA, B, A , F A , A (t, 1 2 3 4

ω) =

⎛ l = ⎜ 4a where A ⎜ c ⎝ (ii) AF FA, B A ,A

,F 1 2 A 3, A 4

l l 4AF FAA, B, A , F−A , −A (2t, 1 2 3 4

∼ ⎛ where A = ⎜⎜ ⎝

c

AF FAA, B, A

(t, ,F 1 2 A 3, A 4

a′ 4 b′

∫

2

2

2ω ) ,

∫

2

f (u)

1 −2π ib3

1 −2π ib3

2

⎡ ⎤ ⎛ a t ⎞ d t 2 −j ⎢ 4 u22 − 1 ⎜⎜ τ 2− 2 ⎟⎟ u2 + 4 τ 2− 2 ⎥ 2⎠ 2 ⎥⎦ b4 ⎝ 2b4 ⎢⎣ 2b4

e

(

∫

2

(- FAA, B, A

,F 1 2 A 3, A 4

(

∫

2

⎛ ⎞ τ KA (τ1, ω1) F A1, A2 ⎜ t + ⎟ ⎜ 2⎟ ⎝ ⎠

(

(

2

(t, ω) =

1 4

∫

2

matrix

⎛ ⎞ ⎛ ⎞ ⎜ε ⎟ ⎜ε t⎟ KA ⎜ 1 , ω1⎟ F A1, A2 ⎜ + ⎟ 2⎟ ⎜ 2 ⎟ ⎜2 ⎝ ⎠ ⎝ ⎠

d2ε

)

⎡ a ⎛ τ ⎞ τ 2⎤ d −i ⎢ 3 u12 − 1 ⎜ t1− 1 ⎟ u1+ 4 t1− 1 ⎥ 2⎠ 2 ⎦ b4 ⎝ 2b4 e ⎣ 2b3

∫

)

⎛ ⎞ ⎜ t⎟ KA (τ1, ω1) F A1, A2 ⎜ τ + ⎟ 2⎟ ⎜ ⎝ ⎠

⎛ ⎞ ⎜ε t⎟ ε F−A3 , −A4 ⎜ − ⎟ KB ( 2 , ω2 ) 2 2 2 ⎜ ⎟ ⎝ ⎠

⎡ a ⎛ τ ⎞ τ 2⎤ d −j ⎢ 4 u22 − 1 ⎜ t 2− 2 ⎟ u2 + 4 t 2− 2 ⎥ 2⎠ 2 ⎦ b4 ⎝ 2b4 e ⎣ 2b4

d2uKB (τ 2, ω2 ) d2τ =

)

⎡ a ⎛ τ ⎞ τ 2⎤ d −i ⎢ 3 u12 − 1 ⎜ t1− 1 ⎟ u1+ 4 t1− 1 ⎥ 2⎠ 2 ⎦ 2 b4 ⎝ 2b4 ⎣ 2b3 d

Using and let parameter ε = 2τ , ∼ ∼ a a A = ( 4 , b , c , 4d ) , B = ( 4′ , b′, c′, 4d′), we obtain

⎛ ⎞ ⎛ ⎞ τ τ KA (τ1, ω1) F A1, A2 ⎜ t + ⎟ F A3 , A4 ⎜ t − ⎟ ⎜ ⎜ 2⎟ 2⎟ ⎝ ⎠ ⎝ ⎠

−2π jb4

−2π jb4

e

⎛ ⎞ ⎜ t⎟ KA (τ1, ω1) F A1, A2 ⎜ τ + ⎟ 2⎟ ⎜ ⎝ ⎠

⎛ ⎞ ⎜t ⎟ F−A3 , −A4 ⎜ − τ ⎟ KB (τ 2, ω2 ) d2τ . ⎜2 ⎟ ⎝ ⎠

⎞ ⎟. ⎠

1

(2t, 2ω).

∫

uKB (τ 2, ω2 ) d2τ =

c′ 4d′ ⎟

KB (τ 2, ω2 ) d2τ =

1

f (u)

Proof. For (i)

⎛ ⎞ > FAA, B, A , F A , A ⎜ t, ω⎟ = ⎟ 1 2 3 4⎜ ⎝ ⎠

2

∫

for

∼ ⎛ t ω⎞ 1 ∼ ⎜ , ⎟, > A ,B 4 F A1, A2, F−A3, −A4 ⎝ 2 2 ⎠

⎞ ∼ ⎛ ⎟, B = ⎜⎜ 4d ⎟ ⎠ ⎝

⎡ ⎤ d′ ⎢ ⎥ j ⎢ 4a′ ε 22− 2 ω 2 ε 2+ 4 ω 22 ⎥ b′ 2b′ 2b′ ⎦ d 2ε e⎣

⎞ ⎛ ⎞ ⎛ ⎜ ⎜ t⎟ t⎟ KA (τ1, ω1) F A1, A2 ⎜ τ + ⎟ F A3 , A4 ⎜ τ − ⎟ 2⎟ 2⎟ ⎜ ⎜ ⎠ ⎝ ⎠ ⎝

∫

ω) =

KB (τ 2, ω2 ) d2τ =

⎞ ⎛ ⎞ l = ⎜ 4a′ b′ ⎟. , B ⎜ c′ d′ ⎟ d⎟ ⎝ 4⎠ 4 ⎠

a 4 b

+ ε)

For (ii)

b⎟

(t, ω) =

2π jb′

A1, A2 (t

)

⎛ ⎞ τ KA (τ1, ω1) F A1, A2 ⎜ t + ⎟ ⎜ 2⎟ ⎝ ⎠

⎛ ⎞ τ F−A3 , −A4 ⎜ − t⎟ KB (τ 2, ω2 ) d2τ . ⎜2 ⎟ ⎝ ⎠ Taking the change of variable ε = 2τ , and let parameter matrix d d l A = (4a, b , c , 4 ) , Bl = (4a′, b′, c′, 4′ ), we obtain

⎡ a ⎢ 4

2

1 ω1

4d

ε − ε + i 1 1 ⎢ 2b 1 b 2 1 2b e⎣ = 4 2 2π ib ⎛ ⎞ ⎛ ⎞ ⎜t ⎜ε t⎟ ε⎟ ⎜ + ⎟ F−A3 , −A4 ⎜ − ⎟ 2⎟ 2⎟ ⎜2 ⎜2 ⎝ ⎠ ⎝ ⎠

∫

1 2π jb′

2⎤

( ω2 ) ⎥⎥⎦ F 1

⎡ a′ ⎤ ω ω 2 2⎥ j ⎢ 4 ε 22− 1 2 ε 2+ 4d′ ⎢ 2b′ ⎥ 2 2 2 b b 2 ′ ′ ⎦d ε e⎣

( )

⎞ ⎛ ⎜ t ω⎟ ∼ 1 ∼ A ,B = > F A , A , F−A , −A ⎜ , ⎟. □ 1 2 3 4 2 4 2⎟ ⎜ ⎠ ⎝

A1, A2

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

where z is the generalized analytic signal (GAS) [24] of the -valued signal f. It is calculated in the time space as follows,

5. Connection with generalized analytic signals 5.1. Definitions

z (t1, t2 )≔f (t1, t2 ) + iHli, A1 (f )(t1, t2 ) + Hrj, A2 (f )(t1, t2 )

The quaternion Wigner distributions and quaternion ambiguity functions have been investigated by several authors [18,15] as a tool for time–frequency signal analysis in quaternion setting. The correct use of the one dimensional WD for time–frequency signal analysis requires use of the analytic signal [7,12]. This version is often referred to as the Wigner–Ville distribution (WVD). This section presents new notions of WVD and corresponding AF associated with correlation products of generalized analytic signal in quaternion algebra setting. Definition 5.1 (QLWVD). Let A1 =

( ), A = ( ) ∈  a1 b1 c1 d1

2

a2 b2 c 2 d2

2×2

be a

matrix parameter such that det (A1 ) = det (A2 ) = 1. The quaternion Wigner–Ville distribution associated with the TQLCT of a -valued signal f (t1, t2 ) ∈ L2 (2 ; ), namely QLWVD, is defined by

= (t, ω)≔

∫

2

137

⎛ τ⎞ ⎛ τ⎞ K Ai 1 (τ1, ω1) z ⎜ t + ⎟ z ⎜ t − ⎟ K Aj 2 (τ 2, ω2 ) d2τ , 2⎠ ⎝ 2⎠ ⎝

j + i/ iA,1j , A2 (f )(t1, t2 ) j,

where the Left-sided, Right-sided and the Two-sided quaternion Hilbert transforms of a signal f : 2 →  are defined [24], respectively, by

1 Hli, A1 (f )(t1, t2 )≔ π

1 Hrj, A2 (f )(t1, t2 )≔ π

/ iA,1j , A2 (f )(t1, t2 )≔

p. v. p. v.

e

∫ ∫

1 p. v. π2

a i 1 (x12 − t12 ) 2b1 f

(x1, t2 ) dx1, t1 − x1 a j 2 (x22 − t 22 )

f (t1, x2 ) e 2b 2 t2 − x2

∫

dx2,

a i 1 (x 2 − t 2 ) e 2b1 1 1 f

2

a j 2 (x22 − t 22 )

(x1, x2 ) e 2b 2 (t1 − x1)(t2 − x2 )

dx1dx2,

provided these integrals exist as principal values. Fig. 1 represents the calculation of GAS and example of Gaussian function 2

(24)

(25)

2

f (t1, t2 ) = e−π (t1 + t2 ) is applied in Fig. 2. The GAS can be calculated in the TQLCT domains as [24]

3 iA,1j , A2 (z )(u1, u2 ) ⎡ ⎛ u ⎞⎤⎡ ⎛ u ⎞⎤ = ⎢ 1 + sgn ⎜ 1 ⎟ ⎥ ⎢ 1 + sgn ⎜ 2 ⎟ ⎥ 3 iA,1j , A2 (z )(u1, u2 ). ⎝ b1 ⎠ ⎦ ⎣ ⎝ b2 ⎠ ⎦ ⎣

(26)

By (24), the QLWVD is defined by the TQLCT of correlation product Rz, z (t, τ ) with respect to τ , represented in Fig. 3,

= (t, ω) =

∫

2

K Ai 1 (τ1, ω1) R z, z (t, τ ) K Aj 2 (τ 2, ω2 )

d2τ = 3iT, j(R z, z (t, ·))(ω).

Fig. 4. Diagram of TQLCT relations between the correlation products and the QLWDs and QLAFs.

(27)

While the corresponding quaternion ambiguity function of GAS

Fig. 5. The first row is the correlation product Rz, z and the second row is a2 = 0, b2 = − 1, c2 = 1, d2 = 0 . (a) Magnitude. (b) Real part. (c) i-part. (d) j-part. (e) k-part.

the

cross-sections

= (0, − 0.9, ω1, ω2 ) ;

a1 = 0, b1 = − 1, c1 = 1, d1 = 0 ;

138

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

Fig. 6. The first row is the correlation product Rz, z and the second row is the cross-sections a2 = − 1, b2 = 1, c2 = 0, d2 = 1. (a) Magnitude. (b) Real part. (c) i-part. (d) j-part. (e) k-part.

Fig. 7. The first row is the correlation product Rz, z and the second row is the cross-sections a2 = 1, b2 = − 1, c2 = 1, d2 = 0 . (a) Magnitude. (b) Real part. (c) i-part. (d) j-part. (e) k-part.

associated with TQLCT, namely QLAVF, is defined by the TQLCT of correlation product Rz, z (t, τ ) with respect to t. That is,

< (τ , ω)≔

∫

2

d2 t. = 3iT, j(R z, z (· , ω))(τ ).

(28)

= (0, − 0.9, ω1, ω2 ) ;

a1 = − 1, b1 = 1, c1 = 0, d1 = 1;

a1 = 01, b1 = − 1, c1 = 1, d1 = 0 ;

By (27), the correlation product Rz, z (t, τ ) can be considered as the inverse TQLCT of = (t, ·). Its substitution into (28) yields

< (τ , ω) =

K Ai 1 (t1, ω1) R z, z (t, τ ) K Aj 2 (t2, ω2 )

= ( − 0.9, 0, ω1, ω2 ) ;

∫

2

K Ai 1 (t1, ω1)

K Aj 2 (t2, ω2 ) d2 t

{∫

2

}

K −A1i (τ , x) = (t, x) K −A2j (τ , x) d2 x

(29)

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

139

Fig. 8. The first row is the correlation product Rz, z and the second row is the cross-sections = (0, 0, ω1, ω2 ) ; a1 = 0, b1 = − 1, c1 = 1, d1 = 0 ; a2 = 0, b2 = − 1, c2 = 1, d2 = 0 . (a) Magnitude. (b) Real part. (c) i-part. (d) j-part. (e) k-part.

Fig. 9. The first row is the correlation product Rz, z and the second row is the cross-sections = (0, 0, ω1, ω2 ) ; a1 = 1, b1 = − 1, c1 = 1, d1 = 0 ; a2 = 1, b2 = − 1, c2 = 1, d2 = 0 . (a) Magnitude. (b) Real part. (c) i-part. (d) j-part. (e) k-part.

140

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

It shows the relationship between QLWVD and QLAVF. 5.2. TQLCT Frequency calculations of the QLWVD and QLAVF Fig. 4 shows the relationships between the correlation products R and P, QLWVD and QLAVF. Solid lines represent direct and inverse TQLCT between functions of neighbor corners, → and ↓ mean taking the TQLCT while ← and ↑ mean taking the inverse TQLCT. The dotted lines are the corresponding TQLCT pairs between the functions of opposite corners. For example, the path R → = is given by (27) and the path R → < by (28), as well as the path = → R → < by (29). Note that the correlation product P is defined indirectly using the path R → = → P

P (μ, ω) =

∫

2

K Ai 1 (t1, μ1)

K Aj 2 (t2,

μ2

{∫

2

K Ai 1 (τ1, ω1) R z, z (t, τ ) K Aj 2 (τ 2, ω2 ) d2τ

}

∫

2

6. Detection of quaternion-valued linear frequency-modulated signal Frequency modulation is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave [12]. In this section, the newly proposed TQLWD is applied to form the Linear Frequency Modulation (LFM) of quaternion-valued signal. Let A =

) d2 t

K Ai 1 (τ1, ω1)

( ) and ab cd

{∫

2



}

K Ai 1 (t1, μ1) R z, z (t, τ ) K Aj 2 (t2, μ2 ) d2 t

K Aj 2 (τ 2, ω2 ) d2τ .

g (t) =

t 2+ t 2 1 2 e− 2 ,

> Af ,,gB (t,

e−

1 2πb′ j

=

⎛ a 2 1 d 2⎞ 2 2 f (t1, t2 ) = e−π (t1 + t 2 ) cos ⎜ t − t1ω11 + ω11⎟ ⎠ ⎝ 2b 1 b 2b ⎛ a′ 2 1 d′ 2 ⎞ cos ⎜ t − t2 ω12 + ω12 ⎟. ⎝ 2b′ 2 ⎠ b′ 2b′

e

2 1

1 11+

2 2

=(t, ω) =

2 a + iπb

2 12 +

d′ ω 2 2b′ 12

).

2

2 iπb − 3a/2 ⎤ ⎡ 2 (a′t2)2 ⎥ ⎢ − (a′t2− 2ω 2) − ⎢ e 2b′(πb′− ja′) − e 2b′(πb′− ja′) ⎥. ⎥ ⎢ ⎦ ⎣

d′ 2 2 1 2 − b′ ω 2 τ 2+ 2b′ ω 2

(

(

a t 2− t1 u + d u 2 2b 1 b 0 2b 0

1

a′ t 2− t 2 v + d′ v 2 2b′ 2 b′ 0 2b′ 0

e

−i

(

2

In general, the QLWVDs = are quaternion-valued functions. Notice

)e

2

t i d ω12 − 1 2 2b

)e

1 2π∣b′∣

2

t j d′ ω 22 − 2 2 2b′

a t 2− t1 u + d u 2 2b 1 b 0 2b 0

1

2b2 − 3abi

− 2π b j

)e e

2

t i d ω12 − 1 2 2b

−j

(

e

2b′ 2 − 3a′b′ j

2b2 − 3abi

+ d ω12 2b

2b′ 2 − 3a′b′ j

2 3a + bi

a′ t 2− t 2 v + d′ v 2 2b′ 2 b′ 0 2b′ 0

[b′t 2+ j (v0 − a′t 2− 2ω 2 )]2

[b′t 2+ j (v0 − a′t 2− 2ω 2 )]2

e d′

1 −2πb′ j

b′− 3a′ j 2 b′t 2+ j (v0 − a′t 2− 2ω 2 ) τ2 + τ2 dτ 2 8b′ 2b′

2

e

e j 2b′ ω 2 a′ + jπb′

)

b − 3ai τ 2+ bt1+ i (u0 − at1− 2ω1) τ 1 dτ 1 8b 1 2b

[bt1+ i (u0 − at1− 2ω1)]2

a′ + jπb′

ω2 1 a 2 2t1ω1 + 3d ω 2) − b 2b 1 e b(2πb + i3a)

( 2ab′′ τ

j

3a′ + b′ j

⎤ ⎡ (a′t 2)2 ⎥ ⎢ − (a′t 2− 2ω 2)2 − ′ ′ ′ ′ ′) ′) ( − ( − b b j a b b j a π π 2 2 +e ⎥ ⎢e ⎥ ⎢ ⎦ ⎣ 2

)

−2πbi

[bt1+ i (u0 − at1− 2ω1)]2

e

d′ 2 e j 2b′ ω 2

e−2π (t1 + t2 )e i( b t1 −

d 2 2 1 1 − b ω1τ1+ 2b ω1

τ2

−2πbi

Moreover, the corresponding QLWVD = is given by (27). 2 2 2 2 − (at1− 2ω1) d e−2π (t1 + t2 )e i 2b ω1 e 2b(πb − ia)

( 2ab τ

τ 2 2 t 2+ 2 − v0 + d′ v02 ] 2 d τ 2b′ b′

1

=

) e j ( 2ab′′ t − b1′ t ω

d ω2 2b 11

−j

∫ e−

Then the corresponding GAS is

( 2ab t − 1b t ω

e

1 −i e 2π∣b∣

∫ e−

signal with a Gaussian envelop of the form [18]

i

i

)

a′

a′ c′ b′ d′

2

e

KBj (τ 2, ω2 ) d2τ

e−j [ 2b′ (t 2+ 2 )

( ) and B = ( ), consider the modulated bandpass

2

1 2πbi

(

In this section, we compare the features of the QLWVDs using the concepts of cross-section of a 4D distribution [18] with fixed values of the signal domain variables ( t1 = t10 , t2 = t20 ), i.e., = (t10, t20, ω1, ω2 ).

z (t1, t2 ) ≈ e−π (t1 + t 2 ) e

2

2

2

5.3. Examples on specific signals

ab cd

2

⎞ ⎛ ⎞ ⎟ ⎜ ⎟ τ1 τ2 − j + , u0 ⎟ KB ⎜ t2 + , v0 ⎟ 2 2 ⎜⎜ ⎟⎟ ⎟⎟ ⎠ ⎝ ⎠

τ1 ⎡ ⎤ τ 2 t1+ 2 ⎢ ⎥ −i ⎢ a t1+ 1 − u0 + d u02 ⎥ τ12 τ 22 t12 t1 t 22 t 2 2 2b 2b b ⎦ e− 2 + 2 τ1− 8 e− 2 + 2 τ 2− 8 e ⎣

K Aj 2 (t2, μ2 ) K Aj 2 (τ 2, ω2 ) = K Aj 2 (τ 2, ω2 ) K Aj 2 (t2, μ2 ).

Let A =

⎜⎜ ⎝

τ τ (t1− 1 ) 2 +(t 2− 2 ) 2

∫

=

and

⎛ ⎜

ω1) KA−i ⎜ t1

∫

The above two expressions for correlation product P are equal because

K Ai 1 (τ1, ω1) K Ai 1 (t1, μ1) = K Ai 1 (t1, μ1) K Ai 1 (τ1, ω1)

a′ b′ c′ d′

it follows from (17) that

K Ai (τ1, 2

ω) =

( ), consider the quaternion-valued

B=

f (t) = KA−i (t1, u0 ) KB−j (t2, v0 ) and Gaussian signal

Chirp signal

or the path R → < → P

P (μ, ω) =

that the GAS has single quadrant spectra. However, the crosssections of Figs. 5–9 have two-quadrant support. The presented examples illustrate the features of the quaternion-valued distribution = . The GAS have single-quadrant spectra. However the corresponding cross-sections of = have half-plane supports in the first and third quadrant, respectively (see Figs. 5–9).

= 2 3a′ + b′ j

2 3a + bi

)e

2

t j d′ ω 22 − 2 2 2b′

KA−i (t1, u0 )

KB−j (t2, v0 )

2 2 + d′ ω 22 − t1 + t 2 2b′ e 2 .

The above QLWD of f (t) and g (t) is a quaternionic LFM in (t, ω) plane. From this fact, we propose the following algorithm for the detection of quaternion-valued linear frequency-modulated signal. Step 1. Compute the QLWDs of two quaternionic signals f (t) , g (t). Step 2. Search for the peak values in the time–frequency plane, then estimate the instantaneous frequency. Step 3. Apply the least-squares approximation to the instantaneous frequency and obtain the final estimation value.

X.-L. Fan et al. / Signal Processing 130 (2017) 129–141

7. Conclusion The paper presents novel concepts of 2D quaternion Wigner distribution and quaternion ambiguity functions in the QLCT setting. Based on the properties of QLCT and generalized analytic signal, the relationship between these two notations are presented. Important properties such as nonlinearity, dilation, Moyal's formula are derived. Moreover, connection with corresponding analytic signal are considered. Examples with bandpass analytic signals illustrate the features of the proposed distributions. Finally, a novel algorithm on detection of quaternion-valued linear frequency-modulated signal is given. The authors are currently attempting to perform the advantages for the detection of linear frequency-modulated signals and apply to realize the energy accumulation in the time–frequency domain. They will be reported in forthcoming papers.

Acknowledgments The research was financially supported by Guangdong Natural Science Foundation (Grant Nos. 2014A030307016, 2014A030313422). The second author acknowledges financial support from the National Natural Science Foundation of China under Grant (Nos. 11401606 and 11501015), University of Macau (No. MYRG2015-00058-FST) and the Macao Science and Technology Development Fund (No. FDCT/099/2012/A3).

References [1] T. Alieva, M.J. Bastiaans, Powers of transfer matrices determined by means of eigenfunctions, J. Opt. Soc. Am. A 16 (10) (1999) 2413–2418. [2] R.F. Bai, B.Z. Li, Q.Y. Cheng, Wigner–Ville distribution associated with the linear canonical transform, J. Appl. Math. 2012 (Article ID 740161) (2012) 14. [3] M. Bahri, On two-dimensional quaternion Wigner–Ville distribution, J. Appl. Math. 2014 (Article ID 139471) (2014) 13. [4] E. Bayro-Corrochano, N. Trujillo, M. Naranjo, Quaternion Fourier descriptors for the preprocessing and recognition of spoken words using images of spatiotemporal representations, J. Math. Imaging Vis. 28 (2) (2007) 179–190. [5] T.W. Che, B.Z. Li, T.Z. Xu, The ambiguity function associated with the linear canonical transform, EURASIP J. Adv. Signal Process. 138 (2012) 1–14. [6] L.P. Chen, K.I. Kou, M.S. Liu, Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform, J. Math. Anal. Appl. 423 (1) (2015) 681–700. [7] L. Cohen, Time–Frequency Analysis: Theory and Applications, Prentice Hall, New York, 1995. [8] M.F. Erden, M.A. Kutay, H.M. Ozaktas, Repeated filtering in consecutive fractional fourier domains and its application to signal restoration, IEEE Trans. Signal Process. 47 (1999) 1458–1462. [9] Y.X. Fu, L.Q. Li, Generalized analytic signal associated with linear canonical transform, Opt. Commun. 281 (2008) 1468–1472. [10] S. Georgiev, J. Morais, Bochner's theorems in the framework of quaternion analysis, in: Quaternion and Clifford–Fourier Transforms and Wavelets Trends in Mathematics, 2013, pp. 85–104. [11] S. Georgiev, J. Morais, K.I. Kou, W. Sprößig, Bochner–Minlos theorem and quaternion Fourier transform, in: Quaternion and Clifford–Fourier Transforms and Wavelets Trends in Mathematics, 2012, pp. 105–120. [12] K. Grochenig, Foundations of Time–Frequency Analysis, Birkhauser, Boston,

141

2000. [13] K. Gürlebeck, W. Sprössig, Quaternionic Analysis and Elliptic Boundary Value Problems, Akademie Verlag, Berlin, 1989. [14] K. Gürlebeck, W. Sprössig, Quaternionic Calculus for Engineers and Physicists, John Wiley and Sons, Chichester, 1997. [15] S.L. Hahn, Multidimensional complex signals with single-orthant spectra, Proc. IEEE 80 (8) (1992) 1287–1300. [16] E. Hitzer, Quaternion fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebras 17 (3) (2007) 497–517. [17] J.J. Healy, M.A. Kutay, H.M. Ozaktas, J.T. Sheridan, Linear Canonical Transforms: Theory and Applications, Springer, New York, 2016. [18] S.L. Hahn, K.M. Snopek, Wigner distributions and ambiguity functions of 2D quaternionic and monogenic signals, IEEE Trans. Signal Process. 53 (8) (2005) 3111–3128. [19] J.A. Johnston, Wigner distribution and FM radar signal design, Proc. Inst. Electr. Eng. F Radar Signal Process. 136 (April) (1989) 81–88. [20] K.I. Kou, J. Ou, J. Morais, Uncertainty principles associated with quaternionic linear canonical transforms, Math. Methods Appl. Sci. 39 (10) (2016) 2722–2736. [21] K. Kou, J. Morais, Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis, Math. Methods Appl. Sci. 36 (2013) 1028–1041. [22] K.I. Kou, Jian-Yu Ou, J. Morais, On uncertainty principle for quaternionic linear canonical transform, Abstr. Appl. Anal., 2013(Article ID 725952) (2013)14 pp. http://dx.doi.org/10.1155/2013/725952. [23] K.I. Kou, J. Morais, Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem, Appl. Math. Comput. 247 (2014) 675–688. [24] K.I. Kou, M. Liu, J. Morais, C. Zou, Envelope detection using generalized analytic signal in 2D QLCT domains, Multidimens. Syst. Signal Process., http://dx.doi.org/10.1007/s11045-016-0410-7. [25] K.I. Kou, M. Liu, S. Tao, Herglotz's theorem and quaternion series of positive term, Math. Methods Appl. Sci., http://dx.doi.org/10.1002/mma.3945. [26] V. Kravchenko, M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics. Research Notes in Mathematics, Pitman Advanced Publishing Program, London, 1996. [27] V. Kravchenko, Applied Quaternionic Analysis. Research and Exposition in Mathematics, vol. 28, Heldermann Verlag, Lemgo, 2003. [28] Y.L. Liu, K.I. Kou, I.T. Ho, New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms, Signal Process. 90 (3) (2010) 933–945. [29] B. Mawardi, E. Hitzer, A. Hayashi, R. Ashino, An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl. 56 (9) (2008) 2411–2417. [30] B. Mawardi, E. Hitzer, R. Ashino, R. Vaillancourt, Windowed Fourier transform for two-dimensional quaternionic signals, Appl. Math. Comput. 216 (2010) 2366–2379. [31] Soo-Chang Pei, Jian-Jiun Ding, Ja-Han Chang, Efficient implementation of quaternion fourier transform, convolution, and correlation by 2-D complex FFT, IEEE Trans. Signal Process. 49 (11) (2001) 2783–2797. [32] S.J. Sangwine, T.A. Ell, Hypercomplex Fourier transforms of color images, IEEE Trans. Image Process. 16 (1) (2007) 22–35. [33] R. Tao, T.E. Song, Z.J. Wang, Y. Wang, The ambiguity function based on the linear canonical transform, IET Signal Process. 6 (2012) 568–576. [34] K.B. Wolf, Canonical transforms, in: Integral Transforms in Science and Engineering, vol. 11, Plenum Press, New York, NY, USA, 1979 (Chapter 9). [35] Y. Yang, K.I. Kou, Uncertainty principles for hypercomplex signals in the linear canonical transform domains, Signal Process. 95 (2014) 67–75. [36] Z.H. Zhang, M.K. Luo, New integral transform for generalizing the Wigner distribution and ambiguity function, IEEE Signal Process. Lett. 22 (4) (2015) 460–464. [37] Z.H. Zhang, New Wigner distribution and ambiguity function based on the generalized translation in the linear canonical transform domain, Signal Process. 118 (2016) 51–61. [38] Z. Zalevsky, D. Mendlovic, M. Alper Kutay, H.M. Ozaktas, J. Solomon, Improved acoustic signals discrimination using fractional Fourier transformbased phasespace representations, Opt. Commun. 190 (1–6) (2001) 95–101. [39] C. Zou, K.I. Kou, Y. Wang, Quaternion collaborative and space representation with application to color face recognition, IEEE Image Process. (2016).