Novel Wigner distribution and ambiguity function associated with the linear canonical transform

Optik 127 (2016) 4995–5012

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Optik journal homepage: www.elsevier.de/ijleo

Novel Wigner distribution and ambiguity function associated with the linear canonical transform Zhi-Chao Zhang ∗ College of Mathematics, Sichuan University, Chengdu 610065, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 8 January 2016 Accepted 10 February 2016

The Wigner distribution (WD) and ambiguity function (AF) associated with the linear canonical transform (LCT) attract much attention in the literature. For this, many generalized time-frequency distributions are currently derived, for example the WD and AF in the LCT domain (WDL, AFL), the unified WD and AF in the LCT domain (UWA, WAL), and the WD and AF based on the generalized convolution in the LCT domain (LWD, LAF). However, there are two issues associated with these generalized distributions, which the first one relates to the fact that the marginal properties of these distributions do not have the elegance and simplicity comparable to those of the WD and AF and the other issue relates to the fact that there has no affine transformation relationship between the LCT and WDL, AFL, LWD, and LAF, and the affine transformation relationships between the LCT and UWA as well as the LCT and WAL do not generalize very nicely and simply the classical results for the WD and AF. Focusing on the above issues, this paper deduces a kind of novel WD and AF associated with the LCT, which has the elegance and simplicity in marginal properties and affine relations associated with the LCT comparable to the WD and AF. Then some essential properties, and relations with other classical time-frequency representations of the newly defined WD and AF are investigated. Applications of the new WD and AF are also performed to show the advantage of the theory. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction The linear canonical transform (LCT) has been widely used in optical system simulation, radar and sonar system analysis, pattern recognition, digit watermarking, and signal processing [1–7]. It is a four parameters class of linear integral transform and includes the classical Fourier transform (FT), the fractional Fourier transform (FRFT), the Fresnel transform (FST), and the scaling operations as its special cases [8–12]. With the rapid development of the modern signal processing technology, in the literature of recent past different scholars have proposed a series of novel time-frequency representations through combining the classical signal processing tools with the LCT [13–25]. The conventional Wigner distribution (WD) and ambiguity function (AF) are two kinds of important methods in optical system  analysis   and non-stationary signal processing [26–32] and defined as the FT of the classical instantaneous autocorrelation function f t + 2 f ∗ t − 2 for  and t, respectively, i.e.,





+∞

WDf (t, ω) =

f

t+

−∞

 2





f∗ t −

 2



e−jω d

(1)

and





+∞

AF f (, ω) =

f −∞

t+

 2





f∗ t −

 2



e−jωt dt.

∗ Tel.: +86 028 18782994260. E-mail addresses: [email protected], zhangzhichao [email protected] http://dx.doi.org/10.1016/j.ijleo.2016.02.028 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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Z.-C. Zhang / Optik 127 (2016) 4995–5012

The LCT has a great flexibility in terms of picking and choosing parameters, then the WD and AF associated with the LCT have attracted much attention because of their preponderance for processing non-stationary signals under low signal-to-noise ratios (SNRs) [3,19–25]. To be specific, Pei et al. formulated the linear canonical WD (LCWD) and linear canonical AF (LCAF) as the LCT is equivalent to an affine transformation for the WD and AF [19]. The LCWD and LCAF take the forms



WDFA (u, v) = and



+∞

 2

FA u + −∞

 AF FA (, v) =



+∞

 2

FA u + −∞







FA∗ u −



 2

 2

FA∗ u −





e−jv d

(3)

e−jvu du,

(4)

respectively, where FA (u) denotes the LCT of f(t) with parameter matrix A = (a, b ; c, d). Then the affine transformation relationships between the LCWD and WD as well as the LCAF and AF are given by [19] WDFA (u, v) = WDf (du − bv, −cu + av)

(5)

AF FA (, v) = AF f (d − bv, −c + av),

(6)

and

respectively, which reveal that the detection performance of the LCWD and LCAF is inherited from the WD and AF, respectively. Nevertheless, the relations shown in the above two equations are particularly useful for the separation of multi-component time-frequency signals in the LCT domain [13]. Following with this direction, similar definitions of the LCWD and LCAF were proposed by Zhao et al. [20], which can be seen from equations reproduced here,





+∞

WDA (u, v) =

FA v + −∞

and





+∞

AF A (, v) =

FA u + −∞

 2





v 2



 2

FA∗ v −



FA∗ u −

v





2

eju d

(7)

eju du.

(8)

Since both of these definitions of the WD and AF associated with the LCT are derived from the FT for the instantaneous autocorrelation function, and then they are not general enough. For this, Li et al. and Tao et al. proposed the WD in the LCT domain (WDL) [21] and the AF in the LCT domain (AFL) [22,23] by means of directly substituting the FT kernel with the LCT kernel, which can be considered as the generalized WD and AF, respectively, i.e., WDLAf (t, ω) =





1

f

j2b

and AFLAf (, ω) =





+∞

−∞



1



+∞

f

j2b

t+

t+

−∞

 2

 2







 2

f∗ t −



f∗ t −

 2

 d e

j

2b

 d e

j

2b

ω2 − 1 ω+ a  2 b

2b

ω2 − 1 ωt+ a t 2 b



d

(9)



2b

dt.

(10)

From (9) and (10), the WDL and AFL can be regarded as the decomposition form of a signal based on the non-orthogonal basis and hence have great advantages and flexibility than the WD and AF. Meanwhile, they are closely related and have their own merits for the detection of non-stationary signals, especially the linear frequency-modulated (LFM) signals [3,24–27]. The former shows energy distribution in the time and LCT-frequency domain, but the latter shows correlation in the time delay and LCT-frequency  shift  domain.  Due to this  consideration,    based on the generalized instantaneous autocorrelation functions associated with the LCT FA1 t + 2 FA∗ t − 2 and FA t + 2 f ∗ t − 2 , 2 the authors presented some novel integral transformations for unifying the WDL and AFL, which are simplified as the UWA [3] and WAL [24], those are, UWAAf 1 ,A2 ,A3 (t, ω)

=



j2b3

and 0 (t, ω) = WALA,A f





1



1

j2b0



+∞

−∞

FA1



+∞

FA t + −∞

t+

 2



 2



FA∗



2



f∗ t −

 2

t−

 2

 j d3

 j d0 e

2b0

e

2b3

a ω2 − 1 ω+ 3  2 b3

a ω2 − 1 ω+ 0  2 b0

2b0

2b3



d

(11)

 d.

(12)

The UWA and WAL are the generalization of the WD, AF, WDL, and AFL and then combine the energy accumulation of the WD and WDL with the autocorrelation of the AF and AFL. This indicates that the UWA and WAL applied to the detection of LFM signals are more effective than the previous definitions. However, the UWA and WAL, respectively, contain nine and six free parameters, and then both of them have a bit of complexity in the theory and application. In reality, the WAL which has less parameters is a special case of the UWA, and then it is a simpler time-frequency analysis tool. With this idea, the authors introduced another two special cases of the UWA, those are, the WD and AF associated with the generalized convolution in the LCT domain [25], which are simplified as the LWD and LAF, respectively. Here, we show them below



LWDAf (t, ω)

+∞

=2 −∞

2

FA (u + bω)F ∗¯ (u − bω)e−jbdω KA∗ (u, 2t)du A

(13)

Z.-C. Zhang / Optik 127 (2016) 4995–5012

and

 LAF Af (, ω)



+∞

=





4997



bd 2 b b ω F ∗ u − ω e−j 4 ω KA∗ (u, )du, A 2 2

FA u + −∞

(14)

where A¯ = (a, −b; −c, d),  A = (−a, b; c, −d), and KA (u, t) stands for the LCT kernel with parameter matrix A = (a, b ; c, d). Due to (12) and (13), the LWD and LAF only contain three free parameters and then are two kinds of simpler integral transformations. Moreover, it is shown in [25] that the LWD and LAF achieve better detection performance than the WD and AF, respectively, as well as the WDL and AFL, respectively. Although the aforementioned time-frequency distributions, such as the WDL, AFL, UWA, WAL, LWD, and LAF, are more powerful for processing non-stationary signals under low SNR compared with the conventional WD and AF, but there are two issues associated with them. The first one relates to the distributions’ marginal properties which are one kind of the most fundamental properties in the theoretical analyses and practical applications. Based on the definitions of those generalized WD and AF, the classical marginal properties for the WD and AF, which present the instantaneous energy (time marginal property) and the energy density spectrum (frequency marginal property), cannot be preserved exactly, that is, the marginal properties for those distributions do not generalize very nicely and simply the classical results for the WD and AF. Therefore, some applications of them are subjected to certain restrictions. The other issue associated with those generalized WD and AF is the affine transformation relationship between the LCT and them. It is well-known that the affine transformation relationships between the LCT and WD as well as the LCT and AF are particularly powerful for the designing of multiplicative filters in the LCT domain. However, there is no affine transformation relationship between the LCT and WDL, the LCT and AFL, the LCT and LWD, as well as the LCT and LAF. Meanwhile, although there has an affine transformation relationship between the LCT and UWA as well as the LCT and WAL, but the derived relations have a bit of complexity in terms of the required conditions, and then there has no elegance and simplicity in the expression form comparable to that of the classical results given by (5) and (6). Therefore, it is inconvenient to implement in filter design in the LCT domain through combining those distributions with the LCT. In this paper, we propose a new kind of WD and AF associated with the LCT through extending the kernels of FT to the kernels of LCT in a different way. The marginal properties of the newly defined WD and AF are obtained and have the elegance and simplicity comparable to those of the WD and AF. In addition, the affine transformation relationships between the LCT and the new WD as well as the LCT and the new AF are also derived and generalize very nicely and simply the classical results for the WD and AF shown in (5) and (6). The remainder of this paper is structured as follows. Section 2 reviews definition and some important properties of the LCT. In Section 3, a new kind of WD and AF associated with the LCT is formulated. Meanwhile, some main properties, and the relationships between the newly derived distributions and other classical time-frequency representations are also discussed in this section. In Section 4, applications of the new WD and AF for the detection of LFM signals, and the designing of multiplicative filters in the LCT domain for the separation of multi-component time-frequency signals are investigated. Finally, Section 5 concludes this paper. 2. Linear canonical transform The LCT can be considered as a quadratic phase system (QPS), which is one of the most important optical systems and is implemented with an arbitrary number of thin lenses and propagation through free space in the Fresnel approximation or through sections of graded-index media [15]. Then it can be defined as the output light field of the QPS [16]

FA (u) = LA [f ](u) =

⎧ ⎪ ⎨

+∞

f (t)KA (u, t)dt,

⎪ ⎩√

cd 2 u de 2 f (du), j

where the LCT kernel is given by KA (u, t) =

1



b= / 0

−∞

e

j

d

2b

u2 − 1 ut+ a t 2 b

(15)

, b=0

 (16)

2b

j2b

with parameter matrix A = (a, b ; c, d) and the parameters a, b, c, d are real numbers satisfying ad − bc = 1. Owing to (15), the LCT is essentially a scaling and chirp multiplication operations when b = 0, and then it is of no particular interest to / 0 in this paper. The LCT has many essential and useful properties our research. Therefore, we merely discuss the LCT in the case of b = [33,34], for example the additivity, the reversibility, and the time, frequency and time-frequency shift properties. Here, we list them in the following. The additivity of LCT: Let A = A2 A1 , thus LA2 [LA1 [f ]](u) = LA [f ](u). The reversibility of LCT: A signal f(t) can be expressed by the LCT of FA (u) with parameter matrix LA−1 [LA [f ]](t) = f (t).

(17) A−1

= (d, − b ; − c, a), that is, (18)

The time shift property of LCT: The LCT of f¯ (t) = f (t − t0 ) has the form j LA [f¯ ](u) = e



ct 0 u− ac t2 2



LA [f ](u − at 0 ).

0

(19)

f (t) = f (t)eju0 t can be expressed as The frequency shift property of LCT: The LCT of  LA [ f ](u) = e



j du0 u− bd u2 2 0



LA [f ](u − bu0 ).

(20)

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Z.-C. Zhang / Optik 127 (2016) 4995–5012

f (t) = f (t − t0 )eju0 t takes the form The time-frequency shift property of LCT: The LCT of −j LA [ f ](u) = ej(ct 0 +du0 )u e

 ac

t 2 +bct 0 u0 + bd u2 2 0 2 0



LA [f ](u − at 0 − bu0 ).

(21)

3. New WD and AF associated with the LCT In this section, unlike the previously generalized definitions of the WD and AF a kind of novel WD and AF associated with the LCT is deduced by virtue of generalizing the FT kernels to the LCT kernels in a different way. These new definitions generalize very nicely and simply the classical results for the WD and AF, including the marginal properties and affine relations associated with the LCT. Then some main properties, and the relations with other traditional time-frequency analysis tools of the newly defined WD and AF are derived. 3.1. Definition of the new WD and AF Thanks to (1) and (2), we obtain different expressions for the WD and AF as follows





+∞

=

WDf (t, u)

f

t+

−∞





+∞

=

f −∞



 t+ 2



+∞

=

 2

fu t + −∞





 2

f∗ t −



 

e−ju d





   ju 2 f∗ t −  e 2

−ju t+

e

 2





fu∗ t −

 2



t−

 2 d

(22)

 d,

where fu (t) = f (t)e−jut , and

 Af f (, u)

(23)



+∞

=

f

t+

−∞





+∞

=

f −∞



+∞

= −∞

t+



 2



  2

 f¯u t + 2



f∗ t − u e 2



−j



 2



t+

e−jut dt



   −j u 2 f∗ t −  e 2 2



t−



 2 dt

(24)

  fu∗ t −  dt, 2

where u

f¯u (t) = f (t)e−j 2 t ,

(25)

fu (t) = f (t)e

(26)

ut j2

.

By substituting the kernels of FT shown in (23), (25) and (26) with the kernels of LCT, we get fu,A (t)

= f (t)KA (u, t)

=

f¯u,A (t)

1



f (t)e

d 2 1 a 2 u − ut + t 2b b 2b

 (27) ,

j2b

= f (t)KA

= and



 j

u 

1

2

,t

 j

f (t)e

a 2 d 2 1u t+ u − t 8b b2 2b



(28)

j2b

  fu,A (t) = f (t)KA − u , t 2

=



1 j2b

 j

f (t)e

d 2 1u a 2 u + t t+ 8b b2 2b



(29) .

Z.-C. Zhang / Optik 127 (2016) 4995–5012

4999

Thus, we derive a kind of new WD associated with the LCT by taking place fu (t) shown in (22) with fu,A (t), i.e.,



WDAf (t, u)



+∞

=

 2

fu,A t + −∞



1 2|b|

=



+∞

f





 2

∗ fu,A t−

 

t+

2

−∞

 d





 

f∗ t −

2

−

j

e

a 1 u + t b b



(30) d.

fu (t) shown in (24) with f¯u,A (t) and Similarly, a kind of new AF associated with the LCT can be derived through taking place f¯u (t) and 

fu,A (t), respectively, that is,  +∞  AF Af (, u)

 f¯u,A t + 2

=

−∞

1 2|b|

=





+∞

f



  ∗ fu,A t− dt 2

t+

−∞

  2





 

f∗ t −

2

−

j

e

1 a ut + t b b



(31) dt.

With the above two equations, we have the following definition. Definition 1.

The new WD and AF associated with the LCT for the parameter matrix A = (a, b ; c, d) are defined as

WDLC Af (t, u) = and



+∞

f

 2

t+

−∞



1 2|b|

AFLC Af (, u) =



1 2|b|



+∞

f

 2

t+

−∞







f∗ t −



 2

 2

f∗ t −

  e

j − 1 u+ a t b

  e



d

b

j − 1 ut+ a t b

b

(32)

 dt.

(33)

In order to make different from the previous results about the WD and AF associated with the LCT, we denote the new WD and AF as f f WDLC A (t, u) and AFLC A (, u) and simplify them as the WDLC and AFLC, respectively. Then the relationship between the WDLC and AFLC is WDLC Af (t, u) = Proof.

1 2|b|



+∞ −∞

AFLC Af (, v)e

j 1 (vt−u) b

dvd.

(34)

From (32) and (33), we obtain 1 2|b|

=



−∞

1 j (vt − u) AFLC Af (, v)e b dvd

 

1

f

2

1 = 2|b|

1 2|b|



+∞

42 |b|

1 = 2|b|

=

+∞

ε+

−∞





+∞

f

ε+

−∞





+∞

f −∞





+∞

f

  2

 ε+ 2

t+

−∞



  2

f

f

  2

∗

∗







f∗ ε−

2

 ε−

  2

 ε− 2



f∗ t −

  2

−

j

e

 

−

j

e

 j

e



 

−

j

e



−

a 1 vε + ε b b

a 1 u + ε b b

1 a u + ε b b

1 a u + t b b



1 j (vt − u) e b dεdvd

 1 2|b|





+∞

1 (t − ε)v e b dv j

 dεd

−∞

(35)

ı(t − ε)dεd

 d

= WDLC Af (t, u). Thus, the proof of (34) is completed. This relation indicates that the WDLC is a two-dimensional FT of the AFLC, simply differing in a 1 constant 2|b| .䊐 Moreover, relationships between the WDLC and WD as well as the AFLC and AF are given by WDLC Af (t, u) =

1 WDf 2|b|

and AFLC Af (, u) =

1 AF 2|b| f

 1 t,

 1 ,

b

b

u−

u−

a t b



(36)



a  . b

(37)

5000

Z.-C. Zhang / Optik 127 (2016) 4995–5012

Because of (36) and (37), the WDLC and AFLC, respectively, reduces to the WD and AF if the parameter matrix has a special form A = (0, 1 ; −1, 0), i.e., WDLC Af (t, u) = AFLC Af (, u) =

1 WDf (t, u), 2

(38)

1 AF (, u). 2 f

(39)

3.2. Properties of the WDLC and AFLC Since the WDLC and AFLC are the generalization of the WD and AF, respectively, and then it is interesting and meaningful to extend some important properties of the WD and AF to those of the WDLC and AFLC, such as the conjugation symmetry property, the marginal properties, the shift properties, the energy distribution and correlation properties, and the Moyal formula. Meanwhile, detailed proofs are given for some complicated properties in this subsection. Conjugation symmetry property: The WDLC and AFLC of f* (t) and f (t) = f(− t) have the forms ¯

¯

WDLC Af∗ (t, u) = WDLC Af (t, u),

AFLC Af∗ (, u) = AFLC Af (−, −u),

WDLC Af (t, u) = WDLC Af (−t, −u),

AFLC Af (, u) = AFLC Af (−, −u),

where A¯ = (a, −b; −c, d). Moreover, the relation



(40)

∗

WDLC Af (t, u) = WDLC Af (t, u) ,



(41)

∗

AFLC Af (, u) = AFLC Af (−, −u)

(42)

holds. Time and time delay marginal properties: The time marginal property of WDLC and the time delay marginal property of AFLC have the forms



+∞

−∞



2

WDLC Af (t, u)du = |f (t)| ,

+∞

−∞

Proof.

AFLC Af (, u)du = f

  2

(43)

 

f∗ −

2

.

(44)

Based on (32) and (33), we have



+∞

−∞

WDLC Af (t, u)du

=

1 2|b|



  f





+∞

f

 t+ 2 t+

−∞

= |f (t)|

t+

−∞

−∞

=



+∞

f

+∞

=



 2

 f



∗



 2





f∗ t −

 t− 2



f∗ t −

 2

 2



j

e

  j a t 1 b e

+∞

−∞

AFLC Af (, u)du

=

1 2|b|



 j a t eb



ddu



1 u e b du −j

d

−∞

(45)

ı()d

f



+∞

f



+∞

=

f −∞

  2

 

+∞

t+

−∞

−∞

=f

+∞



2

=





2|b|

and



1 a − u + t b b

 t+ 2 t+

 2

 f



 

f∗ −

2

∗



  2



f∗ t −

 t− 2



f∗ t −

 2

  2

j

−

e

  j a t 1 b e

 j a t eb

a 1 ut + t b b

2|b|



+∞

 dtdu



1 ut e b du −j

dt

−∞

(46)

ı(t)dt

.

Thus, the proofs of (43) and (44) are completed. 䊐 Remark 1. The time marginal property of WDLC given by (43) and time delay marginal property of AFLC given by (44), respectively, exactly preserves corresponding results for the WD and AF. Linear canonical marginal property: The linear canonical marginal property of the WDLC and AFLC are given by



+∞

−∞

WDLC Af (t, u)dt = |FA (u)|2

(47)

Z.-C. Zhang / Optik 127 (2016) 4995–5012

and



+∞

−∞

Proof.

AFLC Af (, u)d = FA

u 2

 u

FA∗ −

2

5001

.

(48)

According to (32), we obtain



+∞

−∞

WDLC Af (t, u)dt

1 = 2|b|





+∞

f −∞

 2

By making the change of variables x = t +



+∞

−∞

WDLC Af (t, u)dt

1 2|b|

=



=



+∞

 t+ 2

1

f



 t− 2



j −

f (x)f ∗ (y)e



+∞

j

f (x)e

j2b

∗

  e

j − 1 u+ a t b

 ddt.

b

and y = t − 2 , (49) turns into

−∞







1 a 2 u(x − y) + (x − y2 ) b 2b

a 2 d 2 1 u − ux + x 2b b 2b

(49)

dxdy

 dx ×



−∞



1 −j2b

 +∞

−j ∗

f (y)e

a 2 d 2 1 u − uy + y 2b b 2b



(50) dy

−∞

= |FA (u)|2 . Then, the proof of (47) is completed. As for (48), we have from (33)



+∞

−∞

1 2|b|

AFLC Af (, u)d =





+∞

f −∞

By making the change of variables x = t +



+∞

−∞

AFLC Af (, u)d

1 2|b|

=



=

= FA



+∞

 2

1

2

 2

  e

j − 1 ut+ a t b

 dtd.

b



j

f (x)f ∗ (y)e



+∞

j

f (x)e

j2b

u



f∗ t −

and y = t − 2 , (51) becomes

−∞





 2

t+

FA −

2



1u a 2 (x − y2 ) − (x + y) + b2 2b

d 2 1u a 2 x+ u − x 8b b2 2b

dxdy

 dx 

−∞

 u ∗

(51)



1 −j2b

 −j

+∞

∗

f (y)e

d 2 1u a 2 y+ u + y 8b b2 2b

 (52) dy

−∞

.

Thus, the proof of this property is completed.䊐 Remark 2. As shown in (47) and (48), the linear canonical marginal property of the WDLC and AFLC generalize very nicely and simply the frequency marginal property of the WD and AF, respectively. With Remark 1 and this remark, the marginal properties of the WDLC and AFLC have the elegance and simplicity comparable to those of the WD and AF. Time shift property: The WDLC and AFLC of f¯ (t) = f (t − t0 ) can be expressed as WDLC Af¯ (t, u) = WDLC Af (t − t0 , u − at 0 ) and AFLC Af¯ (, u) = ej Proof.



− 1 ut 0 + a t0 b



b

(53)

AFLC Af (, u).

(54)

Due to (32), the relation

WDLC Af¯ (t, u)

=

1 2|b|

1 = 2|b|



 

+∞

f −∞





+∞

f −∞

t+







  − t0 f ∗ t − − t0 e 2 2







  t + − t0 f ∗ t − − t0 e 2 2

−

j

 j −

1 a u + t b b

 d

a 1 (u − at 0 ) + (t − t0 ) b b



(55) d

= WDLC Af (t − t0 , u − at 0 ) holds. Thus, the proof of (53) is completed. As for (54), we have from (33) f¯ AF A (, u)

1 = 2|b|





+∞

f −∞

t+





 

  j − t0 f ∗ t − − t0 e 2 2

− 1 ut+ a t b

b

 dt.

(56)

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Z.-C. Zhang / Optik 127 (2016) 4995–5012

By making the change of variable ε = t − t0 , the above equation turns into

AFLC Af¯ (, u)

=

1 2|b|



=e



f

ε+

−∞

a 1 ut 0 + t0 b b

−

j



+∞

1 a ut 0 + t0 b b

−

j





=e



 2







 2

f∗ ε−

1 2|b|





j −

e

 

+∞



a 1 u(ε + t0 ) + (ε + t0 ) b b

f

ε+

−∞

  2



f∗ ε−

  2



−

j

dε

e

1 a uε + ε b b

(57) dε

AFLC Af (, u).

Thus, the proof of this property is completed. 䊐 Frequency shifting property: The WDLC and AFLC of  f (t) = f (t)eju0 t can be presented by WDLC A (t, u) = WDLC Af (t, u − bu0 )

(58)

AFLC A (, u) = eju0  AFLC Af (, u).

(59)

f

and

f

Proof.

Owing to (32) and (33), the relations

WDLC A (t, u)

f

=

1 2|b|

1 = 2|b|





+∞

f

t+

−∞





+∞

f

t+

−∞

  2

 e

ju0



t+

   −ju0 2 f∗ t −  e 2



  2

f

∗



t−

  2

j −

e







t−

 j 2 e

1 a (u − bu0 ) + t b b

−

1 a u + t b b

 d



(60) d

= WDLC Af (t, u − bu0 ) and

AFLC A (, u)

f

=

1 2|b|





+∞

f

t+

−∞

 2



 e

ju0



   −ju0 2 f∗ t −  e 2

t+





t−

 j 2 e

 −

a 1 ut + t b b

 dt

(61)

= eju0  AFLC Af (, u) hold. Thus, the proofs of (58) and (59) are completed. 䊐 Energy distribution and correlation properties: The energy distribution property of WDLC and the correlation property of AFLC take the forms



−∞

and





+∞

WDLC Af (t, u)dtdu

=

−∞

AFLC Af (, u)ddu = 2

2

+∞

|f (t)| dt = −∞



+∞



+∞

|FA (u)|2 du

(62)

−∞

+∞

 f ()f ∗ (−)d = 2

−∞

+∞

−∞

FA (u)FA∗ (−u)du.

(63)

Remark 3. The energy distribution property of WDLC given by (62) can be easily derived from (43) and (47). Also, it is easy to obtain the correlation property of AFLC shown in (63) by use of (44) and (48). Moyal formula: The Moyal formula of the WDLC and AFLC can be presented as





+∞ −∞



∗ WDLC Ag (t, u) dtdu

WDLC Af (t, u)



1  = 2|b| 

+∞

−∞

2  1 2 |f, g| f (x)g (x)dx = 2|b| ∗

(64)

and



+∞ −∞



AFLC Af (, u)



∗ AFLC Ag (, u) ddu



1  = 2|b| 

+∞

−∞

2  1 2 f (x)g (x)dx = |f, g| . 2|b| ∗

(65)

Z.-C. Zhang / Optik 127 (2016) 4995–5012

Proof.

5003

Thanks to (32), the relation

 +∞ −∞



∗

WDLC Af (t, u) WDLC Ag (t, u)

dtdu

=

=

=

=

 

1

f

42 |b| 1 2|b| 1 2|b| 1 2|b|



WDLC Af (t, u) −∞



−∞

 



+∞

f

t+

−∞



+∞

f



ε 2

t+

1 a j (ε − )u j t( − ε)  ∗  ∗ ε  ε e b ddεdtdu f t− g t+ g t− ×e b 2 2 2 2

 a  ∗  ∗ ε  ε  j t( − ε) 1 × f t− g t+ g t− e b 2 2 2 2 2|b|  ∗ 2

f

t−

 ∗ 2

g

t+

ε 

ε

2

2

g t−



+∞

 1 j (ε − )u du ddεdt e b

−∞

a t( − ε) ı(ε − )ddεdt e b j

ε ∗ ε ∗ ε  ε f t− g t+ g t− dεdt 2 2 2 2

t+

−∞

(66)

and y = t − 2ε , (66) becomes



1  2|b| 

+∞

f (x)g ∗ (x)f ∗ (y)g(y)dxdy

−∞



=

t+

−∞

 

1 = 2|b|

∗ WDLC Ag (t, u) dtdu



+∞

f

holds. By making the change of variables x = t +

 +∞



+∞

2

+∞

−∞

2 

f (x)g ∗ (x)dx

(67)

1 2 |f, g| . = 2|b| Thus, the proof of (64) is completed. Similarly, we can obtain (65) on the basis of (33) and the same variable substitutions. 䊐 3.3. Relationships between the WDLC (AFLC) and other time-frequency analysis tools In this subsection, we investigate the relationships between the WDLC (AFLC) and some classical time-frequency representations, including the WD and AF, the LCT, the short-time Fourier transform (STFT) [35,36], and the short-time linear canonical transform (STLCT) [14,17]. 3.3.1. Relationships between the WDLC (AFLC) and the WD (AF) Theorem 1.

a

WDLC Af¯ (t, u) = AFLC Af¯ (, u) = Proof.

2

−j t Assume f¯ (t) = f (t)e 2b , then the WDLC and AFLC of f¯ (t), respectively, can be expressed by the WD and AF of f(t) as follows:

1 WDf 2|b|

 u t,

b

 u

1 AF 2|b| f

,

b

(68)

,

.

(69)

We derive from (32) and (33) that

WDLC Af¯ (t, u)

=

1 2|b|

1 = 2|b|





+∞   −j a t +   2 f t+ e 2b 2 −∞





+∞

f −∞

1 WDf = 2|b|

 t+ 2

 f

∗



 u t,

 t− 2



2



  j a t−  2 f∗ t − e 2b 2



2

j

e

=

1 2|b|

=

1 2|b|



 f

t+

−∞

1 AF = 2|b| f

 u ,

b

d

1 u e b d

+∞   −j a t +   2 f t+ e 2b 2 −∞ +∞



(70)

b

 

1 a u + t b b

−j

and

AFLC Af¯ (, u)

−

 2





f∗ t −

 2

.

Thus, the proofs of (68) and (69) are completed. 䊐



2



  j a t−  ∗ 2 f t− e 2b 2

1 −j ut e b dt



2

j

e

1 a − ut + t b b

 dt (71)

5004

Z.-C. Zhang / Optik 127 (2016) 4995–5012

Remark 4. By contrasting (68) and (69) with (36) and (37), Theorem 1 presents another version of relationships between the WDLC and WD as well as the AFLC and AF. Similarly, when the parameter matrix becomes A = (0, 1 ; −1, 0), (68) and (69) reduce to (38) and (39), respectively. 3.3.2. Relationships between the WDLC (AFLC) and the LCT Theorem 2. Assume FA1 (u) is the LCT of f(t) with parameter matrix A1 = (a1 , b1 ; c1 , d1 ), then the WDLC and AFLC of FA1 (t), respectively, can be expressed by the WDLC and AFLC of f(t) as follows: WDLC AFA (t, u) = WDLC Af ( dt − bu, − ct + au),

(72)

AFLC AFA (, u) = AFLC Af ( c + au), d − bu, −

(73)

1

1

where the parameters take a = a1 − ba b1 , b= Proof.

b1 , b

c = aa1 + bc1 − ad1 − ab b1 , and d = d1 + ba b1 , satisfying a d − b c = 1. 2

Owing to (5) and (6), the following relations

WDFA (t, u) = WDf (d1 t − b1 u, −c1 t + a1 u)

(74)

AF FA (, u) = AF f (d1  − b1 u, −c1  + a1 u)

(75)

1

and 1

hold. Based on (36) and (37), we have WDLC AFA (t, u) = 1

AFLC AFA (, u) = 1

 1

1 WDFA 1 2|b|

1 AF FA 1 2|b|

t,

 1 ,

b

b

u−

u−

1

=



= where a 1 ω− b b and then





ω=−

d1 +



1 WDf d1 t − b1 2|b| WDLC Af





d1 +

1

 a

b

b1

b

a = a1 − ba b1 , b= If we denote

b1 , b



a t , −c1 t + a1 b



1 b

u−

a t b

 (78)

b1 u, ω , t− b





(77)

u−

a b1 b1 t − u = −c1 t + a1 b b

a2 b1 aa1 + bc 1 − ad1 − b

(76)

a  . b

With (36), (74) and (76), we obtain WDLC AFA (t, u)



a t , b



t + a1 −

1 b

u−



a t , b

(79)



a b1 u. b

(80)

c = aa1 + bc1 − ad1 − ab b1 , and d = d1 + ba b1 , we derive from (78) and (80) 2

WDLC AFA (t, u) = WDLC Af ( dt − bu, − ct + au), 1

where the parameters satisfy

a d − b c = a1 d1 − b1 c1 = 1.

(81)

Then, the proof of (72) is completed. Similarly, with (37), (75) and (77) the relation (73) holds. Thus, the proof of this theorem is completed. 䊐 Remark 5. Theorem 2 shows that the LCT is equivalent to an affine transformation for the WDLC and AFLC, which generalizes very nicely and simply the classical result for the WD and AF shown in (5) and (6). Therefore, the WDLC and AFLC applied to the designing of multiplicative filters in the LCT domain for the separation of multi-component time-frequency signals are very useful and effective, as discussed in [13]. When the parameter matrix becomes A = (0, 1 ; −1, 0), (72) and (73), respectively, reduces to (74) and (75). In addition, when the parameter matrix has a special form A1 = A−1 = (d, − b ; − c, a), we obtain an interesting result as shown in the following corollary. Assume FA−1 (u) is the LCT of f(t) with parameter matrix A−1 = (d, − b ; − c, a), then the relations are established as follows:

Corollary 1.

−1

WDLC AF −1 (t, u) = WDLC Af (u, t), A

−1

AFLC AF −1 (, u) = AFLC Af (u, ). A

(82) (83)

Z.-C. Zhang / Optik 127 (2016) 4995–5012

Proof.

5005

b = −1, d = 0, and then the relations Let A1 = A−1 = (d, − b ; − c, a) in Theorem 2, we have a = a + d, c = 1, and

WDLC AF −1 (t, u) = WDLC Af [u, −t + (a + d)u]

(84)

AFLC AF −1 (, u) = AFLC Af [u, − + (a + d)u]

(85)

A

and A

hold. According to (32) and (33), the above two equations turn into WDLC AF −1 (t, u) A

= WDLC Af [u, −t + (a + d)u]

=

=

1 2|b|

1 2|b|





+∞

f

u+

−∞





+∞

f

u+

−∞

 2

 2





 

f∗ u−



f∗ u−

 2

 2



1 a − (−t + au + du) + u b b

j

e

 

j

e

1 d t − u b b

 d



(86) d

−1

= WDLC Af (u, t) and AFLC AF −1 (, u) A

= AFLC Af [u, − + (a + d)u]

=

=



1 2|b|



+∞

f

t+

−∞



1 2|b|



+∞

f

t+

−∞

 u 2

 u 2

 

f∗ t −



f∗ t −

 u 2

j −

e



 u 2

j

e

a 1 (− + au + du)t + ut b b

1 d t − ut b b

 dt



(87) dt

−1

= AFLC Af (u, ). Thus, the proof of this corollary is completed. 䊐 Remark 6. Corollary 1 reveals that the WDLC and AFLC of FA−1 (t) with parameter matrix A, respectively, is equal to the WDLC and AFLC of f(t) with parameter matrix A−1 , simply differing in variables exchanging. 3.3.3. Relationships between the WDLC (AFLC) and the STFT The STFT of a signal f(t) is defined as [35,36]



+∞

Sf (t, u) =

f ()g ∗ ( − t)e−ju d,

(88)

−∞

where g(t) is the window function. Then we obtain a relationship between the WDLC and STFT in the following theorem. Theorem 3. i.e.,

The WDLC of a signal f(t) with parameter matrix A can be expressed by the STFT of f(t) added with window function g(t) = f(− t),

WDLC Af Proof.

t b ,

2 2

u+

a t 2



1 j 1 tu e 2 Sf (t, u). |b|

=

(89)

From (32), we have by means of making the change of variable ε = t +

WDLC Af (t, u) =

1 |b|



+∞



f (ε)f ∗ (2t − ε)e



j − 2 u(ε−t)+ 2a t(ε−t) b

WDLC f

 a

, u+ t 2 2 2

dε,

(90)

−∞

and then

t b A

b

 2

=

1 |b|



 +∞

j −

f (ε)f ∗ (t − ε)e

2 b

b 2

u+

a t 2



ε−

t 2



+



a t t ε− 2 b

−∞

1  +∞ 1 j tu e2 f (ε)f ∗ (t − ε)e−juε dε. = |b| −∞

 dε (91)

Meanwhile, based on (88) the STFT of f(t) with the window function g(t) = f(− t) has the form



+∞

Sf (t, u) = −∞

f (ε)f ∗ (t − ε)e−juε dε.

(92)

5006

Z.-C. Zhang / Optik 127 (2016) 4995–5012

By combining (91) with (92), we obtain the relation WDLC Af

t b ,

2 2

u+

a t 2



=

1 j 1 tu e 2 Sf (t, u), |b|

and therefore, the proof of (89) is completed. 䊐 Similarly, a relationship between the AFLC and STFT is presented as follows. Theorem 4. is,

The AFLC of a signal f(t) with parameter matrix A can be expressed by the STFT of f(t) added with window function g(t) = f(t), that 1 1 ej 2 u Sf (, u). 2|b|

AFLC Af (, bu + a) = Proof.

(93)

From (33), we have by virtue of making the change of variable ε = t +

AFLC Af (, u) =



1 2|b|

+∞



f (ε)f ∗ (ε − )e

j − 1 u(ε− 2 )+ a  (ε− 2 ) b

b



 2

dε,

(94)

−∞

and then

AFLC Af (, bu + a)

=

=

1 2|b| 1 2|b|





+∞



1  (bu + a) ε − 2 b

j −

f (ε)f ∗ (ε − )e



+



a   ε− 2 b

 dε

−∞

 1 j u e2

(95) +∞

f (ε)f ∗ (ε − )e−juε dε.

−∞

Meanwhile, due to (88) the STFT of f() with the window function g() = f() can be expressed as



+∞

Sf (, u) =

f (ε)f ∗ (ε − )e−juε dε.

(96)

−∞

With (95) and (96), we derive the relation 1 1 ej 2 u Sf (, u), 2|b|

AFLC Af (, bu + a) =

and therefore, the proof of (93) is completed. 䊐 3.3.4. Relationships between the WDLC (AFLC) and the STLCT The STLCT [14,17], which substitutes the FT kernel with the LCT kernel, is a generalization of the STFT. It takes the form SfA (t, u)

=





1

j2b

+∞

f ()g ∗ ( − t)e

j

d

2b

u2 − 1 u+ a  2 b

2b



d,

(97)

−∞

where g(t) stands for the window function. Then we obtain a relationship between the WDLC and STLCT in the following theorem. Theorem 5.

The WDLC of a signal f(t) with parameter matrix A can be expressed by the STLCT of f(t) with parameter matrix A added with

window function g(t) = f (−t)e WDLC Af Proof.

 t u ,

2 2



=

j a t2 2b

, i.e.,

2 j e −jb

1

2b

tu− d u2



2b

SfA (t, u).

The relation (90) can be rewritten as

 t u A

WDLC f

, 2 2

=

=

1 |b|



1 e |b|

 =

+∞





f (ε)f ∗ (t − ε)e

1 d 2 tu − u 2b 2b

2 e −jb

 j

−

j

−∞

j

(98)





1 a a 2 1 uε + ut + tε − t b 2b b 2b

 dε

 +∞

a j (ε − t)2 −j f (ε)f ∗ (t − ε)e 2b e

d 2 1 a 2 u − uε + ε 2b b 2b

−∞

1 d 2 tu − u 2b 2b





1 j2b



 +∞

a j (ε − t)2 −j f (ε)f ∗ (t − ε)e 2b e



d 2 1 a 2 u − uε + ε 2b b 2b

SfA (t, u) =



1

j2b

+∞

−∞

f (ε)f ∗ (t − ε)e

−j a (ε−t)2 j 2b

e

d

 dε.

−∞

According to (97), the STLCT of f(t) with parameter matrix A related to the window function g(t) = f (−t)e



(99)

dε

u2 − 1 uε+ a ε2 2b b 2b



dε.

j a t2 2b

can be expressed as (100)

Z.-C. Zhang / Optik 127 (2016) 4995–5012

5007

By combining (99) with (100), we have the relation WDLC Af

 t u ,

2 2

 =

2 j e −jb

1 2b

tu− d u2

 SfA (t, u).

2b

Thus, the proof of (98) is completed. 䊐 Similarly, a relationship between the AFLC and STLCT is presented as follows. Theorem 6. The AFLC of a signal f(t) with parameter matrix A can be expressed by the STLCT of f(t) with parameter matrix A added with window function g(t) = f (t)e AFLC Af (, u) =

Proof.

j a t2 2b

, that is, 1



−j2b

e

j

1 2b

u− d u2

 SfA (, u).

2b

(101)

The relation (94) can be rewritten as

AFLC Af (, u)

=

=

=

1 2|b|





1 e 2|b|

+∞

−∞

 j

1 d 2 u − u 2b 2b

 j

1



j

f (ε)f ∗ (ε − )e

−j2b

e





1 a a 2 1 u + ε −  − uε + b 2b b 2b

 dε

 a j +∞ −j (ε − )2 f (ε)f ∗ (ε − )e 2b e

a 2 d 2 1 u − uε + ε 2b b 2b

−∞

d 2 1 u − u 2b 2b





1 j2b



 a j +∞ (ε − )2 −j f (ε)f ∗ (ε − )e 2b e



a 2 d 2 1 u − uε + ε 2b b 2b





1 j2b

+∞



f (ε)f ∗ (ε − )e

 dε.

−∞

Based on (97), the STLCT of f() with parameter matrix A related to the window function g() = f ()e SfA (, u) =

(102)

dε

d 2 1 a 2 −j a (ε−)2 j 2b u − b uε+ 2b ε 2b e

j a 2 2b

can be expressed as

 dε.

(103)

−∞

With (102) and (103), we derive the relation AFLC Af (, u) =



1 −j2b

e

j

1 2b

u− d u2



2b

SfA (, u).

Thus, the proof of (101) is completed. 䊐

4. Applications of the WDLC and AFLC In this section, we use the WDLC and AFLC to detect the LFM signals and design the filters in the LCT domain with the aim of showing the advantage of the theory.

4.1. Detection of the LFM signals The LFM signals are a kind of important non-stationary signals which are widely applied in the communications, radar, and sonar systems [26,27]. Then the detection of LFM signals is one of the most important research topics in the engineering. Here, we show that the WDLC and AFLC can be used to the detection of the LFM signals. An one-component LFM signal is chosen as 2

f (t) = ej(k1 t+k2 t ) (k2 = / 0),

(104)

where k1 and k2 represent the initial frequency and frequency rate of f(t), respectively. Then, we obtain the WDLC and AFLC of f(t) as shown in the following two theorems. Theorem 7.

2

The WDLC of f (t) = ej(k1 t+k2 t ) (k2 = / 0) has the form

WDLC Af (t, u) = ı[bk1 + (2k2 b + a)t − u].

(105)

5008

Z.-C. Zhang / Optik 127 (2016) 4995–5012

Proof.

Due to (32), we have

WDLC Af (t, u)

=

=

=

1 2|b|

1 2|b|

1 2|b|







+∞

f −∞



 

t+

 

+∞ j k1

2

e −∞





f∗ t −

 2

t+







a b

e

2

 +k2

+∞ j k1 + 2k2 +

 

t+

−

j

e

a 1 u + t b b

2 

 2

  −j k1

t−

e

 2

 d



 +k2

t−

 2

2   1 j

−

e

b

u +

a t b

 d

(106)



 t−

1 u b



d

−∞

= ı[bk1 + (2k2 b + a)t − u], and then the proof of (105) is completed. 䊐 Theorem 8.

2

The AFLC of f (t) = ej(k1 t+k2 t ) (k2 = / 0) can be presented as

AFLC Af (, u) = ejk1  ı[(2k2 b + a) − u]. Proof.

(107)

Owing to (33), we derive

AFLC Af (, u)

=

=

1 2|b|

1 2|b|







+∞

f −∞



t+

 

+∞ j k1

  2

e −∞

1 = ejk1  2|b|



f∗ t −

 2

t+





2k2 +

e

2

 +k2

 +∞ j

 

a b

t+

 2

−

j

e

1 a ut + t b b

2 

  −j k1

t−

e



 2

dt



 +k2

t−

 2

2   1 j

−

e

b

ut +

a t b

 dt

(108)



 −

1 u b

t

dt

−∞

= ejk1  ı[(2k2 b + a) − u], and then the proof of (107) is completed. 䊐 From Theorems 7 and 8, we can conclude that the WDLC and AFLC of a single component LFM signal are able to generate impulses at a straight line u − (2k2 b + a)t − bk1 = 0 in (t, u) plane and u − (2k2 b + a) = 0 in (, u) plane, respectively. Therefore, the WDLC and AFLC applied to the detection of one-component LFM signals are very useful and effective. Furthermore, since the bi-component LFM signal can be expressed by the summation of two single component LFM signals, i.e., f (t) = g(t) + h(t),

(109)

2

2

/ 0), h(t) = ej(m2 t+ω2 t ) (ω2 = / 0), and ω1 = / ω2 , and hence the WDLC and AFLC of can be similarly deduced as where g(t) = ej(m1 t+ω1 t ) (ω1 = follows: 2 2 The WDLC of f (t) = g(t) + h(t) = ej(m1 t+ω1 t ) + ej(m2 t+ω2 t ) has the form WDLC Af (t, u)

=

1 2|b|

1 = 2|b| g



 

+∞

f

t+

−∞



+∞

 

  2

g t+

−∞

f∗ t −

  2





  2

+h t+ g,h

j

e

−

a 1 u + t b b

 d

    

 

2

2

g t−



+h t−

∗  2

j

e

−

1 a u + t b b



(110) d

h,g

= WA (t, u) + WAh (t, u) + WA (t, u) + WA (t, u) = ı[bm1 + (2ω1 b + a)t − u] + ı[bm2 + (2ω2 b + a)t − u] + WDLC Ag,h (t, u) + WDLC Ah,g (t, u),

Z.-C. Zhang / Optik 127 (2016) 4995–5012

5009

in which the first two terms stand for the auto terms, whereas the rest represent the cross terms that are given by

WDLC Ag,h (t, u)

=

=

=

=

WDLC Ah,g (t, u)

=

=

=





1 2|b|



+∞

 2

g t+ −∞



1 2|b|

 

+∞ j m1



 2

t+

e

1 |b|



1 2|b|



+∞

 

h t+ −∞



1 2|b|

 

+∞ j m2

2

1 = 2|b|

t+



+∞

  −j m2

e

+∞ ω1 − ω2 2 j  j 4 e e

−∞



d



 2

t−





g∗ t −

 2

t+

e



 

e

 +ω2

  −j m1

t−

e

ω2 − ω1 2 j  j 4 e e

−∞



+ ej(m2 t+ω2 t

  2

 

f∗ t −

 

g t+

2

−∞



  2

e



d

m1 + m2 2

t+



(111)



d

2

m1 + m2 b 2

,

d



 2

 +ω1

t−

 2

2   1 −

j

e



−



a 1 u + ω1 + ω2 + b b

b



u +

t+

a t b

 d

m1 + m2 2



(112)



d

2

m1 + m2 b 2

u − (bω1 + bω2 + a)t − b2 (ω2 − ω1 )

.

 dt

   

 

2

2

g,h

b

a t b

has the form

1 a ut + t b b

g t−

e

u +







+h t+

g

−

j

2)

−

j

1 a u + ω1 + ω2 + b b

−j j 2 ej[(ω2 −ω1 )t +(m2 −m1 )t] e (ω2 − ω1 ) 2)

2   1

b2 (ω1 − ω2 )

 +∞

 2

u − (bω1 + bω2 + a)t −

−∞



t−



−

a 1 u + t b b

2 

 2

t+

−

j

2

 +ω2



−∞

2 

 2

t+



1 2 ej[(ω2 −ω1 )t +(m2 −m1 )t] 2|b|

 f

 +ω1





AFLC Af (, u) +∞

e



−j j 2 ej[(ω1 −ω2 )t +(m1 −m2 )t] e (ω1 − ω2 )

1 2|b|





 2

1 a u + t b b

−∞

The AFLC of f (t) = g(t) + h(t) = ej(m1 t+ω1 t

=



1 2 ej[(ω1 −ω2 )t +(m1 −m2 )t] 2|b|

1 = |b|





h∗ t −

−

j



+h t−

∗  2

−

j

e

1 a ut + t b b



(113) dt

h,g

= AF A (, u) + AF hA (, u) + AF A (, u) + AF A (, u) = ejm1  ı[(2ω1 b + a) − u] + ejm2  ı[(2ω2 b + a) − u] + AFLC Ag,h (, u) + AFLC Ah,g (, u), in which the first two terms stand for the auto terms, whereas the rest represent the cross terms that are given by

AFLC Ag,h (, u)

=

=

=

=

1 2|b|

1 2|b|





+∞

g t+ −∞





 

+∞ j m1

 2



t+

e



 2

h∗ t −

 2



 +ω1

t+



j

e  2

1 a − ut + t b b

2 

  −j m2

t−

e

 dt

 2



 +ω2

−∞

j 1 e 2|b|

1 2|b|



ω − ω 1 2 4

m1 + m2 2 +  2

j e (ω1 − ω2 )

j

ω − ω 1 2 4





 +∞

−∞

2 +

j 2

ej(ω1 −ω2 )t e



t−

 2

2   1



j

e

a − ut + t b b

1 a − u + ω1 + ω2 + b b



 dt

  + (m1 − m2 )

(114) t

dt 2

[u − (bω1 + bω2 + a) − b(m1 − m2 )] m1 + m2  −j 2 4b2 (ω1 − ω2 ) e ,

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Z.-C. Zhang / Optik 127 (2016) 4995–5012

AFLC Ah,g (, u)

=

=

=

1 2|b|

1 2|b|



h t+ −∞





 

+∞ j m2

  2 t+

e



 

g∗ t −

 2



2

 +ω2

t+

 2

−

j

e

a 1 ut + t b b

2 

  −j m1

t−

e

 dt

 2



 +ω1

−∞

j 1 e 2|b|

1 = 2|b|

 

+∞

ω − ω 2 1 4

m1 + m2 2 +  2

j j e (ω2 − ω1 )

ω − ω 2 1 4





 +∞

−∞

2 +

j 2

ej(ω2 −ω1 )t e

t−

 2

2   1 j

e

−

b

ut +



a 1 − u + ω1 + ω2 + b b





a t b

 dt

  + (m2 − m1 )

(115) t

dt 2

[u − (bω1 + bω2 + a) − b(m2 − m1 )] m1 + m2  −j 2 4b2 (ω2 − ω1 ) e .

As shown in (110) and (113), since the first two auto terms are able to generate impulses which the cross terms cannot generate, and therefore, although the existence of cross terms has a certain influence on the detection performance, but the bi-component LFM signal still can be detected. This indicates that the WDLC and AFLC are also useful and powerful for detecting bi-component LFM signals. 4.2. Designing of multiplicative filters in the LCT domain Based on the method introduced in [13], the WD and AF can be used to the separation of multi-component time-frequency signals on account of the affine transformation relationships between the LCT and WD as well as the LCT and AF. Theorem 2 shows that the affine transformation relationships between the LCT and WDLC as well as the LCT and AFLC have the elegance and simplicity comparable to those of the WD and AF. Therefore, in this subsection we show that with Theorem 2, the WDLC and AFLC also can be applied to the designing of multiplicative filters in the LCT domain for the separation of multi-component time-frequency signals. Assume a bi-component time-frequency signal f(t) comprises of two signals g(t) and h(t), and the WDLC of g(t) and h(t) have no overlapping or minimal overlapping in (t, u) plane but possess a strong coupling in the time and frequency domain. Since the LCT is equivalent to an affine operation of the WDLC, and then the WDLC of FA1 (t) can be regraded as an affine transformation for the WDLC of f(t). Therefore, the WDLC of FA1 (t) can realize decoupling in the pathognomonic LCT domain through picking and choosing an appropriate parameter matrix A1 . Thus, the bandpass filter in the LCT domain for the separation of g(t) and h(t) can be derived. Here, an example is discussed as follows: The WDLC of f(t) = g(t) + h(t) is shown in Fig. 1, where the region in right-upper-corner represents the distribution of the WDLC of g(t), and the region in left-lower-corner stands for the distribution of the WDLC of h(t). Furthermore, by taking cut lines with slope  that separate these two regions in Fig. 1, and then we obtain Fig. 2, where ti (i = 1, 2, 3) are abscissa for the intersections of the cut lines and the t axis. Then equations of the cut lines in the (t, ω) plane (distribution of the WDLC of f(t)) can be expressed as ω − (t − ti ) = 0,

i = 1, 2, 3.

(116)

Based on the affine transformation relationship given by (72), the following matrix equation



t ω



 =

d − b − c a



u

 (117)

v

holds. By combining (116) with (117), the cut lines’ equations turn into ( a +  b)v − ( c +  d)u + ti = 0,

i = 1, 2, 3,

(118)

Fig. 1. The WDLC of f(t).

Z.-C. Zhang / Optik 127 (2016) 4995–5012

5011

Fig. 2. The parallel cut lines in the WDLC of f(t).

which can be seen as corresponding forms in the (u, v) plane (distribution of the WDLC of FA1 (u)). In order to implement in decoupling, we select the parameter matrix A1 in which the parameters satisfy a +  b = 0, that is equivalent to  = u = ati =



a1 −

 a b

b1 ti = −

b1 t , b i

ab1 −ba1 . b1

i = 1, 2, 3.

In this case, (118) becomes (119)

Thus, the transfer function of multiplicative filter for the separation of g(t) and h(t) can be selected for

 b  ⎧ b1 1 ⎪ ⎨ 1, u ∈ − b t2 , − b t1 Mg (u) =   ⎪ ⎩ 0, u ∈/ − b1 t , − b1 t b

or

2

b

 b  ⎧ b1 1 ⎪ ⎨ 1, u ∈ − b t3 , − b t2 Mh (u) =  b  . ⎪ b1 1 ⎩ 0, u ∈ / − t , − t b

3

b

(120)

1

(121)

2

When the transfer function is given by (120), the signal g(t) can be separated out as follows: g(t) = LA−1 [FA1 (u)Mg (u)],

(122)

1

where A−1 = (d1 , −b1 ; −c1 , a1 ). 1 When the transfer function is given by (121), the signal f(t) can be separated out as follows: h(t) = LA−1 [FA1 (u)Mh (u)],

(123)

1

where A−1 = (d1 , −b1 ; −c1 , a1 ). 1 Similarly, we also can use the AFLC to implement in filter design in the LCT domain for the separation of multi-component time-frequency signals in view of the affine transformation relationship between the LCT and AFLC shown in (73). 5. Conclusion Focusing on two problems for the existing WD and AF associated with the LCT, this paper formulates a kind of new WD and AF through substituting the FT kernels with the LCT kernels. The newly defined WD and AF, respectively, includes the conventional WD and AF as its special case and is simplified as the WDLC and the AFLC. Then we study some important properties of the WDLC and AFLC, such as the conjugation symmetry property, the marginal properties, the shift properties, the energy distribution and correlation property, and the Moyal formula, which are the extension of corresponding properties of the WD and AF. The derived marginal properties generalize very nicely and simply those of the WD and AF. Meanwhile, we discuss the relationships between the new distributions and other common time-frequency representations, including the WD and AF, the LCT, the STFT, and the STLCT. The derived affine transformation relationships associated with the LCT have the elegance and simplicity comparable to the classical results for the WD and AF. We also investigate the applications of the WDLC and AFLC, which indicate that the WDLC and AFLC applied to the detection of LFM signals and the designing of multiplicative filters in the LCT domain are very useful and effective. Conflict of interest statement The author declares that there is no conflict of interests to this work.

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