Reversible Joint Hilbert and Linear Canonical Transform Without Distortion
Abstract
Generalized analytic signal associated with the linear canonical transform (LCT) was proposed recently [1]. However, most real signals, especially for baseband real signals, cannot be perfectly recovered from their generalized analytic signals. Therefore, in this paper, the conventional Hilbert transform (HT) and analytic signal associated with the LCT are concerned. To transform a real signal into the LCT of its HT, two integral transforms (i.e., the HT and LCT) are required. The goal of this paper is to simplify cascades of multiple integral transforms, which may be the HT, analytic signal, LCT or inverse LCT. The proposed transforms can reduce the complexity when realizing the relationships among the following six kinds of signals: a real signal, its HT and analytic signal, and the LCT of these three signals. Most importantly, all the proposed transforms are reversible and undistorted. Using the proposed transforms, several signal processing applications are discussed and show the advantages and flexibility over simply using the analytic signal or the LCT.
Analytic signal, fractional Hilbert transform, generalized analytic signal, Hilbert transform, linear canonical transform
I Introduction
The Hilbert transform (HT) is a linear operator connecting the real and imaginary parts of an analytic function. The HT plays an important role in various subjects of signal processing, image processing and optics. One of the most important subjects is the construction of analytic signals. The analytic signal (AS) of a realvalued signal is defined as
(1) 
where is the HT of ,
(2) 
Although contains only nonnegative frequencies of , one can recover from the real part of without any distortion due to the Fourier transform Hermitian property of . This explains why analytic signals are commonly used in modulation and demodulation [2, 3]. The analytic signal can also be expressed in terms of complex polar form, i.e., . Accordingly, analytic signals arise in wide signal processing applications involving amplitude envelope and phase, such as phase retrieval [4], instantaneous frequency estimations [5, 6], time delay and group delay estimations [7, 8], quadratic timefrequency distributions [8], the HilbertâHuang transform [9, 10], QRS detection from ECG [11, 12], and so on.
The LCT, first introduced in [13, 14], is a parameterized general linear integral transform. Many wellknown signal processing operations, such as the Fourier transform, the fractional Fourier transform (FRFT), the Fresnel transform, and scaling and chirp multiplication operations, are the special cases of the LCT [16, 17, 18]. The LCT is an important tool in optics because the paraxial light propagation through a firstorder optical system can be modeled by the LCT [19, 20, 16]. Besides, the LCT is very useful for filter design, radar system analysis, signal synthesis, timefrequency analysis, phase reconstruction, pattern recognition, graded index media analysis, encryption, modulation, and many other applications [21, 22, 23, 24, 25, 26].
Due to the practicality of the analytic signal over the real signal and the flexibility of the LCT over the Fourier transform, the main goal of this paper is to derive lowcomplexity, reversible and undistorted transforms which combine the analytic signal and the LCT. The HT and analytic signal associated with the LCT were first introduced in the generalization of the HT. In 1996, Lohmann et al. [27] introduced the fractional HT (FHT). Instead of applying a phase shifter in the frequency domain (i.e., a sign function) as in the conventional HT, the FHT operates in the fractional Fourier domain using the FRFT. The extensions of the FHT include discrete version of the FHT [28, 29], factional analytic signal [30], and the generalization of the FHT [31]. It has been found that the FHT is useful for image compression and edge enhancement [27, 28], and secure singlesideband (SSB) modulation [29, 31]. Another generalization of the HT, called generalized HT (GHT), was proposed by Zayed [32]. Instead of using the FRFT as in the FHT, the GHT uses a chirp function of the form . In [1], Fu and Li extended the GHT to the LCT domain, which is termed parameter Hilbert transform (PHT). The PHT uses a chirp function of the form , and thus is in fact equivalent to the GHT when . Since this paper is focused on the LCT, only the PHT is discussed in the following.
Replacing the HT in (1) by the PHT, the resulting signal is termed generalized analytic signal (GAS). If the parameter in the PHT is , the PHT is reduced to the conventional HT, and the GAS is reduced to the conventional analytic signal. If , the GAS no longer contains only the nonnegative components in the Fourier domain; and thus many properties and applications of the conventional analytic signal do not hold for the GAS. Most importantly, as , most real signals, especially for baseband real signals, cannot be recovered from their GASs without distortion. For example, consider a real signal with timefrequency distribution (TFD) as shown in Fig. 1(a), which is symmetric about the vertical axis due to . The cutoff lines in Fig. 1(b) and (c) separate the positive and negative portions of in the Fourier domain and the LCT domain, respectively. It is obvious that the conventional analytic signal shown in Fig. 1(b) contains the whole information of , and thus can be used to reconstruct perfectly. However, for the GAS depicted in Fig. 1(c), cannot be recovered losslessly. As has energy more concentrated in the baseband, the distortion will be greater.
When , the GAS is reduced to the conventional analytic signal and irrelevant to the LCT. When , the GAS is irreversible. Accordingly, the conventional analytic signal and HT associated with the LCT are considered for designing reversible and undistorted transforms. Denote as the LCT with parameter matrix , which will be introduced in the next section, and define the LCT of an arbitrary signal as ,
(3) 
Consider a real signal . For a comprehensive understanding of the analytic signal and HT associated with the LCT, all the relationships among the following six kinds of signals are investigated: , , , , and . It can be found that for some relationships, two or more integral transforms are required. For example, to obtain from , two integral transforms (i.e., the analytic signal and LCT) are required; and to obtain from , three integral transforms (i.e., the inverse LCT, HT and LCT) are used. Therefore, the main objective of this paper is to simplify cascades of multiple integral transforms into the socalled joint transforms in this paper. Using the joint transforms to realize the relationships can reduce computational complexity. Besides, all the joint transforms are reversible without any distortion. All the joint transforms are also verified by numerical simulations. These simulations show that the numerical differences between the joint transforms and the cascades of integral transforms are down to or less, which may be caused by numerical roundoff error.
Since the joint transforms are related to the analytic signal, HT and LCT, several signal processing applications of the analytic signal, HT and LCT can be extended to the joint transforms. For the joint transform combining the advantages of the analytic signal and the flexibility of the LCT, it can be expected that using the joint transform is preferred than simply using the analytic signal or the LCT.
This paper is organized as follows. Section II provides some useful integrals involving exponential functions. The definitions and some properties of the HT, analytic signal and LCT are also introduced in this section. In Section III, the definitions and derivations of all the joint transforms associated with the HT, analytic signal and LCT are presented. Some simulations are given in Section IV to verify the joint transforms and depict the advantage of them. Several signal processing applications of the joint transforms are discussed in Section V. Finally, conclusions are made in Section VI.
Ii Preliminaries
Iia Some Useful Integrals Involving Exponential Functions
It has been indicated in [33] that for ,
(4) 
Accordingly, the Fourier transform of a chirp function is also a chirp,
(5) 
where . The symbol PV, called the Cauchy principle value, is used to assign values to the improper integral in (5). In [33], it has also been shown that for ,
(6) 
where is the error function. Consider a function of form
(7) 
where denotes the unit step function. Based on (6), the inverse Fourier transform of is given by
(8) 
where is also known as the Faddeeva function [34] with input , and
(9) 
It will be shown later that and are widely used in the joint transforms. Note that PV is also used in the inverse Fourier transform of (IIA). However, throughout the rest of this paper, symbol PV is omitted to simplify formula expressions.
IiB Hilbert Transform and Analytic Signal
The Hilbert transform (HT) [35, 36] on the real line is defined as
(10) 
where denotes convolution. Taking the Fourier transform of both sides of (10) with respect to yields
(11) 
where denotes the sign function, i.e., is for , for and for . This equation implies that applying the HT twice to yields ; and thus, the inverse HT (IHT) can be written symbolically as
(12) 
A list of properties, extensions and applications of the HT have been organized in [36, 37].
For a real signal , it is well known that its Fourier transform is guaranteed to be Hermitian, i.e., . Accordingly, the nonnegative frequencies contain the whole information of . Discarding the negative frequencies of leads to a complex signal, , with Fourier transform given by
(13) 
From (11) and (13), it is obvious that can be obtained from the HT of ; that is,
(14) 
Although is complexvalued, it occupies only half bandwidth of that of . Besides, can be easily recovered from through . Since contains no negative frequencies, it is called an analytic signal.
(24) 
IiC Linear Canonical Transform
In this paper, the definition of the linear canonical transform (LCT) with four parameters [16, 38] is adopted,
(15) 
where is the LCT operator with
(16) 
Some properties of the LCT, which will be used later, are listed below:

Conjugation property [18]
The conjugate of the LCT of is equivalent to the LCT of the conjugate of , i.e.,(17) 
Equivalent expressions of the LCT ()
If , the LCT in (IIC) can be rewritten as(19) (20) For , the LCT in (IIC) can be expressed as three equivalent forms. For ease of distinguishing these LCT expressions, the original definition of the LCT with given in (IIC) is called LCT Form I, and the three equivalent forms are called LCT Forms II, III and IV, respectively:
LCT Form II (21) LCT Form III (22) LCT Form IV (23) where in (21) has been defined as in (IIA). Note that is in general not true because may be negative. The derivations of (21)(23) are presented in Appendix A.
Iii Joint Transforms Associated With the HT, Analytic Signal and LCT
In order to benefit from the advantages of the analytic signal and the flexibility of the LCT, we want to derive lowcomplexity, reversible and undistorted transforms which combine the analytic signal (or the HT) and the LCT. As discussed in the introduction, a kind of analytic signal associated with the LCT is the generalized analytic signal (GAS) [1]. However, when the parameter in the GAS is , the GAS is irreversible. When , the GAS is reduced to the conventional analytic signal and irrelevant to the LCT. Therefore, the conventional analytic signal and HT associated with the LCT are considered.
For a comprehensive understanding of the analytic signal and HT associated with the LCT, all the relationships among the following six kinds of signals are examined and illustrated in Fig. 2: , , , , , and (denote a real signal, HT of , analytic signal of , LCT of , LCT of , and LCT of , respectively). Using the operators HT, analytic signal, LCT, ILCT, real part, and imaginary part (denoted by , , , , Re, and Im, respectively), one can easily transform one of the six signals into another without any information loss. However, it can be found that two or three integral transforms are required for the relationships between and , between and , and so on. Therefore, in this section, the cascades of multiple integral transforms are simplified to our socalled joint transforms.
All possible joint transforms regarding these six kinds of signals include joint LCTAS (), joint LCTHT (), joint ASILCT (), joint HTILCT (), joint LCTHTILCT () and joint LCTconjugationILCT (). The reader is reminded that AS is abbreviated from analytic signal. The definitions of these joint transforms are listed in (24), where is used in and . Use the joint LCTAS () as an example. In order to generate from , conventionally, we have to find the analytic signal () of first, and then apply the LCT () to ; that is, . In our method, is obtained directly from by , and the intermediate is not generated. All the relationships involving multiple integral transforms can be equivalently carried out by one of the above joint transforms. The joint LCTAS and joint LCTHT can be deemed as the generalizations of the analytic signal and the HT, respectively. As where the LCT is equivalent to identity operator, the joint LCTAS is reduced to the conventional analytic signal, and the joint LCTHT is reduced to the conventional HT. In the following, the derivations of the joint transforms are presented. The case of the LCT/ILCT with is ignored since it is simply a timescaled version of multiplied by a linear chirp, i.e., .
Iiia Joint LCTAS ()
The joint LCTAS, denoted by , transforms into . It is equivalent to calculating the analytic signal (AS) of first and then generating from by the LCT,
(25) 
Alternatively, the relationship between and can be expressed as , where the Fourier transform of is . Accordingly, if , from (20), the joint LCTAS is give by
(26)  
(27) 
If , based on the LCT Form IV in (23), it follows that
(28) 
According to (7) and (IIA), formula (IIIA) can be rewritten as
(29) 
where and are defined in (IIA). The joint LCTAS is also used in the transformation from to .
IiiB Joint LCTHT ()
The joint LCTHT, denoted by , transforms into . It is equivalent to transforming into by the HT first and then transforming into by the LCT,
(30) 
The joint LCTHT can be derived from the following alternative relationship:
(31) 
Here, (19) and the LCT Form II in (21) are adopted for , while has been derived in (27) and (29). Therefore, the joint LCTHT is given by
(32) 
where is given in (IIA). The joint LCTHT is also applied to the transformation from to .
IiiC Joint ASILCT ()
The joint ASILCT, denoted by , transforms into . It is equivalent to transforming into by the ILCT first and then calculating the analytic signal (AS) of ,
(33) 
The joint ASILCT can also be expressed as
(34) 
where can be determined by according to (20) and (23). For , formula (20) implies that
(35) 
Substituting (35) into (34) results in
(36) 
For , recall LCT Form IV in (23) with , and then is given by
(37) 
Again, substituting (37) into (34), it follows that
(38) 
where has been defined in (7). From (IIA), the inverse Fourier transform of is . From the definitions in (IIA), it is obvious that , and since for any complex number . Therefore, (IIIC) can further simplify to
(39) 
When realizing the transformation from to , the joint ASILCT can also be adopted.
IiiD Joint HTILCT ()
The joint HTILCT, denoted by , transforms into . It is equivalent to transforming into by the ILCT first and then calculating the HT ,
(40) 
Alternatively, the following relationship is used,
(41) 
For , has been given in (IIIC), while can be obtained by substituting for in (IIC). Therefore, for , the joint HTILCT is given by
(42) 
Similarly, for , the in (IIIC) and the LCT Form III in (22) with replaced by lead to
(43) 
where is given in (IIA). The negative joint HTILCT, i.e., , can be used to transform into .
(50) 
IiiE Joint LCTHTILCT ()
The joint LCTHTILCT, denoted by , transforms into . It is equivalent to transforming into by the ILCT first, then calculating the HT , and finally transforming into by the LCT,
(44) 
Since , now the problem is how to determine from . It has been shown in (26) and (IIIA) that can be expressed in terms of , and (35) and (37) show that can be expressed in terms of . Accordingly, for , substituting (35) into (26) results in
(45) 
For , substituting (37) into (IIIA) leads to