Reversibility Property

Next, we examine the NMSE of the reversibility property:

NMSE =

∑

n

h_i[n]− O^MDLCT⁻¹O^MDLCT{hi[n]}²

∑

n |hi[n]|² . (5.85)

Again, let the parameters in M be uniformly distributed random numbers on the interval (−2, 2). The NMSEs resulting from 200 simulation runs are sorted in ascending order and displayed in Fig. 5.10. The CDDHFs-based method doesn’t satisfy the reversibility property perfectly. Although the proposed DLCT doesn’t has perfect additivity, it satisfies the reversibility property perfectly. In Fig. 5.10, all the NMSEs of the proposed DLCT are below 10⁻²⁵and numerically verify the proofs. With the reversibility property, it is unnecessary to develop the inverse DLCT additionally because it can be realized by the forward DLCT with M⁻¹.

At the end, comparisons between the CDDHFs-based DLCT and the proposed DLCT are summarized in Table 5.2.

50 100 150 200

Figure 5.10: Normalized mean-square errors (NMSEs) of the reversibility property for 200 different M’s. The NMSEs are sorted in ascending order. The input signals are (a) h₁[n], (b) h₂[n], (c) h₃[n] and (d) h₄[n] depicted in Fig. 5.7. The parameters in each M are uniformly distributed random numbers on the interval (−2, 2).

5.6 Conclusion

In this chapter, we develop a discrete LCT (DLCT) which is irrelevant to the sampling periods and doesn’t involve oversampling operation. The proposed DLCT is based on the well-known CM-CC-CM decomposition, which decomposes the LCT to two chirp mul-tiplications (CMs) and one chirp convolution (CC). One advantage of this decomposition over many other decompositions is no scaling operation involved because scaling oper-ation will change the sampling period or introduce interpoloper-ation error. The CM-CC-CM decomposition is invalid for B = 0. Accordingly, we modify the decomposition and the proposed DLCT fit for the B = 0 case. We also investigate special cases of the proposed DLCT. The proposed DLCT can be implemented by three discrete CMs and two FFTs (three for B = 0), which yield lower computational complexity than the previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs) [1]. The relation between the proposed DLCT and the continuous

LCT is also derived to approximate the samples of the continuous LCT. Compared with the CDDHFs-based method, the proposed method has somewhat higher approximation accuracy. Besides, simulation results show that approximate additivity property can be achieved with error as small as the CDDHFs-based method. Most importantly, the pro-posed method has perfect reversibility, which is proved mathematically and by numerical examples. With the reversibility property, the inverse transform of the proposed DLCT can be realized by the forward DLCT.

Table 5.2: Comparisons between the CDDHFs-based DLCT [1] and proposed DLCT CDDHFs-based Proposed

Sampling periods ∆x= ∆u =^√1/N ∆x= ∆u

Complexity O(N²) O(N log₂N )

Accuracy Worse Better

Additivity Approximate Approximate Reversibility Approximate Perfect

Chapter 6

Reversible Joint Hilbert and Linear Canonical Transform Without

Distortion

Generalized analytic signal associated with the linear canonical transform (LCT) was pro-posed recently [108]. However, most real signals, especially for baseband real signals, cannot be perfectly recovered from their generalized analytic signals. Therefore, in this chapter, 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 trans-forms (i.e., the HT and LCT) are required. The goal of this chapter 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.

6.1 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 process-ing, image processing and optics. One of the most important subjects is the construction of analytic signals. The analytic signal (AS) of a real-valued signal x(t) is defined as

x_A(t), A{x(t)} = x(t) + jˆx(t) (6.1)

where ˆx(t) is the HT of x(t),

ˆ

x(t), H{x(t)}. (6.2)

Although x_A(t) contains only non-negative frequencies of x(t), one can recover x(t) from the real part of x_A(t) without any distortion due to the Fourier transform Hermitian prop-erty of x(t). This explains why analytic signals are commonly used in modulation and demodulation [109, 110]. The analytic signal can also be expressed in terms of com-plex polar form, i.e., x_A(t) = a(t)e^jϕ(t). Accordingly, analytic signals arise in wide signal processing applications involving amplitude envelope and phase, such as phase retrieval [111], instantaneous frequency estimations [40, 41], time delay and group de-lay estimations [70,112], quadratic time-frequency distributions [70], the Hilbert–-Huang transform [113, 114], QRS detection from ECG [115, 116], and so on.

Due to the practicality of the analytic signal over the real signal and the flexibility of the linear canonical transform (LCT) [74, 75] over the Fourier transform, the main goal of this chapter is to derive low-complexity, 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. [117] introduced the fractional HT (FHT). Instead of applying a π/2 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 fractional Fourier transform (FRFT). The extensions of the FHT include discrete version of the FHT [118, 119], factional analytic signal [120],

image compression and edge enhancement [117, 118], and secure single-sideband (SSB) modulation [119, 121]. Another generalization of the HT, called generalized HT (GHT), was proposed by Zayed [122]. Instead of using the FRFT as in the FHT, the GHT uses a chirp function of the form e^{−jcot(α)2 t2}. In [108], Fu and Li extended the GHT to the LCT domain, which is termed parameter (a, b)-Hilbert transform (PHT). The PHT uses a chirp function of the form e^−j2bat2, and thus is in fact equivalent to the GHT when ^a_b = cot(α).

Since this chapter is focused on the LCT, only the PHT is discussed in the following.

Replacing the HT ˆx(t) in (6.1) by the PHT, the resulting signal is termed generalized analytic signal (GAS). If the parameter a in the PHT is a = 0, the PHT is reduced to the conventional HT, and the GAS is reduced to the conventional analytic signal. If a ̸= 0, the GAS no longer contains only the non-negative 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 a ̸= 0, most real signals, especially for baseband real signals, cannot be recovered from their GASs without distortion. For example, consider a real signal x(t) with time-frequency distribution (TFD) as shown in Fig. 6.1(a), which is symmetric about the vertical axis due to X(−f) = X^∗(f ). The cutoff lines in Fig. 6.1(b) and (c) separate the positive and negative portions of x(t) in the Fourier domain and the LCT domain, respectively. It is obvious that the conventional analytic signal shown in Fig. 6.1(b) contains the whole information of x(t), and thus can be used to reconstruct x(t) perfectly. However, for the GAS depicted in Fig. 6.1(c), x(t) cannot be recovered losslessly. As x(t) has energy more concentrated in the baseband, the distortion will be greater.

When a = 0, the GAS is reduced to the conventional analytic signal and irrelevant to the LCT. When a ̸= 0, the GAS is irreversible. Accordingly, the conventional ana-lytic signal and HT associated with the LCT are considered for designing reversible and undistorted transforms. DenoteL^M as the LCT with parameter matrix M , which will be introduced in the next section, and define the M -LCT of an arbitrary signal z(t) asL^Mz (ω),

L^Mz (ω), L^M{z(t)}. (6.3)

(a) (b) (c)

Figure 6.1: Time-frequency distributions of (a) a real signal x(t), (b) conventional analytic signal of x(t), and (c) generalized analytic signal of x(t) using the PHT. The cutoff lines in (b) and (c) separate the positive and negative portions of x(t) in the Fourier domain and LCT domain, respectively.

Consider a real signal x(t). 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: x(t), ˆx(t), xA(t), L^Mx (ω),L^Mxˆ (ω) andL^MxA(ω). It can be found that for some relationships, two or more integral transforms are required. For example, to obtain L^MxA(ω) from x(t), two integral transforms (i.e., the analytic signal and LCT) are required;

and to obtainL^Mxˆ (ω) fromL^Mx (ω), three integral transforms (i.e., the inverse LCT, HT and LCT) are used. Therefore, the main objective of this chapter is to simplify cascades of multiple integral transforms into the so-called joint transforms in this chapter. Using the joint transforms to realize the relationships can reduce computational complexity. Be-sides, 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 dif-ferences between the joint transforms and the cascades of integral transforms are down to 10⁻⁷ or less, which may be caused by numerical round-off 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.