Fractional Hilbert Transformer Research Articles

Microwave fractional Hilbert transformer implemented in a dispersion-tailored few-mode fiber

We experimentally demonstrate, for the first time to our knowledge, a microwave fractional Hilbert transformer in a few-mode fiber using a transversal filtering approach. The filter taps are provided by a tunable true-time delay line that is realized by exploiting the spatial dimension of a dispersion-engineered double-clad step-index few-mode fiber. Both the fractional order and operational bandwidth of the fractional Hilbert transformer can be continuously tuned by adjusting the tap coefficients and varying the operational optical wavelength, respectively. The magnitude and phase response for different fractional orders, ranging from 0.17 to 1.00 that correspond to phase shifts of 15° to 90°, are measured. Operational bandwidths of 7.4 to 10.6 GHz are demonstrated for a classical Hilbert transformer. Real-time temporal fractional Hilbert transform of a Gaussian-like pulse is also performed. Our results are in good agreement with theory, validating the viability of our approach for implementation of microwave fractional Hilbert transformers.

The Fractional Hilbert Transform of Generalized Functions

The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry to a space of generalized functions known as Boehmians. Moreover, we introduce a new fractional convolutional operator for the fractional Hilbert transform to prove a convolutional theorem similar to the classical Hilbert transform, and also to extend the fractional Hilbert transform to Boehmians. We also produce a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate the convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.

The Fractional Hilbert Transform on the Real Line

This paper examines some special properties and important results of the fractional Hilbert transform (FHT) on the real line ℝ . In this rigorous study, we modify certain theorems of classical Hilbert transform for FHT and develop new theorems. Moreover, FHT of some common functions is given and eigenfunctions are also studied. We prove that FHT, denoted by H α , is an isomorphism on L 2 ℝ . We also show that in L 2 ℝ , FHT is an isometry. Furthermore, we investigate Riesz inequality on L p ℝ for p > 1 to establish Hilbert formulae for FHT.

Highly Versatile Broadband RF Photonic Fractional Hilbert Transformer Based on a Kerr Soliton Crystal Microcomb

We demonstrate an RF photonic fractional Hilbert transformer based on an integrated Kerr micro-comb source featuring a record low free spectral range of 48.9 GHz, yielding 75 microcomb lines across the C-band. By programming and shaping the comb lines according to calculated tap weights, we demonstrate that the Hilbert transformer can achieve tunable bandwidths ranging from 1.2 to 15.3 GHz, switchable centre frequencies from baseband to 9.5 GHz, and arbitrary fractional orders. We experimentally characterize the RF amplitude and phase response of the tunable bandpass and lowpass Hilbert transformers with 90 and 45-degree phase shift. The experimental results show good agreement with theory, confirming the effectiveness of our approach as a powerful way to implement the standard as well as fractional Hilbert transformers with broad and switchable processing bandwidths and centre frequencies, together with high reconfigurability and greatly reduced size and complexity.

Optical super-resolution for two unequally bright point sources based on the fractional Hilbert transform

We propose to utilize fractional Hilbert transform (FHT) to fulfill the optical super-resolution for two unequally bright point sources. For such an asymmetric case, we compare the FHT scheme with both the direct imaging and conventional Hilbert transform schemes. We calculate the Fisher information about the separation and the relative intensity of two light sources. Based on single and multiple parameter estimation, we demonstrate that the FHT scheme has its unique advantages in estimating the separation of the unequally bright point sources. An unconditional enhancement of the measurement precision can be achieved by the FHT scheme. We believe that the FHT can be a good candidate in fulfilling the optical super-resolution in realistic scenarios.

Generalized entropic uncertainty principles on the complex signals in terms of fractional Hilbert transform

Uncertainty principle is of much signification to physics and so on. Hilbert transform (HT) plays the role of changing the real signal into the complex signal, which is related to the Fourier transform (FT) in traditional time-frequency domain. The fractional Hilbert transform (FRHT) plays the same role as the traditional counterpart in the fractional Fourier transform (FRFT) domain as well as in optics. Different from the general complex signals, the complex signals derived from HT and FRHT have the new properties such as the absence of negative frequency and so on. In this paper, the generalized entropic uncertainty relations in terms of Shannon entropy, Rényi entropy and Tsallis entropy with respect to the complex signals derived from FRHT are demonstrated for the first time based on the mathematical principles, including the special cases for the traditional HT. Interestingly, these novel derived uncertainty bounds proved to be different from and much lower (e.g., always less by ln2) in some cases than that of the traditional complex (and real) signals, which reveals the new properties of these complex signals derived from FRHT for the further potential valuable application. Moreover, the relationships between the uncertainty principles from the two types of complex signals are established. Finally, the numerical simulation is performed in this paper to show the efficiency of the proposed principles.

Kerr Micro-combs for Radio Frequency Photonics -INVITED

We review applications of Kerr micro-combs in RF photonic systems including fractional differentiators, Hilbert Transformers and many other functions.

Microwave and RF Photonic Fractional Hilbert Transformer Based on a 50 GHz Kerr Micro-Comb

We report a photonic microwave and RF fractional Hilbert transformer based on an integrated Kerr micro-comb source. The micro-comb source has a free spectral range (FSR) of 50GHz, generating a large number of comb lines that serve as a high-performance multi-wavelength source for the transformer. By programming and shaping the comb lines according to calculated tap weights, we achieve both arbitrary fractional orders and a broad operation bandwidth. We experimentally characterize the RF amplitude and phase response for different fractional orders and perform system demonstrations of real-time fractional Hilbert transforms. We achieve a phase ripple of < 0.15 rad within the 3-dB pass-band, with bandwidths ranging from 5 to 9 octaves, depending on the order. The experimental results show good agreement with theory, confirming the effectiveness of our approach as a new way to implement high-performance fractional Hilbert transformers with broad processing bandwidth, high reconfigurability, and greatly reduced size and complexity.

Particle swarm optimization-based kurtosis maximization in fractional Hilbert transform for bearing fault diagnosis

This paper explores the application of Fractional Hilbert Transform (FrHT) for bearing fault diagnosis. Though recent publications portray the use of FrHT and show that kurtosis varies significantly with the fractional parameter (p), no attempt is made to calculate the optimum value of p. We propose the use of particle swarm optimization to maximize the kurtosis and get a better performing FrHT for bearing fault diagnosis. Simulation results show improvement of kurtosis over traditional Hilbert transform. Bearing fault sizes of 7 mil, 14 mil and 21 mil are more separable using the proposed method than the conventional Hilbert transform.

Integrated reconfigurable photonic filters based on interferometric fractional Hilbert transforms.

In this paper, we present integrated reconfigurable photonic filters using fractional Hilbert transformers (FrHTs) and optical phase tuning structure within the silica-on-silicon platform. The proposed structure, including grating-based FrHTs, an X-coupler, and a pair of thermal tuning filaments, is fabricated through the direct UV grating writing technique. The thermal tuning effect is realized by the controllable microheaters located on the two arms of the X-coupler. We investigate the 200GHz maximum bandwidth photonic FrHTs based on apodized planar Bragg gratings, and analyze the reflection spectrum responses. Through device integration and thermal modulation, the device could operate as photonic notch filters with 5GHz linewidth and controllable single sideband suppression filters with measured 12dB suppression ratio. A 50GHz instantaneous frequency measuring system using this device is also schematically proposed and analyzed with potential 3dB measurement improvement. The device could be configured with these multiple functions according to need. The reconfigurable structure has great potential in ultrafast all-optical signal processing fields.

Fractional Hilbert Transform Sampling Method and Its Filter Bank Reconstruction

In this paper, conventional Hilbert transform sampling method is generalized to fractional Hilbert transform sampling method. First, a frequency-domain analysis method is applied to derive the fractional Hilbert transform sampling theorem of a band-limited signal. Then, an analog filter bank method is presented to recover the original continuous-time signal from the discrete-time sampled signals. Because the analog filter bank is not easy to be implemented, a digital filter bank method is proposed to solve the reconstruction problem. Next, a sparse design of digital reconstruction filters is studied to reduce the arithmetic implementation complexity using sparse FIR filter design techniques. Finally, several numerical examples are illustrated to show the effectiveness of the proposed fractional Hilbert transform sampling method and its filter bank reconstruction.

Wideband digital fractional Hilbert transformer designs based on interlaced and derivative sampling methods

In this paper, the design of wideband digital fractional Hilbert transformer (FHT) is presented. First, the wideband digital FHT is designed by using the reconstruction formula of interlaced sampling method and derivative sampling method. The closed-form design is obtained, so the filter coefficients are easily calculated. Then, the Lp-norm minimization method is applied to design digital wideband FHT whose solution can be obtained by using convex optimization tool. Next, the orthogonal matching pursuit method is used to design sparse digital FHT for reducing arithmetic complexity. Finally, several examples are demonstrated to show that the proposed FHT has smaller design error in high frequency band than the conventional digital FHT.

On the Relationship between Generalized Fractional Hilbert Transform with Some Classical Transforms

The generalized Fractional Hilbert Transform is the generalization of the Hilbert transform which has many applications in image reconstruction, optics, signal analysis and so on. In this paper, we established the relationship between generalized fractional Hilbert transform with Fourier transform, Laplace transform, Mellin transform, Hartley transform, Hilbert transform, and Stieltjes transform.

Designs of Discrete-Time Generalized Fractional Order Differentiator, Integrator and Hilbert Transformer

In this paper, the design of a generalized fractional order differentiator (FOD) whose magnitude and phase responses can be controlled independently is investigated. First, a relation between conventional FOD and generalized FOD is studied such that the design tools of conventional FOD in the literature can be used to design variable generalized FOD directly. Then, the similar method is applied to design generalized fractional order integrator (FOI). Next, the proposed generalized FOD and FOI are used to generate a secure single side band (SSB) signal for saving the transmission bandwidth. The parameters of variable generalized FOD and FOI can be used as the secure keys for construction and reconstruction. Finally, the relation between fractional Hilbert transformer and generalized FOD is studied and the edge detection application is demonstrated to show the flexibility and effectiveness of the proposed generalized FOD.

Terahertz-bandwidth photonic fractional Hilbert transformer based on a phase-shifted waveguide Bragg grating on silicon.

We propose and experimentally demonstrate the first THz bandwidth on-chip photonic fractional Hilbert transformer. The reported design uses a novel approach, based on a uniform and nonapodized single phase-shifted integrated waveguide Bragg grating on silicon, where the fractional order P can be engineered by simply varying the effective index modulation δn. Experimental results for P=1.5 show very low processing error for a broad range of pulse bandwidths between 77GHz and 2.07THz, corresponding to a time-bandwidth product as high as 27.

A Fractional Hilbert Transform for 2D Signals

The Hilbert transform is an important tool in image processing and optics. The Hilbert transform can be generalized to a fractional Hilbert transform. The generalization is driven by optics and image processing. We will generalize the fractional Hilbert transform into 2 dimensions by rotating the Hilbert transform in \({\mathbb{R}^{3}}\). The definition of the Hilbert transform as well as of the rotations will be done by quaternions.

The Local Fractional Hilbert Transform Based on Fractal Theory

In this paper, the local fractional Hilbert transform in fractal space is established. The characteristics of this local fractional transform are discussed in the following. Considering the basic properties of the local fractional Hilbert transforms, a kind of transform for local fractional is derived and analyzed.

On the Designs of Variable Fractional Hilbert Transformers

In this brief, the designs of variable fractional Hilbert transformers (VFHTs) are presented. First, the definition and properties of FHT are briefly reviewed. Then, the fact that the frequency response of FHT with order $v$ can be obtained from the linear combination of those of the two FHTs with different orders $p$ and $q$ will be shown. Based on this fact, a new VFHT structure is derived, in which the phase response of FHT can be easily changed by adjusting the linear combination coefficients without redesigning the FHT. Finally, numerical examples are demonstrated to compare the proposed method with conventional Farrow structure method in terms of design error, computational complexity, and memory storage requirements.

Continuously tunable photonic fractional Hilbert transformer using a high-contrast germanium-doped silica-on-silicon microring resonator.

We propose and experimentally demonstrate a continuously tunable fractional Hilbert transformer (FHT) based on a high-contrast germanium-doped silica-on-silicon (SOS) microring resonator (MRR). The propagation loss of a high-contrast germanium-doped SOS waveguide can be very small (0.02 dB/cm) while the lossless bend radius can be less than 1 mm. These characteristics lead to the fabrication of an MRR with a high Q-factor and a large free-spectral range (FSR), which is needed to implement a Hilbert transformer (HT). The SOS MRR is strongly polarization dependent. By changing the polarization direction of the input signal, the phase shift introduced at the center of the resonance spectrum is changed. The tunable phase shift at the resonance wavelength can be used to implement a tunable FHT. A germanium-doped SOS MRR with a high-index contrast of 3.8% is fabricated. The use of the fabricated MRR for the implementation of a tunable FHT with tunable orders at 1, 0.85, 0.95, 1.05, and 1.13 for a Gaussian pulse with the temporal full width at half-maximum of 80 ps is experimentally demonstrated.

Design of digital Feller fractional order integrator

In this paper, the design problems of the digital Feller fractional order integrator (FOI) are presented. First, the continuous-time fractional order integrals are reviewed briefly. Then, the fact that the frequency response of Feller FOI is linear combination of frequency responses of the conventional FOI is shown. Based on this fact, the conventional FOI is applied to design fixed Feller FOI including trapezoidal integration rule and Simpson integration rule methods. Next, the fractional sample delay is used to improve the designed Feller FOI and the discussion on the variable Feller FOI design is made. Finally, the relation between Feller FOI and fractional Hilbert transformer is studied and the application examples are demonstrated to show the effectiveness of the proposed digital Feller fractional order integrators.