The Linear Canonical Transform and Time-Frequency Representations

The linear canonical transform (LCT) has been shown to be useful and powerful in signal processing, optics, etc. Many results of this transform are already known, including sampling theory. Most existing sampling theories of the LCT consider the class of bandlimited signals. However, in the real world, many analog signals arising in engineering applications are non-bandlimited. In this correspondence, we propose a sampling and reconstruction strategy for a class of function spaces associated with the LCT, which can provide a suitable and realistic model for real applications. First, we introduce definitions of semi- and fully-discrete convolutions for the LCT. Then, we derive necessary and sufficient conditions pertaining to the LCT, under which integer shifts of a chirp-modulated function generate a Riesz basis for the function spaces. By applying the results, we present a more comprehensive sampling theory for the LCT in the function spaces, and further, a sampling theorem which recovers a signal from its own samples in the function spaces is established. Moreover, some sampling theorems for shift-invariant spaces and some existing sampling theories for bandlimited signals associated with the Fourier transform (FT), the fractional FT, or the LCT are noted as special cases of the derived results. Finally, some potential applications of the derived theory are presented.

In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. We introduce several new transforms. They are all the generalization of the cosine, sine, or Hartley transform. We first derive the fractional cosine, sine, and Hartley transforms (FRCT/FRST/FRHT). They are analogous to the FRFT. Then, we derive the canonical cosine and sine transforms (CCT/CST). They are analogous to the LCT. We also derive the simplified fractional cosine, sine, and Hartley transforms (SFRCT/SFRST/SFRHT). They are analogous to the SFRFT and have the advantage of real-input-real-output. We also discuss the properties, digital implementation, and applications (e.g., the applications for filter design and space-variant pattern recognition) of these transforms. The transforms introduced in this paper are very efficient for digital implementation. We can just use one half or one fourth of the real multiplications required for the FRFT and LCT to implement them. When we want to process even, odd, or pure real/imaginary functions, we can use these transforms instead of the FRFT and LCT. Besides, we also show that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided (t /spl isin/ [0, /spl infin/]) signal processing.

It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp Modulation Transform (CMT) are all subsets of the more general LCT. Using the <i>ABCD</i> Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin's matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating fields of varying curvature.

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

Generalized Heisenberg uncertainty relations for complex-valued signals out of Hilbert transform associated with LCT

The wave packet transform associated with the linear canonical transform

The vibration signals analysis is a very effective and reliable method for detecting the gear failures. Because the vibration signals acquired from the gear in the variable speed condition often contain more useful fault information, the analysis of the gear vibration signals during the variable speed condition has been a hot research topic. In this paper, a method based on the multiscale chirplet path pursuit (MSCPP) and the linear canonical transform (LCT) has been applied to diagnose the gear fault in the variable speed condition for the first time. First, by using the MSCPP method to estimate the instantaneous meshing frequency, the suitable signal segment approximation to the acceleration or deceleration process can be selected. Then, because the LCT is a novel and efficient nonstationary signals analysis tool, the optimal LCT spectrum of the selected signal has been attainted to diagnose the gear faults based on the properties of the LCT. In addition, the simulations and the experimental evaluation are provided to verify the effectiveness of the proposed method.

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.

Linear canonical transform (LCT) is very useful and powerful in signal processing and optics. In this paper, discrete LCT (DLCT) is proposed to approximate LCT by utilizing the discrete dilated Hermite functions. The Wigner distribution function is also used to investigate DLCT performances in the time-frequency domain. Compared with the existing digital computation of LCT, our proposed DLCT possess additivity and reversibility properties with no oversampling involved. In addition, the length of input/output signals will not be changed before and after the DLCT transformations, which is consistent with the time-frequency area-preserving nature of LCT; meanwhile, the proposed DLCT has very good approximation of continuous LCT.

A New Discretization Algorithm of Linear Canonical Transform

Linear canonical transform play an important role in many fields of optics and signal processing society. Well known transforms in these fields such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be looked at as special cases of the linear canonical transform. In this paper we obtain new sampling formulae for reconstructing signals that are band-limited or time-limited in the linear canonical transform sense. In the new sampling formulae, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate of linear canonical transform domain

In recent years, the Hilbert - Huang transform and the linear canonical transform draw extensive attention in signal processing community, especially in Radar, optical signal processing and so on. This paper investigates the Hilbert - Huang transform associated with the linear canonical transform and the algorithm for the non-stationary signal processing is also provided. And finally, the simulation results are proposed to show the correctness of the proposed methods.

Analysis of A-stationary random signals in the linear canonical transform domain

It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the linear canonical transform (LCT). Indeed it can be shown that many well know optical transforms such as the optical fourier transform (OFT), optical fractional fourier transform (OFRT), the effect of a lens or chirp modulation transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT&apos;s (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin&apos;s matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating holds of varying curvature.