Fractional cosine, sine, and Hartley transforms

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.

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.

The linear canonical transform(LCT), which is a generalization of the classical Fourier transform and fractional Fourier transform(FRFT), was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful.In this paper, we briefly introduce the LCT and a number of its properties and then discuss the LCT's relationships with time-frequency representations such as the Wigner distribution and the short-time Fourier transform.These relationships have a very simple and natural form and show that the LCT performs a homogeneous linear mapping in the time-frequency plane.Finally, an example of the application of the LCT is given.

Discrete quaternion linear canonical transform

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

As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical 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 <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.

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

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.

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

Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution

Speech recovery based on the linear canonical transform

The linear canonical transform (LCT) is a kind of integral transforms with wide applications in signal analysis. There have been numerous studies in the literature to generalize the LCT by making use of the quaternion algebra. In this paper, we first define the octonion linear canonical transform (OCLCT). Based on the definition of OCLCT, we extend the relationship between the LCT and the Fourier transform (FT) to the OCLCT and the octonion Fourier transform (OFT). Then explore related properties for the OCLCT such as shift property, inversion formula, isometry and Riemann-Lebesgue lemma. The relation between OCLCT and 3-D LCT is also builded. Moreover, based on these properties, we obtain Heisenberg’s uncertainty principle and Donoho-Stark’s uncertainty principle associated with the OCLCT. Finally, some potential applications are presented to show the effectiveness of the OCLCT.

Taking advantage of the high speed arithmetic offers by the parallelism of the digital signal processor DSP TMS320C6711 of TI, we implemented two algorithms to find the Fourier transform of fractional order of two dimensional signals ("The discrete fractional Fourier transform" and "Fractional Fourier Transform calculation through the fast-Fourier-transform algorithm"). As a result, we have two algorithms for calculate the fractional Fourier transform of images using a DSP, the images are captured using a camera CCD and the fractional Fourier transform is display in a monitor, both are connected to the DSP device directly. Finally, the work focuses in the evaluation of this algorithms, in order to find which is best performance related to four criteria determined from the properties of the Fourier transform of fractional order: The inverse transform, fractional Fourier transform of order one or standard Fourier transform, fractional Fourier transform of order zero or operator identity and order additivity.

In this paper, we propose a novel fractional Fourier transform (FrFT) based multi-carrier order division multi-access communication system, in which each user is uniquely identified by an FrFT order. Transform domain communication system (TDCS) with FrFT scheme is also proposed to synthesize the wide-band baseband waveforms in all FrFT domains with different users' FrFT orders, which enhances the interference avoidance capability of this system under most of interference. Therefore, multiple independent data streams can be transmitted by using FrFT---OFDM in the same time and different FrFT domains. However, chirp bases, as the new kind of carriers with different modulated rates, are merely mutually approximately orthogonal. There is a problem of energy leakage between multiple chirp carriers, which possibly causes the multiple chirp carriers inter-shielding to influence the FrFT---OFDM demodulation performance. An efficient allocation algorithm of multiple chirp carriers by presetting the carrier parameters is proposed to solve this problem. Based on MC-CDMA, a variable bit rate system structure is proposed for TDCS with FrFT scheme under different channel environments. In order to simplify the process of modulation and demodulation of TDCS with FrFT scheme, a whole new cyclic shift key modulation mode in FrFT domain is also proposed. Both theories and simulations confirm strictly the validity of the proposed system.