Nonuniform multiresolution analysis associated with linear canonical transform

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 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.

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

In the paraxial limit, optical systems can be well described as ABCD systems, which are linear, lossless systems and can be well modeled using the Linear Canonical Transform (LCT). In theory, their effects are perfectly reversible. In practice, finite component sizes mean that a system designer must be aware that a waveform passing through a practical implementation of such a system may lose information due to walk-off and apertures, and make allowances accordingly. Such limitations also place restrictions on the bandwidth of a waveform which may be propagated through a system. These considerations are also very important from the point of view of attempting to simulate such systems for design or analysis. We consider the parameters of a system which result in low loss due to this factor, demonstrating the parameters which result in lossless systems for particular types of signal. We offer mathematical proof that certain classes of two-parameter LCTs preserve bandwidth or compact support, or transform one into the other. We propose a matrix-based methodology for minimizing aperture effects in ABCD systems.

The linear canonical transforms (LCTs) are a parameterised group of linear integral transforms. The LCTs encompass a number of well-known transformations as special cases, including the Fourier transform, fractional Fourier transform, and the Fresnel integral. They relate the scalar wave fields at the input and output of systems composed of thin lenses and free space, along with other quadratic phase systems. In this paper, we perform a systematic search of all algorithms based on up to five stages of magnification, chirp multiplication and Fourier transforms. Based on that search, we propose a novel algorithm, for which we present numerical results. We compare the sampling requirements of three algorithms. Finally, we discuss some issues surrounding the composition of discrete LCTs.

Digital speckle photography can be used in the analysis of surface motion in combination with an optical linear canonical transform (LCT). Previously [D. P. Kelly et al. Appl. Opt.44, 2720 (2005)] it has been shown that optical fractional Fourier transforms (OFRTs) can be used to vary the range and sensitivity of speckle-based metrology systems, allowing the measurement of both the magnitude and direction of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. This requires two bulk optical systems. We extend the OFRT analysis to more general LCT systems with a single limiting aperture. The effect of a limiting aperture in LCT systems is examined in more detail by deriving a generalized Yamaguchi correlation factor. We demonstrate the benefits of using an LCT approach to metrology design. Using this technique, we show that by varying the curvature of the illuminating field, we can effectively change the output domain. From a practical perspective this means that estimation of the motion of a target can be achieved by using one bulk optical system and different illuminating conditions. Experimental results are provided to support our theoretical analysis.

Digital holography is an electronic means of measuring the complex amplitude of an optical wavefield using CCD or CMOS arrays. With an appropriate reconstruction algorithm the intensity values registered by the pixels produce images of a particular (object) scene. These holographic systems are very sensitive to even small vibrations or deformations of an object, mainly due to the phase information that is also recovered by the measurement. Hence these systems are useful in a wide array of different metrology problems. It is important that we somehow quantify the information that can be recovered with such a detection scheme; better if we can provide a theoretical framework to optimize an optical system design for a given metrology problem. In this manuscript we show how the Linear Canonical Transform can fulfill this role and can optimally match the space-bandwidth product (SBP) of a signal to be measured with the SBP of a CCD/CMOS detector array. We provide formulae that determine the performance of generalized holographic optical systems (containing lenses and sections of free space), taking into account the finite extent of the CCD array, the size of the pixels, and the spacing between them. Some illustrative examples are presented, with associated numerical simulations.

Fabry-Perot cavities are central to many optical measurement systems. In high-precision experiments, such as aLIGO and AdVirgo, coupled cavities are often required, leading to complex optical behavior. We show, for the first time to our knowledge, that discrete linear canonical transforms (LCTs) can be used to compute circulating optical fields for cavities in which the optics have arbitrary apertures, reflectance and transmittance profiles, and shape. We compare the predictions of LCT models with those of alternative methods. To further highlight the utility of the LCT, we present a case study of point absorbers on the aLIGO mirrors and compare it with recently published results.

NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session: 2711 Teaching Hardware Design of Fixed-Point Digital Signal Processing Systems David V. Anderson1 and Tyson S. Hall2 1 Georgia Institute of Technology, Atlanta, GA 30332–0250, dva@ece.gatech.edu 2 Southern Adventist University, Collegedale, TN 37315–0370, tyson@southern.edu Abstract Signal processing theory and practice are enabling and driving forces behind multimedia de- vices, communications systems, and even such diverse fields as automotive and medical sys- tems. Over 90% of the signal processing systems on the market used fixed-point arithmetic because of the cost, power, and area savings that fixed-point systems provide. However, most colleges and universities do not teach or teach only a very little fixed-point signal process- ing. This issue is being addressed slowly around the country but now a new challenge or opportunity presents itself. As reconfigurable logic technology matures, field-programmable gate arrays (FPGAs) are increasingly used for signal processing systems. They have the ad- vantage of tremendous throughput, great flexibility, and system integration. The challenge is that signal processing in FPGAs is a much less constrained problem than signal processing in special purpose microprocessors. The opportunity arises in that it is now possible to explore more options and, more especially, to take a more systems-level approach to signal processing systems. In short, designing a signal processing system using FPGAs provides opportunities to look at many system design issues and trade-offs in a classroom setting. We have developed a course to teach signal processing in FPGAs at Georgia Institute of Technology and in this paper we consider the challenges and methods of teaching fixed- point system design in this course. We discuss the topics chosen and how they differ from traditional microprocessor-based courses. We also discuss how systems engineering concepts are woven into the course. 1 Motivation This paper describes the early development of a system design curriculum that uses field programmable gate arrays (FPGAs) to implement digital signal processing (DSP) systems. The goals of this curriculum are several-fold, including: • to prepare students for system design problems with which they are likely to be pre- sented upon graduation; Proceedings of the 2007 American Society for Engineering Education Annual Conference & Exposition Copyright c 2007, American Society for Engineering Education

The statistical properties of speckles in paraxial optical systems depend on the system parameters. In particular, the speckle orientation and the lateral dependence (x and y) of the longitudinal speckle size can vary significantly. For example, the off-axis longitudinal correlation length remains equal to the on-axis size for speckles in a Fourier transform system, while it decreases dramatically as the observation position moves off axis in a Fresnel system. In this paper, we review the speckle correlation function in general linear canonical transform (LCT) systems, clearly demonstrating that speckle properties can be controlled by introducing different optical components, i.e., lenses and sections of free space. Using a series of numerical simulations, we examine how the correlation function changes for some typical LCT systems. The integrating effect of the camera pixel and the impact this has on the measured first- and second-order statistics of the speckle intensities is also examined theoretically. A series of experimental results are then presented to confirm several of these predictions. First, the effect the pixel size has on the measured first-order speckle statistics is demonstrated, and second, the orientation of speckles in a Fourier transform system is measured, showing that the speckles lie parallel to the optical axis.

The availability of high quality Spatial Light Modulators (SLMs) and high-resolution cameras have made it technologically feasible to perform many optical signal-processing functions in a relatively inexpensive manner. Furthermore, recent fast algorithms have been proposed to compute Linear Canonical Transforms (LCTs), which may be used to simulate paraxial optical systems. Optical Encryption offers the possibility of high-speed parallel encryption of image data. Such encryption/decryption may involve the capture of full field information, i.e. phase and intensity. Applications of Digital Holography go beyond applications to optical encryption. They include Holographic Interferometry (metrology) and Holographic displays. In this paper we discuss the design and analysis of systems used in the capture, encryption/decryption and display of 2-D (image) data and present our recent experimental and theoretical results.

In this paper, we investigate the physical layer security of multiuser multi-input single-output (MU-MISO) communication and colocated multi-input multi-output (MIMO) radar coexistence systems, in which the strong radar signals are exploited as inherent jamming signals to disrupt mali-cious receptions. The transmit beamformers of communication and radar systems are jointly designed to ensure the secure transmission by minimizing the maximum eavesdropping signal-to-interference-plus-noise ratio (SINR) on multiple legitimate users, while satisfying the quality-of-service (QoS) of legitimate transmission, the requirement of radar target detection, and the transmit power constraints of radar and communication systems. An efficient fractional programming (FP) and semi-definite relaxation (SDR) based algorithm is proposed to solve the non-convex optimization problem. Simulation results verify the advancement of the proposed joint transmit beamforming on secure transmission for radar and communication coexistence systems and the effectiveness of the associate design algorithm.

In the last decades many possible applications of nonlinear dynamics in communication systems and signal processing have been reported. Conversely, techniques usually employed by the signal processing and communication systems communities, as correlation, power spectral density analysis, and linear filters, among others have been used to characterize chaotic dynamical systems. This chapter presents four works that aim to use tools from both fields to generate new and interesting results: (1) a message authentication system based on chaotic fingerprint; (2) a study of the spectral characteristics of the chaotic orbits of the Henon map; (3) an investigation of the chaotic nature of the signals generated by a filtered Henon map, and (4) a communication system that presents equalization and a switching scheme between chaos-based and conventional modulations.

In this letter, the coexistence of a multiple-input multiple-output (MIMO) radar and a distributed MIMO communication systems is studied. Target returns contributed from both the radar transmitters and communication transmitters are employed to complete the radar task, leading to a hybrid active-passive MIMO radar network. For the communication task, not only are the communication signals received directly from communication transmitters, but also those bounced off from the target are exploited to extract useful information. The target localization Cramer–Rao bound and mutual information are derived for the radar and communication systems, respectively. It is shown that there is a performance gain due to the cooperation between the radar and communication systems.

In this paper, we propose an interference alignment (IA) method for interference elimination in co-existing Phased Array (PA) radar and multi-user multiple-input multiple-output (MIMO) communication systems. Although ergodic IA method has been proposed to cancel the mutual interferences, it will definitely reduce the degrees-of-freedom (DoFs) of communication systems. Differently, our proposed IA method allows to improve the DoFs. It is examined by the Neyman-Pearson (NP) test for radar functionality, while the communication performance is evaluated by DoFs and bit error rate (BER) analysis. Both theory analysis and numerical results are provided to validate the effectiveness of the proposed IA method.