Sampling and Reconstruction in a Shift Invariant Space with Multiple Generators

In this research letter, we investigate the sampling set for the shift-invariant spaces, where the generator of shift-invariant spaces belongs to Wiener-amalgam space. Furthermore, a sufficient condition of sampling set for general shift-invariant spaces is derived, which is the partial solution of Aldroubi-Grochenig conjecture (Aldroubi and Grochenig, 2000).

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Generalized sampling in shift-invariant spaces with multiple stable generators

Optimal shift invariant spaces and their Parseval frame generators

In many applications only the low frequency components of a signal can be measured due to the lowpass behavior of many physical systems. Nevertheless, if additional information on the structure of the signal is known, it might still be possible to reconstruct the signal from its low-frequency content. This paper studies signals in shift-invariant spaces with multiple generators and derives necessary conditions on the bandwith of the lowpass filter as well as sufficient conditions on the generators such that signal recovery is possible. If the signal can not be recovered from its low frequency components, an appropriate pre-processing of the signal is proposed which improves the reconstruction ability. In particular, it is shown that modulating the signal with one or more mixing functions prior to lowpass filtering can ensure the recovery of the signal in many cases.

We present a new method for the fast reconstruction of a function f from its samples f(xj) under the assumption that f belongs to a shift-invariant space $V(\varphi)$. If the generator $\varphi$ has compact support, then the reconstruction is local, quite in contrast to methods based on band-limited functions. Using frame theoretic arguments, we show that the matrix of the corresponding linear system of equations is a positive-definite banded matrix. This special structure makes possible the fast local reconstruction algorithm in O(S2J) operations, where J is the number of samples and S is the support length of the generator $\varphi$. Further optimization can be achieved by means of data segmentation. Ample numerical simulation is provided.

Recovery of missing samples for oversampling in shift invariant spaces

Today, many existing types of video transmission and storage infrastructure are not able to handle UHD uncompressed video in real time. To reduce the required bit rates, a low-latency lightweight compression scheme is needed. To this end, several standardization efforts, such as Display Stream Compression, Advanced DSC, and JPEG XS, are currently being made. Focusing on screen content use cases, this paper provides a comparison of existing codecs suited for this field of application. In particular, the performance of DSC, VC-2, JPEG 2000 (in low-latency and low-complexity configurations), JPEG and HEVC Screen Content Coding Extension (SCC) in intra mode are evaluated. First, quality is assessed in single and multiple generations. Then, error robustness is evaluated by inserting one-bit errors at random positions in the compressed bitstreams. Unsurprisingly, the most complex algorithm, HEVC SCC intra, achieves the highest compression efficiency on screen content. JPEG 2000 performs well in the three experiments while HEVC SCC does not provide multi-generation robustness. DSC guarantees quality preservation in single generation at high bit rates and VC-2 provides very high error resilience. This work gives the reader an overview of the objective quality assessment that will be conducted as part of JPEG XS evaluation procedures.

Recovery of compressive samples from a randomly modulated converter

This article discusses the structure of various shift-invariant spaces of cardinal functions generated by a single kernel, and their role in some approximate sampling methods. Particularly, conditions on the generating kernel φ are given which imply that the associated L p shift-invariant space coincides with the shift-invariant space for the cardinal function L φ generated by the kernel. Additionally, we discuss how such spaces are used in summability methods for the classical sampling series.

Preface. 1. The Physics of Electricity. 1.1 Basic Quantities. 1.1.1 Introduction. 1.1.2 Charge. 1.1.3 Potential or Voltage. 1.1.4 Ground. 1.1.5 Conductivity. 1.1.6 Current. 1.2 Ohm's law. 1.2.1 Resistance. 1.2.2 Conductance. 1.2.3 Insulation. 1.3 Circuit Fundamentals. 1.3.1 Static Charge. 1.3.2 Electric Circuits. 1.3.3 Voltage Drop. 1.3.4 Electric Shock. 1.4 Resistive Heating. 1.4.1 Calculating Resistive Heating. 1.4.2 Transmission Voltage and Resistive Losses. 1.5 Electric and Magnetic Fields. 1.5.1 The Field as a Concept. 1.5.2 Electric Fields. 1.5.3 Magnetic Fields. 1.5.4 Electromagnetic Induction. 1.5.5 Electromagnetic Fields and Health Effects. 1.5.6 Electromagnetic Radiation. 2. Basic Circuit Analysis. 2.1 Modeling Circuits. 2.2 Series and Parallel Circuits. 2.2.1 Resistance in Series. 2.2.2 Resistance in Parallel. 2.2.3 Network Reduction. 2.2.4 Practical Aspects. 2.3 Kirchhoff's Laws. 2.3.1 Kirchhoff's Voltage Law. 2.3.2 Kirchhoff's Current Law. 2.3.3 Application to Simple Circuits. 2.3.4 The Superposition Principle. 2.4 Magnetic Circuits. 3. AC Power. 3.1 Alternating Current and Voltage. 3.1.1 Historical Notes. 3.1.2 Mathematical Description. 3.1.3 The rms Value. 3.2 Reactance. 3.2.1 Inductance. 3.2.2 Capacitance. 3.2.3 Impedance. 3.2.4 Admittance. 3.3 Power. 3.3.1 Definition of Electric Power. 3.3.2 Complex Power. 3.3.3 The Significance of Reactive Power. 3.4 Phasor Notation. 3.4.1 Phasors as Graphics. 3.4.2 Phasors as Exponentials. 3.4.3 Operations with Phasors. 4. Generators. 4.1 The Simple Generator. 4.2 The Synchronous Generator. 4.2.1 Basic Components and Functioning. 4.2.2 Other Design Aspects. 4.3 Operational Control of Synchronous Generators. 4.3.1 Single Generator: Real Power. 4.3.2 Single Generator: Reactive Power. 4.3.3 Multiple Generators: Real Power. 4.3.4 Multiple Generators: Reactive Power. 4.4 Operating Limits. 4.5 The Induction Generator. 4.5.1 General Characteristics. 4.5.2 Electromagnetic Characteristics. 4.6 Inverters. 5. Loads. 5.1 Resistive Loads. 5.2 Motors. 5.3 Electronic Devices. 5.4 Load from the System Perspective. 5.4.1 Coincident and Noncoincident Demand. 5.4.2 Load Profiles and Load Duration Curve. 5.5 Single- and Multiphase Connections. 6. Transmission and Distribution. 6.1 System Structure. 6.1.1 Historical Notes. 6.1.2 Structural Features. 6.1.3 Sample Diagram. 6.1.4 Topology. 6.1.5 Loop Flow. 6.1.6 Stations and Substations. 6.1.7 Reconfiguring the System. 6.2 Three-Phase Transmission. 6.2.1 Rationale for Three Phases. 6.2.2 Balancing Loads. 6.2.3 Delta and Wye Connections. 6.2.4 Per-Phase Analysis. 6.2.5 Three-Phase Power. 6.2.6 D.C. Transmission. 6.3 Transformers. 6.3.1 General Properties. 6.3.2 Transformer Heating. 6.3.3 Delta and Wye Transformers. 6.4 Characteristics of Power Lines. 6.4.1 Conductors. 6.4.2 Towers, Insulators, and Other Components. 6.5 Loading. 6.5.1 Thermal Limits. 6.5.2 Stability Limit. 6.6 Voltage Control. 6.7 Protection. 6.7.1 Basics of Protection and Protective Devices. 6.7.2 Protection Coordination. 7. Power Flow Analysis. 7.1 Introduction. 7.2 The Power Flow Problem. 7.2.1 Network Representation. 7.2.2 Choice of Variables. 7.2.3 Types of Buses. 7.2.4 Variables for Balancing Real Power. 7.2.5 Variables for Balancing Reactive Power. 7.2.6 The Slack Bus. 7.2.7 Summary of Variables. 7.3 Example with Interpretation of Results. 7.3.1 Six-Bus Example. 7.3.2 Tweaking the Case. 7.3.3 Conceptualizing Power Flow. 7.4 Power Flow Equations and Solution Methods. 7.4.1 Derivation of Power Flow Equations. 7.4.2 Solution Methods. 7.4.3 Decoupled Power Flow. 7.5 Applications and Optimal Power Flow. 8. System Performance. 8.1 Reliability. 8.1.1 Measures of Reliability. 8.1.2 Valuation of Reliability. 8.2 Security. 8.3 Stability. 8.3.1 The Concept of Stability. 8.3.2 Steady-State Stability. 8.3.3 Dynamic Stability. 8.3.4 Voltage Stability. 8.4 Power Quality. 8.4.1 Voltage. 8.4.2 Frequency. 8.4.3 Waveform. 9. System Operation, Management, and New Technology. 9.1 Operation and Control on Different Time Scales. 9.1.1 The Scale of a Cycle. 9.1.2 The Scale of Real-Time Operation. 9.1.3 The Scale of Scheduling. 9.1.4 The Planning Scale. 9.2 New Technology. 9.2.1 Storage. 9.2.2 Distributed Generation. 9.2.3 Automation. 9.2.4 FACTS. 9.3 Human Factors. 9.3.1 Operators and Engineers. 9.3.2 Cognitive Representations of Power Systems. 9.3.3 Operational Criteria. 9.3.4 Implications for Technological Innovation. 9.4 Implications for Restructuring. Appendix: Symbols, Units, Abbreviations, and Acronyms. Index.

Transgenic technologies provide a promising means by which desirable traits can be introduced into cultured fish species within a single generation thus accelerating the production of genetically superior broodstock for aquaculture. However, before such fish are allowed to be marketed as food they must receive government regulatory approval. Two pivotal regulatory requirements are: (1) complete characterization of the genomically integrated transgene and, (2) demonstration that the transgene remains stable over multiple generations. We have generated a stable line of growth hormone (GH) transgenic Atlantic salmon (Salmo salar) using an "all fish" gene construct (opAFP-GHc2) containing a growth hormone cDNA from chinook salmon whose expression is regulated by the 5' promoter and 3' termination regions derived from an ocean pout antifreeze protein (AFP) gene. In this study we show that a reorganized form of the opAFP-GHc2 construct (termed EO-1alpha) integrated as a single functional copy into a 35 bp repeat region of the genomic DNA. PCR based mapping revealed that the linear sequence of the EO-1alpha integrant was organized as follows: base pairs 1580-2193 of the ocean pout promoter region followed by the intact chinook salmon GH cDNA, the complete ocean pout antifreeze 3' region, and the first 1678 bp of the ocean pout antifreeze 5' region. Sequence analysis of the EO-1alpha integrant and genomic flanking regions in F2 and F4 generation salmon revealed that they were identical. In addition, apart from the disruption at the integration sites, the consensus sequences of the integrant in these two generations of salmon were identical to the sequence of the opAFP-GHc2 construct. These results indicate that the EO-1alpha transgene codes for the chinook salmon GH, and that the transgene and the integration site have remained stable over multiple generations.

Stabilized finite element method for viscoplastic flow: formulation and a simple progressive solution strategy

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Erratum to: Periodic operators and elements. Shift operators. Shift invariant spaces