Quadratic Phase Function Research Articles

Laplace Transforms of a Family of Functions Related to Quadratic Phase Function

A parametric family of complex valued functions in the form of quadratic exponents are defined starting from the real valued Gaussian function. The family of functions include the quadratic phase function, which is also called the two-sided complex chirp. It is proven using contour integrals on the complex plane that the Laplace transforms of these functions are also complex valued quadratic exponents; the region of convergence of the Laplace transform include the entire <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> -plane for finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$|s|$ </tex-math></inline-formula> for a range of values of the defining parameter of the family. Fourier transforms are also presented as the special cases of the Laplace transform.

Stabilizer rank and higher-order Fourier analysis

We establish a link between stabilizer states, stabilizer rank, and higher-order Fourier analysis – a still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemerédi's theorem \cite{gowers1998new}. We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of Fpn where p is the dimension of the qudit) which are fundamental objects in higher-order Fourier analysis. This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in \cite{peleg2021lower} it was shown that the n-qubit magic state has stabilizer rank &amp;#x03A9;(n). Here we show that the qudit analog of the n-qubit magic state has stabilizer rank &amp;#x03A9;(n), generalizing their result to qudits of any prime dimension. Our proof techniques use explicitly tools from higher-order Fourier analysis. We believe this example motivates the further exploration of applications of higher-order Fourier analysis in quantum information theory.

A two-step approach to the processing of high-resolution SAR satellite data based on the chirp z-transform

ABSTRACT The processing of high resolution spotlight synthetic aperture radar (SAR) data suffers from Doppler spectrum aliasing because of the limited pulse repetition frequency (PRF). A modified two-step approach (TSA) based on the chirp z-transform (CZT) to high resolution spaceborne SAR data processing is proposed to deal with the spectral aliasing problem in this paper. The range frequency dependent quadratic phase function applied in modified TSA, which can remove the Doppler modulation of the signal completely, gives rise to the effect that the Doppler bin size varies with the range frequency. Thus, the extra spectrum resampling procedure is required to obtain a uniformly sampled signal spectrum. A spectrum resampling approach based on the chirp z-transform is proposed, which can overcome the range frequency dependent distortion. Experiments are performed on simulated point and experimental SAR data to validate the presented approach.

Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary point

The ( 2 × 2 ) (2\times 2) matrix conjugacy problem (the Riemann–Hilbert problem) with rapidly oscillating off-diagonal entries and quadratic phase function is considered, specifically, the case when one of the diagonal entries vanishes at a stationary point. For solutions of this problem, the leading term of the asymptotics is found. However, the method allows us to construct complete expansions in power orders. These asymptotics can be used, for example, to construct the asymptotics of solutions of the Cauchy problem for the nonlinear Schrödinger equation for large times in the case of the so-called collisionless shock region.

Edge and Contrast Enhancement Using Spatially Incoherent Correlation Holography Techniques

Image enhancement techniques (such as edge and contrast enhancement) are essential for many imaging applications. In incoherent holography techniques such as Fresnel incoherent correlation holography (FINCH), the light from an object is split into two, each of which is modulated differently from one another by two different quadratic phase functions and coherently interfered to generate the hologram. The hologram can be reconstructed via a numerical backpropagation. The edge enhancement procedure in FINCH requires the modulation of one of the beams by a spiral phase element and, upon reconstruction, edge-enhanced images are obtained. An optical technique for edge enhancement in coded aperture imaging (CAI) techniques that does not involve two-beam interference has not been established yet. In this study, we propose and demonstrate an iterative algorithm that can yield from the experimentally recorded point spread function (PSF), a synthetic PSF that can generate edge-enhanced reconstructions when processed with the object hologram. The edge-enhanced reconstructions are subtracted from the original reconstructions to obtain contrast enhancement. The technique has been demonstrated on FINCH and CAI methods with different spectral conditions.

Depth-of-field engineering in coded aperture imaging.

Extending the depth-of-field (DOF) of an optical imaging system without effecting the other imaging properties has been an important topic of research for a long time. In this work, we propose a new general technique of engineering the DOF of an imaging system beyond just a simple extension of the DOF. Engineering the DOF means in this study that the inherent DOF can be extended to one, or to several, separated different intervals of DOF, with controlled start and end points. Practically, because of the DOF engineering, entire objects in certain separated different input subvolumes are imaged with the same sharpness as if these objects are all in focus. Furthermore, the images from different subvolumes can be laterally shifted, each subvolume in a different shift, relative to their positions in the object space. By doing so, mutual hiding of images can be avoided. The proposed technique is introduced into a system of coded aperture imaging. In other words, the light from the object space is modulated by a coded aperture and recorded into the computer in which the desired image is reconstructed from the recorded pattern. The DOF engineering is done by designing the coded aperture composed of three diffractive elements. One element is a quadratic phase function dictating the start point of the in-focus axial interval and the second element is a quartic phase function which dictates the end point of this interval. Quasi-random coded phase mask is the third element, which enables the digital reconstruction. Multiplexing several sets of diffractive elements, each with different set of phase coefficients, can yield various axial reconstruction curves. The entire diffractive elements are displayed on a spatial light modulator such that real-time DOF engineering is enabled according to the user needs in the course of the observation. Experimental verifications of the proposed system with several examples of DOF engineering are presented, where the entire imaging of the observed scene is done by single camera shot.

On-chip dispersive phase filters for optical processing of periodic signals.

We develop a dispersive phase filter design framework suitable for compact integration using waveguide Bragg gratings (WBGs) in silicon. Our proposal is to utilize an equivalent "discrete" spectral phase filtering process, in which the original continuous quadratic spectral phase function of a group velocity dispersion (GVD) line is discretized and bounded in a modulo 2π basis. Through this strategy, we avoid the phase accumulation of the GVD line, leading to a significant reduction in device footprint (length) as compared to conventional GVD devices (e.g., using a linearly chirped WBG). The proposed design is validated through numerical simulations and proof-of-concept experiments. Specifically, using the proposed methodology, we demonstrate 2× pulse repetition-rate multiplication of a 10 GHz picosecond pulse train by dispersion-induced Talbot effect on a silicon chip.

Lagrangian Distributions and Fourier Integral Operators with Quadratic Phase Functions and Shubin Amplitudes

We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as the composition of a Weyl pseudodifferential operator and a metaplectic operator and derive a characterization of its Schwartz kernel in terms of phase space estimates. Extending the conormal distributions in the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.

The Fast Physical Optics Method on Calculating the Scattered Fields From Electrically Large Scatterers

In this article, we propose the fast physical optics (FPO) method to calculate the scattered fields from the electrically large scatterers. The quadratic patches are adopted to discretize the surfaces of the scatterers. Invoking the quadratic amplitude and phase functions approximations, the closed-form formulas could be obtained with the complementary error function. Hence, the FPO method is time-saving compared with the conventional methods. Moreover, the accuracy and the physical insight of the closed-form formulas are expounded. Numerical examples demonstrate the efficiency of the FPO method with respect to both the accuracy and CPU time.

Bargmann-Type Transforms and Modified Harmonic Oscillators

We study some complete orthonormal systems on the real line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real line.

Parameters estimation for chirp signal based on QPF-FRFT

Accuracy and speed of parameters estimation for radar signal is very important in optical wireless communications. In order to implement high accuracy and fast speed of parameters estimation, a new estimation algorithm is presented in the paper. The method is based on quadratic phase function (QPF) and fractional Fourier transform (FRFT), which is used to estimate parameters for the chirp signal. In order to avoid two-dimensional searching during FRFT, chirp rate is directly estimated by QPF method, and then the optimum transform order of FRFT is calculated by estimating the chirp rate. Finally, center frequency and amplitude of the chirp signal are estimated by FRFT. Simulation results show that the proposed QPF-FRFT method is faster than the FRFT method and two step maximum modulus searching method. Otherwise, when the SNR is higher than -4dB, the center frequency, chirp rate and amplitude of chirp signal can be estimated by QPF-FRFT method more accurately than FRFT method. In conclusion, results on experimental of chirp signal illustrate the efficient performance of the algorithm.

Intrapulse modulation type recognition for pulse compression radar signal

The existing modulation recognition algorithms for a pulse compression radar (PCR) signal can hardly adapt to complex modulation types and low signal-to-noise ratio (SNR). To solve the problems, with respect to the seven kinds of widely used PCR signals—including linear frequency modulation signal, Baker code, Frank code, P1 code, P2 code, P3 code, and P4 code—a modulation type recognition algorithm based on integrated quadratic phase function (IQPF) and fractional Fourier transform (FrFT) is proposed. First, signals are preclassified according to their chirp rates (CRs) estimated through IQPF. Then, FrFT is carried out depending on the order, which is correlated to the estimated CR. Finally, signals in each class are subdivided and modulation recognition is accomplished according to the features of the FrFT spectrum. The simulation results validate the feasibility of the algorithm. They also demonstrate that, compared against existing research, the proposal achieves better correct recognition performance for various modulation types under low SNR condition.

Coded aperture correlation holography system with improved performance [Invited

Coded aperture correlation holography (COACH) is a recently introduced technique for recording incoherent digital holograms of general three-dimensional scenes. In COACH, a random-like coded phase mask (CPM) is used as a coded aperture. Even though the CPM is optimized to reduce background noise, there is still a substantial amount of noise, mitigating the performance of COACH. In order to reduce the noise, we first modify the hologram reconstruction method. Instead of computing the correlation between a complex hologram of the entire object and a hologram of a source point, in this study the numerical correlation is performed with a phase-only filter. In other words, the phase function of the Fourier transform of the source point hologram is used as the spatial filter in the correlation process. Furthermore, we propose and demonstrate two additional methods for reducing the background noise in COACH. The first is based on the integration of a quadratic phase function, as used in Fresnel incoherent correlation holography (FINCH), with the CPM of COACH. This hybrid COACH-FINCH system enables a dynamic trade-off between the amount of background noise and the axial resolution of the system. The second method is employed by recording COACH holograms with multiple independent CPMs and averaging over the reconstructed images. The results of the above two techniques are compared with FINCH and with a regular imaging system.

Quasi-classical asymptotics of solutions to the matrix factorization problem with quadratically oscillating off-diagonal elements

The paper investigates the asymptotic behavior of solutions to the 2 × 2 matrix factorization (Riemann-Hilbert) problem with rapidly oscillating off-diagonal elements and quadratic phase function. A new approach to study such problems based on the ideas of the stationary phase method and M. G. Krein’s theory is proposed. The problem is model for investigating the asymptotic behavior of solutions to factorization problems with several turning points. Power-order complete asymptotic expansions for solutions to the problem under consideration are found. These asymptotics are used to construct asymptotics for solutions to the Cauchy problem for the nonlinear Schrodinger equation at large times.

SAR Imaging of Moving Target Based on Quadratic Phase Function

In this paper, a novel signal processing technique hasbeen developed to refocus moving targets image from their smeared responses in the Synthetic Aperture Radar (SAR) image according to the characteristics of the received signals for moving targets. Quadratic Phase Function is introduced to the parameters estimation for moving target echo and SAR imaging. Our method is available even under a low SNR environment and acquiring an exact SAR image of moving targets. The simulated results demonstrated the validity of the algorithm proposed.

A Parameter Estimation Method for Poly-Phase Coded Signal Based on the Modified Fractional Fourier Transform

To estimate parameters of poly-phase coded radar signals, a modified Fractional Fourier transform (FrFT) is advanced. This algorithm employs Integrated Quadratic Phase function (IQPF) to estimate frequency rate with which the best degree for Frft is computed. Its computational cost is lower than that of Radon-Wigner transform (RWT) and Radon-Ambiguity transform (RAT). When the signal-to-noise ratio (SNR) is -7 dB, Simulations show its parameter estimation performance closes to the Cramer-Rao Lower Bounds (CRLB).

Fresnel incoherent correlation holography (FINCH): a review of research

Abstract In this review, we describe our method for creating holograms of incoherent objects, dubbed Fresnel incoherent correlation holography (FINCH). FINCH creates holograms by a single-channel on-axis incoherent interferometer process. Like any Fresnel hologram, the object is correlated with quadratic phase functions, but the correlation is carried out without any movement. Generally, in the FINCH system, light is reflected, or emitted, from a three-dimensional (3D) object, propagates through a spatial light modulator (SLM), and is recorded by a digital camera. The SLM is used as a beam-splitter of the single-channel incoherent interferometer, such that each spherical beam originated from each object point is split into two spherical beams with two different curve radiuses. Incoherent summing of the entire interferences between all the couples of the spherical beams creates the Fresnel hologram of the observed 3D object. When this hologram is reconstructed in the computer, the 3D properties of the object are revealed. In this review, we describe various aspects of FINCH which have been described recently, including FINCH of reflected white light, FINCH of fluorescence objects, a FINCH-based holographic fluorescence microscope, a FINCH configuration which capitalizes on the polarization sensitivity of the SLM and finally FINCH is analyzed in view of linear system theory.

ON THE HERMAN–KLUK SEMICLASSICAL APPROXIMATION

For a subquadratic symbol H on ℝd× ℝd= T*(ℝd), the quantum propagator of the time dependent Schrödinger equation [Formula: see text] is a Semiclassical Fourier-Integral Operator when Ĥ = H(x, ℏDx) (ℏ-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when ℏ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by H, with a full uniform asymptotic expansion in ℏ for the amplitude.In this paper, our goal is to revisit this well-known and fundamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global L2estimates when ℏ is small and time t is large. One of the simplest choice of the phase is known in chemical physics as Herman–Kluk formula. Moreover, we prove that the semiclassical expansion for the propagator is valid for [Formula: see text] where δ &gt; 0 is a stability parameter for the classical system.

AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM

AbstractThere has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms $U^d(G)$, $d=1,2,3,\dots$, on a finite additive group $G$; in particular, to detect arithmetic progressions of length $k$ in $G$ it is important to know under what circumstances the $U^{k-1}(G)$ norm can be large.The $U^1(G)$ norm is trivial, and the $U^2(G)$ norm can be easily described in terms of the Fourier transform. In this paper we systematically study the $U^3(G)$ norm, defined for any function $f:G\to\mathbb{C}$ on a finite additive group $G$ by the formula\begin{multline*} \qquad\|f\|_{U^3(G)}:=|G|^{-4}\sum_{x,a,b,c\in G}(f(x)\overline{f(x+a)f(x+b)f(x+c)}f(x+a+b) \\ \times f(x+b+c)f(x+c+a)\overline{f(x+a+b+c)})^{1/8}.\qquad \end{multline*}We give an inverse theorem for the $U^3(G)$ norm on an arbitrary group $G$. In the finite-field case $G=\mathbb{F}_5^n$ we show that a bounded function $f:G\to\mathbb{C}$ has large $U^3(G)$ norm if and only if it has a large inner product with a function $e(\phi)$, where $e(x):=\mathrm{e}^{2\pi\ri x}$ and $\phi:\mathbb{F}_5^n\to\mathbb{R}/\mathbb{Z}$ is a quadratic phase function. In a general $G$ the statement is more complicated: the phase $\phi$ is quadratic only locally on a Bohr neighbourhood in $G$.As an application we extend Gowers's proof of Szemerédi's theorem for progressions of length four to arbitrary abelian $G$. More precisely, writing $r_4(G)$ for the size of the largest $A\subseteq G$ which does not contain a progression of length four, we prove that$$ r_4(G)\ll|G|(\log\log|G|)^{-c}, $$where $c$ is an absolute constant.We also discuss links between our ideas and recent results of Host, Kra and Ziegler in ergodic theory.In future papers we will apply variants of our inverse theorems to obtain an asymptotic for the number of quadruples $p_1\ltp_2\ltp_3\ltp_4\leq N$ of primes in arithmetic progression, and to obtain significantly stronger bounds for $r_4(G)$.

Direct measurement of spectral phase for ultrashort laser pulses

We present an intuitive pulse characterization method that provides an accurate and direct measurement of the spectral phase of ultrashort laser pulses. The method requires the successive imposition of a set of quadratic spectral phase functions on the pulses while recording the corresponding nonlinear spectra. The second-derivative of the unknown spectral phase can be directly visualized and extracted from the experimental 2D contour plot, without any inversion algorithm or mathematical manipulation.