Quadratic Doppler Research Articles

Coarse-to-fine coherent integration method for radar maneuvering target with complex motion

In recent years, target detection has drawn increasing attention in the field of radar signal processing. We consider the problem of coherent integration for detecting a highly maneuvering target with complex motion, involving range migration (RM), linear Doppler frequency migration, and quadratic Doppler frequency migration within the coherent interval. Owing to the different sensitivity of the envelope and phase to the target motion, the parametric and nonparametric models are, respectively, established. A coarse-to-fine integration algorithm is correspondingly proposed. Specifically, the keystone transform and fold factor searching are first utilized to remove the parametric RM and to achieve coarse integration. Then, the embedded phase gradient autofocus is applied to estimate the nonparametric nonlinear Doppler phase and realize refined integration. Compared with several representative algorithms, the proposed method could achieve a good trade-off between computational complexity and detection performance. Moreover, this method can be easily extended to detect a high-order maneuvering target. Finally, both simulations and real data experiments are presented to demonstrate the effectiveness of the proposed method.

A Fast Coherent Integration Algorithm for Maneuvering Target Detection

This paper investigates the coherent integration problem in the maneuvering target detection. We propose a new coherent integration algorithm based on the short-time Fourier transform (STFT) and the cubic phase function (CPF), i.e., STFT-CPF. First, the STFT is employed to construct a special compensation function for the linear range migration. Then, the CPF is applied to estimate the target’s acceleration and jerk, thereby compensating the linear Doppler frequency migration and the quadratic Doppler frequency migration. After that, the coherent integration can be achieved by the FT. The proposed algorithm involves only 1-D search. It reaches a good balance between the computational cost and detection performance in a certain range of acceleration and jerk. Simulations and real data demonstrate the effectiveness of the proposed algorithm.

Ground-Based Radar Detection for High-Speed Maneuvering Target via Fast Discrete Chirp-Fourier Transform

Detection of high-speed maneuvering targets has attracted a great deal of attention recently. There are two main problems to be solved: improving detection ability under the condition of complicated range migration and Doppler frequency migration effects, and reducing computational load. Different from most existing fast algorithms which are at the cost of detection ability, this paper devises a computationally attractive method with excellent detection performance. First, the keystone transform is carried out to remove linear range migration. Thereafter, a fast discrete chirp-Fourier transform (FDCFT) based on radix-4 decomposition is proposed to compensate the undersampled linear Doppler frequency migration and quadratic Doppler frequency migration. Because of exploiting inherent symmetry and periodicity as in the fast Fourier transform (FFT), the FDCFT can largely reduce the computational complexity without performance loss. The novelty of the proposed algorithm lies in combining linear transform with the concept of decimation-in-time FFT, which avoids the demanding multi-dimensional search and severe performance loss via introducing nonlinear transforms. It is shown that the proposed method has an approximately optimal detection performance but with relatively low computational cost.

Radar maneuvering target detection based on two steps scaling and fractional Fourier transform

• The problems of quadratic range migration and Doppler frequency migration for the maneuvering target are addressed. • A method based on TSS-FrFT is proposed to achieve the coherent integration and it has superior detection performance and low computational complexity. • The acceleration matched filter is presented to remove the DFRC caused by the sudden change of targets acceleration. This paper considers the detection problem of radar maneuvering target with the range migration (RM) and Doppler frequency migration (DFM) and proposes an coherent integration method based on two steps scaling and fractional Fourier transform, i.e., TSS-FrFT. To be specific, this method corrects the quadratic RM (QRM) and DFM based on FrFT and a scaling processing of the rotation angle firstly. Then, it eliminates the coupling between the range frequency variable and FrFT domain (FrFD) variable via a second scaling processing. In the end, the coherent integration is achieved by the inverse Fourier transform (IFT) with respect to the range frequency variable. Compared with the existing methods, the proposed TSS-FrFT method has superior detection performance and low computational complexity. Moreover, considering the sudden change of the target acceleration within the coherent integration interval, this paper also presents an acceleration matched filter (AMF) to remove the Doppler frequency rate change (DFRC) caused by this acceleration change. Computer simulation results are also given to demonstrate the effectiveness of the proposed method from different aspects, i.e., computational complexity, coherent integration for a single target and multiple targets and detection performance for a uniform acceleration and non-uniform accelerations.

Mass defect effects in atomic clocks

We consider some implications of the mass defect on the frequency of atomic transitions. We have found that some well-known frequency shifts (the gravitational shift and motion-induced shifts such as quadratic Doppler and micromotion shifts) can be interpreted as consequences of the mass defect in quantum atomic physics, i.e. without the need for the concept of time dilation used in special and general relativity theories. Moreover, we show that the inclusion of the mass defect leads to previously unknown shifts for clocks based on trapped ions.

Effects of mass defect in atomic clocks

We consider some implications of the mass defect on the frequency of atomic transitions. We have found that some well-known frequency shifts (such as gravitational and quadratic Doppler shifts) can be interpreted as consequences of the mass defect, i.e., without the need for the concept of time dilation used in special and general relativity theories. Moreover, we show that the inclusion of the mass defect leads to previously unknown shifts for clocks based on trapped ions..

A low complexity coherent integration method for maneuvering target detection

This paper considers the coherent integration problem for detecting a maneuvering target with complex motions, where the velocity, acceleration and jerk result in respectively the range migration (RM), linear Doppler frequency migration (LDFM) and quadratic Doppler frequency migration (QDFM) within the coherent pulse interval. A new coherent integration algorithm based on keystone transform (KT) and generalized dechirp process (GDP), i.e., KTGDP, is proposed. In this method, KT and fold factor searching are first employed to correct the RM, and then GDP is applied to estimate the target's radial acceleration and jerk. With the estimated motion parameters, LDFM and QDFM can be compensated and the coherent integration can be achieved via Fourier transform. In addition, at the cost of some performance loss, a fast coherent integration method combing KT and cubic phase function (CPF), i.e., KTCPF, is also introduced to further reduce the computational complexity. Compared with the generalized Radon–Fourier transform (GRFT) method, the proposed algorithms can avoid the blind speed side lobe (BSSL) effect and have much lower computational burden. Finally, we evaluate the performance via some numerical simulations.

Spaceborne/airborne bistatic synthetic aperture radar focusing on an analytical bistatic point target reference spectrum

An analytical bistatic point target reference spectrum (BPTRS) is proposed. The BPTRS is derived from the one-to-one relationship between slow time and Doppler frequency with no Taylor series expansions. Based on the BPTRS, a two-dimensional frequency-domain focusing algorithm for spaceborne/airborne bistatic synthetic aperture radar is developed. The algorithm exploits a range-dependent sub-blocks processing method to perform range cell migration correction and quadratic Doppler term compensation, and then a scaled inverse Fourier transformation and a sinc interpolation are used to correct the geometric distortions in azimuth and range, respectively. Some simulation experiments are also presented to verify their performance.

Joint Data Detection and Channel Estimation for Fading Unknown Time-Varying Doppler Environments

This work considers a joint channel estimation and data detection technique for Multiple Space-Time Trellis Codes (MSTTCs) operating over unknown time-varying channels with large Doppler spread. We propose an algorithm, called Doppler Adaptive Smoothed Data Detection and Kalman Estimation (DA-SDD-KE), that jointly detects data and estimates the channel as well as the time-varying Doppler. In this scheme, an Adaptive Kalman Predictor (AKP) consisting of a KP and a covariance-based Doppler estimator is incorporated into a Per-Survivor Processing (PSP)-based algorithm that utilizes the past, present and future received symbols for smoothed data detection. For comparison purposes, we also develop a Doppler Adaptive version of the Delayed Mixture Kalman Filtering (DMKF) technique, referred to as DA-DMKF, where the adaptive estimations of the channel and the Doppler shift are based on sequences of importance samples. Moreover, we propose a model for generating a Rayleigh fading process with time-varying Doppler using the sum of sinusoids method. The performance of the DA-SDD-KE and DA-DMKF algorithms over channels with constant, linear and quadratic Doppler functions is evaluated using computer simulations, revealing that the DA-SDD-KE algorithm performs well for all considered Doppler functions, and provides a considerably gain over the DA-DMKF algorithm.

Compensation of systematic errors in a compact atomic beam frequency standard due to evanescent field inside a Ramsey cavity

Systematic errors due to evanescent field inside an E-plane type Ramsey cavity are corrected by introducing an extended square pulse profile in our Ramsey spectral analysis. This procedure is crucial for the proper operation of thermal beam based atomic frequency standards with short cavities. By accounting for the spatial extent of the field inside the cavity, the method enables a correct calculation of transition probability for atoms at the interaction region. The authors demonstrate that the systematic errors in quadratic Doppler and end-to-end cavity phase shifts can be reduced by an order of magnitude.

Interaction of a standing frequency-modulated wave with a gas in a flight region

A formula is obtained for the power of the nth harmonic absorbed upon interaction of a standing frequency-modulated wave with a gas for an arbitrary modulation frequency and amplitude of the frequency deviation. The flight effects, the quadratic Doppler effect, and the recoil effect are taken into account.

On the Cross‐Correlation between Dark Matter Distribution and Secondary Cosmic Microwave Background Anisotropies of the Ionized Intergalactic Medium

We investigate a cross-correlation between the weak gravitational lensing field of the large-scale structure κ and the secondary temperature fluctuation field Δ of the cosmic microwave background (CMB) induced by Thomson scattering of CMB photons off the ionized medium in the mildly nonlinear structure. The cross-correlation is expected to observationally unveil the biasing relation between the dark matter and ionized medium distributions in the large-scale structure. We develop a formalism for calculating the cross-correlation function and its angular power spectrum based on the small angle approximation. As a result, we find that the leading contribution to the cross-correlation comes from the secondary CMB fluctuation field induced by the quadratic Doppler effects of the bulk velocity field v of the ionized medium since the cross-correlations with the O(v) Doppler effect and the Ostriker-Vishniac effect of O(vδ) are suppressed on relevant angular scales because the linear dependence on the bulk velocity field leads to cancellations between positive and negative contributions among the scatterers for the ensemble average. The magnitude of the cross-correlation can be estimated as κΔ ~ 10-10 on small angular scales (l 1000) under the currently favored cold dark matter model of the structure formation and the simplest scenario of homogeneous reionization after a given redshift of zion 5. Although the magnitude turns out to be small, we find several interesting aspects of the effect. One of them is that the density-modulated quadratic Doppler effect of O(δev2) produces the cross-correlation with the weak lensing field that has linear dependence on the electron density fluctuation field δe or, equivalently, on the biasing relation between the dark matter and electron distributions. In other words, the angular power spectrum could be either positive or negative, depending on the positive biasing or antibiasing, respectively. Detection of the cross-correlation thus offers a new opportunity to observationally understand the reionization history of intergalactic medium connected to the dark matter clustering.

Post‐Newtonian Theory for Precision Doppler Measurements of Binary Star Orbits

The determination of velocities of stars from precise Doppler measurements is described here using a relativistic theory of astronomical reference frames to determine the Keplerian and post-Keplerian parameters of binary systems. Seven reference frames are introduced: (1) the proper frame of a particle emitting light, (2) the star-centered reference frame, (3) the barycentric frame of the binary, (4) the barycentric frame of the Galaxy, (5) the barycentric frame of the solar system, (6) the geocentric frame, and (7) the topocentric frame of observer at the Earth. We apply successive Lorentz transformations and the relativistic equation of light propagation to establish the exact treatment of Doppler effect in binary systems both in special and general relativity theories. As a result, the Doppler shift is a sum of (1) linear in c-1 terms, which include the ordinary Doppler effect and its variation due to the secular radial acceleration of the binary with respect to observer; (2) terms proportional to c-2, which include the contributions from the quadratic Doppler effect caused by the relative motion of binary star with respect to the solar system, the motion of the particle emitting light and diurnal rotational motion of observer, orbital motion of the star around the binary's barycenter, and the orbital motion of the Earth; and (3) terms proportional to c-2, which include the contributions from redshifts due to fields of the star, the star's companion, the Galaxy, the solar system, and the Earth. After parameterization of the binary's orbit, we find that the presence of periodically changing terms in the Doppler shift enables us to disentangle different terms and measure, along with the well-known Keplerian parameters of the binary, four additional post-Keplerian parameters, which characterize (1) the relativistic advance of the periastron; (2) a combination of the quadratic Doppler and shifts associated with the orbital motion of the primary relative to the binary's barycenter and the companion's field, respectively; (3) the amplitude of the gravitational lensing contribution to the Doppler shift; and (4) the usual inclination angle of the binary's orbit, i. We briefly discuss the feasibility of practical implementation of these theoretical results, which crucially depends on further progress in the technique of precision Doppler measurements.

Frequency shifts of a methane standard

Frequency shifts are calculated for a methane standard the reproducing the frequency of the molecular transition generating λ = 3.39 μm radiation by the methods of absorption saturation and dispersion. Allowance is made for the simultaneous influence of the quadratic Doppler effect, of splitting because of the recoil effect, of collisional and transit broadening, and of a magnetic hyperfine structure of the spectral line.

Transit-time effects and quadratic Doppler effect in saturated absorption and dispersion spectroscopy

An investigation is made of the resonance profile in saturated absorption and dispersion spectroscopy for arbitrary saturation of the absorption, subject to allowance for transit-time effects and the quadratic Doppler effect. Expressions are obtained for saturated absorption and dispersion signals in the rate approximation for a beam of rectangular shape in the case of two opposite waves with very different intensities and a standing wave. The widths of the resonances and the maximum resolving power of saturated absorption and dispersion spectroscopy are considered. General expressions are obtained for the frequency shifts, due to the quadratic Doppler effect, of stabilized lasers in the case of arbitrary values of the absorption saturation, frequency deviation, and number of the signal harmonic used in frequency stabilization. An analysis of these expressions is made in some limiting cases.

High-resolution high-contrast excitation of calcium in separated fields

A procedure is suggested for compensating the broadening due to the quadratic Doppler effect and for band shifts caused by phase mismatch in the observation of a Ramsey signal in an atomic calcium beam. The relative error in the frequency stabilized by this method is estimated to be 2 × 10− 12.

Influence of the quadratic Doppler effect on the shift of a dispersion resonance in the transit limit

An expression is obtained for the shift of a dispersion resonance because of the quadratic Doppler effect in the transit limit.

Second-order Doppler corrections to the optical susceptibility for two counter-propagating beams

The authors consider the effect of the quadratic Doppler term on the inhomogeneous complex optical susceptibility of a two-level system for a weak beam in the presence of a saturating, counter-propagating wave, with both waves at the same frequency. There is a red shift of the resonance frequency in the absorption as well as in the dispersion. That red shift is a function of the temperature of the medium and the degree of saturation. An asymmetry of the absorption and dispersion curves is also observed which becomes stronger when the authors approach relativistic velocities of the particles. They also study for comparison purposes the effect of the quadratic Doppler term in the particular case of only one strong beam, without the weak one. They show the different behaviour of the red shift between the two cases.

Nonlinear resonances in a system of separated optical fields under the influence of the recoil and quadratic Doppler effects

A theoretical analysis is made of a resonant interaction of two-level atoms with the field of three strong spatially separated standing waves. It is shown that this interaction should give rise to a resonance of width equal to the reciprocal of the transit time between the fields and this resonance should be observed for all rational values of the ratio of the distances by which the waves are separated in space. Calculations are made of the corrections to the flux of particles excited in an atomic beam and to the power absorbed by a gas because of the interaction of atoms with the three fields. It is found that because of recoil a resonance of this kind splits into an infinite series of components separated by intervals of the order of the recoil frequency. The results are given of a numerical calculation of the strongest components of this resonance. The shift and deformation of lines due to quadratic Doppler effect are considered. The results of the calculations are compared with the available experimental data.

Collinear laser spectroscopy of the Balmer-α line

A pilot study on the usability of collinear laser spectroscopy in fast beams for metrological purposes is reported. The method combines high resolution and high sensitivity, and allows the measurements on free atoms without perturbing interactions. The Balmer-α line was choosen for this purpose. Possible applications include the determination of the Rydberg constant, the quadratic Doppler effect and high precision measurements of high voltages.