Amplitude-phase Method Research Articles

Near/far-side angular decompositions of Legendre polynomials using the amplitude-phase method

A decomposition of Legendre polynomials into propagating angular waves is derived with the aid of an amplitude-phase method. This decomposition is compared with the ’Nussenzveig/Fuller’ so called near/far-side decomposition of Legendre polynomials. The latter decomposition requires the Legendre function of the second kind. This is not the case with the amplitude-phase decomposition. Both representations have the same asymptotic expressions for large values of (l+1/2)sin theta , where l and theta are the polynomial degree and the angle respectively. Furthermore, both components of both representations satisfy the Legendre differential equation. However, we show the two representations are not identical.

Coherence-based phase unwrapping for broadband signals

In this paper, a coherence-based method for unwrapping the relative phase between microphones are investigated. For broadband signals, this method has the potential to lead to more accurate intensity vector estimations using the Phase Amplitude and Gradient Estimator (PAGE) method [D. C. Thomas et al., J. Acoust. Soc. Am. 137, 3366-3376 (2015)] Simple unwrapping methods function by detecting phase jumps above a threshold, which works well for frequencies associated with high signal coherence. However, since unwrapping for these methods is triggered by only one previous frequency data point, frequency regimes of low coherence are often unwrapped incorrectly. By including coherence in a phase unwrapping algorithm, these errors can be reduced. Regions of relatively low coherence are given less weight in phase unwrapping, and are checked for unwrapping errors. For broadband signals with continuous relative phase, using both the coherence and multiple data points to unwrap, frequencies associated with low coherence result in fewer unwrapping errors. Phase values for jet noise data with low coherence (<0.1) have been successfully unwrapped using this method, and have resulted in more reliable PAGE intensity estimates. This paper also investigates unwrapping in interference nulls produced by coherent, radiating sources. [Funded by the NSF.]

Analiza drgań oraz wyważanie urządzeń wirnikowych metodą amplitudowo-fazową z wykorzystaniem czujników i oscyloskopu cyfrowego

Analiza drgań oraz wyważanie urządzeń wirnikowych metodą amplitudowo-fazową z wykorzystaniem czujników i oscyloskopu cyfrowego

Analysis methods of determining velocity distribution waves in the cavitation zone ultrasound field

The main parameters of cavitation processing technology environments is the speed of sound in a cavitation medium pressure, the duration, size: cavitation region, bubbles and core. The most successful use of ultrasound associated with handling liquid media, because in them the phenomenon of ultrasonic cavitation, which is an effective mechanism for concentration of energy of sound waves in the low-density high energy density, which is caused by pulsations and cavitation bubbles popping. An important component in ensuring an effective process of cavitation treatment process fluids is to calculate the ultrasonic technological devices that hold for precavitational mode with the following specification obtained by previous studies. However, reliable results associated with the methods of measuring the velocity of ultrasonic waves in a developed cavitation because it is a part of almost all the defining parameters of ultrasonic cavitation processing technology environments. Measuring the velocity of ultrasonic waves carried pulse, optical, amplitude-phase methods. Pulse methods have a wider dynamic range, measurement accuracy independent of the distance between the transmitter and the object opportunity emitter distance from the subject a considerable distance. Amplitude-phase methods require extended frequency band transceiver emitting element and amplifier path that reduces security and selectivity. Frequency and phase methods have the narrow bandwidth receiving radiating path that provides good value energy radiated and received signals, high noise immunity. Is an effective method for determining the rate for power characteristics, the essence of which is to determine the energy flux density and energy density relative to the ultrasound field. It has been found that the accuracy of measuring the speed of sound depends on whether it should receive the absolute value or relative value can restrict the speed of sound when changing any external parameters. Absolute measurement accuracy is about 10-5, while the relative change as accuracy is much higher, it reaches the order of 10-7. The choice of measurement due to the conditions of the experiment. The essential difference in the numerical values of the speed, complicates the acceptance of an opinion on the actual numerical value of speed and requires more informed research in determining the accepted assumptions and preconditions in the formulation and conduct research.

Non-Relativistic Phase Shifts for Scattering on Generalized Radial Yukawa Potentials

Non-relativistic phase shifts for a generalized Yukawa potential V(r) = − V0(e−αr/r) − V1(e−2αr/r2) are studied by the amplitude-phase method and by a frequently used analytic method based on a Pekeris-type approximation of power-law potential terms. Small variations of V1 seem to have marginal effects on the effective potential and on exact phase shifts. However, as pointed out in this study, a Pekeris-type approximation in scattering applications often implies serious distortions of both effective potentials and phase shifts. The Pekeris-type based analytic approximation in this study seems to give low-quality scattering results for this model potential at low energies.

Tunneling through a barrier with the phase–amplitude method

A previous study (Rawitscher, 2015) of the solution of Milne’s non linear equation for the phase and amplitude of a one-dimensional wave function is extended to the case where the incident energy is less than the potential (Barrier region). The numerical method again consists in implementing a spectral expansion of the amplitude in terms of a number of Chebyshev polynomials. The method is applied to a Morse-type potential, for energies in a resonance region, and for one energy below the resonance region, and a strong repulsive Coulomb potential. The results are compared with highly accurate direct solutions of the Schrödinger equation, and were found to be accurate to 1:10−6.

진폭-위상 복합비교 기법과 상관형 위상비교 기법의 방향탐지 정확도 분석

In this paper, we present the direction finding accuracy of correlative interferometer method and amplitude-phase comparison method. Spiral antennas are used for amplitude-phase comparison method and blade antennas are used for correlative interferometer method. Those are made for uniform circular array (UCA) direction finding antenna systems. We simulate the accuracy of azimuth angle with 3 antennas UCA when SNR is 20 dB and baseline is 0.5 wave length. Correlative interferometer method has better accuracy than amplitude-phase comparison method.

Exact solution to the Coulomb wave using the linearized phase-amplitude method

The author shows that the amplitude equation from the phase-amplitude method of calculating continuum wave functions can be linearized into a 3rd-order differential equation. Using this linearized equation, in the case of the Coulomb potential, the author also shows that the amplitude function has an analytically exact solution represented by means of an irregular confluent hypergeometric function. Furthermore, it is shown that the exact solution for the Coulomb potential reproduces the wave function for free space expressed by the spherical Bessel function. The amplitude equation for the large component of the Dirac spinor is also shown to be the linearized 3rd-order differential equation.

Applications of the amplitude-phase method to symmetric double-well potentials

An amplitude-phase method is used to derive general quantization conditions for energy levels in smooth double-well potentials. The resulting quantization condition is applied to symmetric double-well potentials, where the two types of symmetry levels are shown to be determined by separate quantization conditions.

A spectral Phase–Amplitude method for propagating a wave function to large distances

The phase and amplitude (Ph–A) of a wave function vary slowly with distance, in contrast to the wave function that can be highly oscillatory. Hence the Ph–A representation of a wave function requires far fewer computational mesh points than the wave function itself. In 1930 Milne presented an equation for the phase and the amplitude functions (which is different from the one developed by Calogero), and in 1962 Seaton and Peach solved these equations iteratively. The objective of the present study is to implement Seaton and Peach’s iteration procedure with a spectral Chebyshev expansion method, and at the same time present a non-iterative analytic solution to an approximate version of the iterative equations. The iterations converge rapidly for the case of attractive potentials. Two numerical examples are given: (1) for a potential that decreases with distance as 1/r3, and (2) a Coulomb potential ∝1/r. In both cases the whole radial range of [0–2000] requires only between 25 and 100 mesh points and the corresponding accuracy is between 10−3 and 10−6. The 0th iteration (which is the WKB approximation) gives an accuracy of 10−2. This spectral method permits one to calculate a wave function out to large distances reliably and economically.

Uniform axial intensity distributions of long-focal-depth cylindrical micromirrors realized by an amplitude-phase modulation method

An amplitude-phase modulation method (APMM) is applied to realize the uniform long-focal-depth (LFD) function of metallic cylindrical focusing micromirrors (MCFMs). Based on rigorous electromagnetic theory and the boundary element method, focal properties of the designed MCFMs are characterized, which contain the actual focal depth, the focal spot size, the normalized encircled energy, the real focal length, and the axial intensity uniformity. On appropriately choosing the incident amplitude distribution with apodization, the designed MCFMs demonstrate the LFD properties with uniform intensity distributions. The superiority of the APMM to the pure phase modulation method is revealed in achieving a higher normalized encircled energy.

Hodograph method to estimate the latitudinal profile of the field-line resonance frequency using the data from two ground magnetometers

The hodograph method enables estimating the latitudinal profile of the field-line resonance (FLR) frequency (f R ) using the data from two ground magnetometers. This paper provides the full details of this method for the first time, and uses a latitudinal chain of ground magnetometers to examine its validity and usefulness. The hodograph method merges the widely-used amplitude-ratio and cross-phase methods in a sense that the hodograph method uses both the amplitude ratio and the phase difference in a unified manner; further than that, the hodograph method provides f R at any latitude near those of the two ground magnetometers. It is accomplished by (1) making a complex number by using the amplitude ratio (phase difference) as its real (imaginary) part; (2) drawing thus obtained complex numbers (one number for one frequency) in the complex plane to make a hodograph; and (3) fitting to thus obtained hodograph a model satisfying the FLR condition, which is a circle with the assumption that the resonance width is independent of the latitude. To examine the validity and usefulness of the hodograph method, we apply it to a Pc 4 event observed by the Scandinavian BEAR array. We also apply the amplitude-phase gradient method (Pilipenko and Fedorov, 1994; Kawano et al., 2002) to the same event, and compare the results; this is the first article applying the both methods to the same dataset.

Coupled radial Schrödinger equations written as Dirac-type equations: application to an amplitude-phase approach

The classical amplitude-phase method due to Milne, Wilson, Young and Wheeler in the 1930s is known to be a powerful computational tool for determining phase shifts and energy eigenvalues in cases where a sufficiently slowly varying amplitude function can be found. The key for the efficient computations is that the original single-state radial Schrödinger equation is transformed to a nonlinear equation, the Milne equation. Such an equation has solutions that may or may not oscillate, depending on boundary conditions, which then requires a robust recipe for locating the (optimal) ‘almost constant’ solutions for its use in the method. For scattering problems the solutions of the amplitude equations always approach constants as the radial distance r tends to infinity, and there is no problem locating the ‘optimal’ amplitude functions from a low-order semiclassical approximation. In the present work, the amplitude-phase approach is generalized to two coupled Schrödinger equations similar to an earlier generalization to radial Dirac equations. The original scalar amplitude then becomes a vector quantity, and the original Milne equation is generalized accordingly. Numerical applications to resonant electron–atom scattering are illustrated.

Scattering states of modified Pöschl-Teller potential in D-dimension

We present a new approximation scheme for the centrifugal term, and apply this new approach to the Schrödinger equation with the modified Pöschl—Teller potential in the D-dimension for arbitrary angular momentum states. The approximate analytical solutions of the scattering states are derived. The normalized wave functions expressed in terms of the hypergeometric functions of the scattering states on the k/2π scale and the calculation formula of the phase shifts are given. The numerical results show that our results are in good agreement with those obtained by using the amplitude-phase method (APM).

On relativistic shifts of negative-ion resonances

Results from a Dirac approach to electron- (neutral) atom resonance calculations are compared with those from nonrelativistic ones. A recently developed amplitude-phase method for the Dirac equations that is valid also in the non-relativistic limit is applied. The main conclusion is that significant energy differences are expected for long-lived resonances.

Amplitude-phase calculations of Regge poles obtained from coupled radial Dirac equations

A recently developed amplitude-phase method for spinor-wave solutions is applied to the calculations of Regge pole positions and residues of Dirac particles. At a given energy the Dirac spin causes two sets of Regge poles that tend to coalesce in the non-relativistic limit. For the particular case of equal Lorentz-type vector and scalar potentials there is only one pole string, located very close to the non-relativistic pole string.

Adaptive Nulling with Time-Modulated Antenna Arrays Using a Hybrid Differential Evolution Strategy

A flexible approach based on a hybrid differential evolution strategy was developed for adaptive pattern nulling with time-modulated linear antenna arrays. The hybrid differential evolution strategy adjusts only the least significant bits of the digital phase shifters and the on—off time sequence to minimize the total output power; deep nulls can then be placed in the direction of the interferences without much compromise of the shape of the main beam. Thanks to the new design freedom introduced in time-modulated antenna arrays, this adaptive nulling approach exhibits better performance over the traditional phase-only, amplitude-only, and phase-amplitude adaptive nulling methods, in terms of the practical implementation, nulling depth, and pattern deterioration. Numerical examples are presented to demonstrate the effectiveness and robustness of the proposed approach.

Resonances in electron-capture total cross sections forC4+andB5+collisions with H(1s)

Quantal calculations of electron-capture and elastic cross sections have been carried out for collisions of ${\mathrm{C}}^{4+}$ and ${\mathrm{B}}^{5+}$ with H($1s$) at collision energies $0.00025lEl0.02$ eV, where the cross sections show numerous resonances. Positions and widths of the resonances are studied by using the phase-amplitude method.

Dirac resonance energies for central potentials with different Lorentz-type potential couplings

The amplitude-phase method is applied to relativistic Dirac-particle resonances related to electron–atom collisions. Complex-energy resonance poles of the S matrix in central potentials of differing Lorentz couplings are studied near the non-relativistic limit. It is confirmed that an equal mixture of a Lorentz vector-type potential (with a single time component) and a Lorentz scalar-type potential of the same radial shape makes the interaction essentially spin independent, as if spin does not couple to orbital angular momentum. Hence, resonance poles of the S matrix depend simply on the orbital angular momentum and the radial quantum number in a similar way as in the Schrödinger limit and in the Klein–Gordon equation. In a Lorentz-vector potential model, there is a splitting of pole positions, but the splitting may be surprisingly small, as demonstrated for one of the potentials considered. The numerical method used automatically assigns a vibrational (radial) quantum number to the resonance state, which is usually a characteristic feature of semiclassical methods.

Amplitude-phase formula for the S-matrix derived from invariants of the reduced first-order radial Dirac equation

A formula for calculating the Dirac S-matrix for central Lorentz scalar and vector potentials is derived by use of a new amplitude-phase method. The derivation also makes use of certain invariants of the reduced 2-spinor radial Dirac equations.