Complex Basis Function Research Articles

Multidimensional Magnitude-Selective Affine-Function-Based Behavioral Model for Multiband Digital Predistortion of RF Power Amplifiers

A kind of novel general multiband digital predistortion (DPD) model is proposed for the linearization of the multifrequency transmitters. By the theoretical analysis, we extend the existing instantaneous sample indexed magnitude-selective affine (IMSA) function-based model to multidimensional cases, which includes the 2-D-IMSA model and the 3-D-IMSA model for concurrent dual-band and triband scenarios, respectively. The core of the new models is to use the multidimensional IMSA function to replace the complex basis function of the classic multiband model through the method of segmentation and approximation. After segmentation, the proposed models have a simple linear structure in each small area, which greatly reduces the model complexity. Experiments show that the proposed models can give similar normalized mean square error and DPD performance in comparison with existing classic models when a power amplifier is excited by two 20-MHz signals in the concurrent dual-band scenario and by three 20-MHz signals in the concurrent triband scenario.

Uniform vs Partial Scaling within Resonances via Padé Based on the Similarities to Other Non-Hermitian Methods: Illustration for the Beryllium 1s22p3s State.

Resonance via Padé (RVP) is an efficient method for calculating autoionization resonance states. It is based on the stabilization technique in which the basis set is scaled. The scaling can be uniform (i.e., all basis functions are scaled) or partial. Herein, we compare the two RVP scaling schemes for calculating an autoionization eigenvalue; moreover, the effect of freezing the core electrons is intertwined within this comparison. In order to study the different behavior of the RVP schemes, we associate each RVP scaling scheme with a complex contour of integration. Similarities between RVP and other non-Hermitian methods emerge from the generated contours, which suggest that RVP introduces similar outgoing boundary conditions as the complex scaling (CS), complex basis function (CBF), and reflection-free complex absorbing potential (RF-CAP) methods. A uniform-RVP contour, unlike a partial one, immediately penetrates the complex plane and influences the interaction region. Hence, uniform scaling within RVP destroys the description of the core electrons, as well as the description of the reference state, and yields less reliable results than partial scaling. The 1s22p3s 1P autoionization state of Be, at the equation-of-motion coupled-cluster level, is used as our case study model.

A General Order Reduction Method of Wideband Digital Predistortion Model Using Attention Mechanism

In wireless networks, for the common in‐phase and quadrature‐phase (I/Q) imbalance in the transmitters, the I/Q branch models of digital predistortion (DPD) need to be identified separately, to improve the linearization effects. The existing order reduction methods of the predistorter are based on the contributions of the complex basis function terms, so as not to deal with the different contributions of I/Q components of the complex basis function terms caused by the separate identification of the I/Q branch models. The separate pruning of the I/Q branch models will increase the complexity. Aiming at this issue, this paper proposes a general order reduction method based on the attention mechanism for the predistortion of the power amplifiers (PAs). This method is suitable for pruning both the traditional models and neural network‐based models. In this method, the attention mechanism is used to evaluate the contributions of the real basis function terms to the predistorted output’s I/Q components through offline training, and the influence of the cross terms of the I/Q branch models is considered. The experimental results based on the comparison with other typical methods under 100 MHz Doherty PA and different I/Q imbalance levels show that this method has superior pruning performance and good linearization ability.

K-Shell photoionization cross sections of one-electron uranium cation by relativistic complex basis function method

The complex basis function (CBF) method is a square-integrable (L2) function method that represents continuum wave functions, and has been applied successfully to various problems, including the calculation of resonance state energies and photoionization cross sections. The previous applications of this method have been limited to non-relativistic problems, and no serious attempt has been made for the relativistic Hamiltonian. In this study, we calculated the photoionization cross sections from the imaginary part of the frequency-dependent polarizability by applying the CBF method to the inhomogeneous Dirac equation of the one-electron uranium cation. The results obtained by employing a pure Coulomb potential to the K-shell photoionization cross sections were in good agreement with previous theoretical values and experimental values for the neutral uranium atom. Phase shifts were obtained by matching the CBF solutions with the numerical ones, which were further propagated by the fourth-order Runge–Kutta method, and thus, the photoionization differential and total cross sections were successfully calculated. The calculated results were analyzed from various viewpoints, including the multipole effect of electromagnetic fields, momentum conservation of photon and photoelectron, interference of multiple perturbation terms, and difference in the orbital angular momentum of the small component wave functions. These analyses demonstrate the effectiveness of the CBF method for the inhomogeneous Dirac equation and thus its applicability to multielectron systems.

Phase Transmittance RBF Neural Network Beamforming for Static and Dynamic Channels

This letter proposes a novel beamforming technique for self-organizing wireless networks, such as those encountered in fifth generation (5G) and the Internet of Things (IoT). The proposed beamformer is based on a complex radial basis function artificial neural network (ANN), which allows phase transmittance between the input and output nodes. The beamforming algorithm is applied to a six-element uniform circular array, considering the mutual coupling between elements. The proposed technique is evaluated over critical static and dynamic scenarios. The static scenario approaches a multiuser environment, highly polluted with electromagnetic interference and plenty of severe in-band undesired signals. The dynamic scenario emulates a close air support military operation, in which an aircraft moves around the ground station at a high velocity. In both scenarios, nonlinearities at the receiver analog RF front end with a –10 dB signal-to-interference ratio (SIR) are considered. Results show that the proposed technique presents significantly a superior performance when compared to other solutions. For the static scenario, the novel beamformer is able to focus the array radiation pattern in the direction of the desired signal, achieving a zero symbol error rate. For the dynamic scenario, the beamforming is able to track the moving station, achieving a low symbol error rate for a wide angular range.

A Schwarz inequality for complex basis function methods in non-Hermitian quantum chemistry.

A generalization of the Schwarz bound employed to reduce the scaling of quantum-chemical calculations is introduced in the context of non-Hermitian methods employing complex-scaled basis functions. Non-Hermitian methods offer a treatment of molecular metastable states in terms of L2-integrable wave functions with complex energies, but until now, an efficient upper bound for the resulting electron-repulsion integrals has been unavailable due to the complications from non-Hermiticity. Our newly formulated bound allows us to inexpensively and rigorously estimate the sparsity in the complex-scaled two-electron integral tensor, providing the basis for efficient integral screening procedures. We have incorporated a screening algorithm based on the new Schwarz bound into the state-of-the-art complex basis function integral code by White, Head-Gordon, and McCurdy [J. Chem. Phys. 142, 054103 (2015)]. The effectiveness of the screening is demonstrated through non-Hermitian Hartree-Fock calculations of the static field ionization of the 2-pyridoxine 2-aminopyridine molecular complex.

Calculation of photoionization differential cross sections using complex Gauss-type orbitals.

Accurate theoretical calculation of photoelectron angular distributions for general molecules is becoming an important tool to image various chemical reactions in real time. We show in this article that not only photoionization total cross sections but also photoelectron angular distributions can be accurately calculated using complex Gauss-type orbital (cGTO) basis functions. Our method can be easily combined with existing quantum chemistry techniques including electron correlation effects, and applied to various molecules. The so-called two-potential formula is applied to represent the transition dipole moment from an initial bound state to a final continuum state in the molecular coordinate frame. The two required continuum functions, the zeroth-order final continuum state and the first-order wave function induced by the photon field, have been variationally obtained using the complex basis function method with a mixture of appropriate cGTOs and conventional real Gauss-type orbitals (GTOs) to represent the continuum orbitals as well as the remaining bound orbitals. The complex orbital exponents of the cGTOs are optimized by fitting to the outgoing Coulomb functions. The efficiency of the current method is demonstrated through the calculations of the asymmetry parameters and molecular-frame photoelectron angular distributions of H2+ and H2. In the calculations of H2, the static exchange and random phase approximations are employed, and the dependence of the results on the basis functions is discussed. © 2017 Wiley Periodicals, Inc.

Optimization of complex slater-type functions with analytic derivative methods for describing photoionization differential cross sections.

The complex basis function (CBF) method applied to various atomic and molecular photoionization problems can be interpreted as an L2 method to solve the driven-type (inhomogeneous) Schrödinger equation, whose driven term being dipole operator times the initial state wave function. However, efficient basis functions for representing the solution have not fully been studied. Moreover, the relation between their solution and that of the ordinary Schrödinger equation has been unclear. For these reasons, most previous applications have been limited to total cross sections. To examine the applicability of the CBF method to differential cross sections and asymmetry parameters, we show that the complex valued solution to the driven-type Schrödinger equation can be variationally obtained by optimizing the complex trial functions for the frequency dependent polarizability. In the test calculations made for the hydrogen photoionization problem with five or six complex Slater-type orbitals (cSTOs), their complex valued expansion coefficients and the orbital exponents have been optimized with the analytic derivative method. Both the real and imaginary parts of the solution have been obtained accurately in a wide region covering typical molecular regions. Their phase shifts and asymmetry parameters are successfully obtained by extrapolating the CBF solution from the inner matching region to the asymptotic region using WKB method. The distribution of the optimized orbital exponents in the complex plane is explained based on the close connection between the CBF method and the driven-type equation method. The obtained information is essential to constructing the appropriate basis sets in future molecular applications. © 2017 Wiley Periodicals, Inc.

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method**Project supported by the National Natural Science Foundation of China (Grant Nos. 11171208 and U1433104).

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional (2D) elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.

An Efficient Transceiver Scheme for SC-FDMA Systems Based on Discrete Wavelet Transform and Discrete Cosine Transform

In this paper, the discrete cosine transform (DCT) is utilized for single carrier frequency division multiple access (SC-FDMA) transmission. Firstly, an improved DCT-based SC-FDMA (DCT SC-FDMA) system is introduced. Then, the paper presents a new transceiver architecture design for the uplink SC-FDMA system that implements a cosine basis function or a complex exponential Fourier basis function. The proposed architecture uses the discrete wavelet transform and a hybrid companding and clipping method for peak-to-average power ratio (PAPR) reduction. From the simulation results, it is shown that the DCT SC-FDMA system achieves a superior bit error rate (BER) performance than that of the discrete Fourier transform based SC-FDMA (DFT SC-FDMA) system. The results also show that the proposed wavelet based transceiver architectures for the DFT SC-FDMA system and the DCT SC-FDMA system can provide a better BER performance and a lower PAPR than the conventional DFT SC-FDMA system.

Complex basis functions revisited: implementation with applications to carbon tetrafluoride and aromatic N-containing heterocycles within the static-exchange approximation.

The method of complex basis functions for computing positions and widths of molecular resonances is revisited. An open-ended and efficient implementation is described. The basis set requirements of the complex basis are investigated within the computationally inexpensive static-exchange approximation, and the results of this investigation lead to a hierarchy of basis sets for complex basis function calculations on small molecules. These basis sets are then applied in static-exchange calculations on some larger molecules with multiple low energy shape resonances: carbon tetrafluoride, benzene, pyridine, pyrimidine, pyrazine, and s-triazine. The results indicate that more sophisticated methods using complex basis functions are worth pursuing in the search for accurate and computationally feasible methods for computing resonance energies in molecular systems.

Construction of complex STO-NG basis sets by the method of least squares and their applications

Electronic resonance state energies and photoionization cross sections of atoms and molecules are calculated with the complex basis function method by using mixture of appropriate complex basis functions representing one-electron continuum orbitals and the usual real basis functions for the remaining bound state orbitals. The choice of complex basis functions has long been a central difficulty in such calculations. To address this challenge, we constructed complex Slater-type orbital represented by N-term Gaussian-type orbitals (cSTO-NG) basis sets using the method of least squares. Three expansion schemes are tested: (1) expansion in complex Gaussian-type orbitals, (2) expansion in real Gaussian-type orbitals, and (3) expansion in even-tempered real Gaussian-type orbitals. By extending the Shavitt–Karplus integral transform expression to cSTO functions, we have established a mathematical foundation for these expansions. To demonstrate the efficacy of this approach, we have applied these basis sets to the calculation of the lowest Feshbach resonance of H2 and the photoionization cross section of the He atom including autoionization features due to doubly excited states. These calculations produce acceptably accurate results compared with past calculations and experimental data in all cases examined here.

Complex configuration interaction calculations of the cross section for the dissociative electron attachment process e− + F2 → F 2− → F + F− using the complex basis function method

The F(2)(-) molecule and the corresponding dynamic processes dealing with electron scattering on the neutral F(2) species have been the subject of many theoretical and experimental investigations in the past. In the context of the Born-Oppenheimer approximation, one of the best theoretical descriptions of the electronic states involves the use of complex basis functions together with configuration interaction (CI) methods. In this work, multireference CI calculations using the complex basis function method have been carried out for the autoionizing ground state of the F(2)(-) molecule. Potential curves and vibrational levels have been obtained for the ground and various excited states of both F(2) and F(2)(-), as well as the variation of the line width of the anionic ground state for the bond distance region in which it is metastable. Cross sections for the dissociative electron attachment process e(-) + F(2) → F(2)(-) → F + F(-) have also been computed within the framework of the boomerang model, and good agreement with available experimental data has been found. In addition, some calculations for the process of vibrational excitation are included which also give good agreement with experiment.

Use of complex configuration interaction calculations and the stationary principle for the description of metastable electronic states of HCl−

The complex multireference single- and double-excitation configuration interaction method has been employed to compute potential curves for the anion of the hydrogen chloride molecule. First, conventional CI calculations with real basis functions have been carried out to determine the potential curves of both HCl and its anion over a large range of internuclear distance. It is shown that adding basis functions with very small exponents leads to sharply avoided crossings for the HCl(-) potentials that greatly complicate the search for resonance states thought to be responsible for features observed in electron collision experiments. By limiting the number of such diffuse-type functions it is possible to describe resonance states at a highly correlated level and still account for their interaction with the continuum in which they are embedded. In the present study of the HCl(-) anion the complex basis function technique of Moiseyev-Corcoran and McCurdy-Resigno is employed to calculate the energy positions and line-widths of the resonance states. Two states of (2)Sigma(+) symmetry are calculated which have potentials that have significantly different shapes than that of the neutral ground state and thus contribute to the cross section for vibrational excitation of the neutral HCl molecule induced by low-energy electron collisions. The lower of these (1 (2)Sigma(+)) correlates smoothly with the bound anionic ground state at large internuclear distances and is seen to be responsible for the sharp peaks observed in the low-energy region of the spectrum. The upper state (3 (2)Sigma(+)) has a much larger bond length and is assigned to the broad bands observed with a maximum in the 2.5-3.0 eV range. The present calculations thus stand in contradiction to earlier claims that the above peaks are caused by so-called virtual states without a definite autoionization lifetime.

Photoionization cross sections of H2+ and H2with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.

Photoionization cross sections with optimized orbital exponents within the complex basis function method

We show a new direction to expand the applicability of the complex basis function method for calculating photoionization cross sections through the imaginary part of the frequency-dependent polarizability. Based on the variational stability of the frequency-dependent polarizability, we made nonlinear optimizations of complex orbital exponents in basis functions representing continuum wave functions, and obtained fairly accurate results for H atom with only one or two complex basis functions particularly with dipole velocity gauge. Results were almost independent of whether Slater-type or Gaussian-type orbitals are used, implying the applicability to general many electron problems. The method was also applied to the (1)S (1s)(2) --> (1)P (1s)(1)(kp)(1) cross section of He atom and the optimized complex orbital exponents were related to those of H atom through the scaling property. The nonlinear optimizations have converged smoothly and the cross sections were in excellent agreement with experiment throughout wide photon energies, which suggest the effectiveness of the approach for many-electron systems.

Calculations of photoionization cross-sections with variationally optimized complex Gaussian-type basis functions

Photoionization cross-sections of various simple systems were calculated with variationally optimizing the complex orbital exponents in Gaussian-type basis functions within the framework of the complex basis function method. The optimizations were performed with the numerical Newton–Raphson method based on the variational principle for the frequency-dependent polarizabilities. Accurate cross-sections were obtained for the Rydberg autoionization of He and Be, and the photodetachment of H − with the optimizations of only one complex basis function. These results show the effectiveness of the optimization of the complex orbital exponents to reduce the number of basis functions significantly and imply the applicability to general problems by adapting many existing techniques developed in bound state theories.

Complex self-consistent field and multireference single- and double-excitation configuration interaction calculations for the Πg2 resonance state of N2−

Self-consistent field and multireference single- and double-excitation configuration interaction calculations employing the complex basis function technique are carried out for the (2)Pi(g) resonance state of the N(2) (-) molecule as well as several other anionic resonance states in the neighboring energy region. The results of calculations employing the same method for the (1)S (2s(2)) state of the He atom and the (1)Sigma(g) (+) (sigma(u) (2)) state of the H(2) molecule are found to be in good agreement with those of earlier work. The present theoretical treatment has succeeded for the first time in satisfying the rigorous criterion of the complex variational principle in computing the N(2) (-) resonance states, namely, a cusp in the plots of real versus imaginary components of the corresponding complex energies has been located at each internuclear distance. On this basis, it is found that the open-shell orbital in the lowest-energy adiabatic N(2) (-) resonance state of (2)Pi(g) symmetry changes its character from quite compact at large internuclear distance to relatively diffuse for r<2.3a(0). This is in contrast to all previous theoretical treatments of this system that have not rigorously satisfied the complex variational principle in their determination of this wave function.

Valence photoionization and autoionizing states of acetylene studied by the complex basis function method in the random phase approximation

The photoionization cross-section of acetylene has a prominent double-hump structure, the peak positions of which are 13.3 and 15.3 eV. The assignment for the lower peak has long been controversial. In this Letter, we settle the problem by calculating the valence photoionization cross-section with the complex basis function method including the channel coupling in the random-phase approximation. At the neutral ground-state geometry, we assigned the lower peak to the 3σ g→3pσ u Rydberg-type autoionizing state, which is perturbed by the 3σ g→3σ u intravalence excited state. We also studied the geometry dependence of the autoionizing state.

Complex-valued radial basic function network, Part I: Network architecture and learning algorithms

The paper proposes a complex radial basis function network. The network has complex centres and connection weights, but the nonlinearity of its hidden nodes remains a real-valued function. This kind of network is able to generate complicated nonlinear decision surfaces or to approximate an arbitrary nonlinear function in complex multi-dimensional space, and it provides a powerful tool for nonlinear signal processing involving complex signals. The paper is divided into two parts. The first part introduces the network architecture and derives both block-data and recursive learning algorithms for this complex radial basis function network. The complex orthogonal least squares algorithm is a batch learning algorithm capable of constructing an adequate network structure, while a complex version of the hybrid clustering and least squares algorithm offers real-time adaptation capability. The identification of a nonlinear communications channel model is used to illustrate these two learning algorithms. In the second part of the paper, a practical application of this complex radial basis function network is demonstrated using digital communications channel equalisation.