Complex-valued Vector Research Articles

Three-dimensional experimental spatial dynamics modelling of a reciprocating Freon compressor housing

Experimental spatial dynamics modelling encompasses the modelling, postprocessing, and visualization of 3D spatial dynamic response of structures as statistically-qualified complex-valued continuous vector fields derived from experimental structural dynamics testing. A weighted least-squares discrete finite element formulation has been developed to reconstruct the continuous 3D velocity response fields from experimental dynamics data. Dynamic 3D rotations, displacements, and accelerations were naturally derived from the full field velocity response model. Also, dynamic strain and stress fields across the shell elements were extracted by directly postprocessing the dynamic response. A Scanning Laser Doppler Vibrometer (SLDV) was used to measure a single frequency of the 3D dynamic response in an in situ test of an operating reciprocating Freon compression. The SLDV measured the velocity response at thousands of locations on the compressor housing from multiple, positionally-registered locations. The geometry from a preexisting finite element model provided the shape model of the compressor. An experimentally derived spatial dynamics model of the compressor housing was solved and postprocessed results were obtained.

Polynomial predictors for complex-valuedvector signals

Polynomial FIR predictors for complex-valued vector signals are derived. It is shown that the generalisation of the polynomial predictor for complex-valued signals results in the same filter as that for real-valued signals. The application of complex-valued vector polynomial prediction is illustrated by prediction of the components of a signal transmitted through a multipath Rayleigh fading channel.

Time-dependent scalar Beltrami-Hertz potentials in free space

We have solved the Beltrami-Maxwell equations for free space in terms of time-dependent scalar functions, the so-called scalar Beltrami-Hertz potentials. The two Beltrami fields have been represented in terms of scalar Beltrami-Hertz potentials. While the method is formulated for general sources, it is at its most powerful when the impressed source current densities are unidirectional: each Beltrami field, a complex-valued vector, can then be derived from a single scalar Beltrami-Hertz potential. We have calculated the corresponding scalar Green function explicity and given closed-form solutions for dipolar sources. Finally, the connection between the Beltrami-Maxwell formalism and conventional electromagnetic theory has been re-affirmed.

The complex Hopfield model

The complex Hopfield model, i.e., with the complex-valued interconnections and the complex-valued vectors, is discussed. Computer simulations in comparison with the Hopfield model are given, which show that the complex Hopfield model can recall the entire complex-valued stored vector (including both the real part and the imaginary part) by only its real part or its imaginary part with high probability (> 95% according to the calculations). The second-order complex Hopfield model, and its computer simulations in comparison with the second-order Hopfield model are also presented.

Topological invariance of the solvability of complex-valued vector fields in the plane

Topological invariance of the solvability of complex-valued vector fields in the plane

Simultaneous Confidence Regions for the Frequency Analysis of Multiple Time Series

Abstract In the frequency analysis of several time series observed simultaneously, it is often necessary to adapt classical results of linear models and multiple regression to situations involving complex-valued variates. Given a real-valued stochastic time series, Y(t), and a suitable vector-valued fixed series, X(t), with components Xi (t) (j = 1, 2, …, r, t = 0, ± 1, …), it may be of interest to assess the effect, in the frequency domain, of each component series of X(t) on the behavior of Y(t). If values of Y(t) and X(t) are made available for t = 0, 1, …, T − 1, then under certain conditions and for large T at frequency λ the finite Fourier transform of Y(t) is approximately linearly related to the finite Fourier transform of X(t) through a complex-valued vector A(λ). In an asymptotic sense, A(λ), with entries Aj (λ) (j = 1, 2, …, r), may be interpreted as a vector of complex regression coefficients of Y(t) on X(t) at frequency λ The amount of amplification and phase shift by which the Xi (t) series must be modified in order to give an optimum fit, on average, to the Y(t) series at frequency λ is, therefore, determined by the real-valued gain, Gj (λ) = |Aj (λ)|, and phase, φ j (λ) = arg Aj (λ), respectively, of Y(t) over Xj (t). After estimating Aj (λ), and also Gj (λ) and φ j (λ), using results from multiple regression analysis adapted to complex-valued variates, a gain plot and a phase plot can be made for each j = 1, 2, …, r, with appropriate confidence regions drawn on the plot at each frequency λ to assess the degree of association between Y(t) and the Xj (t) series. In this article new asymptotic simultaneous confidence regions for gains and phases are derived, and we show theoretically that they are shorter than similar confidence regions currently available in the statistical literature. The tools used here for constructing these improved regions include a differential identity for the complex multivariate Gaussian distribution and a multivariate probability inequality for the amplitudes of components of a complex-valued Gaussian random vector. Numerical computations covering a variety of possible situations show the actual amount of asymptotic improvement. Empirical investigations of the finite sample behavior of the proposed new simultaneous confidence regions indicate good overall performance.

Simultaneous Confidence Regions for the Frequency Analysis of Multiple Time Series

Abstract In the frequency analysis of several time series observed simultaneously, it is often necessary to adapt classical results of linear models and multiple regression to situations involving complex-valued variates. Given a real-valued stochastic time series, Y(t), and a suitable vector-valued fixed series, X(t), with components Xi (t) (j = 1, 2, …, r, t = 0, ± 1, …), it may be of interest to assess the effect, in the frequency domain, of each component series of X(t) on the behavior of Y(t). If values of Y(t) and X(t) are made available for t = 0, 1, …, T − 1, then under certain conditions and for large T at frequency λ the finite Fourier transform of Y(t) is approximately linearly related to the finite Fourier transform of X(t) through a complex-valued vector A(λ). In an asymptotic sense, A(λ), with entries Aj (λ) (j = 1, 2, …, r), may be interpreted as a vector of complex regression coefficients of Y(t) on X(t) at frequency λ The amount of amplification and phase shift by which the Xi (t) series ...

On Logarithmic Norms

This paper discusses some normlike properties of the logarithmic norm and related efficiency questions. Some computational questions are raised and some examples of use are indicated.