The position of the excitation bump (x, Fig.5.4.c) is important in order to achieve a good matching. Therefore, the structure in Fig. 5.4 is simulated again for varying x with both the F3D simulator in FDFD method and CST MWS and the metal block height of 35 µm. It means that the internal port representing the excitation bump is moved along the symmetry axis (Fig. 5.3a) with various distances from the centre point of the antenna, the crossing point of the diagonals. The simulation results of reflection coefficients at the internal port for different positions of it are shown in Fig. 5.7. From the results of both the FDFD and FDTD procedures it is evident that by changing the position of the excitation the impedance of the internal port (input impedance, 50Ω ) can be matched with the characteristic impedance of the slot resulting in a higher level of radiation (i.e. high return loss). However, shifting the bump causes also a slight shift in resonance **frequency**, with different level of radiation depending on the actual position. The shift in the port position in either direction by 300 µm results in more than 5 dB change in radiation level as well as more than 0.2 GHz shift in **frequency**. Thus, the slot length has to be adapted iteratively in order to fix the resonant **frequency** to the specified value.

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The discretization of time derivatives can be done by forward (explicit) or backward (implicit) **finite** differences. Implicit methods are unconditionally stable, but they re- quire solving a System of Linear Equations (SLE) in each time step. Explicit methods are much lighter in computation, since they can be written as a matrix-vector multipli- cation in each time step. A major drawback of explicit methods is that they are only conditionally stable and some are even always unstable. The most commonly used ex- plicit method is the so-called ‘leap-frog’ method, introduced by Yee [93] in 1966. It consists of a (staggered) central **difference** quotient featuring second order accuracy. The stability of the scheme is connected to the grid dispersion relation, which describes the velocity of a plane wave on the grid as dependent on the direction of the wave vector. It reads for a particular Cartesian cell (∆x, ∆y, ∆z) (see e.g. [99])

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In the article [19], the Maxwell’s equations are discretized in space by a node-based and edge-based **finite** element method, and an efficient solver is described both for the **frequency** and time **domain** formulations. The paper [24] describes a general way to investigate the stability of temporal discretization schemes such as backward **difference**, forward **difference** and central **difference** methods in electromagnet- ics. The stability is determined by analyzing the root locus map of a characteristic equation and evaluating the spectral radius of **finite** element system matrix. Stability properties are given in [3] for simulations of transient electromagnetic phe- nomena for the Maxwell’s equations. In the article [25], time **domain** **finite** element methods based on Whitney elements are proposed for solving transient response problems on tetra- hedral meshes. One of the proposed schemes is uncondition- ally stable, another scheme is explicit but does not require matrix inversions. An explicit time **domain** **finite** element algorithm is presented for the Maxwell’s equations in [30], for complex media in [28] (only numerical result). An energy conserving method for 3D Maxwell’s equations is obtained in [37], based on an exponential operator splitting approach. Many papers have been written about time **domain** discon- tinuous Galerkin (TDDG) methods in computational electro- magnetics [38]–[49]. Other time **domain** methods to solve either the Maxwell’s equations or the vector wave equation can be found in [50]–[62]. Several books for electromagnet- ics [63]–[68] are available for analysis and simulation.

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Applied on reconstruction theory this distinction does not play such a big role. Nevertheless it highlights one fundamental **difference** between classical and new approach: In the classical case the reconstruction is applied pointwise on the in- put. Hence it is self-evident to use causal filters. This is more common because this allows to transform a reconstruction in real-time. Anyhow it is also possible to use non-causal filters, but this enforces a more complex reconstruction process. Working in **frequency** space the reconstruction is done for the whole system at once. Hence it does not play any role whether the reconstruction function is causal or not. Summarized one can say that both kinds of filters (causal and non-causal) can be applied in both reconstruction approaches. But only for the classical case the differ- ence between causal and non-causal reconstruction becomes apparent. The reason is that the classical approach has to make a **difference** in appliance whereas the reconstruction process in **frequency** space is not affected by the filter-causality. More relevant to reconstruction theory becomes the distinction between FIR and IIR filters. FIR filter producing -as mentioned by its name- an output that has a **finite** length. This is because their output only depends on a limited range of points around the actual position. Hence the output has to vanish if all inputs within the range of the filter vanish. In literature one finds often a causal representation of this behaviour for which only the N-th last point affect the actual output but the definition can be easily extended for the non-causal case:

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Temporal output is the most commonly used output type in FDTD and consists of a single value for a given quantity recorded as a function of time. This includes one or more ﬁeld components measured at a single fixed point in space, the energy density in the monitor **domain**, the power ﬂow through the monitor **domain**, or the result of an overlap integral. In contrast, spatial output comprises measured values which are recorded as a function of space at fixed values of time. This includes measurements of selected ﬁeld components, the amplitude of the Pointing vector, the energy density, or the power ﬂow along a speciﬁed axis, all in dependence of the spatial coordinates. Spatial output requires much more disk storage space than temporal output, but is often necessary to obtain desired data. [88] To obtain the spectral characteristics of a given index structure, a **frequency** analysis has to be performed on the recorded time-**domain** data. If, for example, the **frequency** response over a broadband spectrum is of interest, a broadband pulse or an impulse excitation can provide this response in a single run. Mathematically, the conversion from the time-**domain** to the **frequency**-**domain** is achieved by means of a Fourier transform (numerical Fourier analysis).

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A global FDTD method for an efficient analy- sis of NLTL is presented. The transmission line losses are included over a wide **frequency** range, and the non-uniform doping profile of the diodes can be considered by applying the FD method to the semiconductor transport equations. This global model also enables an optimization of the NLTL’s and diodes. As a consequence, fewer

Computational electromagnetics gained its momentum ever since the **Finite** Element Method (FEM) [1] started being applied to solving the Maxwell’s equations. Exploration of Yee’s work [2] changed this sce- nario. The method proposed by Yee, later named as **Finite** **Difference** Time **Domain** (FDTD), got a huge following mainly due to its simplicity. It is written for the linear non–dispersive media and Cartesian grid. A few years later Weiland proposed [3] another volume discretization method, termed as **Finite** Integration Technique (FIT). FIT presents a closed theory, which can be applied to the full spectrum of electromagnetics. FIT on a Cartesian grid with appropriate choice of sampling points in the Time **Domain** (TD) is algebraically equivalent to FDTD. In other words, FIT is a more generalized form of FDTD. All the above mentioned methods got refined and applied to various classes of problems [4–16] over the years and each of these methods enjoys its own set of advantages and disadvantages. FEM, more specifically, continuous FEM, has high flexibility in modeling the curved surfaces when employed on un- structured grids. But the TD formulation of the FEM results in a global implicit time marching scheme which requires global matrix inversion at each time step. This restricts FEM practically to **Frequency** **Domain** (FD) simulations. FD methods must solve an algebraic system of equations at all significant frequencies in the desired **frequency** range individually, making them more costly with increasing band- width. Fast **frequency** sweep algorithms can be employed in order to avoid calculating the solution at each and every **frequency** sample. However, these algorithms are memory intensive. When opted for parallel computing, FD methods require significant global operations.

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range. The **frequency** π/16 corresponds to 8 years and separates the financial cycle range from the business cycle range. An initial visual inspection of Figures 4 to 6 delivers at least two noteworthy results.
First, especially for the US and UK the spectral densities of credit, credit to GDP and house prices are substantially shifted to the left in the later period compared to the first one, indicating - at least superficially - that longer cycles became present, see Figures 4 and 5 . Moreover, the peaks of the spectral densities are much more pronounced, suggesting that these longer cycles are also more important for the variation of the process. For Germany, as illustrated in Figure 6 , this is only true for house prices. We obtain no clear results for German credit and credit to GDP. Therefore, German data does not provide much evidence in favor of the postulated financial cycle properties.

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Contribution
The main objective of the current paper is to establish the empirical regularities of the financial cycle on the basis of straightforward time series econometric methods. In particular, our contribution is twofold. First, we provide a complete characterization of the properties of the financial cycle in the **frequency** **domain**. To this end, we estimate univariate time series models for the usual financial variables and use the estimated models to compute their corresponding **frequency** **domain** representations. Compared to **frequency**- based filter methods and the turning point analysis, our approach has the advantage that no a priori assumption on the cycle length is needed. Moreover, also very long cycles can be detected, even if the sample period is limited. Second, a distinguishing feature of our estimation approach is that it allows us to test existing hypotheses on various characteristics of the financial cycle by statistical means. In this respect, we address the following questions: At what **frequency** does the financial cycle mostly operate? Does the financial cycle indeed have a longer duration, as well as a larger amplitude than the classical business cycle? Have the characteristics of the financial cycle changed over time?

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For the US, we find significant time **domain** Granger causality from credit to housing and the other way around. From the **frequency** **domain** we obtain the information that this relation is particularly significant at lower frequencies. This corroborates the results from the coherency analysis of the bivariate VARs in the previous section (see Figure 4 ). While Granger causality remains a reduced form concept, our findings are in line with the widespread notion that house prices were a key factor of the pronounced credit expansion in the US during the 1990s and 2000s which led to 2007/2008 crisis and the following collapse in economic growth. In the time **domain**, where relations at specific frequencies cannot be identified, US GDP appears not to be significantly influenced by housing or credit, at least not at the 10%-level. However, looking in the **frequency** **domain**, we can see that house prices do significantly affect GDP, but they do so only at lower frequencies. Similarly, Figure 7 also shows a significant low **frequency** Granger-causal relationship from the VIX to house prices.

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where tc is the time of the actuation duty cycle and is taken for all investigations to be half of the period, tc = T /2. The pulse train actuation velocity can be applied with smoothing at the discontinuities introduced by Knopp. 34 Signals in the time **domain** can be represented by a series of sines. A common applied algorithm is the Fast Fourier Transform. It takes a signal and breaks it down into sine waves of different amplitudes and frequencies. The discrete Fourier transform is then a set of fundamental and harmonic components. The first jet velocity in Eq. (19) describes a harmonic signal and the second two are related to a discontinuous velocity distribution. An harmonic signal in the **frequency** **domain** is represented entirely by its fundamental **frequency**, so there is no zero **frequency** component. Since the latter two are discontinuous signals, the Gibb’s phenomenon, apparent around t/T = 0, T /2, allows the restoration of the time signal only to a certain degree. For this reason, the reconstruction of the time signals for the half-wave rectified sine is carried out with the following approach: To reproduce an order of magnitude in u(t) (fig. 2(b)), about three harmonics are necessary, for two orders of magnitude ten harmonics, respectively. Accordingly, the pulse train is reconstructed with about seven harmonics for an order of magnitude of u(t), and with about seventeen harmonics for two orders of magnitude. The pulse train reveals clearly that a high amount of frequencies must be resolved to reconstruct a time signal. The half-wave rectified sine and pulse train shaped jet velocity has the analytically inverse discrete Fourier-Transform of

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The **finite** **difference** coefficients calculated as described in Section 3.4 are denoted by ˆ c for the variables between grid points and c for the variables at grid points. Finally, the discrete differential operator A can be formed by applying the appropriate matrix T as diagonal elements (cf. Section 3.2).

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1. INTRODUCTION
Especially in the design and development phase of ships the prediction of underwater sound characteristics is of great interest. In the lower **frequency** range classical numerical methods like the **Finite** Element Method (FEM) are a suitable tool to predict the required physical quantities . In the higher **frequency** range the computational costs for classical numerical methods exceed the commonly available computer capacities. Energy based methods provide an efficient possibility to calculate acoustic quantities in the higher **frequency** range. A widely used energy based method is the Statistical Energy Analysis (SEA). To apply this method the whole model is divided into larger substructures before the calculation. An energy density is than assigned to each substructure as unknown value. For the application of the energy based **Finite** Element Method (EFEM) the substructures are further divided into **finite** elements, which yields the possibility to determine the local distribution of energy density on substructures. The power flow in coupling regions is therefore described more accurately due to this additional discretization.

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Most of the existing insights in the financial cycle are based on either the analysis of turning points ( Claessens et al. , 2011 , 2012 ) or **frequency**-based filter methods ( Drehmann et al. , 2012 ). The turning point approach requires to pre-specify a rule which is applied to an observed time series in order to find local maxima and minima. **Frequency**-based filter methods require to pre-specify a **frequency** range at which the financial cycle is assumed to operate. Therefore, both approaches are rather descriptive and do not allow to test hypotheses on the properties of the financial cycle. Indeed, while there is a broad consensus in the literature concerning the main characteristics of the business cycle (see e.g.

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surface impedance of these materials are estimated using Miki’s model [ 17 ] and mapped to a six pole rational function using a vector fitting algorithm [ 21 ]. The source is located 2 m from the impedance boundary and the receiver is located 1 m from the boundary, at the midpoint between the source and the boundary. A basis order of P = 4 is used and a high spatial resolution is employed, roughly 14 points per wavelength (PPW) at 1 kHz, ensuring minimal numerical errors in the **frequency** range of interest. The initial condition is a Gaussian pulse with spatial variance σ = 0.2 m 2 . The resulting complex pressure is shown in the **frequency** **domain** in Fig. 1 . The simulated pressure matches the analytic solution perfectly, both in terms of amplitude and phase, for both boundary conditions tested, thus confirming the high precision of the LR boundary model.

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Explizite und implizite finite Differenzenmethoden zur Lösung der Diffusionsgleichung in zwei Raumdimensionen Zusammenfassung In diesem Bericht stehen rotationssymmetrische Diffusionspro[r]

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Abstract. Determination of aeroelastic stability boundaries for full aircraft configu- rations by solving the time-accurate unsteady Reynolds-averaged Navier-Stokes (RANS) equations is recognized as extremely computationally expensive or impractical. This is due to the wide range of flight conditions, frequencies, and structural deformation mode shapes that must be examined to ensure a configuration is free from flutter. Nonetheless there is an increasing demand within the aerospace industry for accurate flutter analysis in the transonic regime, which can only be satisfied with the use of high-fidelity RANS codes. Hence we are motivated to seek a more efficient numerical method. By assuming that perturbations to the flow are small and harmonic, we can derive an efficient alternative method by linearization of the RANS equations, a linearized **frequency** **domain** (LFD) solver. With this approach the unsteady simulation reduces to a single non-linear steady computation followed by a single linear simulation in the **frequency**-**domain**. This method is not new, but has principally been applied to turbomachinery so far. The contribution of this paper twofold: firstly to show that LFD is sufficiently accurate and reliable for applications to aeroelastic problems that occur in external aerodynamics, and secondly to demonstrate the speed-up that can be expected over full unsteady computations. Viscous transonic analysis is carried out on complex geometries in three-dimensions. The results show good agreement with full unsteady simulation and experiment, and a reduction in computational costs up to one order of magnitude is demonstrated.

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Die angesprochenen Wellengleichungen werden dabei ¨ uber die, in der seismischen Modellie- rung ¨ublichen **Finite**-Differenzen Methode (FDM) gel¨ost[GPI14]. Die FDM ist ein numeri- sches Verfahren und kann daher auch keine exakten Ergebnisse liefern, sondern ausschließ- lich N¨aherungsl¨osung bis zu einer bestimmten Genauigkeit. Ein Maß f¨ ur die Genauigkeit von numerischen Verfahren ist die Konsistenzordnung, welche die Abh¨angigkeit zwischen den entstehenden Fehlern und der Schrittweite beschreibt [Fel14]. Eine der grundlegends- ten Wellengleichungen ist die vollst¨andig elastische Wellengleichung[Fri14]. Die einfachste FDM, die f¨ur die L¨osung dieser Wellengleichung verwendet werden kann hat eine Kon- sistenzordnung von O(∆t 2 , ∆x 2 ) [Mar89]. Je h¨oher die Ordnung der Konsistenz, desto genauer arbeitet ein numerischer Algorithmus.

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MEG and EEG are important tools for non-invasive functional brain imaging because of their exquisite temporal resolu- tion. However, since the MEG/EEG inverse problem is ill-posed, priors have to be set to obtain a unique source esti- mate. While spatio-temporal priors have been frequently used, the time-**frequency** (TF) characteristics of the signals are rarely taken into account. In this contribution, we present an inverse solver, which is based on structured sparsity in the TF **domain**.

OCNR 314, 6300 Ocean Drive, Texas A & M University – Corpus Christi, Corpus Christi, TX, 78412 U.S.A., E-mail: david.hudgins@tamucc.edu, Phone: 361-825-5574
Abstract. This paper first applies the MODWT (Maximal Overlap Discrete Wavelet
Transform) to Euro Area quarterly GDP data from 1995 – 2014 to obtain the underlying cyclical structure of the GDP components. We then design optimal fiscal and monetary policy within a large state-space LQ-tracking wavelet decomposition model. Our study builds a MATLAB program that simulates optimal policy thrusts at each **frequency** range where: (1) both fiscal and monetary policy are emphasized, (2) only fiscal policy is relatively active, and (3) when only monetary policy is relatively active. The results show that the monetary authorities should utilize a strategy that influences the short-term market interest rate to undulate based on the cyclical wavelet decomposition in order to compute the optimal timing and levels for the aggregate interest rate adjustments. We also find that modest emphasis on active interest rate movements can alleviate much of the volatility in optimal government spending, while rendering similarly favorable levels of aggregate consumption and investment. This research is the first to construct joint fiscal and monetary policies in an applied optimal control model based on the short and long cyclical lag structures obtained from wavelet analysis.

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