In the paper, maximal values xe(τ) of the solutions x(t) of the linear differential equations excited by the Dirac delta function are determined. There are obtained the analytical solutions of the equations and also the maximal positive values of these solutions. The obtained sufficient conditions of the positivity of these solutions are defined by the Theorems. There are also formulated the necessary conditions of the positivity of these solutions. The analytical formulae enable the design of the system with prescribed properties [3].

The paper presents proposal of a model of the fluidized bed boiler adapted for use in model-based controllers e.g. predictive, adaptive or internal model control (IMC). The model has been derived in the form of transfer function matrix which allows its direct implementation in the controller structure. Formulated model takes into consideration the principal cross-coupling between process variables which enables the opportunity to search for feasibility of decoupling control. The results of the identification of the dynamics of the 2 MW industrial bubbling fluidized bed boiler using the proposed model form was presented. According to the experimental data it was found that despite of introduced simplifications presented model allows the boiler behavior prediction.

Source/filter models have frequently been used to model sound production of the vocal apparatus and musical instruments. Beginning in 1968, in an effort to measure the transfer function (i.e., transmission response or filter characteristic) of a trombone while being played by expert musicians, sound pressure signals from the mouthpiece and the trombone bell output were recorded in an anechoic room and then subjected to harmonic spectrum analysis. Output/input ratios of the signals’ harmonic amplitudes plotted vs. harmonic frequency then became points on the trombone’s transfer function. The first such recordings were made on analog 1/4 inch stereo magnetic tape. In 2000 digital recordings of trombone mouthpiece and anechoic output signals were made that provide a more accurate measurement of the trombone filter characteristic. Results show that the filter is a high-pass type with a cutoff frequency around 1000 Hz. Whereas the characteristic below cutoff is quite stable, above cutoff it is extremely variable, depending on level. In addition, measurements made using a swept-sine-wave system in 1972 verified the high-pass behavior, but they also showed a series of resonances whose minima correspond to the harmonic frequencies which occur under performance conditions. For frequencies below cutoff the two types of measurements corresponded well, but above cutoff there was a considerable difference. The general effect is that output harmonics above cutoff are greater than would be expected from linear filter theory, and this effect becomes stronger as input pressure increases. In the 1990s and early 2000s this nonlinear effect was verified by theory and measurements which showed that nonlinear propagation takes place in the trombone, causing a wave steepening effect at high amplitudes, thus increasing the relative strengths of the upper harmonics.

Sound and vibrations are often perceived via the auditory and tactile senses simultaneously, e.g., in a car or train. During a rock concert, the body vibrates with the rhythm of the music. Even in a concert hall or a church, sound can excite vibrations in the ground or seats. These vibrations might not be perceived separately because they integrate with the other sensory modalities into one multi-modal perception.

This paper discusses the relation between sound and vibration for frequencies up to 1 kHz in an opera house and a church. Therefore, the transfer function between sound pressure and acceleration was measured at different exemplary listening positions. A dodecahedron loudspeaker on the stage was used as a sound source. Accelerometers on the ground, seat and arm rest measured the resulting vibrations. It was found that vibrations were excited over a broad frequency range via airborne sound. The transfer function was measured using various sound pressure levels. Thereby, no dependence on level was found. The acceleration level at the seat corresponds approximately to the sound pressure level and is independent of the receiver position. Stronger differences were measured for vibrations on the ground.

The transfer function (TF) method is presently a well-known method used to detect various types of winding damage in power transformers. Although abundant research has been done on this subject using laboratory windings as test objects, it is hard to find one, whose test objects are actual large-power transformer windings. Hence, a 400 kV disc winding consisting of 86 discs is used in this paper to study turn-to-turn short circuit with the help of the TF method. To evaluate the effects of this type of fault on TF curves, some mathematical comparison algorithms are used in this research.

Since a few years ago, there is an increasing interest for utilization of transfer functions (TF) as a reliable method for diagnosing of mechanical faults in transformer structure. However, this paper aims to develop the application of TF method in order to evaluate the drying quality of active part during the manufacturing process of transformer. To reach this goal, the required measurements are carried out on 50 MVA 132 KV/33 KV power transformer when active part is placed in the drying chamber. Two different features extracted from the measured TFs are then used as the inputs to artificial neural network (ANN) to give an estimate for required time in drying process. Results show that this new represented method could well forecast the required time. The results obtained from this method are valid for all the transformers which have the same design.

As it is found in the related published literatures, the transfer function (TF) evaluation method is the most feasible method for detection of winding mechanical faults in transformers. Therefore, investigation of an accurate method for evaluation of the TFs is very important. This paper presents three new indices to compare the transformer TFs and consequently to detect the winding mechanical faults. These indices are based on estimated rational functions. To develop the method, the necessary measurements are carried out on a 1.3 MVA transformer winding, under intact condition, as well as different fault conditions (axial displacement of winding). The obtained results demonstrate the high potential of proposed method in comparison with two other well-known indices. Additionally, two important methods for describing TFs by rational functions are studied and compared in this paper.

The measurement of frequency characteristics, like magnitude and phase, related to a specific transfer function of DC–DC converters, can be a difficult task – especially when the measured signal approaches the boundary of a small-signal model validity (i:e. 1/3 of the switching frequency fS). It is hard to find a paper where authors mention the measurement techniques they use to draw frequency characteristics. Meanwhile the presence of noise in the output signal does not enable to directly measure the gain and the phase shift between the input and output signals. In such situations additional analysis is required in order to achieve a reliable result. This paper contains a description of a few methods that can be used to analyse measured signals in order to determine the gain and the phase shift of a specific transfer function. They enable to verify mathematical models in a wide range of frequencies (up to 1/3 fS). The methods use signals measured in the time domain and can be implemented in mathematical software such as Matlab or Scilab.

This paper analyses the performance of Differential Head-Related Transfer Function (DHRTF), an alternative transfer function for headphone-based virtual sound source positioning within a horizontal plane. This experimental one-channel function is used to reduce processing and avoid timbre affection while preserving signal features important for sound localisation. The use of positioning algorithm employing the DHRTF is compared to two other common positioning methods: amplitude panning and HRTF processing. Results of theoretical comparison and quality assessment of the methods by subjective listening tests are presented. The tests focus on distinctive aspects of the positioning methods: spatial impression, timbre affection, and loudness fluctuations. The results show that the DHRTF positioning method is applicable with very promising performance; it avoids perceptible channel coloration that occurs within the HRTF method, and it delivers spatial impression more successfully than the simple amplitude panning method.

This paper proposes a generalized fractional controller for integer order systems with time delay. The fractional controller structure is so adopted to have a combined effect of fractional filter and Smith predictor. Interestingly, the resulting novel controller can be decomposed into fractional filter cascaded with an integer order PID controller. The method is applied to two practical examples i.e. liquid level system and Shell control fractionator system. The closed- loop responses resulting from the proposed method are compared with that of the available methods in the literature. For quantitative evaluations of the proposed method, Integral Absolute Error (IAE) and Integral Square Control Input (ISCI) performance criteria are employed. The proposed method effectively enhances the closed-loop response by improving the IAE values, reducing the control effort inputs to achieve the desired output. The disturbance rejection and robustness tests are also carried out. The robustness test reveals a significant improvement in the maximum absolute sensitivity measure. That is displayed in numerical simulations of the paper.

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used third-order 3x3 pixels Laplace contour filters including the difference schemes used to derive them. The authors focused on the mathematical properties of the Laplace filters. The basic reasons of the differences of the properties were studied and indicated using their transfer functions and modified differential equations. The relations between the transfer function for the differential Laplace operator and its difference operators were described and presented graphically. The impact of the corner elements of the masks on the results was discussed. This is a theoretical work. The basic research conducted here refers to a few practical examples which are illustrations of the derived conclusions.We are aware that unambiguous and even categorical final statements as well as indication of areas of the results application always require numerous experiments and frequent dissemination of the results. Therefore, we present only a concise procedure of determination of the mathematical properties of the Laplace contour filters matrices. In the next paper we shall present the spectral characteristic of the fifth order filters of the Laplace type.

The head-related transfer function (HRTF) is dependent on the position of the sound source (both direction and distance) and is also affected by individual anatomical parameters. Individualized HRTFs have been shown to affect the perception of sound direction, but have not been considered in distance perception. This work aims to discover, by means of psychoacoustic experiments for a virtual reproduction system through a pair of in-ear headphones, the effect of individualized HRTF on auditory distance perception for a nearby sound source. The individualized HRTFs of six subjects and the non-individualized HRTFs of a mannequin at seven distances between 0.2 and 1.0 m and five lateral azimuths between 45X and 135X in the horizontal plane were processed with white noise to generate binaural signals. Further, the individualized and non-individualized HRTFs were used in the auditory distance perception experiments. Results of distance perception show that the variance of distance perception results among subjects is significant, the reason could be the stimuli are lack of dynamic cue and early reflections, or the auditory difference of distance perception among subjects. However, via the analyses of mean slope of perceptual distance and correlation between the perceptual and real distance, we find that the individualized HRTF cue has insignificant influence on distance perception.

The individual Head-Related Transfer Functions (HRTFs) typically show large left-right ear differences. This work evaluates HRTF left-right differences by means of the rms measure called the Root Mean Square Difference (RMSD). The RMSD was calculated for HRTFs measured with the participation of a group of 15 subjects in our laboratory, for the HRTFs taken from the LISTEN database and for the acoustic manikin. The results showed that the RMSD varies in relation to the frequency and as expected is small for more symmetrical HRTFs at low frequencies (0.3÷1 kHz). For higher frequency bands (1÷5 kHz and above 5 kHz), the left-right differences are higher as an effect of the complex filtering caused by anatomical shape of the head and the pinnae. Results obtained for the subjects and for data taken from the LISTEN database were similar, whereas different for the acoustic manikin. This means that measurements with the use of the manikin cannot be considered as perfect average representation of the results obtained for people. The method and results of this study may be useful in assessing the symmetry of the HRTFs, and further analysis and improvement of how to considered the HRTFs individualization and personalization algorithms.

In the paper, maximal values x_e(τ) of the solutions x(t) of the linear differential equations excited by the Dirac delta function are determined. The analytical solutions of the equations and also the maximal positive values of these solutions are obtained. The analytical formulae enable the design of the system with prescribed properties. The complementary case to the earlier paper is presented. In an earlier paper it was assumed that the roots si are different, and now we consider the case when s₁ = s₂  = … = s_n.

The paper is a structured, in-depth analysis of dual active bridge modeling. In the research new, profound dual active bridge converter (DAB) circuit model is presented. Contrary to already described idealized models, all critical elements including numerous parasitic components were described. The novelty is the consideration of a threshold voltage of diodes and transistors in the converter equations. Furthermore, a lossy model of leakage inductance in an AC circuit is also included. Based on the circuit equations, a small-signal dual active bridge converter model is described. That led to developing control of the input and output transfer function of the dual active bridge converter model. The comparison of the idealized model, circuit simulation (PLECS), and an experimental model was conducted methodically and confirmed the high compatibility of the introduced mathematical model with the experimental one. Proposed transfer functions can be used when designing control of systems containing multiple converters accelerating the design process, and accurately reproducing the existing systems, which was also reported in the paper.

The parameters of sigma-delta audio DAC depend mainly on digital sigma-delta modulator's features, especially on its noise transfer function (NTF). Many methods of design and optimization of the loop filter's coefficients in sigma-delta modulators have been proposed so far. These methods enable the designer to get suitable noise transfer functions for specific application. This paper reviews NTF design and optimization methods which are particularly useful in audio applications.

The fractional order proportional, integral, derivative and acceleration (PI ^λD ^µA) controller is an extension of the classical PIDA controller with real rather than integer integration action order λ and differentiation action order µ. Because the orders λ and µ are real numbers, they will provide more flexibility in the feedback control design for a large range of control systems. The Bode’s ideal transfer function is largely adopted function in fractional control systems because of its iso-damping property which is an essential robustness factor. In this paper an analytical design technique of a fractional order PI ^λD ^µA controller is presented to achieve a desired closed loop system whose transfer function is the Bode’s ideal function. In this design method, the values of the six parameters of the fractional order PI ^λD ^µA controllers are calculated using only the measured step response of the process to be controlled. Some simulation examples for different third order motor models are presented to illustrate the benefits, the effectiveness and the usefulness of the proposed fractional order PI ^λD ^µA controller tuning technique. The simulation results of the closed loop system obtained by the fractional order PI ^λD ^µA controller are compared to those obtained by the classical PIDA controller with different design methods found in the literature. The simulation results also show a significant improvement in the closed loop system performances and robustness using the proposed fractional order PI ^λD ^µA controller design.

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used fifth-order pixels Laplace filters including the difference schemes used to derive them (finite difference method – FDM and finite element method – FEM). The results of the research concerning third-order pixels matrices of the convolution Laplace filters used for digital processing of images were presented in our previous paper: The mathematical characteristic of the Laplace contour filters used in digital image processing. The third order filters is presented byWinnicki et al. (2022). As previously, the authors focused on the mathematical properties of the Laplace filters: their transfer functions and modified differential equations (MDE). The relations between the transfer function for the differential Laplace operator and its difference operators are described and presented here in graphical form. The impact of the corner elements of the masks on the results is also discussed. A transfer function, is a function characterizing properties of the difference schemes applied to approximate differential operators. Since they are relations derived in both types of spaces (continuous and discrete), comparing them facilitates the assessment of the applied approximation method.

In the last decade, infrared imaging detectors trend has gone for smaller pixels and larger formats. Most of the time, this scaling is carried out at a given total sensitive area for a single focal plane array. As an example, QVGA 30 µm pitch and VGA 15 µm pitch exhibit exactly the same sensitive area. SXGA 10 µm pitch tends to be very similar, as well. This increase in format is beneficial to image resolution. However, this scaling to even smaller pixels raises questions because the pixel size becomes similar to the IR wavelength, but also to the typical transport dimensions in the absorbing material. Hence, maintaining resolution for such small pixel pitches requires a good control of the modulation transfer function and quantum efficiency of the array, while reducing the pixel size. This might not be obtained just by scaling the pixel dimensions. As an example, bulk planar structures suffer from excessive lateral diffusion length inducing pixel-to-pixel cross talk and thus degrading the modulation transfer function. Transport anisotropy in some type II superlattice structures might also be an issue for the diffusion modulation transfer function. On the other side, mesa structures might minimize cross talk by physically separating pixels, but also tend to degrade the quantum efficiency due to a non-negligible pixel fill factor shrinking down the pixel size. This paper discusses those issues, taking into account different material systems and structures, in the perspective of the expected future pixel pitch infrared focal plane arrays.