An analysis of the technical possibilities of using the Schauder basis function system for presenting and compressing information, for example, network data, is provided. Advantages compared to other bases have been revealed. On the basis of experimental studies, recommendations are provided for their use for digital processing of audio information at a representative level in computer systems and networks. From our point of view, digital systems that use the representation of sound signals by a system of basic functions have great potential. Further considerations will be directed to justifying the expediency of their use in engineering developments. At the same time, computer modeling is an efective tool for finding rational methods of building digital signal processing equipment. Preliminary modeling of signal analysis and synthesis processes allows at the research stage to determine the complexity of hardware solutions, possible parameters of digital processing (accuracy of transmission and reproduction, degree of elimination of redundancy (compression), speed, and others).
A characteristic feature of modernity is the sharp excess of the growth of the flow of information over
the expansion of data transmission channels [
The ever-growing volume of useful information obtained from sources of various physical origins
requires the search for efective preprocessing methods at the stage of presentation, storage, and
determination of channel transmission methods [
The most important task of this problem is to reduce the redundancy while maintaining the desired transmission quality [14, 15, 16]. Traditional experiments have become methods of reducing the redundancy of sound signals, which involve coding of acoustic oscillations or analysis/synthesis based on hardware and software tools [17, 18, 19]. Despite the fact that a significant amount of research has been conducted in the field of digital processing of sound signals, especially speech signals, there are still many unsolved problems [20, 21, 22]. One of the reserves of improving systems for reducing the redundancy of sound signals is the search for new methods of presentation and processing using programming computing tools of computer technologies [23, 24, 25].
In 1927, Schauder proposed a system of linearly independent basis functions. The use of Schauder functions in digital signal processing systems is the main topic of research. The distribution in a row according to the system of basic functions is generally presented as:
() = ∑︁ (),
=0 where () is the realization of the signal as a random process; () – system of basic functions; are the coeficients of the schedule.
From our point of view, it is appropriate to present the Schauder function in the form [26]: where () is a system of Haar functions defined on the orthogonality interval as follows: () =
1() , ⎧⎪1, ⎪ ⎪ ⎪ ⎪⎪⎪⎪ 1 ∫︁ ⎪⎨⎪ 0
+1 ∫︁ ⎪⎪ 2 2 ⎪ ⎪ ⎪⎪⎪⎪ 0 ⎪⎩⎪0, () = () = = 0, ∈ [0, ], = 1, ∈ [0, ], () () , = 2, 3, . . . , ∈ [0, ],
∈/ [0, ], 1 () = 1, [0, ] , ⎪⎪⎧⎪2 2−1 , ∈ ⎪ ⎪ ⎪ ⎪ ⎨−2 2−1 , ∈ ⎪ ⎪ ⎪ ⎪ ⎪⎪⎩⎪0, ∈ ︂[ ( − 1)
2−1 ︂[ ( − 0.5)
2−1 ︂[ ( − 1) 2−1 ( − 0.5) ]︂ 2−1 ]︂ , 2−1 ]︂ , 2−1 , where = 2−1 + ; = 1, 2, 3, . . .; = 1, 2, . . . , 2−1 — the number of the Haar function group — sets the duration of the zero value of the Haar function, which is equal to /2−1 , and its amplitude, which is equal to 2(−1)/2 . The index determines the position of a non-zero value on the segment [0, ]. One group includes Haar functions with the same duration of a non-zero value. The relationship between single and double numbering of functions is as follows: = 1, 2, 3, . . . , ; = 1, 2, 3, . . . , 2−1 ; = +2−1 . Figure 1 presents graphical images of the Haar (a) and Schauder (b) functions, which are locally defined on the interval [0, ].
(a) (b) Figure 1: Graphic representation of Haar (a) and Schauder (b) functions.
The formulas for calculating the expansion coeficients for = 1, 2, 3 have the following form [26, 27, 28]: using the values of the realization of the signal () at the points ∆ t are given [0, ].
Analyzing the expression for calculating coeficients: = ︂( − 0.5 2−1
︂) − 2
1 [︂ (︂ − 1 2−1 ︂) + ︁(
2−1 ︁) ]︂ value of ( ) respectively for 2, 4, 8, . . . , 2 ; coeficients where = 2−1 +; = 1, 2, . . . , 2−1 ; = 1, 2, . . . , ; = 2 ∆ t ; ∆ t is the signal quantization interval over time, it can be established that for = 0, 2, . . . , 2 there is a regular repeatability, the same () values are used many times to calculate other coeficients.
So, for example, the value of (0) is used to calculate the coeficients 2, 3, 5, 9, . . . , 2(−1) +1, 4, 5, 9, . . . , 7·2 −3 +1, and 1, 4, 8, . . . , 7·2 −3 .
During the research, it was established that the values of () at the points = 2− + (where = 1, 2, 3, . . ., = 0, 1, . . . , 2) are used for calculation of a series of coeficients with serial numbers = 2 + . When using this approach, the calculation time can be significantly reduced. Without taking into account the regularity, each time when calculating the next coeficient, it would be necessary to recalculate the coordinates of the () values on the segment [0, ].
In this way, it is proposed to use the discrete Schauder transformation for the design and research of network encoding/decoding means, which can be implemented at the algorithmic-program level or in a hardware-technological design based on FPGA-type integrated circuits. The advantage of this method is the use of the Schauder transformation, which has certain advantages compared to trigonometric bases, for example, the possibility of local processing on the time and frequency interval when segmenting the incoming sound stream, reducing the time for mathematical calculations of the expansion coeficients, minimizing the amount of memory. All this significantly afects the speed of data delivery to the user. A variant of the structural diagram of the application of this method in the information transmission channel consisting of a Schauder encoder, a transmission channel and a Schauder decoder has been developed.
A method of dynamic sound recognition for a multi-sensor Internet of Things location monitoring sensor network, characterized in that modules are placed on each device or node of the Internet of Things that needs monitoring, which configure and receive operational data and data about the environment of the device or node of the Internet of Things (Figure 2).
The transmitter contains the following modules and blocks: • Message source: Generates an output message, which can be in the form of an analog or digital signal; • Encoder: Converts a message into a digital code that is convenient for transmission. This may include data compression or error correction; • Modulator: Uses a carrier frequency signal generated by a synthesizer to superimpose information (modulation) on a high-frequency signal; • Power Amplifier: Amplifies the modulation signal to a level suficient for transmission; • Transmitting antenna: Radiates an amplified signal into space.
The receiver contains the following modules and blocks: • Receiving antenna: Receives a signal from space; • UHF converter: The received signal is often attenuated and mixed with an intermediate frequency (UHF) for further processing; • Demodulator: Extracts the information signal from the modulation signal; • Decoder: Converts the digital code back to the original form of the message, performs decoding and error correction; • Message Recipient: The end device receiving the recovered message.
A codec using the method of Schauder basis functions performed in the encoder and decoder for the audio recognition system works as follows:
Coding (encoder): 1. Analog-to-digital conversion (ADC): The audio signal is first converted from analog to digital using an ADC. 2. Decomposition into Schauder basis functions: The digital signal is decomposed into basic Schauder functions. These functions form a set of orthogonal bases that can be used to represent the signal. Decomposition is carried out by projecting the signal onto these basis functions. The result is a set of coeficients describing the signal in the space of basis functions. 3. Quantization of coeficients: The coeficients obtained as a result of the decomposition are quantized to reduce the amount of data. This means that the values of the coeficients are rounded to the nearest permissible value from a limited set of levels. 4. Coding of coeficients: Quantized coeficients are encoded for further transmission or storage.
This may include the use of compression algorithms such as Hufman coding or other entropy coding techniques.
Decoding (decoder): 1. Decoding coeficients: The received coded coeficients are decoded to restore the quantized coeficients. 2. Interpolation and dequantization: The quantized coeficients are converted back to continuous form (dequantized) to improve the accuracy of the reconstructed signal. 3. Signal reconstruction: The reconstructed signal is formed by a linear combination of Schauder basis functions using dequantized coeficients. This process is the reverse of decomposition and allows the signal to be restored to its original form (or with allowable losses if it is a lossy system). 4. Digital-to-analog conversion (DAC): The recovered digital signal is converted back to analog form using a DAC if analog output is required.
The working principle of the method is based on the use of Schauder basis functions, which are orthogonal and enable the eficient representation of any signal, including audio signals, with minimal information loss. These basis functions yield coeficients that provide a compact representation of the signal while preserving its essential characteristics. By quantizing and encoding these coeficients, the data volume required for transmission is significantly reduced. The decoding and reconstruction processes then apply inverse operations to those used during encoding, allowing for the accurate recovery of the original signal.
Figures 3–7 illustrate implementations of a fragment of a speech signal processed using the Schauder transformation. The signal is presented in blocks of equal size (2048 samples each). For each block, the decomposition coeficients were calculated with a spacing of 64 samples. The number of calculated decomposition coeficients was 64, 32, 16, 8, and 4. Accordingly, the synthesis of the output signal was performed using the specified number of coeficients (marked as "b" in the figure). In addition, an option for selecting informative coeficients from among the calculated ones based on the "|max|" or "max, min" principles was investigated, as shown in Figs. c and d, respectively.
By comparing the obtained signal implementations, we can conclude that the transformation based on Schauder’s functions reproduces the speech signal well. The calculation of consecutive 64 and 32 coeficients ensures a fairly accurate reproduction of signals.
When reproducing the signal by 16 coeficients, the implementation difers in the smoothing of local peaks in the areas of the high-frequency interval, while the low-frequency areas are reproduced quite well. If 8 coeficients out of 64 possible are used for signal synthesis, then the signalgrams show a clear diference between the output signal and the input signal.
This diference is especially clearly expressed when using 4 coeficients of the decomposition. Similar conclusions can be drawn regarding the number of informative coeficients selected according to the "|max|" principle. or "max, min".
Comparing the images of the output signal obtained by synthesis with the same number of coeficients but selected in diferent ways, we come to the conclusion that these images difer from each other with a certain diference. In order to clarify these changes, it is necessary to conduct an auditory subjective analysis in the future. Carrying out only a visual comparison of the images of the output signals, it is dificult to determine an objective criterion in which case the output signal is of higher quality. In order to make a qualitative assessment of the output signals, you should listen to these signals and compare their static characteristics for diferent options.
In the theory and practice of signal processing, researchers increasingly turn to their representation by a finite system of basis functions, orthogonal or non-orthogonal.
The solution to this problem is relevant in connection with the need to improve the means of analysis, processing and synthesis of real physical processes.
Systems of non-orthogonal linearly independent functions have a number of advantages, for example,
currently known PL-functions and Schauder functions [
Among the advantages of non-orthogonal decompositions should also be added the simplicity of their hardware implementations in comparison with analyzers and synthesizers built on the principle of orthogonal transformations [29, 32, 33].
The method is to include in each group of sensors sound sensors and ultrasound sensors that detect and recognize diferent acoustic waves, while using at least three sensors of each type in such a way that one sensor of each type is in working condition at the same time, and the other sensors were in an inactive state, after a certain period of time, the sensor in the working state is put into the rest state, and the other sensors in the rest state are put into the working state, and each sensor will store the environmental and operational data that it senses.
Information from the sensors is processed and transmitted to the anomaly analysis unit using a system based on the Schauder algorithm and containing information transmission channels consisting of a Schauder encoder, a transmission channel and a Schauder decoder, the data anomaly analysis unit is configured to perform data anomaly analysis based on the received pre-processed data and obtaining the result of the analysis of anomalies, as well as correcting the errors of the result of the analysis of anomalies to obtain the final result of the analysis.
[13] O. Solomentsev, et al., Method of optimal threshold calculation in case of radio equipment maintenance, in: Lecture Notes in Networks and Systems, volume 462, Springer, 2022, pp. 69–79. doi:10.1007/978-981-19-2211-4_6. [14] S. Dutchak, N. Opolska, R. Shchokin, O. Durman, M. Shevtsiv, International aspects of legal regulation of information relations in the global internet network, Journal of Legal, Ethical and Regulatory Issues 23 (2020) 1–7. [15] Y. Daradkeh, L. Guryanova, S. Kavun, T. Klebanova, Forecasting the cyclical dynamics of the development territories: Conceptual approaches, models, experiments, European Journal of Scientific Research 74 (2012) 5–20. [16] R. Brumnik, T. Klebanova, L. Guryanova, S. Kavun, O. Trydid, Simulation of territorial development based on fiscal policy tools, Mathematical Problems in Engineering 2014 (2014) 843976. [17] V. Kalashnikov, S. Dempe, B. Mordukhovich, S. Kavun, Bilevel optimal control, equilibrium, and combinatorial problems with applications to engineering, Mathematical Problems in Engineering 2017 (2017) 7190763. [18] A. Dudnik, O. Trush, V. Kvasnikov, T. Domkiv, Development of distributed multi-segment wireless networks for determining external situations, in: CEUR Workshop Proceedings, volume 2845, 2021, pp. 127–137. URL: https://ceur-ws.org/Vol-2845/Paper_13.pdf. [19] O. Pysarchuk, A. Gizun, A. Dudnik, V. Griga, T. Domkiv, S. Gnatyuk, Bifurcation prediction method for the emergence and development dynamics of information conflicts in cybernetic space, in: CybHyg, volume 2654, 2019, pp. 692–709. URL: https://ceur-ws.org/Vol-2654/paper54.pdf. [20] I. Bakhov, Y. Rudenko, A. Dudnik, N. Dehtiarova, S. Petrenko, Problems of teaching future teachers of humanities the basics of fuzzy logic and ways to overcome them, International Journal of Early Childhood Special Education (INT-JECSE) 13 (2021) 844–854. doi:10.9756/INT-JECSE/V13I2. 211127. [21] A. Dudnik, Y. Kravchenko, O. Trush, O. Leshchenko, N. Dakhno, V. Rakytskyi, Study of the features of ensuring quality indicators in multiservice networks of the wi-fi standard, in: IEEE 3rd International Conference on Advanced Trends in Information Theory (ATIT), Kyiv, Ukraine, 2021, pp. 93–98. doi:10.1109/ATIT54053.2021.9678691. [22] L. Kuzmych, D. Ornatskyi, V. Kvasnikov, A. Kuzmych, A. Dudnik, S. Kuzmych, Development of the intelligent instrument system for measurement parameters of the stress - strain state of complex structures, in: IEEE 4th International Conference on Advanced Trends in Information Theory (ATIT), Kyiv, Ukraine, 2022, pp. 120–124. doi:10.1109/ATIT58178.2022.10024222. [23] O. Trush, A. Dudnik, M. Trush, O. Leshchenko, K. Shmat, R. Mykolaichuk, Mask mode monitoring systems using it technologies, in: IEEE 4th International Conference on Advanced Trends in Information Theory (ATIT), Kyiv, Ukraine, 2022, pp. 219–224. doi:10.1109/ATIT58178.2022. 10024216. [24] A. Dudnik, B. Presnall, M. Tyshchenko, O. Trush, Methods of determining the influence of physical obstructions on the parameters of the signal of wireless networks, in: IT&I Workshops, volume 3179, 2021, pp. 227–240. URL: https://ceur-ws.org/Vol-3179/Paper_21.pdf. [25] N. Dakhno, O. Leshchenko, Y. Kravchenko, A. Dudnik, O. Trush, V. Khankishiev, Dynamic model of the spread of viruses in a computer network using diferential equations, in: IEEE 3rd International Conference on Advanced Trends in Information Theory (ATIT), Kyiv, Ukraine, 2021, pp. 111–115. doi:10.1109/ATIT54053.2021.9678822. [26] V. Geranin, N. Meleshko, L. Sereda, The root-mean-square error of representation of stationary random processes in the basis of schauder’s functions, in: Proceedings of the XI Vses. symposium "Methods of representation and hardware analysis of random processes and fields", L., 1980, pp. 80–85. [27] Methods and microelectronic means of digital transformation and signal processing, Abstracts of conference reports, vol. 1, vol. 2, 1989. [28] I. Krysanova, N. Meleshko, V. Taran, The device for calculating the coeficients of the decomposition of speech signals into a series according to the schauder function system, KPI Electroacoustics and Sound Engineering (1985). [29] O. Popov, Y. Kyrylenko, I. Kameneva, A. Iatsyshyn, A. Iatsyshyn, V. Kovach, A. Kiv, The use of specialized software for liquid radioactive material spills simulation to teach students and postgraduate students, in: CEUR Workshop Proceedings, volume 3085, 2022, pp. 306–322. [30] R. Aleksieieva, A. Fesenko, A. Dudnik, Y. Zhanerke, Software tool for ensuring data integrity and confidentiality through the use of cryptographic mechanisms, in: MoMLeT+ DS, volume 3426, 2023, pp. 259–273. URL: https://ceur-ws.org/Vol-3426/paper21.pdf. [31] A. Dudnik, S. Dorozhynskyi, S. Grinenko, O. Usachenko, B. Vorovych, O. Grinenko, Methods of constructing a lighting control system for wireless sensor network “smart home”, in: Smart Technologies, 2022. doi:10.1007/978-3-031-04809-8_15. [32] A. Dudnik, I. Bakhov, O. Makhovych, Y. Ryabokin, O. Usachenko, Models and methods for improving performance of wireless computer networks based on the decomposition of lower layers of the osi reference model, International Journal of Emerging Technology and Advanced Engineering 12 (2022) 152–162. doi:10.46338/ijetae0122_15. [33] O. Trush, I. Kravchenko, M. Trush, O. Pliushch, A. Dudnik, K. Shmat, Model of the sensor network based on unmanned aerial vehicle, in: IEEE 3rd International Conference on Advanced Trends in Information Theory (ATIT), Kyiv, Ukraine, 2021, pp. 138–143. doi:10.1109/ATIT54053.2021. 9678623.