|Digital Signal Processing||
Vector and spectral digital signal processing in musical acoustics by the method of parametric discrete Fourier transform
A system analysis of the methods of vector and spectral processing of musical signals used in musical acoustics has shown the widespread use of classical methods based on discrete Fourier transform (DFT) and algorithms for its fast calculation - fast Fourier transform (FFT) algorithms. This is explained by the adequacy of the DFT mathematical apparatus to the structure of musical signals, the possibility of visual physical interpretation and interpretation of the results. At the same time, the DFT, in addition to its advantages, also has significant drawbacks, which are manifested in the form of undesirable effects of superposition, stockade, leakage, and scallop effect. When processing musical signals, the effect of the picket is especially undesirable, which impedes the processing of signals due to the mismatch of the frequencies of musical sounds and their overtones with the frequencies obtained as a result of using DFT.
The article considers a generalization of the discrete Fourier transform in the form of a parametric discrete Fourier transform (DFT-P). This conversion allows you to eliminate or significantly reduce the effect of undesirable effects of DFT in vector and spectral processing of signals. The article shows that the use of DFT-P instead of DFT allows one to obtain a more detailed spectral picture, both in statics and dynamics. The fact is that both the vector and spectral patterns obtained using the DFT are only “thinned out” versions of the corresponding pictures obtained on the basis of the DFT-P.
It is proposed to increase the efficiency and effectiveness of the methods of spectral and vector processing of musical signals in musical acoustics through the application of the method of parametric Fourier transform (DFT-P). The problems of using DFT-P to determine the fundamental tone and timbre of musical signals are investigated.
2. Ponomarev V.A., Ponomareva O.V. Teoriya i primenenie parametricheskogo diskretnogo preobrazovaniya Fur'e (Theory and application of the parametric discrete Fourier transform) // Digital signal processing. 2011. no. 1. pp. 2 - 6.
3. Ponomarev V.A. Ponomareva O.V., Ponomarev A.V. Method for Effective Measurement of a Sliding Parametric Fourier Spectrum. Optoelectronics, Instrumentation and Data Processing, vol. 50, no. 2, pp.1-7. (2014).
4. Ponomareva O.V. Osnovy teorii diskretnyh kosvennyh izmerenij parametrov signalov (Fundamentals of the theory of discrete indirect measurements of signal parameters). Izhevsk. Publishing house IzhSTU/ 2016. 172 p.
5. Ponomareva O., Ponomarev A., Ponomarev V. Evolution of Forward and Inverse Discrete Fourier Transform // Proceedings of IEEE East-West Design & Test Symposium (EWDTS 2018). 2018. C. 313-318.
6. Ponomareva O., Ponomarev A., Ponomareva N. Window - presume Parametric Discrete Fourier Transform // Proceedings of IEEE East-West Design & Test Symposium (EWDTS 2018). 2018. C. 364-368.
7. Ponomareva, O.V. Razvitie teorii i razrabotka metodov i algoritmov cifrovoj obrabotki informacionnyh signalov v parametricheskih bazisah Fur'e (The development of theory and the development of methods and algorithms for digital processing of information signals in parametric Fourier bases) Âis .... Dr. tech. Sciences: 05.13.01 / Ponomareva Olga Vladimirovna. - Izhevsk, 2016 .-- 357 p.
8. Aldoshina, I.A. Muzykal'naya akustika: uchebnik / I.A.Aldoshina, R.Pritts (Musical acoustics: textbook / I.A. Aldoshina, R. Pritts) SPb: Kompozitor – Sankt-Peterburg. 2006. 720 p.
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10. Ponomareva, N. V. Komp'yuternaya spektral'naya obrabotka muzykal'no-akusticheskih signalov na osnove parametricheskogo diskretnogo preobrazovaniya Fur'e (Computer-aided spectral processing of musical acoustic signals based on the parametric discrete Fourier transform) // Priborostroenie - 2017: materialy 10-j Mezhdunarodnoj nauchno-tekhnicheskoj konferencii, 1-3 noyabrya 2017 goda, Minsk, Respublika Belarus' / Belorusskij nacional'nyj tekhnicheskij universitet; redkol.: O. K. Gusev [i dr.]. Minsk: BNTU/ 2017. pp. 16-18.
11. Ponomareva, N. V. Problemy komp'yuternoj spektral'noj obrabotki signalov v muzykal'noj akustike (Problems of computer spectral processing of signals in musical acoustics) // Intellektual'nye sistemy v proizvodstve. 2018. vol. 16. no 1. pp.. 26-32.
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Fundamentals of the theory of two-dimensional digital signal processing in Fourier bases with variable parameters
Keywords: two-dimensional discrete Fourier transform, cyclic two-dimensional correlation function, spatial domain, spatial frequency domain, canonical decomposition of a random signal.
The theory under development is a generalization of the classical theory of discrete two-dimensional spectral processing of signals specified on a finite reference plane in Fourier bases. When conducting 2-D correlation analysis, 2-D signal filtering, and 2-D spectral and vector processing of 2 D signals, the operation of supplementing the 2-D signal on the reference plane with zero samples is widely used. Based on the analysis of structural modification of the 2-D DFT matrix, in the case of supplementing the 2-D signal on the reference plane with zero samples, a system of 2-D parametric discrete exponential functions (2-D DEF-P) is introduced. The analytical properties of the 2-D DEF-P system, which is the basis of the newly introduced direct two-dimensional parametric discrete Fourier transform (2-D DFT-P), are considered.
The theoretical foundations of the proposed theory allow us to develop new and improve existing methods and algorithms of two-dimensional Fourier - signal processing, as well as eliminate or significantly reduce the effects of negative effects accompanying the use of standard 2-D DFT. The proposed two-dimensional DFT-P significantly expands the mathematical apparatus of information technology methods for digital spectral processing of signals and images in the spatial and spatial-frequency domains.
Studies of the analytical properties of 2-D DEF-P, which are the basis of 2-D DFT-P, allow us to conclude that the probabilistic properties of 2-D DFT-P of two-dimensional signals - the properties that follow from the two-dimensional version of the Wiener-Khinchin theorem - are promising.
2. Ponomarev V.A., Ponomareva O.V. Vremennye okna pri ocenke energeticheskih spektrov metodom parametricheskogo diskretnogo preobrazovaniya Fur'e (Time windows for estimating energy spectra by the method of parametric discrete Fourier transform // Avtometriya. 1983. no. 4. pp. 39 - 45.
3. Ponomarev V.A., Ponomareva O.V. Modifikaciya diskretnogo preobrazovaniya Fur'e dlya resheniya zadach interpolyacii i svertki funkcij (Modification of the discrete Fourier transform for solving problems of interpolation and convolution of functions) // Radiotekhnika i elektronika. 1984. vol. 29. no 8. pp. 1561 - 1570. 4. Ponomareva O.V. Razvitie teorii spektral'nogo analiza diskretnyh signalov na konechnyh intervalah v bazise parametricheskih eksponencial'nyh funkcij (Development of the theory of spectral analysis of discrete signals at finite intervals in the basis of parametric exponential functions) // Digital Signal Processing. 2010. no. 2. pp. 7 - 11.
5. Ponomarev V.A., Ponomareva O.V. Teoriya i primenenie parametricheskogo diskretnogo preobrazovaniya Fur'e (Theory and application of the parametric discrete Fourier transform) // Digital signal processing. 2011. no. 1. pp. 2 - 6.
6. Ponomareva O.V. Neinvariantnost' skol'zyashchego energeticheskogo parametricheskogo Fur'e-spektra dejstvitel'nyh tonal'nyh signalov (Non-invariance of the moving energy parametric Fourier spectrum of real tonal signals) // Digital signal processing. 2014. no. 2. pp. 7 - 14.
7. Ponomareva O.V. Osnovy teorii diskretnyh kosvennyh izmerenij parametrov signalov (Fundamentals of the theory of discrete indirect measurements of signal parameters). Izhevsk. Publishing house IzhSTU/ 2016. 172 p.
8. Ponomarev A.V. Dvumernaya obrabotka signalov v diskretnyh bazisah Fur'e (Two-dimensional signal processing in discrete Fourier bases) // Intellectual systems in production. 2019. vol.17. no. 1. pp.71-77.
9. Ponomareva O.V., Ponomarev A.V. Interpolyaciya v prostranstvennoj oblasti dvumernyh diskretnyh signalov s pomoshch'yu bystryh preobrazovanij Fur'e (Interpolation in the spatial domain of two-dimensional discrete signals using fast Fourier transforms) // Intellectual systems in production. 2019. vol.17. no. 1. pp.88-94.
10. Ponomareva O., Ponomarev A., Ponomarev V. Evolution of Forward and Inverse Discrete Fourier Transform // Proceedings of IEEE East-West Design & Test Symposium (EWDTS 2018). 2018. C. 313-318.
11. Ponomareva O., Ponomarev A., Ponomareva N. Window - presume Parametric Discrete Fourier Transform // Proceedings of IEEE East-West Design & Test Symposium (EWDTS 2018). 2018. C. 364-368.
12. Ponomarev V.A., Ponomareva O.V., Ponomarev A. V. Izmerenie vremennyh spektrov diskretnyh signalov na konechnyh intervalah (Measurement of time spectra of discrete signals at finite intervals) // Vestnik IzhGTU im. Kalashnikova M.T. 2016. vol.19. no2. pp. 80-83.
13. Ponomareva O.V., Alekseev V.A., Ponomarev A. V. Bystryj algoritm izmereniya spektra dejstvitel'nyh signalov metodom aperiodicheskogo diskretnogo preobrazovaniya Fur'e (Fast algorithm for measuring the spectrum of real signals by the method of aperiodic discrete Fourier transform) // Vestnik IzhGTU im. Kalashnikova M.T. 2014. no2. pp. 106-109.
14. Ponomareva O.V. Izmerenie spektrov kompleksnyh signalov na konechnyh intervalah metodom aperiodicheskogo diskretnogo preobrazovaniya Fur'e (Measurement of spectra of complex signals at finite intervals by the method of aperiodic discrete Fourier transform) // Intellectual systems in production. 2014. no. 1(23). pp.100-107.
15. Ponomarev V.A., Ponomareva O.V. Invariantnost' tekushchego energeticheskogo Fur'e–spektra kompleksnyh diskretnyh signalov na konechnyh intervalah (Invariance of the current energy Fourier spectrum of complex discrete signals at finite intervals) // Izvestiya vysshih uchebnyh zavedenij Rossii. Radioelektronika .2014. no2. pp. 8-16.
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Multi-criteria synthesis for class of digital structurally protected radio signals
To adapt prospective radio communication systems to the action of a combination of additive “white” Gaussian noise (AWGN) and other interfering factors (including narrow-band interference), it is advisable to use multi-criteria synthesis of radio signals with a dip in the power spectral density at frequencies interference action. The combined quality criterion included private criteria responsible for reducing narrow-band interference effects; noise immunity with AWGN, as well as minimizing out-of-band radiation and maximizing signal structural hiding. The application of the last criterion led in radio monitoring systems to a significant increase in the probability of bit error from detectors set of the known signals types under consideration (by up to 25%).
This, in turn, will complicate the recognition of the type of radio signal and will lead to the impossibility of unauthorized interception of the transmitted information or setting an effective (signal-like) interference. To exclude the influence of a particular modulation type recognition algo-rithm, it is advisable to consider a quality indicator that provides an increase in structural hiding in the form of maximizing the probability of a bit error from detector set of the known signals types in the case of a mixture of a synthesized signal and AWGN at a relatively high signal-to-noise ratio.
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The preliminary analysis showed that these signals are located on the “power-bandwidth trade-off” plane in areas that is referred to both phase and frequency shift keying. At the same time the modulation type of RSCCQC retunes by changing only two parameters. The parameters define a depth of intersymbol interferences and interchannel (between in-phase and quadrature components of radio signal) interferences.
Since in practice the radio monitoring of radio signal structure is executed under condition of a prior unknown modulation type and parameters of analyzed radio signal, in this article two classification algorithms provided reliable recognition in these assumptions were selected. To evaluate the ambiguity of algorithm decisions (i.e. the resistance to recognition) the entropy is used. It was shown that the first classification algorithm has a maximum of the decision ambiguity on the set of synthesized RSCCQC at point that corresponds to a local minimum of the decision ambiguity of the second algorithm.
And so, to determine the resistant RSCCQC the combined quality index is chosen. This quality index is a weighted sum of decision entropies of the considering classification algorithms. The RSCCQC maximizing this combined criterion provides approximately equal entropy values of both algorithms. This value is 0.9 bits. It was shown that the resistant to recognition RSCCQC are comparable by basic spectral and energy characteristics with known bandwidth-efficient signals.
The work was executed with support of Russian President's Grant MK-3658.2018.9 (agreement No. 075-02-2018-681 dated 11/16/2018).
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To obtain a possibility of specifying also an arbitrary spectrum sidelobe falloff rate, it is proposed to represent the window w(t) as a product of a halfwave of the cosine function raised to the power μ (where μ is a nonnegative real number) and an initial portion of the generalized Fourier series of m+1 terms (m is a small integer, called the order of the function) in powers of the same cosine function:
where N is the total number of sampling points over the signal duration. The first multiplier has a μth order zero at the ends of the window, which provides the spectrum sidelobe falloff rate 6(μ+1) dB/oct, that is the sidelobe falloff rate can be chosen arbitrarily.
The unknown coefficients bk of the series in the second multiplier are obtained as a solution to the minimax optimization problem so as to provide the minimum possible (at the given μ and m) maximum sidelobe level in the frequency region |f|≥β, where β is the specified main lobe halfwidth measured at the level of the highest sidelobe. It is proved that the functional of the problem is convex and the solution is unique. The problem is reduced to the linear Chebyshev approximation problem, which makes it possible to use the known mathematical theory to develop a criterion for an approximate solution obtained numerically. The criterion evaluates how close is the sidelobe level achieved to the exact minimum value. The functional is shown to have a ravine structure, and an efficient minimization method is developed. For integer μ parameters of optimal window functions are tabulated in our previous papers [8-12].
In this paper, tables of coefficients of the synthesized optimal functions and their parameters are provided for the sidelobe falloff rates 9, 15, and 21 dB/oct down to the sidelobe levels of about minus 200 dB. Simple linear relationships are obtained that give fine estimates of the equivalent noise bandwidth and minus 6-dB main lobe width of the optimum function from its minus 3-dB width. Functions are presented that feature the minimum possible level of the maximum sidelobe at m = 3, 4 and 5 and integer μ in the range from 3 through 12. Characteristics of the windows obtained are discussed and their implementation in the time and frequency domains is illustrated.
The optimal windows developed are simple to calculate and are intended for real-time applications.
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The solution of adaptive space filtering problem under L linear constraints is as follows: W = R-1C(CHR-1C)-1FH, where C is the N x L matrix of linear constraints, F is L-element row vector of linear constraints coefficients. It is shown in the article that this solution can be presented in strictly equivalent numerically stable form W = ÔÍÔCÔ1Ô1HFH, where Ô1 is upper triangular L x L matrix, that orthonormalizes the columns of the matrix Y1 = ÔÑ.
Estimation of the linearly constrained algorithm numerical stability is executed by computer si-mulation as the estimate of required computer word length for numerically stable algorithm (that uses orthogonalization) as compared with two other methods – the first based on direct matrix inversion by sequential elimination (Gauss method), and the second based on its equivalent solution of finding matrix of the form C1 = R-1C by solving the system RC1 = C.
The results obtained show that numerically stable algorithm requires lesser computer word length, the required word lengths may be different as one and a half to twice, the processing costs being comparable in all three cases.
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At the level of an invention the modem of new generation in which the method of statistical modulation based on change of characteristic function of the quasidetermined signal depending on emergence of a logical "0" and logical "1" cable signal is applied is constructed. In work Tikhonov's law is used from statistical radio engineering and on its basis the digital quasidetermined signal is constructed.
The demodulator from two independent channels is developed for reception of modulated signals with distribution under Tikhonov's law, each of which measures assessment only valid or assessment only of an imaginary part of characteristic function and compares its value to a threshold on a predefined algorithm. Qualitatively also the noise stability of both channels of the modem is quantitatively estimated during the work in the channel with "white" noise. It is shown that noise stability of both independent channels of the modem different and is, at least, two orders better than this characteristic of the known device with multiposition carrier shift and also devices with ideal phase shift keying.
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