Digital Signal Processing

Scientific & Technical

“Digital Signal Processing” No. 4-2022

In the issue:

- two-dimensional Fourier transform
- partially adaptive antenna array
- optimal signal ensembles reception
- frequency responses of reference signals
- transmission systems energy efficiency analysis
- mobile sources detection
- echometric signal classification
- videosignal phase-energy functions
analysis of heart rate variability

Crossed complex-conjucated symmetry coefficients of the two-dimensional discrete Fourier transform with variable parameters of real signals
O.V. Ponomareva, e-mail: ponva@

A.V. Ponomarev, e-mail:
N.V. Ponomareva, e-mail:
Kalashnikov Izhevsk State Technical University (Kalashnikov ISTU), Russia, Izhevsk
Sevastopol State University, Russia, Sevastopol

Keywords: real signal, parametric discrete Fourier transform, discrete two-dimensional Fourier transform with variable parameters, cross complex conjugate symmetry.

In many areas of scientific research, the methods and algorithms of digital Fourier processing have applications in solving a wide range of practical problems. The methods and algorithms of this group for processing complex and real discrete finite signals are based on one-dimensional, two-dimensional (generally multidimensional) discrete Fourier transform. The practice of applying the methods of digital Fourier processing of discrete finite signals based on discrete Fourier transforms has revealed both the advantages of these unitary transformations and their disadvantages, which manifest themselves in the form of a number of negative effects. These are, first of all, the picket fence effect, the aliasing effect and the leakage effect, as well as the scalloping effect. The paper considers two new discrete Fourier transforms that are a generalization of the classical discrete Fourier transforms: the parametric discrete Fourier transform (one-dimensional case) (DFT-P) and the discrete two-dimensional Fourier transform with variable parameters (two-dimensional case) (2D DFT-IP). These Fourier transforms, which are a development of the classical discrete Fourier transforms, make it possible to eliminate or significantly weaken the influence of the negative effects inherent in standard discrete Fourier transforms. Due to the wide distribution of real signals, in order to develop effective and efficient methods for the Fourier processing of this class of signals in new bases, the properties of the complex conjugate symmetry of the DFT-P and 2D DFT-IP coefficients are considered in the work. The concept of cross complex-conjugate symmetry of 2D DFT-WT coefficients of real signals is introduced. The properties of the cross complex conjugate symmetry of the 2D DFT-WT coefficients of real signals are confirmed by the results of mathematical modeling. Methods and algorithms for fast calculation of the discrete Fourier transform with variable parameters of real signals for various combinations of variable parameters have been developed.


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Combination of row and column antenna signals
V.I. Djigan, e-mail:

Institute for design problems in microelectronics of Russian Academy of Sciences, Moscow, Russia

Keywords: Adaptive cancellation of interferences, adaptive antenna array, rectangular array, partial adaptation, Recursive Least Squares (RLS) algorithm.


This paper considers a two-dimensional adaptive antenna array, whose antennas are placed in the nodes of the equally spaced rectangular grid. If the number of the antennas in such array is large, then the partial adaptation is used to reduce the arithmetic complexity of its adaptive algorithm. The total signals of the antennas over each row and each column of the array are adaptively processed in base-band. The radiation pattern of such an array does not have the grating lobes. This distinguishes it from a hybrid array, which is composed using the subarrays in the case of a large number of antennas. The radiation pattern of the hybrid antenna array is characterized by the grating lobes. Such lobes restricts the possibility of the interference suppression if the spatial locations of their sources coincide with those of these lobes. The arithmetic complexity of the used adaptive algorithm does not depend on the total number of its antennas in the considered partially adaptive antenna array. It depends on the number of rows and columns of antennas in the array. The examples of the computational procedure based on the Matrix Inversion Lemma (MIL) Recursive Least Squares (RLS) adaptive algorithm for the weight calculation of the partially and fully adaptive arrays are presented. Instead of the above mentioned adaptive algorithm, the gradient adaptive algorithms or other RLS adaptive algorithms can also be used in these adaptive arrays. The simulation demonstrate that the steady-state performance of the partially adaptive array is about the same as that of the fully adaptive array if the number of the received signal sources does not exceed the number of the weights of the partially adaptive array, which equals the sum of its rows and columns. In this case, the dynamic behavior of the compared full and partially adaptive arrays in the terms of the radiation pattern values towards to the sources of the received signals is differed only at the initial stage of the transient response. The considered technology of partial adaptation can be used in the rectangular adaptive arrays with large number of regularly spaced antennas.

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Multidimensional signal ensembles, their structure and methods of optimal reception
Bykhovskiy Mark, e-mail:
The Moscow Technical University of Communications and Informatics (MTUCI), Russia, Moscow

Keywords: optimal multidimensional signals, error-correcting codes, message transmission rate, energy and spectral efficiency, optimal signal reception.

The paper examines the issues related to the structure of multidimensional signal ensembles with hyperphase modulation (HPPM). The author considers two functional designs of optimal demodulators of HPPM signals, one of which is synthesized on the basis of the Neumann-Pearson method, and the second one is based on a method that allows to determine the coordinates of signal points for HPPM signals by using a maximum likelihood method.

The article shows that the complexity of the technical implementation of the first functional circuit increases exponentially with an increase in the number of signals with HPPM, while the second functional circuit only linearly. At the same time, the noise immunity of the second circuit in terms of reception noise immunity is practically not inferior to that synthesized on the basis of the Neumann-Pearson method.

The proposed methods allow to estimate with a high accuracy the probability of error when receiving HPPM signals. The paper contains formulas and charts that allow to estimate the probability of error during signal reception depending on normalized signal duration, as well as on the marginal speed of message transmission and on the signal-to-noise ratio at the demodulator input. The author presents the results of a comparison of the communication system with HPPM to traditional communication systems, in which two-dimensional signal ensembles and error-correcting codes (ECC) are used. It is shown that communication systems with HPPM provide 2 ... 3 dB higher energy efficiency, as well as 1.5 ... 2 times higher spectral efficiency as compared to traditional communication systems. It is noted that the technical implementation of communication systems with HPPM is substantially simpler than communication systems in which two-dimensional signal ensembles and long ECC are utilized.

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Analysis of the frequency responses of the correlation processing procedures for arbitrary andphase shift keying reference signals

E.V. Kuzmin, e-mail:
Siberian Federal University (SibFU), Russia, Krasnoyarsk

Keywords: correlator, frequency response, spread spectrum phase shift keying signal, phase discriminator, early-late time-delay discriminator, signal delay, Fourier transform, Rayleigh identity.

Analytical expressions for dot products of a harmonic signal with arbitrary parameters and quadrature reference signals have been analytically obtained. The Rayleigh identity is used as a methodological basis, which made it possible to calculate the above dot products in the frequency domain. The general case is considered, which assumes reference signals with a known spectrum (but without specifying them), as well as a special case in which analytical solutions are obtained for spread spectrum phase shift keying reference signals generated by a binary pseudo-random sequence. The behavior of dot products – responds of quadrature correlators to an external harmonic signal for one and M > 1 periods of the reference phase shift keying signal is studied. The frequency responses of typical correlation processing procedures are obtained: a quadrature correlation scheme, a family of phase discriminators, and an early-late time-delay discriminator. Simulation modeling was carried out and confirmed the correctness of the analytical solutions obtained in the article, which illustrated graphically.

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9. Kuzmin E.V., Zograf F.G. Povyshenie verojatnosti pravil'nogo poiska shumopodobnogo signala po vremeni zapazdyvanija na fone tonal'noj pomehi (Enhancement of the probability of spread-spectrum signal correct searching in case of narrow-band interference) // Uspekhi sovremennoi radioelektroniki (Journal Achievements of Modern Radioelectronics). 2016. no 11. pp. 137–140.

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11. Ye F., Tian H., Che F. CW interference effects on the performance of GPS receivers // Progress In Electromagnetics Research Symposium – Fall (PIERS – FALL), 19-22 November 2017, Singapore. pp. 66–72.

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13. Du R., Yue L., Yao S., Zhang D., Wang Y. Single-tone interference method based on frequency difference for GPS receivers // Progress In Electromagnetics Research M. 2019. V. 79. pp. 61–69.

14. Kuzmin E.V. O vlijanii kvantovanija po urovnju na jeffektivnost' procedury poiska shumopodobnogo signala po zaderzhke na fone shuma i garmonicheskoj pomehi (Efficiency of the spread spectrum signal searching procedure in case of continuous wave interference and quantization effect) // Tsifrovaya obrabotka signalov (Digital signal processing). 2020. no 2. pp. 41–45.

15. Kulikov G.V., Do Chung Tien. Effektivnost' fazovogo algoritma adaptivnoi fil'tratsii pri prieme signalov s mnogopozitsionnoi fazovoi manipulyatsiei (Efficiency of the phase adaptive filtering algorithm when receiving signals with multiposition phase shift keying) // Zhurnal radioelektroniki [jelektronnyj zhurnal] (Journal of Radioelectronics). 2020. no 4. URL:

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Mobile sources detection by a receiver system
V. K. Klochko, e-mail:
Vu Ba Hung, e-mail:
The Ryazan State Radio Engineering University (RSREU), Russia, Ryazan

radio signals, Doppler receivers, signal detection, paramer estimates, velocity vector estimates, mathematical and computer modeling.

The problem of detecting several mobile sources by a system of several mutually oriented radio receivers at given range boundaries at current moments is solved. Based on the signals received in the receivers, a decision is made on the presence of sources, their number, spatial coordinates and velocity vectors are determined. The solution is based on the algebraic criterion for classifying direction vectors into sources according to the principle of their conjugation.

The purpose of the work – to increase the efficiency of the radio receiver positioning system when detecting useful signals from mobile low-altitude sources in conditions of interference. It is proposed to increase efficiency of sources detection due to location of receivers in a certain way, coordinated operation of receiver-producing station with several auxiliary receivers when processing Doppler frequency spectra of received signals using algebraic criteria.

The problem of distinguishing close velocity vectors in space (respectively, increasing the resolution of the Doppler frequency) is solved as a problem of distinguishing velocity vectors in projections on the planes formed by the sight lines of the receivers. At the same time, the more receivers (projection planes), the better the resolution.

Computer simulation results are provided showing the advantage of operating the system over a single transceiver station. The applied orientation of the work – algorithmic support of radio systems for the protection of small areas and ultrasound diagnostics devices.

1. Bakulev P.A. Radar systems: a textbook for universities. M.: Radio engineering, 2007. 376 p.

2. Klochko V.K. Direction finding of moving objects by a multi-position Doppler system // Radio engineering. 2020. T. 84, ¹ 11 (21). P. 5 - 12.

3. Klochko V.K. Algebraic approach to the direction finding of objects in a multi-position system theme of receivers // Digital signal processing. 2022. ¹ 1. P. 28-33.

4. Klochko V.K., Wu Ba Hung. Algorithms for increasing the resolution over the Doppler frequency in the radio receiver system // Radio engineering and telecommunication systems. 2022. ¹ 3. P. 31–42.

5. Marple Jr. S.L. Digital spectral analysis and its applications. M.: Mir, 1990. 584 p.

6. Methods and algorithms of digital spectral signal analysis: tutorial / V.I. Koshelev. M.: COURSE, 2021. 144 p.

7. Klochko V.K, Kuznetsov V.P, Wu Ba Hung. Estimation of radio signal parameters from mobile low-altitude objects // Bulletin of Ryazan State Radio Engineering University. 2022. ¹ 80. P. 12 - 23.

Analysis and classification of an echometric signal for fluid level detection in oil-producing wells
A.A. Zaikin:

R.A. Minnullina:
Kazan Federal University (KFU), Kazan, Russia

Keywords: noise suppression, peak detection, classification.

This article is devoted to fluid level detection on an echometric signal also known as echogram. Aim of the article is to process the signal of any type and create a classifier to distinguish normal echograms from defective ones.

Echometry is one of the most widely used methods of fluid level detection in oil-producing wells. On the echogram, position of the acoustic wave reflection from the surface is displayed as diminishing peaks with the same distance between them. This study proposes a method for the signal modeling through the solution of the quadratic programming problem (for denoising) and a peak detection algorithm. The algorithm consists of two steps. First step determines position of the starting shift. The second step uses position of the first peak to find distance between remaining peaks. This step is based on the objective function maximization, which describes the total impact of peaks. Knowing the distance between peaks allows determining the amount of peaks in the whole signal.

Peak position detection depends on the quality of the echogram because external sounds and foreign objects also affect it. The paper proposes feature extraction methods from the signal, which help differentiate between normal and damaged signals. These features include number of peaks, the presence of local maxima close to the peak, etc. Logistic regression was chosen as a machine learning model. Despite the large amount of data, many echograms are almost the same, so 60 damaged and 60 normal hand-selected echograms were used for training. The obtained results showed that some normal signals might be identified as defective in single cases. However, the main task of determining pure normal signals has been completed. According to the results of 5-fold cross-validation, the accuracy of the classifier is 97%.

Obtained results show that proposed algorithm is able to process different types of the signal and finds peaks successfully, and the classifier selects normal echograms with high accuracy.

1. Nalimov K.G. Informatsionnaya sistema ekhometrirovaniya mnogoimpul'snymi signalami dlya opredeleniya urovnya zhidkosti v neftedobyvayushchikh skvazhinakh. Dokt, Diss. [Information system of echometry with multipulse signals for determining the liquid level in oil-producing wells. Dokt, Diss.] Tomsk, 2007. 132 p.

2. Zholmagambetova B.R., Mazakov T.ZH., Bukenov M.M., Izat E.Z. Electrocardiogram R-peak detection and noise reduction with hybrid linearization and principal component analysis. Trudy universiteta [Proc. of the university”], ¹3 (80) 2020; p. 157-162.

3. Suyi Li, Shanqing Jiang, Shan Jiang, Jiang Wu, Wenji Xiong, Shu Diao1. A Hybrid Wavelet-Based Method for the Peak Detection of Photoplethysmography Signals. Hindawi Computational and Mathematical Methods in Medicine Volume 2017. 9468503, 8 p.

4. Filipa Esgalhado, Arnaldo Batista, Valentina Vassilenko, Sara Russo, Manuel Ortigueira. Peak Detection and HRV Feature Evaluation on ECG andPPG Signals. Symmetry 2022, 14, 113 p.

5. Dakhva M. S., Leukhin A. N. Sravneniye pyati algoritmov dlya obnaruzheniya R-pikov v EKG-signale [Comparison of five algorithms for detecting R-peaks in an ECG signal]. Bulletin of the Volga State University of Technology. Series: Radio engineering and infocommunication systems. 2018. No. 3 (39). p. 39-49.

6. Kavsaoglu, Ahmet & Polat, Kemal & Bozkurt, Mehmet. (2016). An innovative peak detection algorithm for photoplethysmography signals: An adaptive segmentation method. TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES. 24. 1782-1796. 10.3906/elk-1310-177.

Phase-energy functions of a video sequence with a rectangular moving object
S.V. Vasilyev, e-mail:
I.V. Zhigulina,
D.A. Derbush

Military scholastic-scientific center of the Air forces «Air forces academy named by prof. N.E. Zhukovskogo and Y.A. Gagarina», Russian Federation, Voronezh

Keywords: video sequence, dynamic object, two-dimensional image, phase-energy characteristic, phase-energy function.

The problem of motion detection is one of the central ones in the processing of video information. An analysis of publications on this topic indicates the presence of significant successes in certain areas, but its comprehensive effective solution has not yet been found.

The interest in this problem is not accidental. Detection of moving objects is carried out in such areas as security, space control, air navigation, transport and industrial control, monitoring of forest fires, etc. Despite the wide variety of developed methods for detecting movements in the frame, their applicability significantly depends on the conditions in which the search task is solved. Insufficient reliability of the corresponding algorithms in the space of all possible states of the background and objects is a limiting factor on the way to the creation of both universal multifunctional image processing systems and video sequences, and specialized ones included, for example, in the automatic decision-making circuit of critical objects.

In well-known works, the classification of motion search methods in video sequences is given. The authors include correlation methods, statistical segmentation, spatial and spatio-temporal filtering to the group of statistical methods. The methods of matching, tracking edge points, tracking point features, as well as methods based on graph models are referred to the parametric approach. All these groups of methods are characterized to some extent by sensitivity to background uniformity, signal-to-noise ratio, volume of a priori information, as well as significant computational complexity.

Spectral analysis is a powerful tool for detecting motion in video sequence frames. The video sensor generates an image in the form of an ordered set of discrete-analog samples of a random two-dimensional spatial field. The image spectrum is described by a two-dimensional Fourier transform discretized in spatial coordinates. It contains two random functions – the amplitude-frequency and phase-frequency spatial spectra. In linear image processing, the square of the conversion module – the energy spectrum - is usually used to reduce the influence of randomness. However, there are many tasks where it is important to use the phase information contained in the phase spectrum, first of all, it is the determination of the location of objects and the identification of movement.

The mathematical apparatus considered in the article allows to take into account both energy and phase information at the same time. The proposed approach is a development of spatial-temporal filtering methods.

1. Mareyev À.V., Orlov À.À., Ryghkova Ì.N. Metodi lokalizatsii objektov v videopotoke // Radiotehnicheskiye i telekommunikatsionnie sistemi. 2021. no. 3. pp. 48-60.

2. Alpatov B.A., Babayan P.V., Ershov M.D. Podhodi k obnarugeniyu i otsenke parametrov dviguschihsya objektov na videoposledovatelnosti primenitelno k transportnoi analitike // Kompyuternaya optika. 2020. no. 5. pp. 746-756.

3. Favorskaya M.N., Pahirka A.I., Shilov A.S., Damov M.V. Metodi poiska dvigeniya v videoposledovatelnostyah // Vestnik sibirskogo gosudarstvennogo aerokosmicheskogo universiteta im. akademika M.F. Reshetneva. 2009. no. 1-2. pp. 69-73.

4. Bogoslovskiy A.V., Zhigulina I.V., Suharev V.A. Vektornoe pole fazoenergeticheskogo spektra izobrageniya i videoposledovatelnosti // Radiotehnika. 2018. no. 12. pp. 13-17.

5. Zhigulina I.V. Energeticheskiye harakteristiki izobrageniy i videoposledovatelnostey // Televideniye: peredacha i obrabotka izobrageniy: materialy 13-th Megdunarodnoi konferencii. Sankt-Peterburg: S.-Pb. GEU «LETI» im. V.I. Ulyanova (Lenina), 2016. pp. 128-131.

6. Ponomarev A.V., Bogoslovskiy A.V., Zhigulina I.V., Suharev V.A. Osobennosti korrelyatsionnogo analiza izobrageniy i videoposledovatelnostey // Gurnal SFU. Tehnika i tehnologii. 2018. no. 11/7. pp. 811-822.

7. Bogoslovskiy A.V., Suharev V.A., Zhigulina I.V., Pantyuhin M.A. Vektornyiye polya, porogdaemiye preobrazovaniyem Furje videosignalov izobrageniy // Radiotehnika. 2021. Vol. 85. no. 7. pp. 127-139.


Methods of multi-speed signal processing in the problems of analysis of heart rate variability
Vityazeva T.A., e-mail:
The Ryazan State Radio Engineering University (RSREU), Russia, Ryazan

Keywords: : heart rate variability, electrocardiosignal, multi-rate signal processing, narrow-band filtering.

The paper considers the problem of heart rate variability analysis, which allows early diagnosis of incorrect functioning of internal human vital activity systems. Methods of digital signal processing are used for the analysis. Approaches based on multi-rate signal processing are proposed, which significantly reduce the computational costs of implementing heart rate variability analysis devices. The optimal structure of multi-stage processing is designed. The characteristics of the applied decimator filters are calculated. An assessment of computational costs has been carried out, demonstrating the gain of the proposed method by tens of thousands of times.

Algorithms of multi-rate joint processing of electrocardiogram and respiration signals have been developed. Joint processing increases the reliability of the decisions made, especially in non-standard cases of rare breathing and expands the functionality of the analysis devices. The paper proposes a method for increasing the degree of synchronicity of jointly processed electrocardiogram and respiratory signals due to their matched registration. The results of modeling of joint multi-rate processing are given, taking into account their matched registration.

The problem of implementing the proposed algorithms on digital signal processors in real time is considered. The 1967VN028 processor of JSC "PKK "Milander" was used for implementation. Filtration and filtration-decimation programs of the analyzed processes have been developed and applied to the problem under consideration. The achievable processing time is estimated. It is shown that the algorithms proposed in the paper are implemented in practice and are effective from the standpoint of the speed and memory required from the processor.

1. Baevskij R. M., Ivanov G.G. i dr. Analiz variabel'nosti serdechnogo ritma pri ispol'zovanii razlichnyh elektrokardiograficheskih system (Analysis of heart rate variability using various electrocardiographic systems) (chast' 1/ part 1)// Vestnik aritmologii (Bulletin of Arrhythmology). 2001. ¹24. S.65-86.

2. Task Force of the European Society of Cardiology and North American Society of Pacing and Electrophysiology. Heart rate variability. Standards of measurement, physiological interpretation and clinical use// Circulation. 1996.V.93(5). P.1043-1065.

3. Vityazev V. V. Cifrovaya chastotnaya selekciya signalov (Digital frequency selection of signals). – Ì.: Radio i svyaz' (Radio and communications), 1993. – 240 p.: il.

4. Patent RF (Patent of the Russian Federation) 2440023. Sposob vyyavleniya periodicheskih sostavlyayushchih v ritme serdca (A method for detecting periodic components in the heart rhythm). L.V.Demina, O.V.Mel'nik, A.A.Miheev //Opubl.( Publ.)20.01.2012. Byulleten' (Bulletin) ¹2.

5. Vityazeva T.A., Miheev A.A. Primenenie mnogoskorostnoj obrabotki signalov v zadachah analiza variabel'nosti serdechnogo ritma (Application of multi-speed signal processing in heart rate variability analysis tasks )//Vestnik Ryazanskogo gosudarstvennogo radiotekhnicheskogo universiteta (Bulletin of the Ryazan State Radio Engineering University). 2014. ¹3(vypusk 49/ issue 49). pp.14-21.

6. Vityazeva T.A., Vityazev S.V., Miheev A.A. Optimal'noe proektirovanie fil'tra analiza variabel'nosti serdechnogo ritma (Optimal design of the heart rate variability analysis filter)// Cifrovaya Obrabotka Signalov (Digital Signal Processing). 2015. ¹2. pp.18-22.

7. Tatyana Vityazeva; Sergey Vityazev; Anatoly Mikheev, Synchronization of Heart Rate and Respiratory Signals for HRV Analysis, 2018 7th Mediterranean Conference on Embedded Computing (MECO), Year: 2018, Pages: 549-552.

8. Patent RF (Patent of the Russian Federation) ¹ 2722263. Sposob formirovaniya sinhronizovannyh posledovatel'no-stej kardioritmogrammy i pnevmogrammy i ustrojstvo dlya ego osushchestvleniya (A method for forming synchronized sequences of cardiorhythmograms and pneumograms and a device for its implementation)./T.A. Vityazeva, A.A. Miheev// Opubl. (Publ.) 28.05.2020. Byulleten' (Bulletin) ¹16.

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