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“Digital
Signal Processing” No. 3-2021 |
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Algorithms for parametric fast Fourier transform
O.V. Ponomareva, e-mail: ponva@mail.ru
A.V. Ponomarev, e-mail: palexizh@gmail.com
N.V. Smirnova, e-mail: yolkanv@gmail.com
Keywords: fast Fourier transform, finite signal, Fourier processing, problem, negative effects of discrete Fourier transform, basis, parametric DFT.
Abstract
Classical Fourier processing of finite information discrete signals (FID signals) is the most important method of digital analysis, modeling, optimization, improvement of control and decision making. The theoretical basis of classical Fourier processing of FID signals is the discrete Fourier transform (DFT). The practical basis of classical Fourier processing of FID signals is the Fast Fourier Transform (FFT). The practice of using classical Fourier processing of FID signals, having confirmed its effectiveness, revealed a number of negative effects inherent in this type of digital signal processing (DSP). The aliasing effect, scalloping effect, picket fence effect, significantly affect the effectiveness of analysis, modeling, optimization, improvement of management and decision making. To increase the efficiency of Fourier processing of FID signals, the authors of the paper have developed a generalization of DFT in the form of a parametric DFT (DFT-P). Since the direct application of parametric Fourier processing of FID signals (as well as the use of classical Fourier processing of FID signals) requires complex multiplications, fast procedures are required for the practical implementation of this type of FID signals. Purpose of the research is to develop algorithms for the fast parametric discrete Fourier transform (FFT-P). The work developed fast procedures for the implementation of DFT-P by time decimation. Parametric FFT-P with substitution (in place) and without substitution (no place) are proposed. The estimation of the efficiency of the FFT-P algorithms is given. The practical significance of the work is in the fact that developing algorithms for the parametric fast Fourier transform can reduce the computational costs of performing parametric discrete transformations by three or more orders of magnitude.
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6. Ponomareva O.V. Razvitie teorii i razrabotka metodov i algoritmov cifrovoj obrabotki informacionnyh signalov v parametricheskih bazisah Fur'e [Development of the theory and development of methods and algorithms for digital processing of information signals in parametric Fourier bases]: Dissertation of the doctor of technical sciences]: Izhevsk, 2016, 357 p. (in Russ.).
7. Ponomareva O.V., Ponomarev A.V. [Spatial interpolation of two-dimensional discrete signals using fast Fourier transforms]. Intelligent systems in production. 2019, vol. 17, no.1, pp.88-94 (in Russ.).
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Signal-to-noise ratio in multichannel receiving system with strong mutual influence between space channels
M.V. Grachev, e-mail: grachev.m.v@rsreu.ru
Yu.N. Parshin , e-mail: parshin.y.n@rsreu.ru
The Ryazan State Radio Engineering University named after V.F. Utkin (RSREU), Russia, Ryazan
Keywords: multichannel receiving system, signal processing, mutual influence, mutual impedance matrix, scaling factor, optimal weight, load impedance optimization.
Abstract
Spatial signal processing in multichannel receiving systems is an actual scientific and technical problem in various fields of radio engineering. In modern multichannel receiving systems due to restrictions on the space area in which the channels of the receiving system are located the mutual influence of the channels increases. The strong mutual influence of the receiving channels affects both the input signal and the system's noise. Taking into account the mutual influence during the spatial processing of the signal makes it possible to increase the efficiency of the system. The aim of this work is to increase the efficiency of signal processing in multichannel receiving systems with strong mutual influence by rational choice of the aperture's size and the elements' number of the antenna system.
A model of mutual influence between the receiving system's channels is based on changing the distance scale between the elements of a certain reference antenna system for which the matrix of mutual impedances can be calculated analytically is used in this paper. An antenna system in the form of thin vibrators is used as a reference antenna system. The multichannel receiving system considered in the work consists of an antenna system, a matching circuit, a unit of low-noise amplifiers, and a weight signal processing unit. The signal source is located in the far field. There are external antenna noise and amplifiers' inner noises in a multichannel receiving system. The matching circuit solves the problem of transmitting the maximum signal power from the antenna system to the low noise amplifier.
The analysis of the mutual influence between the channels on signal-to-noise ratio in the multichannel receiving system is carried out. An increase in the number of channels at a fixed aperture value leads to an increase in the degree of mutual influence between the channels. The receiving system's channels mutual influence leads to correlation of antenna noise and a decrease in the efficiency of signal processing. With a strong degree of receiving systems channels mutual influence the output signal-to-noise ratio is practically independent of the number of channels and is equal to the signal-to-noise ratio at the output of a single-channel receiving system. It is also noted that the choice of the optimal weight vector allows increasing the stability of the multichannel receiving system to an increase in the degree of mutual influence of the channels.
The influence of the antenna system size aperture on the output signal-to-noise ratio is considered. Increasing the size of the aperture leads to a decrease in the degree of mutual influence and to increase in the efficiency of signal receiving. However with a large number of receiving channels there is a mismatch between the elements of the antenna system and the low-noise-amplifiers.
The analysis showed the need to take into account the mutual influence between the channels when choosing the processing weight vector and the spatial structure of the system. Increasing in the degree of receiving systems channels mutual influence leads to a decrease in the signal-to-noise ratio at the output of the multichannel receiving system. Taking into account signal distortions and noise correlation caused by mutual influence during signal weighting allows increasing the value of the output signal-to-noise ratio.
References
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Mathematical modeling of the processes of information exchange of discrete messages via communication channels based on generating probability functions of a random variable
V.A. Tsymbal, e-mail: tsimbalva@mail.ru
S.E. Potapov, e-mail: 41kaf_rabota@mail.ru
I.D. Potapova
, e-mail: 41kaf_rabota@mail.ru
Military Academy of the Peter the Great Strategic Missile Forces, Russia, Serpukhov
Keywords: markov processes, mathematical models, generating function of distribution moments, duplex communication channels, multi-packet messages, temporary characteristics.
Abstract
One of the important tasks of designing and operating automated control systems for critical objects is to study the probabilistic and temporal characteristics of delivering multi-packet messages through their communication channels. At the same time, in some cases, the traditional mathematical apparatus of absorbing finite Markov chains does not allow obtaining adequate modeling results due to the need to simultaneously take into account the times and probabilities of transitions of inhomogeneous random processes.
The article proposes a new mathematical apparatus for modeling the processes of bringing discrete messages through communication channels with decisive feedback, based on the theory of generating probability functions of a random variable. The substantiation of the validity of the use of this device for modeling a wide range of processes of information exchange of discrete messages and confirmation of the reliability of mathematical models obtained with its help is given.
At the same time, it is shown that the developed method of studying the probabilistic-temporal characteristics of delivering multi-packet messages using generating probability functions does not contradict the results obtained in the traditional way, however, it can significantly reduce computational costs and automate the process of obtaining the result.
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Potential capabilities of communication sistems with limited friquency band channals
V.A. Pakhotin, e-mail: VPakhotin@kantiana.ru
K.V. Vlasova,
S.V. Petrov,
A. N. Aleshchenko,
R.V. Simonov
Physical and Technical Sciences and Information Technologies of the Immanuel Kant Baltic Federal University (IKBFU)
Keywords: communication systems with limited frequency band of channels, uncertainty ratio, Cebyshev's window functions, unorthogonal signals, truth-like functionality, Communication channel seal.
Abstract
The problem of channel multiplexing in communication systems with a limited frequency band is considered. A solution to this problem is proposed on the basis of maximum likelihood filters, which allow filtering channel signals from the group signal in the region of their non-orthogonality. Expressions based on the likelihood functional are derived, which determine the structure, impulse and frequency characteristics of the maximum likelihood filters. This is a new type of filter that contains information about the correlation relationships between signals, which allows you to eliminate errors due to accompanying signals. The distinctive features of these filters and their capabilities for filtering non-orthogonal signals contained in the adopted implementation are discussed. The issues of limitations in communication systems caused by the uncertainty relation, the uncertainty function are discussed. Issues related to guard intervals are considered. The main limitations are related to the limiting data transfer rate in accordance with Shannon's theorem, with the specific data transfer rate. It is shown that the capabilities of maximum likelihood filters make it possible to create communication systems with non-orthogonal carrier frequencies. The data transfer rate in such systems exceeds the limiting speed determined in accordance with Shannon's theorem. The results of model calculations are presented that illustrate the potential for filtering signals with maximum likelihood filters in comparison with matched filters (Fourier filters). Filtering results for signals modulated by Chebyshev window functions are analyzed. Potential filtering capabilities of signal constellations in a two-channel communication system with a small frequency difference of the carrier frequencies of the channels are analyzed. The issues of noise immunity are discussed. Based on the results of the work, a conclusion was made about the possibility of significant channel compaction in communication systems with a limited frequency band.
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Spectral analysis based on the periodogram method of processing a pseudo-ensemble of binary-quantized signal segments using window functions
V.N. Yakimov, Russia, Ryazan, e-mail: yvnr@hotmail.com
A.V. Mashkov, e-mail: mavstu@list.ru
Samara State Technical University, Russia, Samara
Keywords: power spectral density, periodogram, window function, binary stochastic quantization.
Abstract
The article discusses the development of mathematical and algorithmic support for the spectral analysis of signals based on the calculation and averaging of periodogram estimates of the power spectral density (PSD). The aim of the work is to increase the computational efficiency of digital spectral analysis of complex signals. A special feature of this development is that the formation of a pseudo-ensemble of segments to be processed is carried out using binary-sign stochastic quantization of the analyzed signal. In the process of forming a pseudo-ensemble, you can control the overlap of segments in time. Overlapping segments are handled by representing them as a series of non-overlapping sections. This eliminates the need to reprocess overlapping portions of segments.
The use of binary-sign stochastic quantization made it possible to carry out analytical calculation of integral operations in the transition from the analog form of modified periodograms to their calculation in discrete form. As a result, the calculation of the PSD estimate was reduced to processing discrete values of functions which are the result of the integral cosine and sine Fourier transforms for window functions. A set of such functions can be generated analytically depending on the used window functions and the requirements for spectral analysis. The main operations of processing these functions are addition and subtraction operations. The need to perform multiplication operations is practically eliminated, which increases the efficiency of spectral analysis.
The study of metrological properties of computational algorithms was carried out using numerical experiments. In this case, test sets of models of complex signals were used. The procedure of binary-sign stochastic quantization for models of complex numerical code signals was carried out on the basis of discrete-event simulation modeling. The test strategy for numerical experiments was aimed at the ability to detect weak components in complex signals. The results of the numerical experiment showed that the developed approach allows performing spectral analysis at a sufficiently low signal-to-noise ratio.
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14. Dvorkovich V.P., Dvorkovich A.V. Okonnye funkcii dlya garmonicheskogo analiza signalov (Windows functions for harmonic analysis of signals). Ed. 2nd. Moscow: Tekhnosfera Publ., 2016, 208 p. (in Russian).
15. Yakimov V.N., Gorbachev O.V. Firmware of the amplitude spectrum evaluating system for multicomponent processes // Instruments and Experimental Techniques, 2013, vol. 56, no. 5, pp. 540–545.
16. Yakimov V.N., Zaberzhinskij B.E., Mashkov A.V. Bukanova Yu.V. Multi-threaded Approach to Software High-speed Algorithms for Spectral Analysis of Multi-component Signals // Proceedings of IEEE XXI International Conference on Complex Systems: Control and Modeling Problems (CSCMP), 2019. pp. 698-701.
Estimation of the probability density function of QAM signals
Maslakov M.L., Ph.D., Leading Research, JSC «Russian institute for power radioengineering», e-mail: maslakovml@gmail.com
Ternovaya A.K., student, Federal State Budget-Financed Educational Institution of Higher Education The Bonch-Bruevich Saint Petersburg State University of Telecommunications, e-mail: ternovaya_00@bk.ru
Keywords: probability density function, In-phase and Quadrature components, quadrature amplitude modulation, histogram method, maximum likelihood method, signal-to-noise ratio, noise dispersion.
Abstract
Statistical analysis of signal constellations of keyed signals transmitted over channels with additive noise is one of the most important tasks in signal processing and deciding on the transmitted symbol. On the basis of their analysis and processing of signal samples, in particular, estimates of the signal-to-noise ratio (SNR) and the bit error rate are obtained. For quadrature amplitude modulated (QAM) signals, it is necessary to consider the in-phase and quadrature components of the complex envelope Uk of each received symbol.
The purpose of this work is to estimation the probability density function of the complex envelope of the QAM signal.
The histogram method and the maximum likelihood method for estimate the probability density function of the coefficients of the complex envelope of the QAM signal are used. The results of numerical simulation are presented. It is shown that in the case of using the histogram method, a significant sample size is required, which is unacceptable when analyzing signals received from channels with fast changing characteristics. Therefore, the authors considered the application of the maximum likelihood method. An analytical expression is derived of the probability density function of the complex envelope for the case of signals with QAM-4 and QAM-16.
The results obtained can application in the problems of estimating the SNR of information signals with QAM modulation.
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Estimation of signal-to-noise ratio values for short samples data signals
Egorov V.V., Ph.D., Chief Research, JSC «Russian institute for power radioengineering»
Maslakov M.L., Leading Research, JSC «Russian institute for power radioengineering», e-mail: maslakovml@gmail.com
Keywords: estimation, signal-to-noise ratio, noise dispersion, bit error rate.
Abstract
One of the main problems of ensuring the operating of an adaptive radio communication system is to assess the quality indicators or characteristics of the radio link, such as the bit error rate and the signal-to-noise ratio. To obtain these estimates, there are methods using test signals and blind methods.
Moreover, in cases where the SNR is large enough, most of the known methods, both test and blind, do not allow obtaining a "good" estimate with a small size of analyzed data. An increase in the size of the analyzed sample will lead to a decrease in the efficiency when timely deciding on the select of the optimal state of the communication system.
A problem of this kind arises in channels with fast changing propagation conditions, as well as in the functioning of an adaptive data transmission system in a difficult signal-interference environment.
The article proposes method of estimation the signal-to-noise ratio using data signals of short duration. This problem is reduced to solving the inverse problem or the problem of inverse modeling, moreover the noise component dispersion value is estimated.
The proposed method makes it possible to quickly estimate the variance of the noise component, on the basis of which current estimates of the bit error rate and signal-to-noise ratio can be obtained.
References
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The PUCCH format 0 detection algorithm modification in the fifth-generation new radio
Mohammad Assaf, postgraduate, Department of radiophysics, Tomsk State University, e-mail: md.moh1600@gmail.com
Oleg G. Ponomarev, Candidate of physics and mathematics, Assistand Professor, Department of radiophysics, Tomsk State University, e-mail: ponomarev@phys.tsu.ru
Keywords: 5G-NR, PUCCH, UCI, phase rotation, threshold.
Abstract
In this article the authors introduce an overview of the 5G NR physical uplink control channel (PUCCH) format 0 by describing its structure in both transmitter and receiver, th appropriate scenario for this format and its performance under several situations. In format 0, gNode does not have information about transmitting ACK/NACK bits unlike the other formats, so the performance is very sensitive to the threshold that determines the existence of control signal or not, and the threshold should be chosen very carefully to avoid false decoding and to maintain DTX to ACK probability at below 1%. New method of choosing this threshold is proposed in this article. Through link-level simulations, the obtained results show good agreement with the minimum requirements recommended by the 3rd generation partnership project (3GPP) standarts.
References
1. Osseiran A., Monserrat J. F., Marsch P., 5G Mobile and Wireless Communications Technology, Cambridge University Press, Cambridge, UK, 2016. – 438 p.
2. Assaf M., Ponomarev O. G. Sample Clock Offset Compensation in the Fifth-Generation New Radio Downlink.” Journal of Physics: Conference Series, vol. 1889, no. 2, 2021, p. 022091.
3. Chen S., Zhao J., The requirements, challenges, and technologies for 5G of terrestrial mobile communication, IEEE communications magazine, vol. 52, no. 5, May 2014, pp. 36 - 43.
4. Dahlman E., Parkvall S., Skold J. 5G NR: The next generation wireless access technology. Academic Press, 2018. – 466 p.
5. Kundu L., Xiong G., Cho J., Physical uplink control channel design for 5g new radio, in 2018 IEEE 5G World Forum (5GWF), Jul. 2018, pp. 233–238
6. Kim, Young-Hoon, et al. Performance Comparison of DTX Detection Schemes for 5G NR PUCCH. 2020 International Conference on Information and Communication Technology Convergence (ICTC), 2020.
7. Du Y., He W., Long H., An improved semi-blind detection algorithm for nr pucch, in 2019 IEEE 5th International Conference on Computer and Communications (ICCC), Dec. 2019, pp. 66–71.
8. ETSI TS 138 211 V15.2.0 (2018-07). Technical specification. 5G; NR; Physical channels and modulation (3GPP TS 38.211 version 15.2.0 Release 15).
9. ETSI TS 38.104. 5G; NR; Base Station (BS) radio transmission and reception (3GPP, TS 38.104 version 16.4.0 Release 16).
Transmission of discrete messages by narrow-band simplex and orthogonal signals
V.A. Vershinin, e-mail: vershinin-vladimir@yandex.ru
Russia, Rybinsk
Keywords: discrete message, simplex signals, orthogonal signals, narrow-band signals, specific costs of the frequency band, complex envelope, noise immunity.
Abstract
The transmission of discrete messages is considered. Two sets of simplex narrowband signals are used for transmission, and any two signals from these sets are orthogonal. Two message elements are transmitted simultaneously. One element is represented by simplex elementary signals of one set, and the other by simplex elementary signals of the second set. Thus, two elementary signals belonging to different sets are transmitted simultaneously, their separation is achieved due to orthogonality.
Two variants of the formation of the transmitted signal are considered. In the first variant, the number of values of the message element is 2, in the second variant, the number of values of the message element is 4. Each value of the message element corresponds to a simplex signal. The specific bandwidth costs for the first and second options are 1.18 and 1.09, respectively, peak factor 2 and 2.63. It is shown that the second option allows for significantly better noise immunity, but the peak factor of the transmitted signal increases.
The possibility of forming and processing a high-frequency transmitted signal using a complex envelope is considered. At the same time, the main part of the algorithm of formation and processing is carried out in the low-frequency region. The transmitted signal is formed from a complex envelope using a quadrature modulator. When receiving, the complex envelope is formed using a quadrature demodulator and low-pass filters.
When modulating with a minimum shift (a known transmission method), the specific cost of the band is 1.18, the peak factor is 1.41. The noise immunity is the same as in the first variant under consideration. The second option makes it possible to obtain significantly better noise immunity than with modulation with minimal shift, but the peak factor increases.
References
1. Vershinin V.A. Peredacha dvoichnyh soobshhenij simpleksnymi signalami [Transmission of discrete messages by narrow-band simplex and orthogonal signals]// Zhurnal radiojelektroniki: jelektronnyj zhurnal [Journal of radio electronics: electronic journal]. 2013. N11. (in Russian)
2. Vershinin V.A. Ispol'zovanie algoritma Viterbi pri peredache perekryvajushhimisja jelementarnymi signalami [Using the Viterbi algorithm when transmitting overlapping elementary signals]// Cifrovaja obrabotka signalov [Digital signal processing].– 2020.– ¹4. (in Russian)
System identification of narrowband systems using adaptive interpolated IIR filters
Goriushkin R.S.
, e-mail: gorushkin.r.s@tor.rsreu.ru
The Ryazan State Radio Engineering University (RSREU), Russia, Ryazan
Keywords: digital signal processing, adaptive filtering, interpolated IIR filters, system identification, LMS, RLS.
Abstract
Digital adaptive filtering has found application in many areas of digital signal processing: in communication systems, radar, audio and video data processing. Identifying an unknown system is a typical task. In this paper, we investigate the application of digital adaptive interpolated filters with infinite impulse response to solve this problem for narrowband systems.
When the signal passes through the structure of the interpolated adaptive filter, additional spectral components appear. To exclude additional spectral components, a low-order smoothing filter is used. Therefore, the use of an interpolated filter assumes the construction of a two-stage system, including an interpolated IIR adaptive filter and an FIR or IIR smoothing filter.
It was shown that the use of a structure consisting of an adaptive interpolated and smoothing filter gives a significant advantage over classical structures in the direct modeling of narrowband filters. With a slight increase in the computational costs required to implement the smoothing filter and some increase in the data memory, it is possible to achieve a much more accurate convergence and to increase the convergence rate of adaptation algorithms. An additional advantage is an increase in filter stability for some parameters of the adaptation algorithms.
The two-stage scheme makes it possible to construct stable adaptive filters based on classical algorithms with an improvement in the identification accuracy of an unknown system. In this work, the LMS and RLS algorithms were used without any special modifications, the changes made to the structure of the adaptive IIR filter are also minimal.
References
1. Oshana R. DSP Software Development Techniques for Embedded and Real-Time Systems. Elsevier, 2006. 608 pp.
2. Proakis J.G., Digital Communication, 4th edn., McGraw Hill, New York, 2001
3. Widrow B., S.D. Stearns, Adaptive Signal Processing, Prentice Hall, Englewood Cliffs, 1985
4. Wood L.C., Treitel S., Seismic signal processing. Proc. IEEE 63, 1975, pp. 649–661
5. Shynk G J. J., Adaptive IIR filtering, IEEE ASSP Magazine, pp. 4-21, Apr. 1989.
6. Vityazev V.V. Mnogoskorostnaya obrabotka signalov (Multirate signal processing). M.: Gorjachaja linija–Telekom, 2017. – 336 p
7. Goriushkin R.S., Vityazev V.V., Analiz primeneniya adaptivnyh grebenchatyj BIH-fil'trov v zadache vydeleniya uzkopolosnogo signala na fone shirokopolosnogo shuma (Analyzing of the adaptive line enhancement of narrowband signal using adaptive interpolated IIR filters) // Cifrovaya obrabotka signalov, M., 2021. Vol.2. pp. 42-47
8. M. Shafaati, M. Ahmadi and H. Mojallali, "Identification of IIR systems using harmony search algorithm," The 2nd International Conference on Control, Instrumentation and Automation, 2011, pp. 1148-1153
9. Feintuch P.L. An Adaptive Recursive LMS Filter // Proceedings IEEE, Vol. 64, No. 11, November 1976. pp. 1622-1624
10. Diniz P. Adaptive Filtering: Algorithms and Practical Implementation. Springer, 2020. 495 pp.
11. S. Koshita, Y. Kumamoto, M. Abe, M. Kawamata, Adaptive IIR Band-Pass/Band-Stop Filtering Using High-Order Transfer Function and Frequency Transformation, Interdisciplinary Information Sciences, 2013, Vol. 19, Issue 2, 2013, pp. 163-172
Autofocus algorithm for increasing the resolution of on-board radar imaging systems by the sharpness optimization method
D.A. Dvoryankov, e-mail: dvoryankov.d.a@mail.ru
V.V. Androsov
S.V. Vityazev , e-mail: vityazev.s.v@tor.rsreu.ru
The Ryazan State Radio Engineering University named after V.F. Utkin (RSREU), Russia, Ryazan
Keywords: autofocus, radar image, sharpness optimization, resolution, phase errors, image pixels.
Abstract
Forming the radar image by synthesizing the antenna aperture, you can often face the phase errors problem due to shortage of information about motion parameter of the airborne radar station carrier. This may lead to image distortion. The paper discusses the effectiveness of sharpness optimization method using the sum of various pixel intensity degrees as the sharpness metric of radar image. To recovery the image it is necessary to estimate the introduced phase component β. One of the main ways to solve such problem is to maximize the sharpness metric of the radar image.
The preliminary results of the research show that the sharpness optimization radar image autofocusing method gives a better efficiency result compared to traditional image focusing algorithms. Using the methods of the golden-section search and Fibonacci numbers allows to reduce the autofocusing time of radar image by more than 5 times in relation to the direct search method. The type of image sharpness metric significantly affects the accuracy of radar image sharpness recovery. Following an in-depth analysis of preliminary research results, some recommendations on the choice of determine intensity image pixels degree parameter also are given. In particular, it was found that according to the criterion «quality/recovery time», the best metric is [I(x,y)]2 (β = 2).
References
1. R.L. Morrison, M.N. Do, D.C. Munson, SAR image autofocus by sharpness optimization: a theoretical study, IEEE Trans. Image Process, vol. 16 (9) (2007), pp. 2309-2321.
2. C.V. Jakowatz, Jr., D.E. Wahl, P.H. Eichel, D.C. Ghiglia, and P.A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach., Kluwer Academic Publishers, Boston, 1996.
3. J.R. Fienup and J. J. Miller, «Aberration correction by maximizing generalized sharpness metrics», J. Opt. Soc. Amer. A, vol. 20, no. 4, pp. 609-620, April 2003.
4. L. Xi, L. Guosui, and J. Ni, «Autofocusing of ISAR images based on entropy minimization», IEEE Transactions on Aerospace and Electronic Systems, vol. 35, no. 4, pp. 1240-1252, October 1999.
5. Schulz, T.J.: «Optimal sharpness function for SAR autofocus», IEEE Signal Process. Lett., 2007, 14, (1), pp. 27-30.
6. Gao, Yang; Yu, Weidong; Liu, Yabo; Wang, Robert: «Autofocus algorithm for SAR imagery based on sharpness optimization», Electronics Letters, 2014, 50, (11), pð. 830-832.
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