Digital Signal Processing

Scientific & Technical

“Digital Signal Processing” No. 1-2021

In the issue:

- discrete-frequency Fourier transform
- beamformer regularization procedure
- high-speed communication systems
- space-time block coding
- parallel transmission of binary messages
- optimization of rejection filters
- correlation coefficient estimate distribution
- interference reflectograms processing

Discrete-frequency Fourier transform determination by the method of discrete Fourier transform with a variable parameter in the time domain
A. V. Ponomarev, PhD, Econ., Associate Professor, e-mail:
O. V. Ponomareva, Dr. Sc., Tech., Professor,
N.V. Smirnova, PhD, Tech., Associate Professor,

Keywords: digital signal processing, discrete-frequency Fourier transform, discrete-time Fourier transform, picket fence effect in the time domain, fast discrete Fourier transform with a variable parameter in the time domain.

In digital signal processing (DSP) discrete-time Fourier transform (DTFT) is widely used in theoretical and applied research. In this paper the new form of Fourier transform - discrete-frequency Fourier transform (DFFT) is introduced into DSP. The aim of the work is to study a discrete Fourier transform with a variable parameter in the time domain, which allows efficiently find the values of the discrete-frequency Fourier transform not only for integers, but also for rational values of the time variable. The discrete Fourier transform method with a variable parameter in the time domain allows you to effectively deal with the effect has called by the authors the picket fence effect in the time domain. The paper introduces discrete exponential functions with a variable parameter in the time domain and investigates in detail their main properties. The paper has proposed algorithms for fast discrete Fourier transform with a variable parameter in the time domain (FFT-VPTD) for both complex and real sequences. A graph of the FFT-VPTD algorithm with decimation in time and with replacement (natural order of samples at the input of the algorithm, binary-inverse order of samples at the output of the algorithm) has been developed.


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6. Khanyan G. S. Sampling theorem in frequency domain for the finite spectrum. Proceedings of 2018 IEEE East-West Design and Test Symposium, EWDTS 2018. 2018, pp. 8524822. doi:10.1109/EWDTS.2018.8524822

7. Ponomarev A. V. Systems Analysis of Discrete Two-Dimensional Signal Processing in Fourier Basis. In: Advances in Signal Processing. Theories, Algorithms, and System Control-7. Favorskaya M. N., Jain L. C. (eds). Springer, Cham. Vol. 184. Pp. 87–96.

8. Ponomareva O. V., Ponomarev A. V., Smirnova N. V. Sliding Spatial Frequency Processing of Discrete Signals. In: Advances in Signal Processing. Theories, Algorithms, and System Control-8. Favorskaya M. N., Jain L. C. (eds). Springer, Cham. Vol. 184. Pp. 97–110.

9. Prozorov D., Trubin I. Detection of a signal in the simo system with spatial correlation of noise. 2018 Proceedings of 7th Mediterranean Conference on Embedded Computing, MECO 2018 – Including ECYPS 2018, 2018, pp. 1–5. doi:10.1109/MECO.2018.8405965

10. Ponomareva O., Ponomarev A., Ponomareva N. Windov-Presum parametric discrete Fourier Transform // Proceedings of IEEE East-West Design & Test Symposium (EWDS’2018). 2018.Ñ.364-368.

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Regularization for the blind LCMP beamformer
M.S. Polyanin, Academician M.F. Reshetnev Information Satellite Systems, Russia, Zheleznogorsk, e-mail:

Keywords: space-time adaptive processing (STAP), regularization procedure, LCMP beamformer, adaptive phased antenna array.


In practice, the input data for a non-recursive adaptive algorithm is usually ill-conditioned or singular sample covariance matrix (SCM) that leads to an incorrect solution. There are various methods for evaluating the SCM that increase the robustness of the optimal beamformer:
1. Diagonal loading ([1], [2], [3], [4], [5]).
2. Probabilistic approach ([6], [7], [8]).
3. Minimax estimate ([9], [10], [11], [12]).
4. Structured estimate ([13], [14]).
5. Hybrid estimate ([15], [16], [17], [18]).
The article discusses one of the widely used and effective methods to reduce the influence of negative factors - the introduction of a regularization coefficient (RÑ) in a sample matrix or a SCM [19].

Calculating the value of the RÑ of the sample matrix is one of the key issues for ensuring robustness of the optimal beamformer of adaptive phased array in the equipment. The article discusses the existing methods for estimating the optimal value of the RC, suitable for a blind beamformer, which have different efficiency and complexity of hardware implementation.

TSVD-based methods are computationally intensive due to the need SVD transformations. Of interest are direct algorithms based on Tikhonov's regularization, which are not based on the statistical characteristics of the input signals. Blind LCMP beamformer was simulated under conditions of poorly conditioned SCM. Comparison of regularization methods depending on different signal-noise situations and the number of samples are shown in the graphs. The OAS and RBLW methods have lower computational complexity compared to LW, due to the lack of calculation of the average estimate of the SCM error. The results of modeling a narrow-band LCMP beamformer using linear shrinkage algorithms are presented.

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Possibilities of creating high-speed communication systems with high spectral and energy efficiency (Part 1)
M.A. Bykhovskiy, Moscow Technical University of Communications and Informatics, Russia, Moscow
, e-mail:

Keywords: high-speed communication systems, signal transmission methods, spectral and energy efficiency, error correction codes.

In this paper, we studied the spectral (SE) and energy (EE) efficiency of two different high-speed communication systems designed to transmit messages with a high specific speed Rf≥1. In the first of them, two-dimensional signals with Quadrature Amplitude Modulation (QAM) are used to transmit messages over the communication channel, and error correction codes (ECCs) with the maximum achievable code distance (MACD). MACD codes include Reed-Solomon (RS) codes or similar low-density parity-check codes (LDPC).

It is shown that with the optimal choice of its parameters - the mode of demodulation of sig-nals from QAM and ECC, it is possible to create a communication system that will have sufficiently high EE and SE. The study showed that relative to the “ideal” communication system, its SE (μs) and EE (μen) with a rational choice of parameters will be, respectively, μs=0.87 and EE μen=-2...-5 dB for 3≤ Rf≤10 bit/sec·Hz. Another ECC also considered, in which multidimensional optimal signal ensembles (ES) are used, which have a relatively small normalized duration.

A communication system was also investigated in which N-dimensional optimal ES are used for signal transmission, and ECCs are not used. It is shown that with a sufficiently long normalized signal duration of such an ES, it can provide high reliability of communication with high SE and EE coefficients (μs=1 and μen≈0), i.e. the characteristics of such a system are close to those of the “ideal” Shannon system.

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Space-time block coding scheme design for the single carrier modulation

Yaroslav Gagiev, Senior Research Engineer, e-mail:
Anastasia Aderkina, Research Engineer, e-mail:
Radio Gigabit LLC, Nizhny Novgorod, Russia

Keywords: space-time block coding; Alamouti scheme; maximal ratio combining; guard intervals; single carrier modulation.

This paper presents a Space-Time block coding scheme based on the Alamouti scheme for the Single Carrier (SC) modulation with frequency domain equalization. In the proposed scheme coding is applied only to symbols of guard intervals independently for each SC block allowing to keep cyclic structure of the SC frame required for equalization in the frequency domain. It was shown that joint equalization of data and guard interval signals at the receiver distorts guard intervals due to absence of space-time structure. This paper proposes a method for independent equalization of data signals and guard interval signals allowing to have optimal processing for both signal types with insignificant complexity increase of the receiver. To analyze optimality of the developed space-time coding scheme with two transmit and one receive antenna comparison with the maximal ratio combining scheme with one transmit and two receive antennas was completed in two channel models: line of sight model and Rayleigh channel model. It was shown that for fixed transmit power from each antenna both schemes demonstrate the same performance. Gain relatively a configuration with one transmit and receive antennas is 3 dB for line of sight channel. For Rayleigh channel SNR gain is in range 4.1-8.3 dB depending on modulation and coding scheme type.

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Using a complex envelope for parallel transmission of binary messages by narrow-band overlapping signals
V.A. Vershinin Russia, Rybinsk, e-mail:

parallel transmission, overlapping signals, crosstalk, specific costs of the frequency band, complex envelope, noise immunity.

Parallel transmission of binary messages by narrow-band orthogonal signals is considered. Before transmission, the binary message is divided into blocks of L elements. The block elements are received for transmission at the same time with an interval of T/2 and are transmitted using the proposed narrow-band orthogonal signals of duration T. The signals of sequentially transmitted blocks partially overlap in time. The bandwidth occupied by the signal at the level of -30 dB of the power spectral density at L=64 and L=128 is 68/T and 132/T, and the specific band costs are 0.531 and 0.516. The peak factor of the signal for the above L values is 7.92 and 11.0.

The use of partially overlapping signals based on the proposed signals in parallel transmission in comparison with non-overlapping sinusoidal orthogonal signals allows us to obtain better specific band costs, increase the rate of decline of the side lobes of the spectral power density (reduce out-of-band radiation). At the same time, the noise immunity practically does not deteriorate.

The purpose of the work is to consider the formation of the transmitted signal and the processing of the received signal using a complex envelope. At the same time, the main part of the algorithm for forming and processing is carried out in the low-frequency region.

An expression for the complex envelope of the transmitted signal is obtained, and an algorithm for generating the transmitted signal is given. An algorithm for obtaining and processing the complex envelope of the received signal is described.

An analytical expression is obtained for the error probability when processing the received signal using a complex envelope. Modeling of signal generation and processing in the Matlab environment is carried out. The simulation results confirm the obtained expression for determining the error probability.

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Steganographic use of the structure of the signal of the digital image
V.A. Kottsov, e-mail:

P.V. Kottsov, e-mail:
Institute of space research of the Russian Academy of Sciences (IKI RAS), Moscow, Russia

Keywords: steganographic method, videoinformation, hidden transmission.

The article considers a new steganographic approach to the hidden transmission of digital video information. With the advent of computer systems and the development of the Internet, the need to hide the content of a message, to confirm significant information, and to secure authorship has increased. Therefore, the need to use cryptographic and steganographic methods in the transmission of confidential information has increased.

Steganography hides the very fact of information transmission. The transmitted information is masked by transmission as part of other information. A simple tabular encoding method using the binary structure of the digital image signal is proposed. The carrier of hidden information is the number of units in the signal code. Fast decoding is performed by sorting the received binary signal at the rate of its arrival and can be used for rapid information exchange.

The procedures in question do not require complex algorithms to be implemented and can be implemented with an FPGA. Its experimental verification was performed. The described technology is protected by a Russian patent.

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Digital filtering of GNSS reflectometry results
E.V. Kuzmin, Siberian Federal University (SibFU), Russia, Krasnoyarsk, e-mail:
A.V. Sorokin, Federal Research Center “Krasnoyarsk Science Center” of the Siberian Branch of the Russian Academy of Sciences (FRC KSC SB RAS), Russia, Krasnoyarsk, e-mail:

Keywords: interference reflectograms, digital filtering, GNSS reflectometry, GNSS-R, multipath, Fourier transform.

A software procedure for digital filtering has been developed for post-processing of experimentally obtained interference reflectograms. An analytical estimate of the coefficient of reducing the effect of additive noise is given. There has also been carried out the filtration of experimental interference reflectograms obtained as a result of sessions of recording the intensity of GNSS signals at various properties of surfaces adjacent to the receiving site.

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Optimization of rejection filters by the probabilistic criterion
D.I.Popov, The Ryazan State Radio Engineering University (RSREU), Russia, Ryazan, e-mail:

Keywords: adaptation, probabilistic criterion, nonlinear programming, iterative optimization procedure, clutter, rejection filter.

The problem of optimization of non-recursive rejection filters (RF) of high orders by the prob-abilistic criterion is considered. The statement of the optimization problem is formulated and an ex-pression is given for the average probability in the Doppler interval.

The problem of optimizing the weight vector of RF is solved by the method of nonlinear programming. For the convergence of the solution to the unimodal extremum, it is proposed to introduce restrictions on the form of the RF frequency response, setting it in the form of equidistant frequency samples and assuming the phase characteristic to be linear. The amplitude-frequency characteristic outside the rejection band (non-transmission) to be monotonic, taking into account its sym-metry in the range of unambiguity. The inverse discrete Fourier transform of the frequency samples determines the filter weights.

In the cascade form of the RF implementation, it is proposed to optimize the weight coefficients of individual links directly. The corresponding iterative optimization procedure is given. The results of the optimization of the RF on by the energy and probabilistic criteria are compared.

Significant gains in the signal-to-noise threshold ratio were found when optimizing the RF parameters of high orders according to the probabilistic criterion in comparison with the energy criterion. The principles of RF adaptation under a priori uncertainty of clutter parameters are proposed and the corresponding block diagram of adaptive RF is presented.

The considered method of optimizing two or more RF parameters of high orders according to the probabilistic criterion opens up new opportunities in improving the efficiency of detecting signals of moving targets against the background of clutter, providing significant gains in comparison with similar results of optimizing the RF according to the energy criterion.

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On the new representation of the correlation coefficient estimate distribution
Bartenev V. G., Professor, Doctor of Technical Sciences, MIREA-Russian Technological University, e-mail:
Bartenev G. V., post-graduate student of VNIIRT
Bautochko A.V., post-graduate student of VNIIRT

Keywords: distribution of the estimate, maximum likelihood method, correlation coefficient, inter-frequency correlation.

When analyzing the effectiveness of radio engineering systems that use the maximum likelihood estimate of the correlation coefficient module, it may be necessary to apply the distribution of this estimate, for example, when analyzing the effectiveness of adaptive detectors [1] or target type classification systems [2]. In [3], the distribution of such an estimate is obtained from the Wishart distribution, represented as an infinite series. Unfortunately, the convergence of an infinite series deteriorates as the true value of the modulus of the correlation coefficient and its estimate approaches unity, which forces us to use recurrent schemes for calculating the terms of the series and using a very large number of them. A new representation is obtained for the distribution of the estimate of the modulus of the correlation coefficient, using the maximum likelihood method in the form of a finite sum. The properties of the new representation of the estimate of the maximum likelihood of the modulus of the correlation coefficient are investigated in comparison with the previously obtained distribution with an infinite series. An example of using a new formula to evaluate the efficiency of classifying discrete interfering reflections based on the inter-frequency correlation feature is given.

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4. Bartenev V. G. Method of classification and blanking of discrete interference. Patent No. 2710894 on application No. 2018134712 registered in the State Register of the Russian Federation 14.01.2020.

5. Bartenev V. G. On the use of three signal signs for classification and blanking of discrete interfering reflections. 2020. No. 4. pp. 54-57.

6. Bartenev V. G. Model-oriented design of programmable radio engineering devices. Practical course// Hotline-Telecom, Moscow, 2019, C. 48-64.

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