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3Radio Electronic Facilities for Signal Transmission, Reception and Processing
UDC 621.396.62 Original article
https://doi.org/10.32603/1993-8985-2023-26-5-40-49
Parameter Justification of a Signal Recognition Algorithm Based on Detection at Two Intermediate Frequencies
Tran Huu Nghi 1I2, Aleksey S. Podstrigaev 1, Nguyen Trong Nhan 2, Danil A. Ikonenko 1
1 Saint Petersburg Electrotechnical University, Saint Petersburg, Russia 2 Le Quy Don Technical University, Hanoi, Vietnam
121 huunghiht@gmail.com
Abstract
Introduction. The signal recognition task for the purposes of RF spectrum management can be solved using a signal recognition algorithm with detection at two intermediate frequencies. This algorithm is based on time-frequency analysis using fast Fourier transform (FFT) and signal envelope processing. Due to the relative simplicity of transformations, this algorithm is implemented on commercially available field programmable gate arrays and allows processing received signals in near real-time. However, the justification of the algorithm parameters providing effective signal recognition by the criterion of minimizing the signal-to-noise ratio (SNR) has not performed so far. Aim. Justification of parameters of the developed signal recognition algorithm, providing the minimum required SNR at the algorithm input.
Materials and methods. The efficiency of the developed algorithm was estimated by computer simulation in the MATLAB environment.
Results. The influence of the parameters of functional blocks and received signals on the efficiency of the developed algorithm was investigated. For chirp, simple pulse, binary, and quadrature phase shift keying signals, the following parameters are recommended: a pulse duration of 5...20 ^s; a chirp rate of 0.8...24 MHz/^s; a code duration of 0.5.. .1 ^s. For these signal parameters, the parameters of the algorithm ensuring its efficiency according to the given criterion are as follows: the number of FFT points equals 1024; the Hamming weight window; bandwidths of bandpass filters are 4 MHz; signal envelope amplitude averaging coefficient equals 0.15.0.25.
Conclusion. The algorithm with the scientifically valid parameters can be used for recognition of signals at the input minimum SNR for the given types and parameters of signals.
Keywords: time-frequency analysis, fast Fourier transform, weight window, signal envelope, bandpass filter
For citation: Tran Huu Nghi, Podstrigaev A. S., Nguyen Trong Nhan, Ikonenko D. A. Parameter Justification of a Signal Recognition Algorithm Based on Detection at Two Intermediate Frequencies. Journal of the Russian Universities. Radioelectronics. 2023, vol. 26, no. 5, pp. 40-49. doi: 10.32603/1993-8985-2023-26-5-40-49
Conflict of interest. The authors declare no conflicts of interest. Submitted 07.09.2023; accepted 09.10.2023; published online 29.11.2023
40
© Tran Huu Nghi, Podstrigaev A. S., Nguyen Trong Nhan, Ikonenko D. A., 2023
Introduction. RF spectrum management is used to control the radio-electronic situation and regulate the use of radio frequencies [1, 2]. Modern radio sources make use of different types of signals. Therefore, recognizing the signal modulation type is a critical task for RF spectrum management [3-5]. This task can be solved by various algorithms using spectral analysis based on fast Fourier transform (FFT) [6-13], wavelet transform [14, 15], high-order cumulants [16, 17], cyclosta-tionary spectral analysis [18, 19], time-frequency distribution [20, 21], and convolutional neural networks [22, 23].
The advantage of algorithms using FFT-based spectrum analysis [6-13] over other algorithms [14-23] consists in their simplicity in terms of technical implementation [24, 25]. Currently, various approaches are employed to implement a highspeed FFT block on commercially available field programmable gate arrays [26, 27]. A review of such algorithms [6-13] found the algorithm [12, 13] to have an advantage over the algorithms [6, 7] in terms of the number of signal types and the algorithms [8-11] to be beneficial in terms of the signal-to-noise ratio (SNR) values required to recognize phase shift keying signals.
A comparison of the probability of correct recognition of down-chirp, up-chirp, simple pulse (SP), binary (BPSK), and quadrature phase shift keying (QPSK) signals depending on the SNR for the developed algorithm was carried out in [12, 13]. However, the justification of the algorithm parameters providing efficient signal recognition has not been performed. Further, we assume that the efficiency can be estimated by the criterion
of minimizing the SNR required to recognize each of the specified signals.
Given the above, this work aims to justify the parameters of the developed signal recognition algorithm [12, 13], which ensure the minimum required SNR at the algorithm input.
Description of the developed algorithm. The algorithm is briefly described by a structural diagram (Fig. 1), which includes the following functional blocks: signal partitioning block SPB; weight window block WWB; FFT block; carrier frequency determination block CFDB; frequency analysis block FAB; generator G; delay line DL; frequency converter FC; frequency doubling block FDB; bandpass filters BPF1 and BPF2; envelope detectors ED1 and ED2; decision-making block DMB. The ranges of the input signal frequencies, intermediate frequencies and double intermediate frequencies are denoted by A/in, A/jf1, and A/jf2, respectively.
When recognizing signals, a linear least-square approximation of the average carrier frequency values (ACFVs) of the signal falling into the processing windows in FAB is carried out. The signal envelopes at intermediate and double intermediate frequencies in the ED1 and the ED2 are also processed. The ACFV of the signal in the processing window f can be calculated as follows:
M
fi = X fm/M,
m=1
(1)
where fm are frequencies in the processing window, at which the signal spectrum value is above the given detection threshold and above half of the
SPB
WWB
FFT
DL
CFDB
in
G
I
FC
FAB
fFl
FDB 4fF2 BPF2
BPF1 - ED1
ED2
DMB
• Down-chirp - Up-chirp SP
■ BPSK •QPSK
Fig. 1. Structural diagram of the signal recognition algorithm with detection at two intermediate frequencies
spectrum maximum and M is the number of these frequencies. Signal recognition is performed after detecting the signal and calculating the ACFV in the CFDB in at least two processing windows. To recognize signals of different types, the following parameters are used: the slope coefficient of the
approximated straight line a, its variance a , signal envelope dips at intermediate K1 and double intermediate frequencies K2.
For chirp signals, the modulus of the coefficient a characterizes the chirp rate, and its sign -the direction of chirp rate (Fig. 2, a, b). For down-chirp and up-chirp signals, the coefficient a takes a negative and positive value, respectively.
For an SP signal, the coefficient a takes a zero value, since the spectrums of the signal falling into the processing windows have the same shape (Fig. 2, c). In addition, when receiving an SP signal, the envelope at the intermediate frequency at the ED1 output has no dips (Fig. 3, a).
For BPSK and QPSK signals, the coefficient a takes a value close to zero, since the spectrum shape of these signals falling into the processing windows depends on the phase shift value, code du-
ration, and processing window length (Fig. 2, d, e). The signal envelope at the intermediate frequency at the ED1 output has dips for both types of signals (Fig. 3, b, d). After frequency doubling, the BPSK signal envelope at the ED2 output has a smooth shape, and there are dips for the QPSK signal (Fig. 3, c, e).
The threshold values of signal envelope dip detection at two intermediate frequencies at the ED 1 and ED2 outputs are set as
Uthr = kavUi
av>
(2)
where kav is the signal envelope amplitude averaging coefficient, and Uav is the signal average envelope amplitude value.
The principle of the algorithm operation is described in greater detail in [12, 13].
Experimental methodology. To minimize the required SNR at the algorithm input, we used MATLAB to investigate the following algorithm parameters and their influence:
- the number of FFT points NpFT on selecting threshold values of the slope coefficient and variance;
f, MHz 400
200
0
f, MHz 400
200
0
2.5
2.5
_L
7.5 a
_L
7.5 c
f, MHz 400
200
0
_L
Down-chirp
_L
10 12.5 t, (is
SP
10
_L
12.5 t, (is
f, MHz 400
200
0
f, MHz 400
200
0
2.5
_L
2.5 5
7.5 b
10 12.5 t, (is
BPSK
_L
_L
2.5
7.5 d
10
12.5 t, (s
QPSK
_L
_L
7.5 10 12.5 t, (is
Fig. 2. Linear approximation of the average carrier frequency values of the signal falling into the processing windows
5
5
5
5
e
U, V 1
0.5
U, V 1
_L
_L
_L
0.5
л-—A /V—J\ П M BPSK
J , ■ , i ,
U, V 1
0.5 -
2.5 5 7.5 10 12.5 15 t, (s a
BPSK
U, V 1
0.5
2.5
7.5
10
U, V 1
0.5
12.5 15 t, (is
2.5
2.5 5 7.5 10 12.5 15 t, (s b
- (Wi qpsk
,-vv A , I.
7.5 10 d
12.5 15 t, (is
til M M и QPSK
J , I 1 1 I
2.5
7.5 10 12.5 15 t, (is
Fig. 3. Signal envelopes at intermediate (left) and double intermediate frequencies (right)
- the number of FFT points ^FFT and weight window type on the dependencies of the probability of correct recognition of down-chirp, up-chirp, SP, BPSK, and QPSK signals on the SNR;
- threshold values of signal envelope dip detection Uthr and bandwidths of bandpass filters used in the algorithm A/Bpf1 and A/BPF2 on the dependencies of the probability of correct recognition of SP, BPSK, and QPSK signals on the SNR.
In estimating the above influence, the minimum SNR value qhr ensuring the probability of correct recognition not less than 0.9 is used.
Selection of parameter variation ranges for input signals and the algorithm. We performed a statistical simulation of constructing the linear approximation of the signal falling ACFVs into the processing windows to estimate the influence of ^FFT on the selecting threshold values of the slope coefficient and variance. Since the probabilities of correct recognition of down-chirp and up-chirp signals do not differ significantly, only up-chirp signal is investigated as a chirp signal in [12].
The input signal is a mixture of a signal and additive white Gaussian noise (AWGN) with the Parameter Justification of a Signal Recognition Algorithm Based on Detection at Two Intermediate Frequencies
zero mean value and unit standard deviation. It is assumed that the pulse duration exceeds the length of the longest processing window by more than a factor of 2. We set the following initial data.
Parameters of the useful signals: pulse duration of chirp and SP signals - from 5 to 20 ^s in increments of 0.05 ^s; code duration of BPSK and QPSK signals - from 0.5 to 1 ^s in increments of 0.1 ^s; carrier frequency of SP, BPSK, and QPSK signals -from 10 to 490 MHz in increments of 10 MHz (the product of the code duration of BPSK, QPSK signals and their carrier frequency have to be an integer number of waves); amplitude - random from 0 to 5; initial frequency value of chirp signals - 10 MHz; deviation of chirp signals - from 4 to 480 MHz; chirp rate Y = 0.8...24 MHz/^s in increments of 0.05 MHz/^s; initial phase of signals - random; number of codes of BPSK and QPSK signals - 20; phase shift law of BPSK and QPSK signals - random.
Fixed parameters of the developed algorithm: sampling rate - 1 GHz; frequency range of the input signal A/in =(0...500)MHz; number of processing windows - 10; intermediate frequency
range A/IF1 =50 ±A/c MHz,
where A/c -
0
0
0
5
5
c
0
e
FFT-based carrier frequency determination error; double intermediate frequency range A/IF2 =2 (50 ±A/c) MHz. The threshold value of
signal detection against the AWGN background in the processing windows is set based on the probability of false alarm 10-7 [28].
Optimizable parameters of the developed algorithm: number of FFT points in processing window ^FFT = 512; 1024; 2048; weight window types - rectangular, Hamming, Blackman; signal envelope amplitude averaging coefficient kav = 0.05...0.45; bandwidth of bandpass filters BPF1 and BPF2 A/BPF1 = A/BPF2 = 2; 4; 6 MHz.
Linear approximation of the ACFVs is performed after detecting the signal on the AWGN background in at least two processing windows.
The simulation result for 104 measurements is presented in Tab. 1. The threshold values of slope coefficient and variance are selected to ensure that the values of the distribution function of the slope coefficient and its variance be not less than 0.99.
Tab. 1 allows us to draw the following conclusions:
1. For chirp signals, the value c increases as ^fft increases at the fixed sampling rate. This is because an increase in the processing window length leads to a broadening of the chirp signal spectrum width. As a result, the error of linear approximation of the ACFVs increases. The slope coefficient a characterizes the chirp rate; therefore, the value a is constant for a different number of FFT points.
2. For SP, BPSK, and QPSK signals, the values a and c2 decrease as ^fft increases at the
fixed sampling rate. This is because an increase in the processing window length leads to an increase in frequency resolution. As a result, the accuracy of determining the signal falling ACFVs into the processing windows increases.
Influence of the parameters of functional blocks and received signals on the algorithm efficiency
Influence o/ the number o/FFT points. To estimate the influence of the number of FFT points on the probabilities of correct recognition of chirp, SP, BPSK, and QPSK signals, depending on the SNR, Nfft = 512; 1024; 2048 are selected. The required SNR at the algorithm input is generated by changing the corresponding value of the signal amplitude. BPSK and QPSK signals have 20 codes with phase shift laws [0 n n 0 0 0 n n 0 n n n 0 0 n n 0 n 0 n] and
[0 n n ^ 0 ^ n 0 0 n n ^ 0 n 0 n 0 ^ n n], 2222 22 2 2
respectively. The threshold values of signal envelope dip detection at two intermediate frequencies are set by formula (2) based on the coefficient kav = 0.2.
The simulation result for 10 measurements with randomly selected signal parameters from the above ranges is shown in Fig. 4.
The following conclusions can be drawn based on the analysis of the results obtained:
1. For chirp signals, increasing ^FFT requires a higher value qfa. This is due to the increase in the spectrum width of the processed chirp signals in the
qhr, dB 2.5 0 — —в--Chirp —$--SP —e--BPSK —»--QPSK
-2.5 — \
-5 — \ .................' .....
-7.5 1
-10 1
512 1024 2048 NFFT, points
Fig. 4. Dependencies of the minimum SNR value providing the probability of correct recognition not less than 0.9 on the number of FFT points
Tab. 1. Threshold values of slope coefficient and variance for a different number of FFT points
Signal type Threshold values
iVFFT = 512 iVFFT = 1024 iVFFT = 2048
Chirp a > 0.75 <s2 < 1.0 a > 0.75 <s2 < 1.75 a > 0.75 <s2 < 3.0
SP |a| < 0.3 <s2 < 0.5 |a| < 0.15 <s2 < 0.15 < 0.1 <s2 < 0.1
BPSK |a| < 0.45 <s2 < 0.9 a < 0.25 <s2 < 0.3 |a| < 0.15 a2 < 0.2
QPSK |a| < 0.45 <s2 < 0.9 a < 0.25 <s2 < 0.3 |a| < 0.15 a2 < 0.2
processing window. For this reason, the error of determining the ACFV calculated by formula (1) at low SNR increases. As a result, to ensure the same accuracy in determining the ACFV when recognizing chirp signals, a higher SNR value is required.
2. For an SP signal, the value Nfft = 1024 requires the lowest value qthr . This is due to the contradiction between frequency resolution and duration of the processed signal at the fixed number of processing windows. Increasing the frequency resolution allows a more accurate determination of the average signal carrier frequency value. However, the probability of false signal envelope dip detection at the intermediate frequency under the influence of noise increases due to an increase in the duration of the processed signal. Thus, value Nfft = 1024 represents a compromise between the considered contradiction for the SP signal.
3. For an BPSK signal, increasing Nfft requires a higher value q^. This is due to the fact that the number of processing windows is fixed, and increasing Nfft leads to a longer duration of the processed signal. As a result, the probability of false signal envelope dip detection at double intermediate frequency due to noise is increased.
4. For a QPSK signal, increasing Nfft requires a lower value qthr . This is due to the fact that, along with an increase in the processing window length, the energy of the signal falling into the processing window increases, and the signal is detected at a lower SNR. Therefore, signal envelope dips can be formed not only in phase shifts but also elsewhere due to the noise.
Thus, as Nfft decreases, the value qthr for recognizing the QPSK signal increases significantly (when Nfft decreases from 2048 to 512 by
about 11 dB). As Nfft increases, the value qthr increases for chirp signal recognition, while being not significantly different for SP and BPSK signals. Therefore, the value Nfft = 1024 for the considered parameters of the signals and the algorithm is optimal in terms of the SNR value required to recognize each signal.
Influence of the weight window type. To estimate the influence of the weight window type on the probabilities of correct recognition of chirp,
Tab. 2. Dependencies of minimum SNR value providing the probability of correct recognition not less than 0.9 on the weight window types
Signal type <?thr > dB
Rectangular Hamming Blackman
Chirp -2.2 -6.0 -6.7
SP -8.0 -8.0 -8.0
BPSK -1.0 -1.0 -1.0
QPSK -7.8 -7.5 -6.6
SP, BPSK, and QPSK signals depending on the SNR, the input signal is weighted at Nfft = 1024 by the following windows: rectangular, Hamming, and
Blackman. The simulation result for 10 measurements with randomly selected signal parameters from the above ranges is presented in Tab. 2.
Following the analysis of the results obtained, the following conclusions can be drawn:
1. Compared to the rectangular and Hamming windows, the Blackman window allows recognition of chirp signals at a lower SNR. One can explain this by the decrease in amplitude at the beginning and end of the weight window, which reduces the spectrum width of the chirp signal falling into the processing window. As a result, the ACFVs under the influence of noise at low SNR are determined more accurately.
2. When weighting by rectangular, Hamming, and Blackman windows, the required SNR values for recognizing SP and BPSK signals differ insignificantly. This is because SP signal weighting by Hamming and Blackman windows changes only the main lobe width without shifting the spectrum central frequency. For a BPSK signal, a phase shift n between the codes leads to summation or subtraction of their spectrums. Therefore, the weight window does not significantly affect the accuracy of determining the ACFV.
3. Compared to the rectangular window, Hamming and Blackman window weighting requires a higher SNR value to recognize the QPSK signal. This is because the decrease in amplitude at the beginning and end of the weight window at phase shifts on n and n/ 2 leads to a decrease in the accuracy of determining the ACFV.
Thus, when processing a chirp signal, the Hamming window requires a significantly lower SNR than the rectangular window. When processing a QPSK signal, this window requires a lower SNR
compared to the Blackman window. In other cases, the Hamming window is comparable to other windows. Therefore, it is reasonable to apply the Hamming window for the developed algorithm.
Influence of threshold values of signal envelope dip detection. To estimate the influence of threshold values of signal envelope dip detection at two intermediate frequencies determined by (2) on the probabilities of correct recognition of SP, BPSK, and QPSK signals depending on the SNR, kav = 0.05...0.45 in increments of 0.05 at
NFFt = 1024 is selected. At kav < 0.05 , the value ^thr for PSK signals reaches unacceptably high values. At kav > 0.45 , there is a high probability of QPSK signal envelope dip detection at two intermediate frequencies under the influence of noise. At the SNR value of 10 dB and kav = 0.5, the probabilities of correct and false detection of signal envelope dips at the intermediate frequency
are 1 and 3 -10-4, respectively, equaling 1 and 0.1406 at the double intermediate frequency. The
simulation result for 10 measurements with randomly selected signal parameters from the above ranges is shown in Fig. 5.
From the analysis of the obtained results, we can draw the following conclusions:
1. For an SP signal, an increase in the kav coefficient leads to an increase in the qthr value. The reason is that an increase in the coefficient kav leads to a higher probability of signal envelope dip detection at the intermediate frequency due to the influence of noise.
the probability of correct recognition not less than 0.9 on the signal envelope amplitude averaging coefficient
2. For a BPSK signal, the coefficient kav = 0.1 requires the lowest gthr. When the kav coefficient increases from 0.05 to 0.1, the value qthr decreases and increases at kav > 0.1. This is explained by the fact that the presence of envelope dips at an intermediate frequency and the absence of envelope dips at the double intermediate frequency are used to recognize a BPSK signal. As a result, an increase in the kav coefficient from 0.1 to 0.45 increases the probability of signal envelope dip detection at both intermediate and double intermediate frequencies.
3. For a QPSK signal, the qthr value decreases along with an increase in the kav coefficient. This is due to the fact that an increase in the kav coefficient increases the probability of signal envelope dip detection at the two intermediate frequencies. It can also be observed from the graph that at kav > 0.25, the value takes a value approximating -8.0 dB. This is because the envelopes are distorted due to the effects of noise at low SNR values. As a result, signal envelope dips can be formed not only in phase alternations but also elsewhere due to noise.
Thus, to provide a compromise between the values of recognition sensitivity of all three signal types, it is reasonable to select the coefficient kav = 0.15... 0.25.
Influence of bandwidths of bandpass filters used in the algorithm. To estimate the influence of bandwidths of bandpass filters used in the algorithm on the probabilities of correct recognition of SP, BPSK, and QPSK signals Pqr depending on the SNR, AfBPF1 =AfBPF2 = 2; 4; 6 MHz at ^FFT = 1024 are selected. The following parameters of initial signals are set: duration of SP signal -5 ps; code duration of BPSK and QPSK signals -0.5 ps; phase shift law of BPSK and QPSK signals -13-bit Barker's code and 16-bit Frank's code, respectively. The threshold values of signal envelope dip detection at two intermediate frequencies are set by formula (2) based on the coefficient kav = 0.2. We randomly selected the remaining parameters from the ranges given above. The simulation results (103 measurements for each SNR value) are presented in Figs. 6-8.
PCR 0.8 — 0.6 — 0.4 — 0.2 — 0
□ gsjHooooôiîOOOOço
о 0
□ О
-20
-15
9 --- 2 MHz 4 MHz 6 MHz
1
-10 -5 0
SNR, dB
Fig. 6. Dependencies of the probability of correct recognition of SP signal on the SNR for different bandwidths of BPF1 and BPF2
PCR 0.8 0.6 0.4 0.2 0
л-й-в-Ф-^-ео
—в— - 2 MHz
- 4 MHz
-0- - 6 MHz
-20
-15
10
-5
0
SNR, dB
Fig. 7. Dependencies of the of the probability of correct recognition of BPSK signal on the SNR for different bandwidths of BPF1 and BPF2
PCR 0.8 0.6 0.4
0.2
- 2 MHz
- 4 MHz
- 6 MHz
0 о о о о
-20 -15
-10
-5
0
SNR, dB
Fig. 8. Dependencies of the of the probability of correct recognition of QPSK signal on the SNR for different bandwidths of BPF1 and BPF2
From the analysis of the obtained results, we can draw the following conclusions:
1. For all signal types, the value increases
with increasing filter bandwidths A/bpf1 and
A/BPF2. The reason is that an increase in the BPF1 and BPF2 bandwidths increases noise at the ED1 and ED2 outputs. Therefore, a higher SNR value is required to obtain acceptable signal envelopes at the detector outputs.
2. At filter bandwidths A/bpf1 and A/BPF2 equal to the spectrum width of BPSK and QPSK signals (2 MHz), the probability of correct recognition does not reach unity even at an SNR value of 5 dB. This is because the FFT-based carrier frequency determination error leads to a shift in the intermediate frequency of the signals after the frequency converter from the central frequency of BPF1. In addition, doubling the frequency results in a doubling of the specified error at the double intermediate frequency. Therefore, the filters do not allow enough frequency components to pass through, which creates envelope dips at the intermediate and double intermediate frequencies. Accordingly, the signal envelope shape is distorted.
Thus, at the given parameters of the investigated signals for Pcr > 0.9, the bandwidths of BPF1 and BPF2 in the developed algorithm should be not less than the sum of the double FFT frequency resolution and the maximum spectrum width of the recognized signals. In this case, the FFT frequency resolution is 0.98 MHz, the spectrum width of SP, BPSK, and QPSK signals are 0.2, 2 and 2 MHz, respectively. Hence, it is reasonable to select A/bpf1 = A/BPF2 = 4 MHz.
Conclusion. For the developed algorithm to recognize down-chirp, up-chirp, SP, BPSK, and QPSK signals, the influence of the algorithm parameters on the input SNR value required to provide the probability of correct signal recognition Pcr > 0.9 was investigated. As a result, we show that the required input SNR is minimal for all signals at the following parameters of the algorithm: number of FFT points Nfft = 1024; weight window - Hamming; signal envelope amplitude averaging coefficient kav = 0.15...0.25; bandwidths of
bandpass filters A/bpf1 = A/BPF2 = 4 MHz.
Author's contribution
Tran Huu Nghi, computer modeling; processing of modelling results; paper editing; formulating conclusions. Aleksey S. Podstrigaev, supervision of scientific work; setting tasks; paper editing; formulating conclusions. Nguyen Trong Nhan, literature analysis; paper editing.
Danil A. Ikonenko, literature analysis; paper editing.
All authors participated in the discussion of the results and in the preparation of the paper.
References
1. Rembovsky A., Ashikhmin A., Kozmin V, Smolskiy S. Radio Monitoring: Problems, Methods and Equipment. Lecture notes in electrical engineering. Springer, 2009, 530 p. doi: 10.1007/978-0-387-98100-0
2. Kuptsov V., Badenko V., Ivanov S., Fedotov A. Method for Remote Determination of Object Coordinates in Space Based on Exact Analytical Solution of Hyperbolic Equations. Sensors. 2020, vol. 20, no. 19, p. 5472. doi: 10.3390/s20195472
3. Nguyen T. N., Podstrigaev A. S., Leonov I. E. Mathematical Model of Signal Modulation Type Recognizing Algorithm in the Autocorrelation Receiver for Radio Engineering Monitoring Means. Trudy MAI. 2020, no. 113, p. 11. doi: 10.34759/trd-2020-113-09 (In Russ.)
4. Dvornikov S. V., Sivers M. A., Dvornikov A. S., Dvornikov S. S. Signal Recognition Based on the Probabilistic Evaluation of the Dispersion of Their Sign Vectors. Voprosy radioelektroniki, seriya Tekhnika televideniya [Issues of Radio Electronics, a Series of Television Technology]. 2020, no. 3, pp. 81-90. (In Russ.)
5. Utkin V V., Korotkov V A., Voynov D. C. Wavelet Filtration Application While Radio Technical Monitoring. The Herald of the Siberian State University of Telecommunications and Information Science. 2018, no. 1(41), pp. 64-71. (In Russ.)
6. Nikitin N. S., Darovskikh S. N. Synthesis of the Algorithm Identification of Signals with Linear Frequency Modulation. Bulletin of the Ural Federal District. Security in the Sphere of Information. 2019, no. 3(33), pp. 12-19. doi: 10.14529/secur190302 (In Russ.)
7. Lihachev V P., Veselkov A. A., Nguen Ch. N. Ustroistvo dlya izmereniya kharakteristik sluchainykh protsessov [Method for Determining the Types of Radar Signals in an Autocorrelation Receiver]. Pat. RF, 2019, no. 2683791. (In Russ.)
8. Kubankova A. Design and Analysis of New Digital Modulation Classification Method. WSEAS Transactions on Communications. 2009, vol. 8, no. 7, pp. 628-637.
9. Yang J., Wang X., Wu H. Modified Automatic Modulation Recognition Algorithm. 2009 5th Intern. Conf. on Wireless Communications, Networking and Mobile Computing. Beijing, China, 24-26 Sept. 2009. IEEE, 2009, pp. 1-4. doi: 10.1109/WIC0M.2009.5302483
10. Zavadsky A. L., Kazak P. A., Kadantsev S. M. Identification of the Modulation Type of Phase-Manipulated Signals Based on the Analysis of the Even Degree Spectrum Structure. Digital Signal Processing. 2019, no. 1, pp. 20-25. (in Russ.)
11. Kubankova A., Kubanek D. Digital Modulation Recognition Based on Feature, Spectrum and Phase Analysis and its Testing with Disturbed Signals. Recent Researches in Telecommunications, Informatics, Electronics and Signal Processing. 2011, pp. 162-166.
12. Tran H. N., Podstrigaev A., Trong N. N. A Signal Classification Algorithm with Detection at Two Intermediate Frequencies for RF Spectrum Monitoring. 2022 Intern. Conf. on Electrical Engineering and Photonics (EExPolytech). St Petersburg, Russia, 20-21 Oct. 2022. IEEE, 2022, pp. 91-94. doi: 10.1109/EExPolytech56308.2022.9950890
13. Tran H. N., Podstrigaev A. S., Nguyen T. N. Sposob klassifikacii signalov [Signal Classification Method]. Pat. RF, no. 2789386, 2023. (In Russ.)
14. Prakasam P., Madheswaran M. Digital Modulation Identification Model Using Wavelet Transform and Statistical Parameters. J. of Computer Networks and Communications. 2008, vol. 2008, pp. 1-8. doi: 10.1155/2008/175236
15. Dvornikov S. V, Saukov A. M. Signal Identification Method Based on Wavelet-Packets. Nauchnoe Priborostroenie [Scientific Instrumentation]. 2004, vol. 14, no. 1, pp. 85-93. (In Russ.)
16. Stogov A. A., Tereshonok M. V., Chirov D. S., Kuzmin G. V. Modulation Type Recognition Using High Order Cumulants. T-Comm. 2012, no. 1, pp. 5658. (In Russ.)
17. Swami A., Sadler B. M. Hierarchical Digital Modulation Classification Using Cumulants. IEEE Transactions on Communications. 2000, vol. 48, no. 3, pp. 416-429. doi: 10.1109/26.837045
18. Dobre O. A., Rajan S., Inkol R. Joint Signal Detection and Classification Based on First-Order Cyclostationarity for Cognitive Radios. EURASIP J. on Advances in Signal Processing. 2009, vol. 2009, pp. 112. doi:10.1155/2009/656719
19. Chilukuri R. K., Kakarla H. K., Subbarao K. Estimation of Modulation Parameters of LPI Radar Using Cyclostationary Method. Sensing and Imaging. 2020, vol. 21, pp. 1-20. doi: 10.1007/s11220-020-00313-3
20. Milne P. R., Pace P. E. Wigner Distribution Detection and Analysis of FMCW and P-4 Polyphase LPI Waveforms. 2002 IEEE Intern. Conf. on Acoustics, Speech, and Signal Processing. Orlando, USA, 13-17 May 2002. IEEE, 2002, vol. 4, pp. IV-3944-IV-3947. doi: 10.1109/ICASSP.2002.5745520
21. Liu Y., Xiao P., Wu H., Xiao W. LPI Radar Signal Detection Based on Radial Integration of Choi-Williams Time-Frequency Image. J. of Systems Engineering and Electronics. 2015, vol. 26, no. 5, pp. 973-981. doi: 10.1109/JSEE.2015.00106
22. Adzhemov S. S., Tereshonok M. V., Chirov D. S. Type Recognition Of The Digital Modulation of Radio Signals Using Neural Networks. Moscow University Physics Bulletin. 2015, no. 1, pp. 23-28. (In Russ.)
23. Kim N., Kehtarnavaz N., Yeary M. B., Thornton S. DSP-Based Hierarchical Neural Network Modulation Signal Classification. IEEE Transactions on
Neural Networks. 2003, vol. 14, no. 5, pp. 1065-1071. doi: 10.1109/TNN.2003.816037
24. Hazza A., Shoaib M., Alshebeili S. A., Fahad A. An Overview of Feature-Based Methods for Digital Modulation Classification. 2013 1st Intern. Conf. on Communications, Signal Processing, and Their Applications (ICCSPA). Sharjah, United Arab Emirates, 12-14 Febr. 2013. IEEE, 2013, pp. 1-6. doi: 10.1109/ICCSPA.2013.6487244
25. Sejdic E., Djurovic I., Jiang J. Time-Frequency Feature Representation Using Energy Concentration: an Overview of Recent Advances. Digital Signal
Processing. 2009, vol. 19, no. 1, pp. 153-183. doi: 10.1016/j.dsp.2007.12.004
26. Huang B., Jia G., Zhu Z. Multi-Channel RF Signal Analysis Device Based on FPGA and DSP. Sixth Intern. Conf. on Intelligent Computing, Communication, and Devices (ICCD 2023). 2023, vol. 12703, pp. 437-443. doi: 10.1117/12.2682801
27. Ivanova N., Galanina N., Moiseev D. FFT Algorithm Realization Features on FPGA. Bulletin of the Chuvash University. 2018, no. 3, pp. 182-191. (In Russ.)
28. Tsui J. B. Y. Special Design Topics in Digital Wideband Receivers. Artech House, 2010, 440 p.
Information about the authors
Tran Huu Nghi, Specialist in "Radioelectronic Systems and Complexes", Postgraduate Student of the Department of Radio Electronic Means of Saint Petersburg Electrotechnical University. The author of 6 scientific publications. Area of expertise: RF spectrum management.
Address: Saint Petersburg Electrotechnical University, 5 F, Professor Popov St., St Petersburg 197022, Russia
E-mail: huunghiht@gmail.com
https://orcid.org/0000-0002-9222-2502
Aleksey S. Podstrigaev, Cand. Sci. (2016), Associate Professor of the Department of Radio Electronic Means of Saint Petersburg Electrotechnical University. The author of more than 120 scientific publications. Area of expertise: design of complex radio systems; microwave devices; digital signal processing; wideband receivers. Address: Saint Petersburg Electrotechnical University, 5 F, Professor Popov St., St Petersburg 197022, Russia E-mail: ap0d@ya.ru https://orcid.org/0000-0003-4144-222X
Nguyen Trong Nhan, Cand. Sci. (2023), scientific collaborator of Le Quy Don Technical University (Hanoi, Vietnam). The author of more than 25 scientific publications. Area of expertise: radio engineering and telecommunications.
Address: Le Quy Don Technical University, 236, Hoang Quoc Viet St., Bac Tu Liem, Hanoi, Vietnam
E-mail: 10th20th30th@gmail.com
https://orcid.org/0000-0001-6626-893X
Danil A. Ikonenko, Master's Student of the Department of Computer Science and Engineering of Saint Petersburg Electrotechnical University. Area of expertise: radio engineering and telecommunications. Address: Saint Petersburg Electrotechnical University, 5 F, Professor Popov St., St Petersburg 197022, Russia E-mail: dan-ikonenko@mail.ru https://orcid.org/0009-0008-4157-3370
