The paper focuses on radar operation and the results of its simulation. The probing signal of the radar is a set of 16 orthogonal carriers. To determine the range in such radar, the MUSIC algorithm was applied, which relates to super - resolution methods. Findings of research show that the MUSIC algorithm makes it possible to increase the radar range resolution in the signal - to-noise 0-20 dB ratio by 4-8 times as compared with the traditional method based on the Fourier transform. The developed models were experimentally verified

Приведены результаты моделирования работы радиолокатора, зондирующий сигнал которого представляет собой набор из 16 ортогональных несущих. Для определения дальности в таком радиолокаторе применен алгоритм MUSIC, относящийся к методам сверхразрешения. Показано, что алгоритм MUSIC позволяет повысить разрешающую способность радиолокатора по дальности в 4-8 раз по сравнению с традиционным методом, основанным на преобразовании Фурье, в диапазоне 0-20 дБ отношений сигнал - шум. Экспериментальным путем проведена верификация разработанных моделей

One of specific features of unmanned aerial vehicle (UAV) application tactics is a possibility to form a large group (swarm) of UAV drones. Detection and recognition of such a group target demands a high radar resolution by target coordinates, and above all, range resolution. Under the range resolution capacity we imply the minimum distance between two targets lying on the same direction line when these targets can be observed separately.

A radar whose probing signal comprises an ensemble of frequency-spaced orthogonal signals (its commonly used name is OFDM radar, i.e. a radar with orthogonal frequencydivision multiplexing) is believed to be a promising solution for detection of low-flying small-size UAV.

Traditionally [

Frequency band in an OFDM radar contains Ncarriers equidistant to one another frequencywise (hereinafter – subcarriers). The subcarriers are orthogonal relative to one another if frequency difference between adjacent subcarriersis divisible by where T - one radiofrequency pulse duration. To generate a signal for transmission, containing N subcarriers, a matrix of complex modulation symbols is introduced

where an,m - the m-th modulation symbol on the n-th subcarrier;

M - number of symbols in a burst.

A probing signal is shaped by means of inverse discrete Fourier transform of each column of matrix Ftx. For target-reflected and received signal, a symmetrical operation is performed, i.e. direct Fourier transform (matrix Frx of the same dimensionality as Ftx), is shaped), with sub ), - sequent processing of the received radar signal is carried out in the frequency domain.

For further argumentation we shall introduce matrix F , such that:

The elements of matrix F of the signal reflected from H targets can be written as [

where bh - coefficient determining signal amplitude decrease during its propagation to target and back;

- multiplier conditioned by Doppler frequency fD,h (T0 - total duration of a sequence of N symbols);

- multiplier conditioned by phase shift resulting from signal propagation to target and back;

- constant;

(Z)k, l - receiver noise.

Having applied direct Fourier transform to the columns of matrix F, and then inverse Fourier transform to the rows of matrix F, we obtain a two-dimensional periodogram Perf with dimensionality NPer х Mper (Nper ≥ N, Mper ≥ M). The periodogram elements, computed by means of the direct and inverse Fourier transforms, can be written as [

Proceeding from the local maxima of the periodogram, range d and velocity v of the targets are computed:

where fC - central carrier frequency. In addition to that, interpolation and window weighting can be applied in periodogram calculation [

The sought-for coordinates can as well be computed using the MUSIC algorithm, through finding a pseudo-spectrum. Let the number of detectable targets be L, with L<N. Let us determine covariance matrix (with dimensionality N × N), where index H designates Hermitian conjugation. Let λ1, λ2,… λN be eigenvalues of matrix RFF, arranged in descending order: λ1 ≥ λ2 >.. .≥ λN-1. In such a case, it is said that eigenvectors {s1,..., sL } associated with eigenvalues λ1,..., λ L form a signal subspace, whereas eigenvectors {g1,...,gN-L} associated with λL+1,..., λN form a noise subspace. Let usdesignate noise subspace eigenvector matrix G = [g1...gN_L], introducing also designation (ω - frequencies of the sought-for pseudo-spectrum). Then aH (o>)GGH a(ω) = 0 for all frequencies of the pseudo-spectrum, corresponding to the ranges to L targets. By way of illustration, we shall graphically represent the pseudo-spectrum as

Target velocities are calculated in a similar way, but the expression for covariance matrix looks as

A block diagram of the mathematical model is given in Fig. 1. The data of matrix Ftx come to the unit of inverse Fourier transform, after which a guard interval is added so as to prevent inter-symbol interference. After digital-toanalogue conversion (DAC), transfer to the microwave-frequency carrier, and amplification (in transmitter), the signal is aired through the antenna. The signal reflected from targets and local terrain features comes through the antenna system to the receiver, where it is amplified and transferred to the intermediate frequency. After analogue-to-digital conversion (ADC), guard interval is removed, the direct Fourier transform procedure is run, and calculation of the coordinates is performed using periodogram and the MUSIC algorithm.

Рис. 1. Блок-схема модели радиолокатора

Determination of the necessary number of subcarriers is performed according to the following algorithm. Let the preset values of signal bandwidth and maximum Doppler frequency of detectable targets be equal to 20 MHz and 100 kHz, respectively. To ensure orthogonality of the reflected signal’s subcarriers, we shall assume that Doppler frequency must not exceed one-tenth of the frequency difference of adjacent subcarriers. Proceeding from this, the maximum number of subcarriers in a selected frequency band cannot exceed 20. The maximum algebraic integer of degree 2 that does not exceed the maximum number of subcarriers is 16.

The number of symbols M in a burst shall be sufficient for the sample size to ensure consistent solving of the MUSIC algorithm and conform to the specified time of target detection. Values N = 16 and M = 512 are acceptable to satisfy the above requirements. Selection of these and other parameters of the model was determined, in particular, by the necessity of further experimental verification of the capabilities of such radar. Parameters of the model are given in Table 1.

Table 1

Model parameters

Parameter | Designation | Value |
---|---|---|

Central subcarrier frequency | fc | 5,8 GHz |

Number of subcarriers | N | 16 |

Signal bandwidth (DAC/ADC sampling rate) | B = N∆f | 20 MHz |

Number of symbols in burst | M | 512 |

One symbol duration | T | 0,8 µs |

Guard interval duration | Tg | 0,8 µs |

Range of unambiguously measured velocities | vmax | ±8080 m/s |

Receiver temperature | T | 300 K |

Receiver noise factor | 4 |

The signal-to-noise ratio was defined as

where Рпрм - power of the received (reflected) signal;

k – Boltzmann’s constant;

T – receiver temperature;

B – signal bandwidth (DAC, ADC sampling rate);

- receiver noise factor.

To evaluate radar operation, two targets were considered, having the same radar crosssection and the same velocity. The following set of distances between two targets was selected for investigation:

where

The far target was located at a range of 100 m. The checks were performed for signal-to-noise ratios SNR = 0 dB, 10 dB, 20 dB. Fig. 2 shows periodograms (in range – velocity coordinates) for SNR = 0 dB, with two targets spaced by ∆d and 2∆d. It is obvious that the targets can only be observed separately when spaced by 2∆d.

Fig. 2. Periodograms (range – velocity diagrams) calculated for SNR = 0 dB:

а – with two targets spaced by ∆d (7.5 m); b – with two targets spaced by 2∆d (15 m)

Fig. 3 shows pseudo-spectra for SNR = 0 dB, with two targets spaced by ∆d and 2∆d, proceeding from which it can be concluded that a possibility for separate observation of the targetsexists in either case. The best resolution with the use of the MUSIC algorithm was obtained at SNR = 20 dB and amounted to 1.88 m (Fig. 4).

Fig. 3. Pseudo-spectrum calculated by the MUSIC method for SNR = 0 dB:

а – with two targets spaced by ∆d (7.5 m); b – with two targets spaced by 2∆d (15 m)

Fig. 4. Pseudo-spectrum calculated by the MUSIC method for SNR = 20 dB with two targets spaced by ∆d/4 (1.875 m)

A possibility for separate observation of the targets under different conditions is shown in Tables 2, 3. As can be seen from the tables, the MUSIC algorithm makes it possible to increaseradar resolution capacity 2-fold for SNR = 0 dB, 4-fold for SNR = 10 dB, and 8-fold for SNR = 20 dB.

Table 2

Discernibility of two closely located targets in periodogram assessment

SNR, dB | Targets spacing | ||||
---|---|---|---|---|---|

1/8Δd | 1/4Δd | 1/2Δd | Δd | 2Δd | |

0 | о | о | о | о | x |

10 | о | о | о | о | x |

20 | о | о | о | о | х |

Note. х – targets discernible, о – indiscernible |

Table 3

Discernibility of two closely located targets with MUSIC algorithm applied

SNR, dB | Targets spacing | ||||
---|---|---|---|---|---|

1/8∆d | 1/4∆d | 1/2∆d | Δd | 2∆d | |

0 | о | о | о | x | x |

10 | о | о | x | x | x |

20 | о | x | x | x | х |

Note: х – targets discernible, о – indiscernible. |

The experimental work was done using a mockup manufactured at “IEMZ “Kupol” JSC. The mockup featured two spaced directional antennas. Signal shaping for transmission and signal reception was implemented by means of two software-controlled transceivers. The received signal was processed in real time on a PC using the GnuRadio open platform [

In this paper, target was represented by a local object arranged in the immediate proximity (5 m) to the radar. Figs. 5, 6 show periodograms and pseudo-spectra obtained with the use of the developed model and by way of experiments.

Fig. 5. Periodograms: а – obtained with the use of developed model; b – obtained experimentally

Fig. 6. Pseudo-spectra computed by MUSIC method: а – obtained with the use of developed model; b – obtained experimentally

Qualitative conformity between the calculated and experimental data can be plainly seen in the figures. Quantitative difference in the spectra in Fig. 6 is explained by non-ideal isolation between transmitter and receiver, presence of noise in the signal for transmission, etc. On the whole, it can be presumed that the developed models conform with the experimental results.

It has been demonstrated that the MUSIC algorithm makes it possible to increase radar resolution capacity without changing signal bandwidth. At the same time, the algorithm has some shortcomings, one of which consists in the following. When shaping pseudo-spectra corresponding to the range and velocity of several targets, it is impossible to unambiguously determine interrelation between range and velocity of a particular target. The periodogram method is free from this shortcoming. Therefore, in radar tasks requiring simultaneous measurement of range and velocity, the MUSIC algorithm can be regarded as a supplement to the main method of periodograms.

The authors declare that there are no conflicts of interest present.