On range super  resolution in a radar with many carriers. Simulation and experiment
Abstract
For citation:
Nagornykh I.L., Bazhenov N.D. On range super  resolution in a radar with many carriers. Simulation and experiment. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2019;(1):2429. https://doi.org/10.38013/25420542201912429
Introduction
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 frequencyspaced orthogonal signals (its commonly used name is OFDM radar, i.e. a radar with orthogonal frequency
division multiplexing) is believed to be a promising solution for detection of lowflying smallsize UAV.
Traditionally [1], range computing in an OFDM radar is done with the use of Fourier transform. In that case, range resolution improvement can only be achieved by increasing
signal band, which is not always possible. In this paper we consider improvement of the range resolution capacity through application of the MUSIC (multiple signal classification)
algorithm [2].
OFDM radar operating principle
Frequency band in an OFDM radar contains N
carriers equidistant to one another frequencywise (hereinafter – subcarriers). The subcarriers are orthogonal relative to one another if frequency difference between adjacent subcarriers
is 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 a_{n,m}  the mth modulation symbol on the nth 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 F_{tx}. For targetreflected and received signal, a symmetrical operation is performed, i.e. direct Fourier transform (matrix F_{rx} of the same dimensionality as F_{tx}), 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 [1]:
where b_{h}  coefficient determining signal amplitude decrease during its propagation to target and back;
 multiplier conditioned by Doppler frequency f_{D,h} (T_{0}  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 twodimensional periodogram Per_{f} with dimensionality N_{Per} х M_{per} (N_{per} ≥ N, M_{per} ≥ M). The periodogram elements, computed by means of the direct and inverse Fourier transforms, can be written as [1][3]:
Proceeding from the local maxima of the periodogram, range d and velocity v of the targets are computed:
where f_{C}  central carrier frequency. In addition to that, interpolation and window weighting can be applied in periodogram calculation [1].
The soughtfor coordinates can as well be computed using the MUSIC algorithm, through finding a pseudospectrum. 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 λ_{1}, λ_{2},… λ_{N} be eigenvalues of matrix R_{FF}, arranged in descending order: λ_{1} ≥ λ_{2} >.. .≥ λ_{N}_{1}. In such a case, it is said that eigenvectors {s_{1},..., s_{L} } associated with eigenvalues λ_{1},..., λ _{L} form a signal subspace, whereas eigenvectors {g_{1},...,g_{NL}} associated with λ_{L+1},..., λ_{N} form a noise subspace. Let us
designate noise subspace eigenvector matrix G = [g_{1}...g_{N}__{L}], introducing also designation (ω  frequencies of the soughtfor pseudospectrum). Then a^{H} (o>)GG^{H} a(ω) = 0 for all frequencies of the pseudospectrum, corresponding to the ranges to L targets. By way of illustration, we shall graphically represent the pseudospectrum as
Target velocities are calculated in a similar way, but the expression for covariance matrix looks as
Description of radar mathematical model
A block diagram of the mathematical model is given in Fig. 1. The data of matrix F_{tx} come to the unit of inverse Fourier transform, after which a guard interval is added so as to prevent intersymbol interference. After digitaltoanalogue conversion (DAC), transfer to the microwavefrequency 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 analoguetodigital 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. Блоксхема 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 onetenth 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 
T_{g} 
0,8 µs 
Range of unambiguously measured velocities 
v_{max} 
±8080 m/s 
Receiver temperature 
T 
300 K 
Receiver noise factor 
4 
The signaltonoise ratio was defined as
where Р_{прм}  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 signaltonoise 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 pseudospectra 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 targets
exists 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. Pseudospectrum 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. Pseudospectrum 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 increase
radar resolution capacity 2fold for SNR = 0 dB, 4fold for SNR = 10 dB, and 8fold 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. 
Experimental verification of models
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 softwarecontrolled transceivers. The received signal was processed in real time on a PC using the GnuRadio open platform [4].
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 pseudospectra 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. Pseudospectra 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 nonideal 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.
Conclusion
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 pseudospectra 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.
About the Authors
I. L. NagornykhRussian Federation
N. D. Bazhenov
Russian Federation
Review
For citation:
Nagornykh I.L., Bazhenov N.D. On range super  resolution in a radar with many carriers. Simulation and experiment. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2019;(1):2429. https://doi.org/10.38013/25420542201912429