An Estimation of Angular Coordinates and Angular Distance of Two Moving Objects by Using the Monopulse, MUSIC, and Root-MUSIC Methods

Abstract

The paper presents results of extensive simulations carried out in order to assess the precision and angular resolution of subspace methods in real radar system. It has been assumed that such a system uses the 32-element uniform linear array (ULA) and radiates only 3 bursts consisting of 8 pulses in given direction. In order to avoid blind Doppler frequencies, pulse repetition interval (PRI) is different in each burst. It has been shown that change of PRI is not only necessary to avoid blind Doppler frequencies but also allows to avoid false values of angular coordinates when two objects are visible in the same beam, in the same range gate, and their echoes attain maximal values in the same Doppler filter. It has also been shown that precision and angular resolution of both MUSIC and root-MUSIC method can be improved by appropriate preprocessing of signal samples used by these methods.

1. Introduction

The paper studies the problem of detection and estimation of angular coordinates of moving objects by means of MUSIC and root-MUSIC methods. MUSIC and root-MUSIC had been invented almost 30 years ago [1, 2]; however, application of these methods in real radar system has been possible only recently thanks to the progress in the field of active electronically scanned arrays, digital beamforming, and multiprocessor systems containing clusters of high-speed general purpose PowerPC processors and FPGA devices connected by means of high-speed serial bus such as RapidIO. An important reason stimulating the studies on these methods were difficulties concerning effective estimation of angular coordinates of closely spaced moving objects by using the monopulse methods, [3–5]. For example, it is not possible to separate individual objects even with different radial speeds if they are illuminated by the same antenna beam and are in the same range gate. More appropriate to solve problems of this kind are superresolution methods, represented here by MUSIC and root-MUSIC, [6, 7]. In application to estimation of angular coordinates of moving objects, these methods use the spatial correlation (covariance) matrix 𝐑 formulated on the basis of complex samples of received signals. As a rule, these samples are obtained on the outputs of individual matched filters (MF) of the array antenna unit, which have functional scheme similar to that shown in Figure 1.

Figure 1 

Functional diagram of the array antenna unit. A/D: analog-digital converter, QPD: quadrature phase detector, and MF: matched filter (digital).

All computer simulations presented in this paper have been performed under the assumption that this linear equidistant array antenna contains 𝑀=32 identical receiving elements spaced by the distance 𝑑=0.7𝜆₀, where 𝜆₀ is the length of received wave.

This 32-element array antenna has been used to create new 24-element array. Samples 𝑥_𝑖(𝑛), where 1≤𝑖≤32, of signals received by the real, 32-element array antenna are grouped in the following way: 𝑥^𝑘(𝑒)(𝑛)=⁸_𝑙=0𝑤_𝑙𝑥_𝑘+𝑙(𝑛)√9,for1≤𝑘≤24,(1) where 𝑤_𝑙=exp[𝑗𝑙(2𝜋𝑑/𝜆)sin(𝜃_max)] is the 𝑙th coefficient of vector 𝑤 shaping the directive gain characteristics of antenna elements, belonging to the 24-element array, and 𝜃_max determines the direction of maximum directivity of each antenna element. This kind of grouping is often called initial preprocessing. Samples 𝑥^𝑘(𝑒)(𝑛), where 1≤𝑘≤24, corresponding to individual elements of the equivalent, 24-element array are subsequently processed using the MUSIC and root-MUSIC methods.

It has been assumed that the antenna array described above radiates 24 pulses, grouped into three 8-pulse bursts differing by the repetition time (frequency), as shown in Figure 2.

Figure 2 

Sequence of pulses used in radars which employ MTD processing.

The signals (radar echoes) received at the outputs of elements of the array are used for digital beamforming, 𝑠(𝑛)=𝐰^𝐻_sum⋅𝐱(𝑛) and are processed later according to the MTD (moving target detection) technique, [8]. Echo signals 𝑠(𝑛) received after each pulse are subjected to coherent integration using predefined sets of weight coefficients 𝑤_𝑘𝑠_doppl(𝑏)=⁸_𝑘=1𝑤_𝑘𝑠(𝑏+𝑘𝐵),(2) where 𝑏 is the number of range gate determining the distance between the object and the radar, 𝐵 is the number of range gates in each scan, and 𝑘 is the number of pulse belonging to a given pulse sequence called the burst. This integration is called Doppler filtration, because it permits to distinguish between the echo signals from the objects moving with different radial speeds. The signal samples from individual array elements, weighted according to (1), form the vector 𝐱^(𝑒) that is the basis for calculation of the following estimate: 𝐑=1𝑁^𝑁_𝑛=1𝐱^(𝑒)𝐱^(𝑒)𝐻,(3) of the correlation matrix 𝐑, where 𝐱^(𝑒)𝐻 is the Hermitian conjugate with respect to the vector 𝐱^(𝑒). In practical considerations, the factor 1/𝑁 appearing in (3) is usually neglected, because it has no influence on the values of the used eigenvectors, but only decides about the scale of the related eigenvalues of matrix 𝐑 [2, 9].

2. The Multiple Signal Classification (Music) Method

As it was mentioned in the introduction, the MUSIC method belongs to the group of subspace methods in which the spectral functions are determined on the base of eigenvectors of the space correlation matrix 𝐑. Similarly, the matrix 𝐑 is formulated on the base of complex samples of the received signals, [1, 6, 10]. In order to explain the essence of this method in the radar application, assume that the 𝑁-element antenna array receives 𝑃 echo signals, reflected from 𝑃 objects, where 𝑃<𝑁. In a given moment, to each echo of the radiated pulse one can assign the vector of complex samples of the signals received in this moment by individual elements of the array antenna; see Figure 1. Therefore, we assume that to the first element of this array, from the direction 𝜃_𝑘, where 1≤𝑘≤𝑃, arrives the signal 𝑠₁(𝑡)=𝐴cos(𝜔𝑡+𝜙_𝑘) represented in further considerations by its complex amplitude 𝑠₁=𝐴exp(𝑗𝜙_𝑘). It is easy to prove on the base of Figure 1 that complex amplitude of the signal coming from the same direction 𝜃_𝑘 to the 𝑖-element of the array can be written as follows: 𝑠_𝑖=𝑠₁exp[−𝑗𝛽𝑑(𝑖−1)sin(𝜃_𝑘)], where 2≤𝑖≤𝑁, 𝛽=2𝜋𝑓/𝑐 is the propagation constant, 𝑐≈3⋅10⁸m/s is the speed of light, and 𝑓 denotes frequency of the narrow-band received signal. The complex samples of this signal, appearing on the outputs of individual matched filters MF, will be shifted in phase in a similar way. The set of these samples constitutes a 𝑁-dimensional column vector 𝐱_𝑘=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑥¹𝑥₂𝑥₃𝑥⋮_𝑁⎤⎥⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝑒1_{−𝑗𝛽𝑑sin(𝜃𝑘)}𝑒^{−𝑗2𝛽𝑑sin(𝜃𝑘)}𝑒⋮_{−𝑗(𝑁−1)𝛽𝑑sin(𝜃𝑘)}⎤⎥⎥⎥⎥⎥⎥⎥⎦𝑠₁𝜃_𝑘=𝝂𝜃_𝑘𝑠₁𝜃_𝑘,(4) where 𝐯𝜃_𝑘=1,𝑒^{−𝑗𝛽𝑑sin(𝜃𝑘)},𝑒^{−𝑗𝛽2𝑑sin(𝜃𝑘)},…,𝑒1≤𝑘≤𝑃,^{−𝑗(𝑁−1)𝛽𝑑sin(𝜃𝑘)}^𝑇,(5) is the so-called array steering vector, [10]. Each element of the array shown in Figure 1 receives simultaneously the signals from all directions 𝜃_𝑘, where 1≤𝑘≤𝑃, and noise signal with variance 𝜎^2𝑛. It means that the resulting column vector of complex samples of all 𝑃 signals and noise, before further digital processing, can be written as 𝐱=^𝑃_𝑘=1𝐱_𝑘+𝐧=^𝑃_𝑘=1𝜈𝜃_𝑘𝑠₁𝜃_𝑘+𝐧=𝐕𝐬+𝐧,(6) where 𝐕=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑒𝟏𝟏𝟏…𝟏^{−𝑗𝛽𝑑sin(𝜃1)}𝑒^{−𝑗𝛽𝑑sin(𝜃2)}𝑒^{−𝑗𝛽𝑑sin(𝜃3)}…𝑒^{−𝑗𝛽𝑑sin(𝜃𝑃)}𝑒^{−𝑗𝛽2𝑑sin(𝜃1)}𝑒^{−𝑗𝛽2𝑑sin(𝜃2)}𝑒^{−𝑗𝛽2𝑑sin(𝜃3)}…𝑒^{−𝑗𝛽2𝑑sin(𝜃𝑃)}𝑒^{−𝑗𝛽3𝑑sin(𝜃1)}𝑒^{−𝑗𝛽3𝑑sin(𝜃2)}𝑒^{−𝑗𝛽3𝑑sin(𝜃3)}…𝑒^{−𝑗𝛽3𝑑sin(𝜃𝑃)}𝑒⋮⋮⋮…⋮_{−𝑗𝛽(𝑁−1)𝑑sin(𝜃1)}𝑒^{−𝑗𝛽(𝑁−1)𝑑sin(𝜃2)}𝑒^{−𝑗𝛽(𝑁−1)𝑑sin(𝜃3)}…𝑒^{−𝑗𝛽(𝑁−1)𝑑sin(𝜃𝑃)}⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(7)is the 𝑁×𝑃 matrix, the columns of which are the elements of direction vectors (5), 𝐬=[𝑠₁(𝜃₁),𝑠₁(𝜃₂),𝑠₁(𝜃₃),…,𝑠₁(𝜃_𝑃)]^𝑇 is the 𝑃-element, column vector representing the signals received by the first element of the array, and 𝐧 is the 𝑁-element column vector representing received noise signals. The correlation matrix 𝐑 of samples of signals obtained on the outputs of antenna system, see Figure 1, can be written in the form of the sum, using the correlation matrix of desirable signals 𝐑_𝑠and the noise correlation matrix 𝐑_𝑛, [1, 10], namely, 𝐑=𝐸𝐱𝐱^𝐻=𝐕𝐑_𝑠𝐕^𝐻+𝐑_𝑛=𝐕𝐑_𝑠𝐕^𝐻+𝜎^2𝑛𝐈,(8) where operator 𝐸{} denotes assignment of the expected value, 𝐈 is the unitary matrix 𝑁×𝑁, and 𝐱^𝐻 is the Hermitian conjugate of the vector 𝐱. In other words, in this case, vector 𝐱^𝐻 is the 𝑁-element row vector, elements of which are complex conjugate with respect to the corresponding complex elements of the column vector 𝐱. Similarly, the matrix 𝐕^𝐻 is the complex conjugate of the matrix 𝐕. According to [1], the correlation matrix of desirable signals 𝐑_𝑠=𝐸𝐒𝐒^𝐻=diag⋅𝜎²¹,𝜎²²,𝜎²³,…,𝜎^2𝑃(9) is the diagonal matrix 𝑃×𝑃, if the received desirable signals are not correlated. Under the assumption 𝑃<𝑁, matrix 𝐕𝐑_𝑠𝐕^𝐻 is the singular matrix, which means that det𝐕𝐑_𝑠𝐕^𝐻=det𝐑−𝜎^2𝑛𝐈=0.(10) It follows from (10) that 𝜎^2𝑛 is the eigenvalue of the matrix 𝐑, [9]. Space, in which the desirable signals are not defined has the dimension 𝑁−𝑃. Hence, 𝜎^2𝑛 is (𝑁−𝑃)-order eigenvalue of the matrix 𝐑. The matrices 𝐑 and 𝐕𝐑_𝑠𝐕^𝐻 are not negative defined, and therefore, the matrix 𝐑 has also 𝑃 other eigenvalues 𝜆_𝑘, satisfying condition 𝜆_𝑘>𝜎^2𝑛>0, where 1≤𝑘≤𝑃. The eigenvectors 𝐪_𝑘, assigned to these eigenvalues are mutually orthogonal [9, 10]. According to the general definition of matrix eigenvalues problem, (𝐑−𝜆_𝑘)𝐪_𝑘=0, we can write 𝐑𝐪_𝑘=𝐕𝐑_𝑠𝐕^𝐻+𝜎^2𝑛𝐈𝐪_𝑘=𝜆_𝑘𝐪_𝑘for1≤𝑘≤𝑁,(11) where 𝜆_𝑘>𝜎^2𝑛>0 if 1≤𝑘≤𝑃 and 𝜆_𝑘=𝜎^2𝑛 if 𝑃+1≤𝑘≤𝑁. It follows from (11) that 𝐕𝐑_𝑠𝐕^𝐻𝐪_𝑘=𝜆^𝑘0,−𝜎^2𝑛𝐪_𝑘,1𝑃≤+1𝑘≤≤𝑘𝑃,≤𝑁.(12) Relation (12) shows that 𝑁-dimensional space of signals and noise can be divided into two mutually orthogonal subspaces, that is, subspace of signals 𝐐_𝑠≡[𝐪₁,𝐪₂,𝐪₃,…,𝐪_𝑃] and subspace of noise 𝐐_𝑛≡[𝐪_𝑃+1,𝐪_𝑃+2,𝐪_𝑃+3,…,𝐪_𝑁]. According to this partition, the correlation matrix 𝐑 can be written as the following sum: 𝐑=^𝑃_𝑘=1𝜆_𝑘−𝜎^2𝑛𝐪_𝑘𝐪^𝐻𝑘+^𝑁_𝑘=1𝜎^2𝑛𝐪_𝑘𝐪^𝐻𝑘,(13) where 𝜆_𝑘 denotes the eigenvalue corresponding to the vector 𝐪_𝑘 and 𝜎^2𝑛 is the variance of the white noise received from the individual antenna elements. After appropriate grouping of the factors of sum (13), we obtain the relation, in which one can distinguish 𝑃 eigenvectors representing desirable signals and 𝑁−𝑃 eigenvectors belonging to the noise subspace 𝐑=^𝑃_𝑘=1𝜆_𝑘𝐪_𝑘𝐪^𝐻𝑘+^𝑁_𝑘=_𝑃+1𝜎^2𝑛𝐪_𝑘𝐪^𝐻𝑘.(14) Each of vectors (4) containing complex samples of desirable signal belongs to the signal subspace 𝐐_𝑠, and therefore, it can be written in the form of sum of eigenvectors, defined in this subspace, namely, 𝐱_𝑘=𝝂𝜃_𝑘𝑠₁𝜃_𝑘=^𝑃_𝑘=1𝑏_𝑘𝐪_𝑘,(15) where 𝑏_𝑘 is the 𝑘 coefficient of a suitable value. It should be pointed out that each component 𝐪_𝑘 of vector (15) is orthogonal with respect to an arbitrary eigenvector 𝐪_𝑚 from the noise subspace 𝐐_𝑛. Consequently, the whole vector (15) is orthogonal to 𝐪_𝑚.This unique property can be expressed as follows: 𝐱^𝐻𝑘𝐪_𝑚=𝝂(𝜃_𝑘)𝑠₁(𝜃_𝑘)^𝐻𝐪_𝑚=𝑠^𝐻1𝜃_𝑘𝝂𝜃_𝑘^𝐻𝐪_𝑚=0for1≤𝑘≤𝑃<𝑚≤𝑁.(16) Applying equation (16) to all eigenvectors of the noise subspace 𝐐_𝑛, we find that the dot product of the vector 𝐯(𝜃_𝑘), representing the signal received from direction 𝜃_𝑘, and the sum of eigenvectors from the noise subspace 𝐐_𝑛, will also take the value close to zero. In the ideal case (exactly defined correlation matrix 𝐑, exactly evaluated eigenvalues and their corresponding eigenvectors, and precise partition onto the vectors from the subspaces of signal and noise), we have 𝐯^𝐻𝜃_𝑘^𝑁_𝑘=_𝑃+1𝐪_𝑘=0.(17) Using the property described by (17), the following estimate of the spectral power density of the signal can be formulated: 𝑃(𝜃)=1∑^{𝑁𝑘=𝑃+1}||𝐯^𝐻(𝜃)𝐪_𝑘||²=1∑^{𝑁𝑘=𝑃+1}𝐯^𝐻(𝜃)𝐪_𝑘𝐪^𝐻𝑘𝐯(𝜃).(18) This estimate is usually called the spectrum of the MUSIC method, [1, 6]. Placing all eigenvectors 𝐪_𝑘 from the noise subspace into the columns of matrix 𝐐_𝑛, spectrum (18) can be written in the equivalent, simpler form 𝑃(𝜃)=1𝐯^𝐻(𝜃)𝐐_𝑁𝐐^𝐻𝑁𝐯(𝜃).(19) Function 𝑃(𝜃) attains local maximum values for angles 𝜃_𝑘 determining directions of arrival of signals being received.

3. The Root-Music Method

Determination of angular positions 𝜃_𝑘 on the base of spectrum (18) or (19), requires performing calculations for great number of discrete values of angle 𝜃 and next determination of all its maximum values in the given, relatively large scanning range 𝜃_min≤𝜃≤𝜃_max. This task is especially laborious and time consuming when the angular resolution of the order of one tenth or one hundreds of degree is required. Therefore, in order to reduce the amount of calculations, the modified version of the MUSIC method, called root-MUSIC, has been elaborated. In this improved version, the problem of evaluation of the local maximum values of function (19) is replaced by the problem of finding the roots 𝜃_𝑘 of the polynomial 𝐯^𝐻(𝜃)𝐐_𝑁𝐐^𝐻𝑁𝐯(𝜃). Estimated values of angular coordinates of objects can be evaluated for the assumed number of roots that should be equal to the number of received desirable signals multiplied by 2. This number is usually determined by means of special criteria, among which the most known are AIC (akaike information criterion) and MDL (minimum description length), [11].

Denominator of function (19) is in general a polynomial, which can be written as 𝐯^𝐻(𝜃)𝐐_𝑁𝐐^𝐻𝑁𝐯(𝜃)=𝐯^𝐻(𝜃)𝐏𝐯(𝜃)=𝐶(𝑧)=^𝑁−1_𝑛=_−𝑁+1𝑐_𝑛𝑧^𝑛,(20) where 𝐏=𝐐_𝑁𝐐^𝐻𝑁 and 𝑧=exp[−𝑗𝛽𝑑sin(𝜃)]. According to this definition, 𝐏=𝐐_𝑁𝐐^𝐻𝑁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑝¹¹𝑝₁₂𝑝₁₃⋯𝑝_1,𝑁−1𝑝_1,𝑁𝑝₂₁𝑝₂₂𝑝₂₃⋯𝑝_2,𝑁−1𝑝_2,𝑁𝑝₃₁𝑝₃₂𝑝₃₃⋯𝑝_3,𝑁−1𝑝_3,𝑁𝑝⋮⋮⋮⋯⋮⋮_𝑁−1,1𝑝_𝑁−1,2𝑝_𝑁−1,3⋯𝑝_{𝑁−1,𝑁−1}𝑝_{𝑁−1,𝑁}𝑝_𝑁,1𝑝_𝑁,2𝑝_𝑁,3⋯𝑝_{𝑁,𝑁−1}𝑝_𝑁,𝑁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(21) is the Hermitian matrix of degree 𝑁. Coefficients 𝑐_𝑛 of the polynomial (20) can be determined by summing elements 𝑝_𝑘𝑙 of matrix 𝐏 placed on its 𝑛th diagonals, namely, 𝑐_𝑛=1𝑁_{𝑘−𝑙=𝑛}𝑝_𝑘𝑙.(22)

According to this formula, 𝑐𝑐₀=𝑝₁₁+𝑝₂₂+𝑝₃₃+⋯+𝑝_{𝑁−1,𝑁−1}+𝑝_𝑁,𝑁/𝑁,₁𝑐=𝑝₁₂+𝑝₂₃+𝑝₃₄+⋯+𝑝_{𝑁−2,𝑁−1}+𝑝_{𝑁−1,𝑁}/𝑁,₂=𝑝₁₃+𝑝₂₄+𝑐𝑝₃₅+⋯⋯+𝑝_{𝑁−3,𝑁−1}+𝑝_{𝑁−2,𝑁}/𝑁,_𝑁−1=𝑝_1,𝑁/𝑁.(23)

As was mentioned earlier, 𝐏 is the Hermitian matrix, and therefore, the coefficient of polynomial (20) with indexes −𝑛 and 𝑛 are mutually conjugate, so 𝑐_−𝑛=𝑐^∗𝑛 for 1≤𝑛≤𝑁−1. Equation 𝐶(𝑧)=0 is of degree 2𝑁−1 and has 2𝑁−1 roots, and to each root 𝑧_𝑛, there corresponds another root 1/𝑧^∗𝑛. These roots are most frequently evaluated by means of the companion matrix method, [12, 13]. It follows from the literature that this method ensures sufficient accuracy of calculations for all roots, even when polynomial 𝐶(𝑧)=0 is of relatively high degree; for instance, 2𝑁−1=63. Due to this valuable property, the companion matrix method has been implemented in the computational environment MATLAB.

Estimates of 𝑃 angular positions of objects being detected are evaluated on a basis of 2𝑃 roots situated nearest to the unitary circle determined on the complex plane 𝑧=Re(𝑧)+𝑗Im(𝑧), namely, on the basis of 𝑃 pairs (𝑧_𝑛, 1/𝑧^∗𝑛). Of course, each pair chosen in this manner determines only one location. With negligibly small power of the noise, 𝜎^2𝑛≈0, the roots lay exactly on the unitary circle mentioned above. From the substitution 𝑧=exp[−𝑗𝛽𝑑sin(𝜃)] introduced above, it follows that estimates of angular positions 𝜃̂_𝑛 are 𝜃̂_𝑛=arcsin−1𝛽𝑑arg𝑧^𝑛,(24) where 1≤𝑛≤𝑃,𝛽=2𝜋/𝜆 and 𝑑 is the antenna element spacing.

4. Application of Music and Root-Music Methods to the Estimation of Angular Coordinates of Moving Objects

As was already mentioned in the introduction, exact estimation of the angular coordinates of moving objects by means of monopulse methods is in many cases impossible because of their limited angular resolution. For this reason, they cannot distinguish the objects illuminated by the same antenna beam and situated at the same slant distance. In order to compare angular resolution of the monopulse, MUSIC and root-MUSIC methods, some computer simulations have been carried out. In these simulations, angular coordinates of two objects (planes) moving with the speed of 𝑣=100m/s have been evaluated. It has been assumed also that both planes are moving at 45° with respect to the north direction.

The position of the first plane is determined by constant angle 𝜃₁=−14.1^∘, while the second plane changes its position 𝜃₂ gradually from −14.1^∘ to −10.1^∘ preserving the course and speed. In other words, angular separation Δ𝜃=|𝜃₁−𝜃₂| between these planes (objects) changes in the range of Δ𝜃=0^∘÷4^∘. A limited number of pulses radiated by the radar in given direction and consequently limited number of signals vectors 𝐱^(𝑒) cause that estimate of correlation matrix 𝐑 is inaccurate and have to be modified using diagonal loading technique before eigendecomposition and estimation of angular coordinates by means of subspace methods 𝐑=1𝑁^𝑁_𝑛=1𝐱^(𝑒)𝐱^(𝑒)𝐻+𝛿⋅𝐈,(25) where 𝛿=4𝜎^2𝑛− loading factor.

The value of mean pulse repetition interval (mean PRI) has been set to 𝑇_𝑝=𝑇_𝑝−mean=2ms. The real PRI in 𝑖th burst is equal to 0.9⋅𝑇_𝑝−mean≤𝑇_𝑝𝑖≤𝑇_𝑝−mean and is modified in each burst in order to avoid blind Doppler frequencies (𝑇_𝑝1=1.0⋅𝑇_𝑝−mean,𝑇_𝑝2=0.95⋅𝑇_𝑝−mean,𝑇_𝑝2=0.9⋅𝑇_𝑝−mean).

The mean square errors of estimates of angular coordinates of both objects obtained by means of the monopulse method are illustrated in Figure 3.

Figure 3 

Mean square errors of estimates of angular coordinates of two objects determined by means of the amplitude monopulse method.

As it is seen, the second plane changing its angular position causes estimation errors of angular coordinates of the first of them. This effect can be eliminated in some extent by means of the additional Doppler filtration. This filtration allows to attenuate echo signals coming from the second plane when both signals have substantially different Doppler frequency shifts normalized with respect to pulse repetition frequency (PRF) 𝐹_𝑝. The change of angular position of the second plane causes observable change of its radial speed, understood as the plane speed component with respect to the radar station and consequently causes the change of its Doppler frequency shift. The results of simulations have shown that for relatively small velocities (𝑣=100m/s), the maximal difference of Doppler frequencies Δ𝑓_𝑑=𝑓_𝑑1−𝑓_𝑑2 is comparable with −3 dB bandwith of Doppler filter 𝐵_−3dB=60Hz and is smaller than its −18 dB bandwith 𝐵_−18dB=120Hz. The Doppler filters have the sidelobes level located at −18 dB. Therefore, if echo of the first object attain its maximal value in 𝑛th Doppler filter, Doppler frequency of echo of the second object is not located in the stopband of this filter and cannot be sufficiently attenuated. The difference of Doppler frequencies Δ𝑓_𝑑 is proportional to the speed of both objects, and at some point, echoes of these objects can be separated using Doppler filtration. Unfortunately, at higher speeds (𝑣=1000m/s) these echos may have similar normalized Doppler frequencies𝑓_𝑑−norm=𝑓_𝑑1%𝐹_𝑝=𝑓_𝑑2%𝐹_𝑝 despite the fact that their Doppler frequencies are different 𝑓_𝑑1=𝑓_𝑑−norm+𝑛⋅𝐹_𝑝,𝑓_𝑑2=𝑓_𝑑−norm+𝑚⋅𝐹_𝑝. This effect causes errors of estimation of angular coordinates of the objects being detected.

The disadvantageous effect under discussion is well illustrated by the simulation results shown in Figure 4. These simulations have been performed for the planes, the speed of which has been increased to 𝑣=1000m/s. The pulse repetition interval (PRI) has been also increased to 𝑇_𝑝=4ms in order to lower the PRF and increase number of similar normalized Doppler frequencies.

Figure 4 

Mean square errors of estimates of angular coordinates of two objects determined by means of the amplitude monopulse method and Doppler filtration (𝑣=1000m/s, 𝑇𝑝=4ms).

From the results presented above, it follows that the additional Doppler filtration is not an universal solution ensuring proper distinction of the moving objects at all times.

The signal to noise ratio 𝑆/𝑁_noise, given in Figures 3 and 4, determines the value of this parameter on the outputs of matched filters MF, see Figure 1. The power ratio of desired signal and jamming signal 𝑆/𝑁_jamm is calculated for the inputs of pulse compression blocks. This ratio has been defined before digital compression, because the noise signal can be differently correlated to the radiated LFM pulse. The second, very important advantage of MUSIC and root-MUSIC methods is their sufficient immunity to relatively high power jamming signals. In order to confirm this conclusion, the additional narrowband strong signal, situated in the direction defined by 𝜃_jamm=1.4^∘ has been introduced. The surface power density of this signal is 60 dB greater than the corresponding surface power density of both desirable radar signals. In this situation, the radar systems with receiving antennas nonadapted to the interferences, will be jammed and eventual detections may have improper estimates of angular coordinates. Mean square errors of such improper estimates of angular coordinates are illustrated in Figure 5. Of course, a negative influence of the single strong jamming signal can be decreased by using the receiving antenna in a form of multielement array, which can be adapted to this undesirable signal, [14, 15].

Figure 5 

Mean square errors of angular coordinates of two objects obtained by using the monopulse method and Doppler filtration in presence of the jamming signal (𝑣=100m/s).

In other words, the jamming signal should be attenuated by the array antenna in the highest degree. The simulation results presented in Figure 6 confirm an effectiveness of application of the above approach to solve the radiolocation problem under consideration.

Figure 6 

Mean square errors of angular coordinates of two objects obtained using the monopulse method with Doppler filtration in presence of the jamming signal attenuated by the adaptive receiving antenna, (𝑣=100m/s, 𝐿jam=98.6dB).

Next, the same radiolocation problem has been solved by using the MUSIC and root-MUSIC methods. Figures 7 and 9 show mean square errors of estimates obtained by means of MUSIC method for 𝑆/𝑁_noise=0dB and 𝑆/𝑁_noise=30dB in presence of jamming signal determined above.

Figure 7 

Mean square errors of angular coordinates of two objects obtained using the MUSIC method in presence of the jamming signal, and 𝑆/𝑁noise=0dB(𝑣=100m/s).

Similarly, the corresponding mean square errors of estimates evaluated using the root-MUSIC method for 𝑆/𝑁_noise=0dB, 𝑆/𝑁_noise=30dB and the same jamming signal are shown in Figures 8 and 10.

Figure 8 

Mean square errors of angular coordinates of two objects obtained using the root-MUSIC method in presence of the jamming signal and 𝑆/𝑁noise=0dB(𝑣=100m/s).

Figure 9 

Mean square errors of angular coordinates of two objects obtained using the MUSIC method in presence of the jamming signal, and 𝑆/𝑁noise=30dB(𝑣=100m/s).

Figure 10 

Mean square errors of angular coordinates of two objects obtained using the root-MUSIC method in presence of the jamming signal and 𝑆/𝑁noise=30dB(𝑣=100m/s).

The values of angular resolution evaluated for these methods are given correspondingly in Tables 1 and 2.

Table 1 

Angular resolution evaluated for MUSIC method.

Table 2 

Angular resolution evaluated for root-MUSIC method.

For comparison, Tables 3 and 4 contain values of angular resolution obtained, when MUSIC and root-MUSIC methods determine angular coordinates on the basis of signal samples received by the real antenna array with 32-elements.

Table 3 

Angular resolution evaluated for MUSIC method (32 elements array).

Table 4 

Angular resolution evaluated for root-MUSIC method (32 elements array).

In other words, in this process, the stage of initial preprocessing, see introduction, has been omitted. Comparing results given in Tables 1 and 2, as well in Tables 3 and 4, we can see that using the proposed initial preprocessing, see relation (1), gives significant amelioration of precision and angular resolution. According to these results, using of initial preprocessing permits to reduce the 𝑆/𝑁_noise ratio for about 6 dB conserving necessary precision and resolution. Thus, the error magnitudes given in Tables 1 and 2 seem to be acceptable for most similar radiolocation problems encountered in practice. As it has been mentioned in the beginnig of the paper, all simulations have been carried out assuming that PRI is changing in each burst. The following values of PRI have been used: 𝑇𝑇_𝑝1𝑝2=1.00⋅𝑇_𝑝−mean,𝑇_𝑝3=0.95⋅𝑇_𝑝−mean,=0.90⋅𝑇_𝑝−mean.(26)

Estimation of correlation matrix on basis of signals vectors received for 3 different intervals allows MUSIC and root-MUSIC methods to estimate the correct values of angular coordinates despite the fact that signals could have the same or very close normalized Doppler frequency 𝑓_𝑑−norm=𝑓_𝑑%𝐹_𝑝 and be correlated for given PRI. This technique is well known in radar literature, but it has been mainly used to avoid blind speeds. Presented simulation show, that it also helps to mitigate the problem of estimation of angular coordinates of closely spaced highly correlated signals. When objects are moving with very high velocities (𝑣=1000m/s) their angular coordinates could be false when emitted signals have the same PRI in each burst as shown in Figures 11 and 12.

Figure 11 

Values of normalized Doppler frequencies of two objects moving with speed 𝑣=1000m/s (𝑇𝑝=2ms).

Figure 12 

Mean square errors of angular coordinates of two objects obtained using the root-MUSIC method (𝑆/𝑁noise=0dB, 𝑣=1000m/s, 𝑇𝑝=2ms).

This effect is especially apparent when PRF is relatively small for instance when 𝐹_𝑝=250 Hz (𝑇_𝑝=4ms). The values of normalized Doppler frequencies and mean square errors of angular coordinates obtained for this case are illustrated in Figures 13 and 14.

Figure 13 

Values of normalized Doppler frequencies of two objects moving with speed 𝑣=1000m/s (𝑇𝑝=4ms).

Figure 14 

Mean square errors of angular coordinates of two objects obtained using the root-MUSIC method (𝑆/𝑁noise=0dB, 𝑣=1000m/s, 𝑇𝑝=4ms).

The further study of this problem and comparison of these approach with well-known techniques of decoration of signals such as spatial smoothing or redundancy averaging is behind the scope of this paper.

5. Conclusions

The radiolocation problem defined at the beginning of the introduction and in Section 4 has been subsequently solved by different methods, that is, amplitude monopulse method, amplitude monopulse method aided by the coherent Doppler filtration, MUSIC, and root-MUSIC. Diagrams shown in Figures 3 and 10, respectively, illustrate the results of computer simulations obtained by means of these methods. Thus, on the basis of diagrams shown in Figure 3, we can deduce that traditional amplitude monopulse method is inadequate for this purpose. Some amelioration can be obtained using the additional coherent Doppler filtration. Unfortunately, this solution is not always effective, because false estimates can appear in cases when echo signals after the Doppler filtration [8] attain similar values in the same frequency channel or have similar normalized Doppler frequencies. This effect is illustrated on diagram shown in Figure 4. Partial elimination of this undesirable effect is possible using multiburst radar signals with variable pulse repetition time (frequency), that is, similar to that shown in Figure 2. The most effective and sufficiently precise for these applications, see Tables 1 and 2, proved to be MUSIC and root-MUSIC methods. This conclusion is well justified by the corresponding simulation results illustrated in Figures 7 and 10.

The radiolocation problem under consideration becomes especially difficult to solve after introducing the narrowband strong jamming interference. In the paper, it has been assumed that this jamming signal is incoming from the direction 𝜃_jamm=1.4^∘, and its power density is 60 dB greater than the corresponding surface power density of both desirable radar signals. The influence of this jamming signal on results of the similar simulations performed by using the MUSIC and root-MUSIC methods is illustrated by diagrams shown in Figures 7 and 10.

The presented results confirm that MUSIC and root-MUSIC methods are suitable for effective solution of radiolocation problems similar to that discussed here. Moreover, they show that these methods can be treated as reliable for the radar signals, for which the power signal-noise ratio is not less than the signal-noise ratio, at which the signal surpasses a detection threshold. It has been also confirmed that the proposed initial preprocessing, see (1), makes possible significant amelioration of precision and angular resolution of MUSIC and root-MUSIC methods. Consequently, they can be applied for relatively weak radar signals, for which 𝑆/𝑁_noise≈0dB.

All simulation results presented in this paper have been obtained under the assumption that the number 𝑃 of objects being detected is known exactly. This number is required to the appropriate partition of space, spanned on the correlation matrix 𝐑, into the useful signal subspace 𝐐_𝑠 and the noise subspace 𝐐_𝑛. Of course, is not easy to determine this number for the majority of quasireal radiolocation scenarios, and for this reason, it can be a source of potential errors. Thus, the additional algorithm determining this number with highest possible precission, according to AIC or MDL criterion, is required.

6. Summary

The main subject of considerations are MUSIC and root-MUSIC methods used to estimation of the angular coordinates (directions of arrival) and angular distance of two moving objects in presence of uncorrelated noise signal and an external, relatively strong narrowband jamming interference. At the receiving antenna, the 32-element uniform linear array (ULA) is used. Extensive computer simulations have been carried out to demonstrate the sufficient accuracy and good spatial resolution properties of these methods in the scenario defined above. It is also shown that using the proposed initial preprocessing, we can increase the accuracy and angular resolution of the methods under discussion. Most of simulation results, presented mainly in a graphical form, have been compared with the corresponding simulation results obtained by using the traditional amplitude monopulse method and the amplitude monopulse method aided by the coherent Doppler filtration.