# Application of the Total Least Square ESPRIT Method to Estimation of Angular Coordinates of Moving Objects

Authors: Wojciech Rosloniec
Journal: International Journal of Antennas and Propagation
 (2010)

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## Abstract

The TLS ESPRIT method is investigated in application to estimation of angular coordinates (angles of  arrival)  of two moving objects at the presence of  an external, relatively strong uncorrelated signal. As a radar antenna system, the 32-element uniform linear array (ULA) is used. Various computer simulations have been carried out in order to demonstrate good accuracy and high spatial resolution of the TLS ESPRIT method  in the scenario outlined above. It is also shown that accuracy and angle resolution can be significantly increased by using the proposed preprocessing (beamforming). The most of simulation results, presented in a graphical form, have been compared to the corresponding equivalent results obtained by using the ESPRIT method and conventional amplitude monopulse method aided by the coherent Doppler filtration.


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## Body

## 1. Introduction

One of  the important  problems of today’s radar technology is the detection and  estimation of angular coordinates of fast moving objects, the radar echoes of which are received on a background of strong uncorrelated interferences and signals reflected from other objects located nearby. It is obvious that in such  situations the conventional monopulse methods are rather noneffective [1, 2]. This noneffectiveness results from the essence of them, since they enable the correct estimation of angular coordinates merely in situations when just one sufficiently strong useful signal is being received in a specific moment. Unfortunately, in the real electromagnetic environment, besides the useful signal (reflected from a detected object), different spurious signals are also received by the radar antenna unit. For instance, these spurious signals may be reflected from the earth, weather precipitation, or other masking objects flying around.  It follows from the literature that  problems of this kind  can be effectively solved by modern radar systems using the angular superresolution methods. Thus, in this paper, one of them, namely, the Total Least Square ESPRIT (TLS ESPRIT), has been used for this aim [3–6]. The corresponding  theory is presented in Section 2. In the next, that is, Section 3, it is shown how this method can be effectively applied to solve different  variants of the  general problem formulated above.  A novel component of the proposed approach is an algorithm of preprocessing the samples of signals received simultaneously by a linear uniform antenna array. This algorithm creates a new “virtual” linear array with smaller number of radiating/receiving elements. It is  important that signal-to-noise ratio evaluated at each of its virtual elements is greater, thanks to integration of signals from consecutive radiating/receiving elements of the original array. Consequently, the angular coordinates of the detected object can be evaluated more accurately. The angular resolution of the applied method is also increased. A short discussion on the results of the corresponding computer simulations is presented in Sections 3, 4, 5, 6, and 7.


## 2. TLS ESPRIT Method

The TLS ESPRIT method can be treated as an improved version of the ESPRIT method, the theoretical basis of which is described in many publications, for example, [5–7]. To recall the essence of the TLS ESPRIT method, let us assume that the signals arriving from different directions are received by two identical subarrays, X and  Y, mutually displaced by distance Δ,  Figure 1.Figure 1
A uniform linear array antenna composed ofN sensors and their (N-1)-sensor subarrays X and Y.The resultant signals received by the arrays are the sum of theK signals sk(t), k=1,…,K being received from specific directions determined by the angles θk. The vectors of signals received by the first subarray (X) and second subarray (Y), as shown in Figure 1, are described by the following equations, respectively:  (1)x(n)=As(n)+u(n),y(n)=AΦs(n)+v(n),
where(2)A=[a(θ1),a(θ2),…,a(θK)]=[a1(θ1)e-jω0τ1(θ1)a1(θ2)e-jω0τ1(θ2)⋯a1(θK)e-jω0τ1(θK)a2(θ1)e-jω0τ2(θ1)a2(θ2)e-jω0τ2(θ2)⋯a2(θK)e-jω0τ2(θK)⋮⋮⋱⋮aN-1(θ1)e-jω0τN-1(θ1)aN-1(θ2)e-jω0τN-1(θ2)⋯aN-1(θK)e-jω0τN-1(θK)],where ω0 is the angular frequency of carrier wave of the signal received by the antenna array, λ is the wavelength of the carrier wave of the signal received by the antenna array, ai(θk) is the parameter representing the directional gain of the ith radiating element, and τi(θk) is the delay of the signal reaching the ith radiating element,(3)Φ=[e-j2πΔ/λsin⁡(θ1)0⋯00e-j2πΔ/λsin⁡(θ2)⋯0⋮⋮⋱⋮00⋯e-j2πΔ/λsin⁡(θK)]-diagonalmatrixofthescalingcoefficients,
having s(n) as the vector of samples received by the antenna array, s(n)=[s1(n),s2(n),…,sK(n)]T, u(n) as the vector of noise samples in the radiating elements of the first subarray, and v(n) as the vector of noise samples in the radiating elements of the second subarray.The vector of samplesz(n), which is composed of samples of signals received from both subarrays, is defined as follows: (4)z(n)=[x(n)y(n)]=A̅s(n)+n(n),
where(5)A̅=[AAΦ],n(n)=[u(n)v(n)].
For the vector z(n), the spatial correlation matrix is formulated
(6)RZ=A̅SA̅H+σ2𝚺,
where S=E{s(n)s(n)H} and Σ≡I=diag.{1,1,1,…,1} is a unitary matrix of the (2N-2)-th degree. The generalized eigenvectors vi, i=1,…,2N-2 of the matrix RZ create a space, in which the directional vectors of signals sk(n), received by both N-1 element antenna arrays lie (Figure 1). All generalized eigenvectors vi satisfy the equation
(7)(RZ-λi𝚺)vi=0,
where λi denotes the ith eigenvalue of matrix RZ. On the basis of eigenvectors corresponding to the K biggest eigenvalues, the vector matrix of the signal space is created
(8)VS=[v1∣v2∣⋯∣vK].
The vectors included in matrix (8) can be presented as a combination of the directional vectors: (9)VS=A̅T
since the directional vectors of received signals (written into columns of matrix A̅) span over the same signal space. Like in the case of the matrix of directional vectors A̅, the eigenvector matrix VS has the structure that allows us to determine the phase shift between the signals received by  subarrays X and Y, see Figure 1(10)VS=[VXVY]=[ATAΦT].That means that for the matrixVXY=[VX∣VY] exists a matrix
(11)F=[FXFY],F∈C2K×K
fulfilling  the following equation, [6, 8]:
(12)VXYF=VXFX+VYFY=ATFX+AΦTFY=0.
By using the matrix Ψ=-FXFY-1, (12) can be written in the form ATΨ=AΦT, which results in
(13)Ψ=T-1ΦT.Here, it should be noticed that (13) has similar form as equation Ψ=VΛVH, which defines eigenvalues and eigenvectors of matrix Ψ. Therefore, columns of matrix T-1 are equal to eigenvectors of matrix Ψ, and values of diagonal elements of matrix Φ are equal to related eigenvalues of matrix Ψ. When the values of the phase shifts included in the diagonal elements of Φ are known, one can determine the directions represented by the angles θk,  from which the signals sk(t) are received.Unfortunately, the method described above requires a very accurate estimate of the correlation matrixRZ, for which the following equation should be satisfied:(14)VXΨ=VY.
In practice, (14) is not satisfied precisely due to an inaccurate estimate of the correlation matrix RZ determined on basis of limited number of measurements of the vector z(n). The estimate of the correlation (covariance) matrix is(15)R̂Z=1Nret∑k=1Nretz(g+k⋅G)z*(g+k⋅G),
where Nret denotes the number of pulse returns (realizations) employed to estimate the values of the spatial correlation matrix, g is the index of range gate, and G is the number of range gates obtained after sampling of the received pulse return. In such case, the generalized eigenvectors, obtained as the result of the decomposition of matrices RZ and Σ, are inaccurate. The sought matrix Ψ, for which the error of mapping the matrix VX into matrix VY is the least, can be found using the criterion of the total least square error. According to this criterion, matrix F is evaluated, for which the value of square of the Euclidean (Frobenius) norm of the product of the matrices VXY and F, that is
(16)P=‖VXYF‖F2=‖VXFX+VYFY‖F2,
is the least. Additionally, a limitation  as described by the relationship below is imposed  on the value of the matrix F(17)FHF=I.
The solution of the aforediscussed problem is the matrix F of dimension 2K×K  composed of K columns, which are equal to eigenvectors corresponding to K least eigenvalues of the matrix product VXYHVXY. The eigenvalues of the matrix Ψ correspond to the sought values of the elements lying on the diagonal of the matrix Φ. Knowing the values of diagonal elements of matrix Φ, values of angles θk can be found.  To summarize, the TLS ESPRIT method can be implemented in the following way.Step 1.
Determine the estimate of the spatial correlation matrixRZ, based on the measurement of  the signal vector z(n).Step 2.
Decompose the matricesRZ and Σ with respect to their eigenvectors and eigenvalues, see (6) and (7), which satisfy the following condition:
(18)R̂ZV=𝚺VΛ,
where V is a matrix of eigenvectors and Λ is a diagonal matrix of eigenvalues.Step 3.
Determine the number of signal sourcesK.Step 4.
Out of the matrixV, select vectors corresponding to K biggest eigenvalues and create a matrix Vs of them.Step 5.
Divide the matrixVs into matrices VX and VY and create a matrix VXY of them
(19)VS=𝚺[v1,v2,…,vK]⟹[VXVY],VXY=[VX∣VY].Step 6.
Decompose the matrix productVXYHVXY with respect to its eigenvectors and eigenvalues, which satisfy the equation VXYHVXY=EΛE.Step 7.
Split the eigenvector matrixE into four submatrices  of dimensions K×K each
(20)E=[E11E12E21E22].Step 8.
Based on the matricesE12 and E22, the columns of which correspond to the K smallest eigenvalues, determine a matrix Ψ by using the transformation Ψ=-E12E22-1.Step 9.
Find the eigenvaluesϕk of matrix Ψ.Step 10.
Based on complex coefficientsϕk, calculate the angles,
(21)θk=arcsin(c⋅arctg(ϕk)ω0Δ),k=1,…,K,
which determine the directions of arrival of signals sk(t), see Figure 1.


## 3. Application  of  the TLS  ESPRIT  Method  to  Evaluate  Angular Coordinates of Two Moving Objects

In this section, the TLS ESPRIT method has been used to solve the problem of estimation of angular coordinates of two objects, moving with velocityv=100 m/s, Figure 2.Figure 2
Polar coordinates (angles of arrival) of the planes (objects) being detected.Let us assume that both objects bear atα=-45° north and the azimuth angle of the first of them is θ1=-14.1°. The azimuth angle θ2 of the second one is changing in the range of -14.1°≤θ2≤-10.1°, although without changing its bearing and velocity. Gradually the angular distance between these two objects, is changing in the range of Δθ=0°÷4°.  In order to estimate angular coordinates of these objects a radar system with electronically controlled antenna (nonrotating mechanically) has been used. This linear array is composed of  N=32 radiating/receiving elements. The distance between two adjacent elements is equal to Δ=0.7λ, where λ is a length of the received electromagnetic wave. It is also assumed that the radar system performs signal processing using the MTD (moving target detection) algorithm, [2, 9]. The processing is performed on the signal composed of 24 pulses grouped in three 8-pulse bursts which differ by their pulse repetition interval, Figure 3. The number of pulses emitted by the radar in given direction is smaller than the number of elements of the array. In order to reduce an influence of inaccurate estimate of correlation matrix R̂Z on estimation of angular coordinates, the appropriate “diagonal loading” has been used (22) (22)R̂Zdiag-load=1Nret∑k=1Nretz(g+k⋅G)z*(g+k⋅G)+4σ2⋅I,
where σ2is the variance of noise in elements of the array and I is the identity matrix.Figure 3
A structure of a radar signal used in the MTD processing.The signals received by the antenna array are used to form the sum and difference antenna patterns. Thus, the angular coordinates of objects being detected can be evaluated by means of   amplitude monopulse method.The sampless(n) of echoes received by the sum beam Σ are subjected to coherent integration with the use of several sets of weight coefficients. The so-understood integration is called Doppler filtering, because it allows the discrimination of signals, which originate from objects moving with different radial velocities with respect to the radar. The signal obtained as a result of the so-understood integration is described by the general relationship:
(23)sdoppl(g)=∑m=18wms(g+m⋅G),
where sdoppl(g) is the sample of signal in a given Doppler filter, s(g) is the sample of the pulse return received by sum antenna beam, g is the index of the range gate which determines the distance between the object and the radar, G is the number of range gates, m is the index of a pulse within the pulse sequence called “burst”, and wm is appropriately chosen weight coefficient. After detection, that is, after distinguishing the range gates in which the level of detected signal exceeds a decision threshold, the radar system performs an estimation of angular coordinates. Naturally, it can be performed by using the conventional amplitude monopulse method. Moreover, the samples of signals appearing at sum and difference antenna outputs can be additionally filtered by the same Doppler filter. The more efficient solution uses the ESPRIT and TLS ESPRIT methods which both use the complex samples of the signal received by the 32-element antenna array; the samples form the vector x(n). The samples are obtained at outputs of matched filters MF, that is, before the beamforming network, see Figure 4.Figure 4
The functional scheme of an antenna unit including the digital beamformer. A/D: analog to digital converter, QPD: quadrature phase detector, MF: digital matching filter.An attempt to evaluate angular coordinates of two closely spaced objects by means of the conventional amplitude monopulse method was met with failure. This failure resulted from the fact that monopulse methods are not able to discriminate objects seen in the same antenna beam when these objects are located at a similar distance to the antenna. The angular resolution of angular coordinates of these objects is determined by 3 dB beam width of azimuth and elevation receive characteristics. Usually the second object, for instance, an aircraft causes errors of angular coordinates of the first one when it is seen within the same antenna beam. This effect can be partially suppressed using the additional Doppler filtering. It allows the attenuation of signals, which sustain another frequency shift due to Doppler effect. It is well known that a change of angular position of an object causes a change of the corresponding radial velocity and Doppler frequency shift related to it. Unfortunately, at the higher radial velocities, the normalized Doppler frequencies of echoes of these objects may be similar and cause errors of estimation of angular coordinates. This effect is illustrated in Figure5. Thus, Doppler filtering can be used to increase signal-to-interference ratio when a radar receives an echo reflected from a single object together with interferences (weather clutter, ground clutter, etc.). However; it is not an appropriate approach to the resolution of two closely spaced objects. When these objects are moving with relatively small velocities (e.g., v=100 m/s), their Doppler frequencies, normalized with respect to pulse repetition frequency (PRF) fd-norm=fd%Fp, are different, but they may appear in 3 dB passband of the same Doppler filter. In the opposite situation, when two objects are moving with high speed (e.g., v=1000 m/s) and PRF (e.g., Fp=1/Tp=250 Hz) is small in comparison to their Doppler frequencies, the normalized Doppler frequencies of corresponding echoes may be similar or equal. The errors caused by this effect are illustrated in Figure 5.Figure 5
Histogram of polar  coordinates evaluated by means of the amplitude monopulse method aided by the coherent Doppler filtration,v=1000 m/s, Tp=4 ms, andS/Nnoise=30 dB.For comparison, Figure6 presents the diagrams of the estimates of the angular coordinates of two closely spaced objects found using the TLS ESPRIT method.Figure 6
Histogram of polar coordinates evaluated by using the  TLS ESPRIT  method for the   plane  velocityv=1000 m/s, Tp=4 ms, and S/Nnoise=30 dB.


## 4. The Preprocessing Preceding the TLS ESPRIT Method

The results illustrated in Figure6 have been obtained using the TLS ESPRIT method for signal samples recorded at outputs of the 24-element equivalent (virtual) antenna array. This “virtual” antenna array has been created in the manner described below, known as preprocessing. The samples of signals xi(n), where 1≤i≤32, received by the real 32-element antenna array, are used to compute the following samples:
(24)xk(e)(n)=∑i=08wixk+i(n)9,for1≤k≤24.
Samples (24) correspond to particular elements of the 24-element antenna array being formed. According to Figure 1, subarray X includes 23 “virtual” elements, at the outputs of which, samples xk(e)(n) (1≤k≤23) are evaluated. Similarly, samples xk(e)(n)2≤k≤24 are evaluated at the outputs of 23 “virtual” elements of subarray Y. Such preprocessing scheme allows to improve the TLS ESPRIT method applied for weaker signals, at S/Nnoise<6 dB.  The signal to noise ratio S/Nnoise has been determined at outputs of the matched filters MF, see Figure 4. The signal to interference (jamming) ratio S/Njamm has been determined at inputs of matched filters (MF). It is justified by fact, that jamming signals may be correlated to emitted chirp modulation pulse.


## 5. Estimation of Angular Coordinates in the  Presence of Interferences

An important advantage of the TLS ESPRIT method is its high immunity against interference. To illustrate this property, let us assume that a point-like interference source is located in the direction represented by angleθjamm=1.4°. It has been assumed also that the surface power density of the jamming signal is 60 dB higher than the surface power density of both useful signals. In the presence of such interference, conventional radar systems (with the receive patterns nonadapting to the interference) will be jammed. An effective way of suppressing the influence of interference is to use the adaptive antenna with digitally formed receive patterns. Such receive patterns should have deep minima (maximum attenuation) at directions from which the strong interfering signals are received. The corresponding results of simulations are summarized in Table 1.Table 1
The angular resolution and mean-square error of angular coordinates evaluated by means of  the TLS  ESPRIT  method.
Angular resolution, degrees (estimation error<0.23°)Maximum value of estimation error,  degreesS/Nnoise (dB)S/Njamm (dB)0.8250.1990—0.850.2090−300.90.2070−600.3750.1716—0.40.2026−300.4750.2266−600.3250.22610—0.350.16710−300.350.17110−600.20.17620—0.20.14720−300.20.18020−600.1250.11430—0.150.09630−300.1250.11430−60


## 6. Reducing the Errors of Angular Coordinates by Using Bursts with Variable Pulse Repetition Intervals

The presence of the second object within the antenna beam causes errors of angular coordinates when both objects have similar or equal normalized Doppler frequencies. Fortunately, the TLS ESPRIT method under discussion is able to discriminate both objects and sources of interference when a variable pulse repetition interval is applied, see Figure3. This property is confirmed by estimates of angles evaluated by using the TLS ESPRIT method. The corresponding results are illustrated in Figures 8, 10, and 12. All these results have been obtained by using the initial preprocessing described in Section 4. Simulations have been carried out for S/Nnoise=0 dB, S/Nnoise=6 dB, and  S/Nnoise=30 dB, respectively. For comparison, the equivalent results, obtained by means of the ESPRIT method, are illustrated in Figures 7, 9, and 11, [7].Figure 7
Histogram of polar coordinates evaluated by using the  ESPRIT  method  atS/Nnoise=0 dB and S/Njamm=-60 dB.Figure 8
Histogram of polar coordinates evaluated by using the  TLS ESPRIT  method  atS/Nnoise=0dB and S/Njamm=-60dB.Figure 9
Histogram of polar coordinates evaluated by using the ESPRIT method atS/Nnoise=6 dB and S/Njamm=-60 dB.Figure 10
Histogram of polar coordinates evaluated by using the TLS ESPRIT  method  atS/Nnoise=6 dB and S/Njamm=-60 dB.Figure 11
Histogram of polar coordinates evaluated by using the ESPRIT method atS/Nnoise=30 dB and  S/Njamm=-60 dB.Figure 12
Histogram of polar coordinates evaluated by using the TLS ESPRIT method atS/Nnoise=30 dB and  S/Njamm=-60 dB.From the analysis of the diagrams obtained by means of the ESPRIT and the TLS ESPRIT method, it results that TLS ESPRIT offers better accuracy and higher resolution. The corresponding values of mean square errors for angular coordinates are given in Table1.Results presented in Figures5–12 and Table 1 have been obtained for bursts with variable pulse repetition intervals. For comparison, Figure 13 presents mean square errors for angular coordinates occurring when bursts with the same pulse repetition interval are used and when the pulse repetition frequency is small (Tp=4 ms) versus the Doppler frequency of signals reflected from objects moving with high velocity (v=1000 m/s). The maximum of error observed in Figure 13 occurs when normalized Doppler frequencies of both objects are equal.Figure 13
Histogram of mean square error of polar  coordinates evaluated by using the TLS ESPRIT  method  atS/Nnoise=10 dB, in absence of an external noise signal and a constant repetition time of  radar pulses, Tp=4 ms, v=1000 m/s.The comparison of results obtained by the TLS ESPRIT for signal samples from the 32-element antenna array to corresponding results obtained for signal samples formulated for equivalent “virtual” 24-element antenna array states that the use of preprocessing increases efficiency of the TLS ESPRIT for weaker signals, for which theS/Nnoise<6 dB. The results presented in Figures 14 and 15 justify this conclusion.Figure 14
Histogram of polar  coordinates evaluated by means of  the TLS ESPRIT method atS/Nnoise=0 dB, S/Njamm=-60 dB  and without initial preprocessing.Figure 15
Histogram of polar  coordinates evaluated by means of the  TLS ESPRIT method atS/Nnoise=6 dB, S/Njamm=-60 dB,  and without initial preprocessing.From the analysis performed above, results that proposed additional preprocessing significantly improve the effectiveness of the TLS ESPRIT method in application under consideration.


## 7. Conclusions

The results of numerous computer simulations illustrated in Figures6, 8, 10, and 12 confirm the applicability of the TLS ESPRIT method to solve different variants of the problem under consideration. The comparison of these results with their correspondents obtained by using the ESPRIT method, see Figures 7, 9, and 11, indicates that the TLS ESPRIT method offers better angular resolution and more reliable estimation of angular coordinates of two moving objects, especially when weak echoes are being received in the presence of strong jamming signal. Another, also important, result of the performed simulations is the confirmation of the usefulness of the proposed signal preprocessing, the essence of which is expressed by formula (24). From the obtained results, see Figures 8, 10, 14, and 15, it follows that the proposed preprocessing contributes to increasing the accuracy of angular coordinates and consequently makes the TLS ESPRIT method more effective, naturally in the application under consideration. In other words, the preprocessing increases the effectiveness of the TLS ESPRIT method especially for the weaker echoes of useful signals. In the first approximation, this essential improvement can be explained in the following way: as a result of the addition of signals xi(n), see (24), the uncorrelated noise waveforms are summed. Thereby, in elements of antenna array, increasing of a signal-to-noise ratio is observed. In this paper, an emphasis has been put on the fact that the effective use of the TLS ESPRIT method requires an appropriate method of scanning. When scanning signals with identical pulse repetition intervals in all three bursts used, see Figure 3, significant errors may occur for some angle locations. This effect is illustrated by the simulation results shown in Figure 13. This parasitic effect can be reduced by changing pulse repetition interval in subsequent bursts, see Figure 3. Here, it deserves noting that bursts with variable pulse repetition intervals are widely used in order to avoid errors related to the so-called “blind speeds” [2, 9].



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*Source: 548953-2011-01-24.xml*