Direction Finding Technique in Radar Array Signal Processing

Abstract

This invention describes a new Direction Finding (DF) algorithm named as Braided Array Sampling via an Inter-Channel Scheme (BASICS) that can enhance estimation accuracy of the direction of arrival (DOA) to a higher level than existing algorithms. It is originally developed from, and designed for high frequency (HF) radars detecting sea echoes. With appropriate analogical reasoning, it can be applied to all kinds of radars and sonars. It breaks the ordinary belief that an array of N sonars can only generate N pictures of spectral for analysis. Without the need of improvement on the hardware, BASICS assumes virtual movements of sonars in order to produce much more than N spectral for computers to analyze, and therefore provides much more accurate DOA estimation of targets on the sea. This invention presents the principle of BASICS and its theoretical supports, as well as the basic conditions to apply BASICS.

Description

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BACKGROUND OF THE INVENTION

This invention is in the field of radar signal processing. People use an array of radars to obtain the information of location and vector velocity of objects they want to detect. By using multiple antennas located in a compact area, the bearing of objects can be extracted from the phase and amplitude differences of signals received by each antenna. This is the prerequisite of Direction Finding (DF) algorithms. A DF algorithm is a process executed by computers in order to estimate the bearing of targets detected by radars. The bearing is usually called Degree of Arrival (DOA). DOA estimation capability is the key concern in array design and signal processing. There are two categories of DOA estimation techniques, Beam Forming (BF) and Direction Finding (DF). The BF techniques are widely applied in radar systems where analog or digital controlled transmitting/receiving beams are steered to find targets' bearings. J. Capon in 1969 proposed a BF algorithm called Minimum Variance Method (MVM), in which an adaptive beam is formed to minimize the power output of the array while keep the array response to the designated direction as constant. This method is considered as the first to obtain higher resolution than Rayleigh resolution limit of an array. The accuracy of BF detecting results depends on array aperture size, and higher accuracy requires larger array deployment. In cases of limited space for array occupation, DF techniques are applied to estimate DOAs for sparsely distributed targets. DF algorithms exploit the phase and amplitude differences of signals received by each sonar in an aperture limited array. A simple and conventional method of DF technique is to use two orthogonal loop sonars, taking the ratio of signals received by each, then using the arctangent function to extract the single target's DOA. It often fails when there exist more than one targets or when sonar patterns are heavily distorted. Many studies have been carried out on sub-space based DF methods like MUltiple SIgnal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT), etc. They have higher angular resolution than conventional BF methods. MUSIC algorithm by Schmidt applies eigenspace method to the correlation matrix of array signals, so that N sonars can generate up to N−1 signal bearings and a noise signal. It outperforms conventional DOA algorithms since it takes advantage of the orthogonality between signal subspaces and those of noises. An important prerequisite of MUSIC is that the number of targets should be known in advance, which is also the main drawback of MUSIC. By building a relation between Vandermonde matrix of sinusoids and covariance matrix of measured data, ESPRIT derives a matrix containing rotational information with respect to DOA information of targets, then they can be obtained directly from immediate matrix calculations.

The methods mentioned above are suitable for fixed-located sonars in an array. Modern Doppler radars adopt coherent cumulation to obtain spectra of moving targets. The coherent cumulative time lasts from seconds to hundreds of seconds depending on the stationarity of the target's echoes. This letter tries to utilize the cumulative time as a key factor in array signal processing to estimate DOA information, i.e., by letting the sonar in an array “moving” in deliberate manners, we can expand the DOA estimation problem from pure space domain to time-space domain. The additional Doppler shifts caused by the sonar movement would help enhance the accuracy of DOA estimation.

There are real-life examples of arrays of moving sonars, but constructing such an array costs much more than stationary sonars and is hard to achieve. This invention presents a technique called Braided Array Sampling via an Inter-Channel Scheme (BASICS) that can randomly braid inter-channel signals among stationary sonars in order to generate the same effect as moving sonar. By this means, BASICS can take the advantage of the high accuracy of DOA estimation of the moving sonars without actually building them.

SUMMARY OF THE INVENTION

This invention is in the field of radar signal processing. This invention presents a technique called Braided Array Sampling via an Inter-Channel Scheme (BASICS) that can randomly braid inter-channel signals among stationary sonars in order to generate the same effect as moving sonar. By this means, BASICS can take the advantage of the high accuracy of DOA estimation of the moving sonars without actually building them. This invention presents the principle of BASICS and its theoretical supports, as well as the basic conditions to apply BASICS.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1: Flow chart of traditional array signal processing

FIG. 2: An example of braided array signal sampling in BASICS

DETAILED DESCRIPTION OF THE INVENTION

1. Signal Model

Given that P narrow-band far-field echo signals impinge on an M-sonar array from directions θ=[θ₁, θ₂, . . . , θ_P]^T, where (·)^T represents the transpose operation. The array output vector X(t) is described by

X(t)=AS(t)+N(t),t=1,2, . . . ,L  (1)

and S(t)=[s₁(t), s₂(t), . . . , s_P(t)]^T denotes the signal vector with zero mean. N(t) represents the additive noise vector with zero mean and σ_n2 variance, which is supposed to be temporally and spatially white. L is the sampling points number within the cumulative time period. Furthermore, we suppose that the first sonar of the array is the reference sonar and its location is fixed to the origin of co-ordinates.

A is the array manifold matrix, which is given by

A=[a(θ₁),a(θ₂), . . . ,a(θ_p), . . . ,a(θ_P)]  (2)

where

a(θ_p)=[e^jω0t,e^{j(ω0t−kp·r2)}, . . . ,e^{j(ω0t−kp·rm)}, . . . ,e^{j(ω0t−kp·rM)}]^T  (3)

and ω₀ is the center angular frequency of the signals, k_p is the wave vector of pth signal and r_m is the radial vector from the first sonar to the mth sonar, and

$\begin{array}{cc}{k}_p·r_m=\frac{2\pi }{{\lambda }_0}{d}_{\mathrm{mp}}={\begin{array}{cc}[x_m& y_m]\end{array}\left[\begin{array}{cc}\sin {\theta }_p& \cos {\theta }_p\end{array}\right]}^T& \left(4\right)\end{array}$

in which λ₀ denotes the center wavelength of the signals. (x_m, y_m) are co-ordinates of the mth sonar and d_mp is the spatial distance between the mth sonar and the first sonar in the direction of the pth signal.

In traditional array signal processing problems, the sonars are immobile and the e^jω0tt in (3) is the same for all sonar terms thus can be replaced by 1.

In Doppler radar system, a Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT) operation is often imposed on the output time series of each sonar channel to obtain the Doppler spectra of received echoes. For the signal model of (1), its Doppler spectra is expressed as

X(ω_d)=AS(ω_d)+N(ω_d)  (5)

where ω_d is the echo's Doppler shift from ω₀, and the manifold matrix A keeps the same form as in (2) but with the e^jω0t factor disappeared as a result of base band filtering in receiver process. FIG. 1 depicts the flow chart of traditional array signal processing.

Now suppose another case where there is only one moving sonar that receiving echoes from those P narrow-band far-field targets. The sonar's output x(t) is

x(t)=Σ_p=1^Pe^{j[ω0t−kp·r(t)]}s_p(t)+N(t),t=1,2, . . . ,L  (6)

where r(t) is the radial vector from the co-ordinates' origin to the sonar position at moment t.

For the signal model of (6), after the DFT/FFT operation the output Doppler spectra will be the convolution between s_p(ω_d) and [e^−jkp·r(t)], the FFT spectra of e^−jkp·r(t), i. e.

x(ω′_d)=Σ_p=1^P{[e^−jkp·r(t)]*s_p(ω_d)}|_ω′d+N(ω′_d)  (7)

where * denotes convolution operation, and ω′_d stands for the combined Doppler shift containing contributions both from echo's ω_d and the equivalent Doppler shift caused by sonar movement. Equation (7) suggests that if the sonar are designed to move along R different routines, then we will obtain R possibly different Doppler spectra, from which the DOA information in k_p may be extracted in a definite manner by solving a system of linear equations, for R can be designed to be much larger than the sonar number Min equation (1).

2. Principle of BASICS

Now we manage to generate virtual movements of an sonar from an array of immobile sonars, so that the DOA estimation advantages described in equation (7) can be exploited. Because a radar system samples signals at discrete moments, a moving sonar appears at different locations at different sampling moments. When two consecutive samples appear, one cannot tell whether they are the outputs from two separately located immobile sonars at consecutive sampling moments, or they are the outputs of a moving sonar which locates just exactly at positions of that two sonars at the consecutive moments. Thus we can generate equivalent moving sonar samplings from those of an array of immobile sonars.

FIG. 2 demonstrates an example of how to engender time series of a moving sonar from the samplings of an immobile M sonar array. We call this procedure “Braided Sampling,” because rather than analyzing each sonar's data separately, we “braid” them together to generate much more pictures for analysis. For a L points time series, there are R=M^L different sampling braiding routines theoretically. After DFT/FFT of those time series, we will obtain many Doppler spectra in which the DOA information are convoluted by sonar movements as depicted in equation (7).

In practice, there are not as many as M^L braided sampling routines that can be used for DOA estimation. Because of the existence of noises, many slightly different routines makes almost no significant differences in their Doppler spectra and cannot be of help in calculations. The most important reason is the constraint cast by Nyquist-Shannon Sampling Theorem, the combination of virtual movement velocity and the target's real velocity should not exceed the allowed maximum velocity corresponding to the sample interval time.

If no movement is assumed on sonar, then by Nyquist-Shannon Sampling Theorem, we only need to make sure that the Doppler Shift of targets, ω_d, is less than f_s/4π, where f_s is the radar sample rate. However, since the sonar is “moving” in our case, we have to consider to enhance f_s to satisfy Nyquist-Shannon Sampling Theorem.

Note that

$\begin{array}{cc}{e}^{-{\mathrm{jk}}_p·r\left(t\right)}=e^{-j\left[k_p·\frac{r\left(t\right)}{t}\right]t}=e^{-j\left[k_p·\stackrel{\_}{v}\left(t\right)\right]t}& \left(8\right)\end{array}$

where

$\stackrel{\_}{v}\left(t\right)=\frac{r\left(t\right)}{t}$

is the mean velocity at moment t averaged from sample start, and its direction is from co-ordinate origin pointing to the sonar's current position. The k_p·v(t) has the dimension of angular frequency and can be denoted as ω_v. Then we let [e^−jkp·r(t)]=f_p(ω_v) equation (7) changes to

x(ω′_d)=Σ_p=1^P[f_p(ω_v)*s_p(ω_d)]|_ω′d+N(ω′_d)=Σ_p=1^P∫_−∞^∞f_p(ω)s_p(ω′_d−ω)dω+N(ω′_d)  (9)

From (8) ω_v can be positive or negative, suppose its range is [−ω_vmax, ω_vmax]. And target's Doppler range is [0, ω_dmax]. According to definition of convolution, the od in (7) and (9) should be within the range of [−ω_vmax, ω_vmax+ω_dmax]. The Nyquist-Shannon Sampling Theorem requires that the f_s of a real number sampling should satisfy

2πf_s>2(2ω_vmax+ω_dmax)  (10)

In our scheme of engendering virtual movement from immobile sonars array, enhancing f_s means transition interval between sonars is reduced and the ω_vmax may also enhances as well. To keep ω_vmax not enhance as f_s does, the virtual routines as shown in FIG. 2 should not contain steps of jumping between sonars located far apart. And in the same time, the distances between sonars should be reduced somehow. Thus the constraint of (10) suggests the array aperture, configuration, distances between sonars in the array should be deliberately designed as well as the virtual routines design.

The above constraints are not so harassing because in (8) as t becomes larger and larger, the ω_v from virtual movement will decrease linearly. The largest ω_vs normally come from initial steps of virtual movement.

3. Algorithm of BASICS

As discussed in the above sections, BASICS obtain DOA information in Doppler domain. A L points time series corresponds to the same point Doppler spectra. For an M-element array, if we design R different virtual routines, we can get RL linear equations from (9). The unknowns of the equation system are Doppler spectra in every bearing cell. If bearing cells number is D, Doppler spectra point number in every direction cell is L′(L′ normally less than L as discussed in the context of formula (10)), then the total unknowns number is DL′, which is much less than RL. By solving the equation system, we will obtain the Doppler spectra result in every bearing cell, and directly get the DOA information of possible targets.

To sum up, BASICS can be addressed as the following process:

1. Randomly generate an ordered list of M elements {i₁, i₂, . . . , i_l, . . . , i_L}, with each element be in {r_m|m∈{1, 2, . . . , M}}. Each list represents a virtual movement of a single sonar such that it is at r_il when t=1.

2. Check if this movement satisfies the requirement given in inequality (10).

3. If false, regenerate a list; if true, make a time series of L samples corresponding to this movement.

4. Do DFT/FFT on the time series in 3, generating a spectra consists of L points, which corresponds to the left side of equation (9).

5. Discretizing the integral at the right side of equation (9) (by dividing the radar bearing scope into D cells, and L′ unknown spectra points in each cell), to get L linear equations for this movement.

6. Repeat Steps 1-5 to generate more groups of equations with L equations in each group, until group number R reaches ten or a hundred times than M.

7. Solve these RL equations to obtain the Doppler spectra result in every bearing cell.

4. Variations of BASICS Application

In the above sections, we introduce the principles of BASICS by engendering virtual movement from actually immobile sonars, that is a kind of frequency modulation, as described in FIG. 2 and equation (6)˜(8). In fact, the virtual movement, or to say the virtual variation or the artificial Doppler shifts can also be produced by amplitude modulation (switching between pattern-different sonars co-located at the same place), or simultaneously by frequency modulation and amplitude modulation (switching between pattern-different sonars located at different locations in an array). The signal model of (6) and (7) can be expanded as

x(t)=Σ_p=1^PA(θ,t)e^{j[ω0t−kp·r(t)]}s_p(t)+N(t),t=1,2, . . . ,L  (11)

and

x(ω′_d)=Σ_p=1^P{[A(θ,t)e^−jkp·r(t)]*s_p(ω_d)}|_ω′d+N(ω′_d)  (12)

where A(θ, t) depicts the variation of sonar pattern's changing with bearing and time. For this model, the above stated BASICS principle and algorithm can also be applied.

NON-PATENT CITATION

• 1. Capon, J. “High-resolution frequency-wavenumber spectrum analysis.” Proc. IEEE. Vol. 57, pp. 1408-1418, August, 1969.
• 2. Schmidt, R. O. “Multiple emitter location and signal parameter estimation.” IEEE Trans. sonar and Propagat. AP-34(3): 276-280. March, 1986.
• 3. Barrick, D. E., and B. J. Lipa. “Evolution of bearing determination in HF current mapping radars.” Oceanography. Vol. 10. No. 2, pp. 72-75. January, 1997.

Claims

1. A method for enhanced DOA estimation, comprising the steps of: Randomly generate a virtual movement of an imagined sonar from an array of immobile sonars, such that at every sampling moment, the movement arrives at the location of a certain real sonar of the array of sonars.The array configuration, distances between sonars, and radar sampling rate should satisfy the constraint described by 2πfs>2(2ωvmax+ωdmax), where fs is the sampling rate, ωvmax is the maximum artificial Doppler shift caused by virtual sonar movement or variation, ωdmax is the maximum Doppler shift of possible real targets.Save this movement if it satisfies the above requirement. Generate a frequency domain spectrum of this imagined sonar by braiding the sample received by every real sonar.List a series of equations based on the spectra we just got.Repeat all previous steps until we get enough virtual movements, and solve the equations generated by those movements or variations, to obtain the Doppler spectra result in every bearing cell.

2. The method of claim 1, further comprising the method of how to determine the spectrum of imagined sonar.

3. The method of claim 1, further comprising the proof of the required average speed limit of virtual movements.

4. The method of claim 1, with the virtual movement, or to say the virtual variation, or the artificial Doppler shifts which is the essential ideal of this patent, being produced by amplitude modulation, for example, switching between pattern-different sonars co-located at the same place.

5. The method of claim 1, with the virtual movement, or to say the virtual variation, or the artificial Doppler shifts which is the essential ideal of this patent, can also be produced simultaneously by frequency modulation and amplitude modulation, for example, switching between pattern-different sonars located at different locations in an array.