RADAR TARGET DETECTION METHOD BASED ON ESTIMATION BEFORE DETECTION

Abstract

The present invention provides a radar target detection method based on estimation before detection (EBD), which comprises: obtaining pre-detect targets (PDTs) based on conventional pulse-Doppler processing and pre-detection; estimating ranges and speeds of PDTs, i.e., performing parameter EBD; establishing a dimension-reduction observation model of a received signal based on PDTs and parameter thereof; reconstructing a target vector in the dimension-reduction observation model based on a sparse recovery algorithm; and designing a generalized likelihood ratio detector based on the reconstruction result for target detection. The method of the present invention can significantly reduce the radar signal processing loss, and the target detector used in the method has the constant false alarm rate (CFAR) property, so that the weak target detection performance can be greatly improved.

Claims

1. A radar target detection method based on estimation before detection (EBD), comprising: 1) performing pulse compression and coherent integration on received baseband signal to obtain a range-Doppler map, performing pre-detection based on the range-Doppler map to obtain interested pre-detect targets (PDTs), wherein the corresponding ranges and Doppler frequencies of cells, wherein the PDTs are pre-detected, are represented by r.sub.ζ and f.sub.ζ, respectively; 2) estimating ranges and the Doppler frequencies of the PDTs, wherein the estimates are represented by and ; 3) establishing a dimension-reduction observation model of a received signal based on and ; 4) reconstructing a target vector in the dimension-reduction observation model based on a sparse recovery algorithm; and 5) adopting a generalized likelihood ratio detector for target detection based on a reconstruction result and outputting target detection results and their parameters.

2. The radar target detection method based on the EBD according to claim 1, wherein the step 2) comprises: adopting and to represent true ranges and Doppler frequencies of the PDTs, respectively, and letting =[] and θ.sub.ζ=[r.sub.ζ;f.sub.ζ], wherein the symbol ; in the square brackets represents connecting two vectors, the received signal is represented as:
y=β+n  (7) y in the equation (7) represents a received signal of one coherent processing interval, wherein β is a dimension-reduction target vector, an i.sup.th element β.sub.i represents a true complex amplitude of an 2th PDT, Ais an observation matrix, and n is an additive white Gaussian noise vector; based on a maximum likelihood criterion, estimates of and β are given as: $\begin{array}{cc}\left\{,\hat{\beta }\right\}=\arg \underset{\theta ,\beta }{\min }{.Math.y-A\left(\theta \right)\beta .Math.}_2^2,& \left(8\right)\end{array}$ wherein θ=[r;f], a minimum over β is attained for:
{circumflex over (β)}=(A(θ).sup.HA(θ)).sup.−1A(θ).sup.Hy  (9) then, minimizing a cost function in (8) is equivalent to minimizing the function:
g(θ)=∥y−A(θ)(A(θ).sup.HA(θ)).sup.−1A(θ).sup.Hy∥.sub.2.sup.2,  (10) obviously, a minimum value of the equation (10) is obtained at θ=, the derivative of g(θ) at evaluated by the first-order Taylor series:
∇.sub.θg(θ)∇.sub.θg()+∇.sub.θ.sup.2g()(θ−),  (11) obviously, ∇.sub.θg()=0, then:
≈θ−(∇.sub.θ.sup.2g()).sup.−1(∇.sub.θg(θ))  (12) θ is replaced with θ.sub.ζ to obtain an estimate of :
≈0.sub.ζ−(∇.sub.θ.sup.2g(θ.sub.ζ)).sup.−1(∇.sub.θg(θ.sub.ζ)).  (13)

3. The radar target detection method based on the EBD according to claim 2, wherein a simplified method used for the estimate of comprises: according to equation (7), one has:
W.sub.iy=W.sub.i()β+W.sub.in  (14) wherein W.sub.i=diag(w.sub.i)=diag(w.sub.d .Math.w.sub.l,i), w.sub.d represents a normalization window function in slow time domain, represents a normalization window function in fast time domain of the i(i=1, 2, . . . , I).sup.th PDT, and I represents a number of the PDTs; in case of normalization, obviously, ∥w.sub.i∥.sub.2.sup.2=∥W.sub.i∥.sub.F=1, then the estimate of and β is represented based on a least squares criterion as: $\begin{array}{cc}\left\{{\hat{\theta }}_{\varrho },\hat{\beta }\right\}=\arg \underset{\theta ,\beta }{\min }{.Math.W_{i}y-W_{i}A\left(\theta \right)\beta .Math.}_2^2& \left(16\right)\end{array}$ g(θ) is further represented as:
g(θ)=∥W.sub.iy−β.sub.iW.sub.ia.sub.i(n.sub.i)∥.sub.2.sup.2+∥W.sub.iy−W.sub.iA.sub.\i(θ)β.sub.\i∥.sub.2.sup.2−(W.sub.iy).sup.HW.sub.iy  (17) wherein n.sub.i=[r.sub.i,f.sub.i].sup.T, r.sub.i and f.sub.i represent the range and the Doppler, respectively, a.sub.i represents a steering vector of the i.sup.th PDT, A.sub.\i denotes the matrix obtained from A by deleting the i.sup.th column and β.sub.\i denotes the vector obtained from β by deleting the i.sup.th entry β, to minimizing g(θ), u.sub.i(η.sub.i)=∥ W.sub.iy−β.sub.iW.sub.ia.sub.i(η.sub.i)∥.sub.2.sup.2 should attain its minimum, then, the estimate of the i.sup.th PDT, denoted by =[].sup.T, can be obtained by minimizing u.sub.i(η.sub.i)u.sub.i(η.sub.i); minimizing u.sub.i (η.sub.i), the estimate of β.sub.i is given by:
{circumflex over (β)}=a.sub.i.sup.H(η.sub.i)W.sub.i.sup.2y,  (18) inserting equation (18) into u.sub.i (η.sub.i), then, minimizing u.sub.i(η.sub.i) is equivalent to minimizing the following equation:
z.sub.i(η.sub.i)=(a.sub.i.sup.H(η.sub.i)W.sub.i.sup.2y)(y.sup.HW.sub.i.sup.2a.sub.i(η.sub.i))  (19) and referring to equation (13), the estimate of is given by: $\begin{array}{cc}\left(\begin{array}{c}{\hat{r}}_{\varrho ,i}\\ {\hat{f}}_{\varrho ,i}\end{array}\right)\approx \left(\begin{array}{c}{r}_{ϛ,i}\\ f_{ϛ,i}\end{array}\right)-{\left(\begin{array}{cc}\frac{{\partial }^2{z}_i}{\partial r_i^2}& 0\\ 0& \frac{{\partial }^2{z}_i}{\partial f_i^2}\end{array}\right)}_{\left({\eta }_i={\eta }_{ϛ,i}\right)}^{-1}{\left(\begin{array}{c}\frac{\partial z_i}{\partial r_i}\\ \frac{\partial z_i}{\partial f_i}\end{array}\right)}_{\left({\eta }_i={\eta }_{ϛ,i}\right)},& \left(20\right)\end{array}$ wherein η.sub.ζ,i=[r.sub.ζ,if.sub.ζ, i].sup.T; the estimate is iteratively updated using the following equation:
=−(∇.sub.η.sup.2z.sub.i()).sup.−1∇.sub.ηz.sub.i(),  (21) wherein =η.sub.ζ,i and t is a number of iterations.

4. The radar target detection method based on the EBD according to claim 2, wherein the step 3) comprises: obtaining based on ; and representing the received signal as y≈β+n, i.e., the dimension-reduction observation model, based on the estimated .

5. The radar target detection method based on the EBD according to claim 2, wherein the target detector in the step 5) is: $.Math.{\tilde{\beta }}_i.Math.\begin{array}{c}\\ \end{array}\begin{array}{c}{H}_1\\ >\\ H_0\\ <\end{array}\gamma \tilde{\sigma },$ wherein γ is a detection threshold, σ is the noise level in β.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] FIG. 1 is a flowchart of the radar target detection method based on EBD according to the present invention;

[0017] FIG. 2A to FIG. 2C are schematic diagrams of the estimation accuracy of the PDT parameters provided by an embodiment of the present invention;

[0018] FIG. 3A to FIG. 3C are schematic diagrams of the statistical characteristics of the reconstructed output noise based on a dimension-reduction model according to an embodiment of the present invention;

[0019] FIG. 4A and FIG. 4B are comparison diagrams of the target detection performances of 4 detectors under simulation conditions according to an embodiment of the present invention;

[0020] FIG. 5A and FIG. 5B are schematic diagrams of the range-Doppler map of the measured data provided by an embodiment of the present invention; and

[0021] FIG. 6 is a schematic diagram showing the comparison between the performance of the method of the present invention and that of the conventional method using measured data.

DETAILED DESCRIPTION

[0022] The technical scheme of the present invention is further described below with reference to the drawings.

[0023] The present invention is applicable to pulse radar system adopting linear frequency modulation waveform, and a target detection processing process is shown as FIG. 1 and comprises:

[0024] Step 1) performing conventional pulse compression and coherent integration on received baseband signal to obtain a range-Doppler map, and performing pre-detection based on the range-Doppler map to obtain interested PDTs.

[0025] Since the signal model in spatial domain is similar to the slow time domain, the present invention considers the fast time domain and the slow time domain. Assuming that a number of range cells in one coherent processing interval is L, a number of pulses is K, and the received signal can be represented as a matrix Y∈.sup.L×K. Neglecting the range and Doppler migration, the elements thereof can be represented as:

[00001]$\begin{array}{cc}{y}_{l,k}=\underset{p=0}{\overset{P-1}{.Math.}}{\alpha }_p\mathrm{rect}\left(\frac{l{T}_s-2r_p/c}{T_{\mathrm{pul}}}\right)\exp \left(j{\mathrm{\pi \mu }\left(l{T}_s-2r_p/c\right)}^2+j2\pi f_{d,p}k{T}_I\right)+n_{l,k},& \left(1\right)\end{array}$

[0026] wherein y.sup.l,k is an l.sup.th range cell and a k.sup.th pulse echo signal, 0≤k≤K−1, and 0≤1≤L−1. P is a number of the target, T.sub.s is a sampling time interval, α.sup.p is a p.sup.th target amplitude, r.sub.p is a target range, T.sub.pul is a pulse duration, T.sub.I is a pulse interval, μ is a frequency modulation slope, and c is light speed. f.sub.d,p=2v.sub.p/λ is Doppler shift, up is a target speed, and λ is a carrier wavelength. n.sub.l,k is additive white Gaussian noise. rect(.) represents a rectangular function.

[0027] Pulse compression is performed in fast time domain and coherent integration is performed in slow time domain to obtain a range-Doppler map. Then, based on the range-Doppler map, a conventional CFAR detection method is adopted for target pre-detection to obtain interested PDTs, wherein the corresponding ranges and Doppler frequencies of cells, wherein PDTs are pre-detected, are represented by r.sub.ζ and f.sub.ζ, respectively.

[0028] Step 2) estimating the PDT range and the Doppler parameter obtained by pre-detection to obtain estimated values and ;

[0029] assuming that the number of the PDTs is I, and the corresponding range r.sub.ζ and Doppler frequency f.sub.ζ can be calculated according to the range cell and the frequency cell wherein the I PDTs are located, respectively.

[0030] The PDTs comprise the main components of the received signal, thus, based on I, r.sub.ζ and f.sub.ζ, the received signal can be approximately represented as:

y≈A.sub.ζβ+n,  (2)

wherein Y represents the received signal of one coherent processing interval, and is obtained by vectorizing Y through stacking the columns into a vector; β is a dimension-reduction target vector, an i.sup.th element β.sub.i thereof represents the true complex amplitude of an i.sup.th PDT; and if the i.sup.th PDT is a false alarm, then β.sub.i=0. n is an additive white Gaussian noise vector. A.sub.ζ∈.sup.LK×I is an approximate dimension-reduction observation matrix, and an i.sup.th column thereof is represented as:

a.sub.ζ,i=s.sub.d(f.sub.ζ,i).Math.s.sub.i(r.sub.ζ,i),  (3)

wherein the symbol .Math. represents Kronecker product, s.sub.d(f.sub.ζ,i) represents the Doppler domain steering vector corresponding to the PDT with a Doppler frequency of f.sub.ζ,i, and s.sub.i(r.sub.ζ,i) represents the fast-time domain steering vector corresponding to the PDT with a range of r.sub.ζ,i, s.sub.d (f.sub.ζ,i) and s.sub.i (r.sub.ζ,i) are represented respectively as:

s.sub.d(f.sub.ζ,i)=[1, . . . ,exp(j2πf.sub.ζ,ikT.sub.I), . . . ,exp(j2πf.sub.ζ,i(K−1)T.sub.I)].sup.T,  (4)

s.sub.i(r.sub.ζ,i)=[q.sub.i(0), . . . ,q.sub.i(l), . . . ,q.sub.i(L−1)].sup.T  (5)

wherein the superscript T represents transpose;
wherein,

[00002]$\begin{array}{cc}{q}_i\left(1\right)=\mathrm{rect}\left(\frac{{\mathrm{lT}}_s-2r_{ϛ,i}/c}{T_{\mathrm{pul}}}\right)\exp \left(j{\mathrm{\pi \mu }\left(l{T}_s-2r_{ϛ,i}/c\right)}^2\right).& \left(6\right)\end{array}$

[0031] Generally, β is still sparse, and reconstruction of β can be performed based on sparse recovery; assuming that the reconstruction result is represented as {circumflex over (β)}, then target detection can be realized based on {circumflex over (β)}. However, in practical applications, the PDTs are not located in integer cells, that is, r.sub.ζ and f.sub.ζ deviate from the true PDT values, and the reconstruction directly based on the equation (2) is faced with the off-grid problem. In this regard, the present invention firstly estimates the true parameters and of the PDTs based on r.sub.ζ and f.sub.ζ, and then performs the target reconstruction and detection.

[0032] Assuming that the true observation matrix corresponding to the PDTs is , ignoring the effect of missed detection targets during pre-detection, the received signal can be represented as:

y=β+n  (7)

wherein =[], the symbol; is used for connecting two vectors to form a vector. An i.sup.th column of Ais =s.sub.d() .Math.. is unknown, however, is obviously very close to A.sub.ζ, that is, the true target parameters are close to parameters corresponding to the integer cells wherein the PDT are located. Let =[r.sub.ζ: f.sub.ζ], r.sub.ζ and f.sub.ζ are known. Since is very close to A.sub.ζ, then is also very close to θ.sub.ζ, i.e., ∥−θ.sub.ζ∥ is very small. Therefore, in the present invention, it is considered to estimate based on θ.sub.ζ. Then, the observation matrix can be obtained. The specific procedures are as follows:

[0033] based on a maximum likelihood criterion, estimates of and β are given as:

[00003]$\begin{array}{cc}\left\{,\hat{\beta }\right\}=\arg \underset{\theta ,\beta }{\min }{.Math.y-A\left(\theta \right)\beta .Math.}_2^2,& \left(8\right)\end{array}$

[0034] wherein θ=[r; f], r and f represent the range and the Doppler frequency, respectively; the minimum over β is attained for:

{circumflex over (β)}=(A(θ).sup.HA(θ)).sup.−1A(θ).sup.Hy  (9)

then the cost function in the equation (8) can be further represented as:

g(θ)=∥y−A(θ)(A(θ).sup.HA(θ)).sup.−1A(θ).sup.Hy∥.sub.2.sup.2  (10)

[0035] Obviously, a minimum value of the equation (10) is obtained at θ=. A first-order Taylor approximation is performed on a first-order derivative of g(θ) at to obtain:

∇.sub.θg(θ)≈∇.sub.θg()+∇.sub.θ.sup.2g()(θ−),  (11)

[0036] Obviously, ∇.sub.θ g()=0, then,

≈θ−(∇.sub.θ.sup.2g()).sup.−1(∇.sub.θg(θ)).  (12)

[0037] Because is also very close to θ.sub.ζ, θ is substituted with θ.sub.ζ, and in the Hessian matrix (∇.sub.θ.sup.2g()) is substituted with θ.sub.ζ to obtain the estimate of :

≈θ.sub.ζ−(∇.sub.θ.sup.2g(θ.sub.ζ)).sup.−1(∇.sub.θg(θ.sub.ζ)).  (13)

[0038] In the equation (13), the first-order derivative vector and the second-order derivative matrix of g(θ) are required to be calculated, and the calculation is very complicated; therefore, a simplified solution method is further given below.

[0039] The following equation can be obtained from the equation (7):

W.sub.iy=W.sub.iβ+W.sub.in  (14)

wherein W.sub.i=diag(w.sub.i)=diag(w.sub.d.Math.w.sub.l,i), w.sub.d represents a normalized window function in the slow time domain, w.sub.l,i represents a normalized window function in the fast time domain of the i.sup.th PDT, w.sub.l,i=[q.sub.w,0, . . . , q.sub.w,l, . . . , q.sub.w,L−1].sup.T, and

[00004]$\begin{array}{cc}{q}_{w,l}=\mathrm{rcct}\left(\frac{l{T}_s-2r_{ϛ,i}/c}{T_{\mathrm{pul}}}\right)w_c\left(l{T}_s-2r_{ϛ,i}/c\right).& \left(15\right)\end{array}$

w.sub.c(t) is a continuous form of the window function in a time T.sub.pul. In the case of normalization, obviously, ∥w.sub.i∥.sub.2.sup.2=∥W.sub.i∥.sub.F=1, wherein the subscript F represents the Frobenius norm of the matrix. Then, based on the least squares criterion, the estimates of and β is:

[00005]$\begin{array}{cc}\left\{,\hat{\beta }\right\}=\arg \underset{\theta ,\beta }{\min }{.Math.W_{i}y-W_{i}A\left(\theta \right)\beta .Math.}_2^2.& \left(16\right)\end{array}$

[0040] The equation (14) actually weights the data in the fast and slow time domains, and the weighting can be used to decouple the i.sup.th PDT from other PDTs in the echo, that is, the i.sup.th PDT and other PDTs have approximately no mutual influence. Then, g(θ) can be further approximately represented as:

g(θ)=∥W.sub.iy−B.sub.iW.sub.ia.sub.i(η.sub.i)∥.sub.2.sup.2+W.sub.iy−W.sub.iA.sub.\i(θ)β.sub.\i∥.sub.2.sup.2−(W.sub.iy).sup.HW.sub.iy  (17)

wherein η.sub.i=[r.sub.i, f.sub.i].sup.T, r.sub.i and f.sub.i represent the range and the Doppler frequency of the i.sup.th PDT, respectively; a.sub.i represents the steering vector of the i.sup.th PDT which is calculated by the equation (3), A.sub.\i denotes the matrix obtained from A by deleting the i.sup.th column, and β.sub.\i denotes the vector obtained from β by deleting the i.sup.th entry, to minimizing g(θ), u.sub.i (η.sub.i)=∥ W.sub.iy−β.sub.iW.sub.ia.sub.i(η.sub.i)∥.sub.2.sup.2 should attain its minimum. Then, the estimate of the i.sup.th PDT, denoted by =[].sup.T, can be obtained by minimizing u.sub.i(η.sub.i).

[0041] Minimizing u.sub.i(η.sub.i) the estimate of β.sub.i is given by:

{circumflex over (β)}.sub.i=a.sub.i.sup.H(η.sub.i)W.sub.i.sup.2y.  (18)

[0042] Inserting equation (18) into u.sub.i(η.sub.i), then, minimizing u.sub.i(η.sub.i) is equivalent to minimizing the following equation:

z.sub.i(η.sub.i)=(a.sub.i.sup.H(η.sub.i)W.sub.i.sup.2y)(y.sup.HW.sub.i.sup.2a.sub.i(η.sub.i))  (19)

[0043] Referring to equation (13), the estimate of is given by:

[00006]$\begin{array}{cc}\left(\begin{array}{c}{\hat{r}}_{\varrho ,i}\\ {\hat{f}}_{\varrho ,i}\end{array}\right)\approx \left(\begin{array}{c}{r}_{ϛ,i}\\ f_{ϛ,i}\end{array}\right)-{\left(\begin{array}{cc}\frac{{\partial }^2{z}_i}{\partial r_i^2}& 0\\ 0& \frac{{\partial }^2{z}_i}{\partial f_i^2}\end{array}\right)}_{\left({\eta }_i={\eta }_{ϛ,i}\right)}^{-1}{\left(\begin{array}{c}\frac{\partial z_i}{\partial r_i}\\ \frac{\partial z_i}{\partial f_i}\end{array}\right)}_{\left({\eta }_i={\eta }_{ϛ,i}\right)},& \left(20\right)\end{array}$

wherein η.sub.ζ,i=[r.sub.ζ,i, f.sub.ζ,i].sup.T. In order to obtain higher estimation accuracy, the estimate is iteratively updated using the following equation:

=−(∇.sub.η.sup.2z.sub.i()).sup.−1∇.sub.ηz.sub.i(),  (21)

wherein =n.sub.ζ, i, and t denotes the t.sup.th iteration usually, t=2 can meet the actual demand. Parameter estimation is performed for each PDT, and then can be obtained.

[0044] Step 3) establishing a dimension-reduction observation model of a received signal based on the estimated values and ;

[0045] can be obtained based on (i.e., and ), and the received signal, based on the , may be further represented as:

y≈β+n.  (22)

[0046] Usually, the number I of the PDTs is much smaller than a number of the cells corresponding to the range-Doppler map; therefore, the equation (22) can greatly reduce the dimension of the vector to be reconstructed, and the equation (22) is the dimension-reduction observation model of the present invention.

[0047] Step 4) reconstructing a target vector based on the dimension-reduction observation model and the sparse recovery algorithm;

[0048] The present invention reconstructs β based on the generalized approximate message Passing (GAMP) algorithm and the equation (22). Assuming that β follows the i.i.d Bernoulli-Gaussian distribution, the marginal probability density function thereof is:

ρ(β.sub.i)=(1−p)δ(β.sub.i)+ρ(β.sub.i;κ,τ.sup.q)  (23)

wherein δ is Dirac function, ρ denotes the fraction of nonzero components. κ and τ.sup.q represent the mean and variance of the Gaussian components, respectively. ρ, κ and τ.sup.q are all unknown, and expectation-maximum (EM) algorithm can incorporated to iteratively learn them.

[0049] GAMP can find both a sparse estimate and a non-sparse, noisy estimate of β, denoted by {circumflex over (β)} and β, respectively. It can be proved that the modulus of the noise in the β approximately follows The Gaussian distribution, and the simulation result is shown in FIG. 3A to FIG. 3C.

[0050] Step 5) designing a generalized likelihood ratio detector based on the reconstruction result for target detection and outputting detection results and their parameters.

[0051] The present invention is based on β for target detection. Determining whether the i.sup.th PDT is a target or not can be summarized as the following hypothesis tests:

H.sub.0:y=.sub.\iβ.sub.\i+n

H.sub.1:y=+.sub.\iβ.sub.\i+n.  (24)

[0052] H.sub.0 and H.sub.1 are two hypotheses in the hypothesis tests. H.sub.0 represents that the i.sup.th PDT is not the target, and H.sub.1 represents that the i.sup.th PDT is the target. is the true steering vector of the i.sup.th PDT, and is also calculated by the equation (3).

[0053] Based on the generalized likelihood ratio criterion, the detector is given by:

[00007]$\begin{array}{cc}.Math.{\tilde{\beta }}_i.Math.\begin{array}{c}\begin{array}{c}{H}_1\\ >\end{array}\\ \begin{array}{c}<\\ H_0\end{array}\end{array}\gamma \tilde{\sigma },& \left(25\right)\end{array}$

[0054] wherein {tilde over (σ)} is the variance of the noise modulus in {tilde over (B)}, and γ is the detection threshold; since the noise modulus follows approximately the Gaussian distribution, the equation (25) is a CFAR detector.

[0055] The implementation steps of the radar target detection method based on EBD provided by the present invention are described above, and the effectiveness of the method is verified by both and measured tests. The method of the present invention will be referred to as EBD below. FIG. 2A to FIG. 2C show the parameter performance of EBD in the present invention. Simulation parameters are shown in Table 1, 4 targets are added to the received signal and three following cases are considered: case 1: the target is located in the integer Doppler cells, and the range cells are 100, 100.2, 140.3, and 191.5 (the fractional part represents the value of deviation from the integer cells); case 2: the target is located in the integer range cells, and the frequency cells are 10, 15.2, 31.3 and 42.5; case 3: the Doppler and range cells of targets are all off-grid, and are set as (15.2,105.3), (31.2,140.4), (42.3,191.1) and (53.5,270.5), respectively; the target parameter estimation accuracy in all three cases is measured by mean square error (MSE), and the results are shown in FIG. 2A to FIG. 2C. The results show that the method of estimating the parameters before detection can effectively estimate the range and Doppler of the target.

TABLE-US-00001 TABLE 1 Radar simulation parameters Nos. Parameters Values 1 Pulse interval 0.2 ms 2 Pulse width  25 μs 3 Bandwidth   4 MHz 4 Sampling rate   5 MHz 5 Number of pulses   5 6 Number of FFT points  64 7 Window used in the fast Hamming time domain 8 Window used in the slow Chebyshev, −45 dB time domain

[0056] FIG. 3A to FIG. 3C show the statistical characteristics of the noise modulus in the reconstruction result S. The following 2 cases are considered: case 1: the received signal contains no target; case 2: 10 targets are added to the received signal, and the SNRs are 0 dB. When the noise characteristics are counted, the true target samples are removed, that is, only the noise samples are counted. FIG. 3A and FIG. 3B show the quantile-quantile plot (Q-Q plot) curves of the noise samples in both cases, respectively, and are compared with the standard Gaussian distribution. FIG. 3C shows the correlation coefficient of the noise samples in both cases. The results show that the noise approximately follows independent Gaussian distribution.

[0057] In FIG. 4A and FIG. 4B, the performance of the EBD provided by the present invention is compared with 3 other detectors including the conventional detection method (TSPM, windowed), the ideal matched filtering (IMF, not windowed), the method of off-grid sparse recovery (OGSR) based on the equation (2). In the simulation, the parameters shown in Table 1 are still adopted, 10 targets are added in each simulation, the target range and Doppler are randomly generated. Simulation results show that EBD is approximately a CFAR detector within the entire interval of SNR, and the gain of detection performance is about 1.9 dB compared with the conventional detection method (TSPM) under the condition that the false alarm rate is 10.sup.−5.

[0058] FIG. 5A. FIG. 5B and FIG. 6 verify the method of the present invention based on the measured data. The radar parameters corresponding to the adopted data are as follows: the bandwidth is 8 MHz, the sampling rate is 10 MHz, the pulse width is 12 microseconds, the number of single frame pulses is 128, the number of FFT points is 128, and the repetition frequency is 12.5 KHz. In the test, two unmanned aerial vehicles DJI Phantom 3 are used as cooperative targets, 1 frame is selected for analysis, and the range-Doppler map is shown in FIG. 5A. The selected frame contains no other target. Because the SNRs of the two unmanned aerial vehicles are high and are 26.3 dB and 21.3 dB, respectively; in order to verify the performance of the provided algorithm, the white Gaussian noise is added to the original received signal. For example, after increasing the noise level by 6 dB, the range-Doppler map using the conventional processing is as shown in FIG. 5B.

[0059] The performance comparison between the EBD and TSPM is shown in FIG. 6, wherein the abscissa represents the noise increment and is represented by Δσ.sub.n.sup.2. It can be seen from the results that the EBD has a better noise resistance ability, i.e., better target detection performance, than TSPM. For both cooperative targets, the EBD has performance gains of 2.2 dB and 1.5 dB, respectively, at a false alarm rate of 10.sup.−5.