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DOA estimation of spectrally overlapped LFM signals based on STFT and Hough transform
EURASIP Journal on Advances in Signal Processing volume 2019, Article number: 58 (2019)
Abstract
Traditional subspace methods which are based on the spatial timefrequency distribution (STFD) matrix have been investigated for directionofarrival (DOA) estimation of linear frequency modulation (LFM) signals. However, the DOA estimation performance may degrade substantially when multiple LFM signals are spectrally overlapped in timefrequency (TF) domain. In order to solve this problem, this paper proposes singlesource TF points selection algorithm based on Hough transform and shorttime Fourier transform (STFT). Firstly, the signal intersections in TF domain can be solved based on the Hough transform, and multiplesource TF points around the intersections are removed, so that the singlesource TF points set is reserved. Then, based on the Euclidean distance operator, the singlesource TF points set belonging to each signal can be obtained according to the property that TF points of the same signal have same eigenvector. Finally, the averaged STFD matrix is constructed for each signal, and DOA estimation is achieved based on multiple signal classification (MUSIC) algorithm. In this way, the proposed algorithm exhibit remarkable superiority in estimation accuracy and angular resolution over the stateoftheart schemes and can achieve DOA estimation in the underdetermined cases. In addition, the proposed algorithm can still perform DOA estimation when multiple LFM signals intersect at one point. Numerical simulations demonstrate the validity of the proposed method.
Introduction
Directionofarrival (DOA) estimation is widely applied in array signal processing field, such as radar, wireless communications, sonar, and seismic sensing [1,2,3,4]. For the singlefrequency signal, traditional subspacebased algorithms and their improved methods have been verified as efficient estimation techniques [5, 6]. With the rapid development of signal processing technologies, the linear frequency modulation (LFM) signal has many good characteristics, such as carrying a large amount of information and possessing better detection performance, which makes it being widely used in electronic countermeasures. Nevertheless, subspacebased algorithms have limitations to deal with LFM signals which are wideband and nonstationary; thus it is very important to study the DOA estimation of LFM signals. The timefrequency (TF) analysis tools can deal with LFM signals, because they can make full use of the TF energy aggregation property of LFM signals. Belouchrani and Amin proposed the concept of STFD [7], and then a series of DOA estimation algorithms based on the STFD matrix have been proposed [8,9,10,11,12,13,14]. These algorithms mainly use the STFD matrix to replace the covariance matrix in the multiple signal classification (MUSIC) algorithm, which forms the TFMUSIC algorithm. Compared with subspacebased algorithms, the TFMUSIC algorithm can provide better DOA estimation performance because the noise power is spread while localizing the source energy in TF domain. In order to construct the STFD matrix, it is necessary to select the singlesource TF points belonging to each signal in TF domain.
When LFM signals are TFdisjoint in TF domain, there exists efficient singlesource TF points selection algorithms [15, 16]. The reference [15] obtained the singlesource TF points based on the improved clustering algorithm. In [16], Heidenreich firstly applied the morphological image processing method to detect instantaneous frequency (IF) segments. Then, multiple detected IF segments of each signal are combined together for DOA estimation based on a bootstrap resampling technique. Unfortunately, these two methods only can be used in the situation that the signals are TFdisjoint. When the signals are nondisjoint (i.e., spectrally overlapped) in TF domain, the studies of singlesource TF points selection algorithms are not very full. In [17,18,19], the singlesource TF points selection algorithms are proposed based on the WignerVille distribution (WVD). However, considering that the WVD has higher computational complexity and generates crossterm TF points in TF domain, it is very difficult to apply the WVD in practice. The reference [20] proposes the shorttime Fourier transform (STFT) distribution model for DOA estimation of LFM signals, which greatly reduces implementation cost. Based on the STFT distribution, many DOA estimation algorithms have also been proposed [21,22,23,24,25] when LFM signals are nondisjoint in TF domain. However, all of them have some limitations. The singlesource TF points selection algorithms in [21] are only suitable for realvalued array manifold matrix, but the array manifold vector in the DOA estimation is complexvalued. In [22, 23], the signals are required to satisfy the Wdisjoint distribution in TF domain. Zhang proposes singlesource TF points selection algorithm based on the TF points intensity information [24]. Although it can successfully realize the DOA estimation when two LFM signals are nondisjoint, the algorithm will fail when multiple LFM signals intersect at one point. In [25], the author uses the forwardbackward spatial smoothing (FBSS) algorithm to eliminate the effect of multiplesource TF points and then perform DOA estimation. Although this method can achieve DOA estimation, the DOA estimation accuracy is poor.
In order to ease the limitations of existing algorithms for spectrally overlapped LFM signals, this paper propose the singlesource TF points selection algorithm based on Hough transform and STFT when multiple LFM signals are nondisjoint in TF domain. Theoretically, the TF distribution of LFM signal is the straight line in TF domain. However, since the sampling data is discrete and finite, the TF distribution of each LFM signal under STFT has somewhat wide, which is called the TF ridgeline. Therefore, based on the Hough transform in TF domain, two line segments can be detected for every LFM signal. By traversing the TF domain, all line segments of LFM signals can be detected, and the line segments intersections can be calculated according to the endpoint coordinates. The area which is surrounded by these intersections is the multiplesource TF points set. Therefore, by removing multiplesource TF points around the intersections, the singlesource TF points set is reserved. Then, based on the Euclidean distance operator, singlesource TF points set belonging to each signal can be obtained according to the property that TF points of the same signal have same eigenvector. Finally, the averaged STFD matrix for every signal is computed, and DOA estimation can be achieved based on the MUSIC algorithm. In this way, the proposed algorithm has lower computational complexity and can achieve DOA estimation in the underdetermined cases. In addition, when there exists multiple signals at the intersections (i.e., greater than two), the accurate DOA estimation can still be achieved. Due to automatically select singlesource TF points set of each signal, the proposed algorithm exhibits remarkable superiority in estimation accuracy and angular resolution over the stateoftheart schemes.
The remainder of this paper is organized as follows. Section 2 formulates the STFT model of LFM signal based on the ULA and illustrates how to select singlesource TF points in detail. Section 3 carries the numerical simulations, and the proposed algorithm is discussed referring to the simulation results. Section 4 concludes the paper.
Methods
STFT model of LFM signal
In this paper, a uniform linear array (ULA) with M sensors is considered, and the interelement spacing is d. Furthermore, N LFM signals are assumed to impinge on the ULA from θ_{1}, θ_{2}, ⋯, θ_{N}. The array output at time t can be modeled as
where X(t) = [x_{1}(t), x_{2}(t), ⋯, x_{M}(t)]^{T} is the array observation vector, S(t) = [s_{1}(t), s_{2}(t), ⋯, s_{N}(t)]^{T} is the LFM signal vector, and N(t) = [n_{1}(t), n_{2}(t), ⋯, n_{M}(t)]^{T} is the additive white Gaussian noise vector. A(t) = [a_{1}(t), a_{2}(t), ⋯, a_{N}(t)] is the M × N array manifold matrix, and a_{n}(t) = [exp(−jf_{n}(t)τ_{n1}), ⋯, exp(−jf_{n}(t)τ_{nM})]^{T} is the steering vector of the nth signal at time t, where j is the imaginary unit. τ_{nm} = − 2π(m − 1)d cos(θ_{n})/c represents the time delay of the nth signal on the mth element respecting to the reference element, where c is the speed of light. The instantaneous frequency f_{n}(t) is denoted as
where f_{n0} and γ_{n} are initial frequency and modulated frequency of the nth signal, respectively. When γ_{n} = 0, the LFM signal becomes narrowband singlefrequency signal.
In 1946, Gabor proposed the concept of the STFT, which is an effective TF analysis tool for dealing with the LFM signal. The STFT distribution of X(t) in the free of noise can be expressed as
where \( {S}_{S_n}\left(t,f\right) \) is the STFT value of the nth LFM signal.
Then, the STFD matrix can be calculated through the correlation operation of S_{X}, namely,
where D_{XX} and D_{SS} are array output STFD matrix and signal STFD matrix, respectively.
The STFT distribution of LFM signals in TF domain can be described as Fig. 1. Figure 1a denotes that the LFM signals are TFdisjoint in TF domain, which means that signals are independent of each other, and the reference [15] has proposed efficient singlesource TF points selection method.
This paper mainly studies that LFM signals are tiny spectrally overlapped in TF domain. That is to say, multiple signals only have a small number of same TF points in TF domain, as is shown in Fig. 1b. In the TF domain, there exist noiseterm TF points and autoterm TF points, and autoterm contains singlesource TF points and multiplesource TF points. Based on the proposed algorithm, we can remove noiseterm TF points and multiplesource TF points in turn and obtain the singlesource TF points set of each signal. Finally, the STFD matrix can be constructed, and subsequent DOA estimation can also be achieved.
In TF domain, the LFM signals have obvious energy aggregation effect. According to [15], D_{XX} is very large at the TF ridgeline, and D_{XX} is close to zero at other TF points. Let Ω_{X} be the TF support domain, we can obtain
In order to filter out the noise and obtain the TF points which have enough energy in TF domain, we can set an empirical threshold value and perform following criterion at each sampling time instant (t, f) of D_{XX}
where ‖·‖_{F} is the Frobenius norm and ε_{1} is a small empirical threshold value and is related to noise. Typically, ε_{1} = 0.05 when SNR = 10 dB [18].
Then, the new TF points set Ω_{A} which includes singlesource TF points Ω_{S} and multiplesource TF points Ω_{M} can be obtained.
Extracting singlesource TF points
At the signal intersections, LFM signals are coherent owing to same frequency, which brings the rank deficiency of the signal subspace matrix. If these intersected TF points are selected to construct the STFD matrix, then subsequent DOA estimation will fail. Therefore, our goal is to remove multiplesource TF points and obtain singlesource TF points set belonging to each signal.
In theory, two LFM signals only have an intersection in TF domain. However, since the sampling data is discrete and finite, the TF ridgeline of LFM signals under STFT has somewhat wide. Therefore, there exist multiple TF points around the intersections when LFM signals are nondisjoint in TF domain. By converting the TF distribution of LFM signals into ordinary grayscale image, we can use the Hough transform method to detect straight lines in the grayscale image [26]. Hough transform is a parameter estimation technique using the voting principle in TF domain. The basic principle is to map the straight line detecting problem in the plane coordinate domain into the parameter coordinate domain by using the pointline duality property, so that the mapped result is easier to be detected. It is assumed that the curve satisfies the following equation
In the plane Cartesian coordinates, the curve satisfies specific parameters (a_{1}, a_{2}, ⋯, a_{m}), and (t, f) is located at the curve. ρ denotes the distance between the curve and the coordinate origin, and θ is the angle between ρ and the sampling time axis. After mapping into the parameter domain, we use the polar coordinates to describe the parameters, and there exists following relationship
Equation (8) shows that a point in the plane coordinates corresponds to a curve in the parameter coordinates. The different points of the same curve in the plane coordinates intersect at a point in the parameter coordinates, as is shown in Fig. 2.
Figure 2 shows that the three points A, B, C of the same line in the plane coordinate domain can be mapped into three curves and curves intersect at the point D. By integrating along with different θ, we can obtain the Hough transform. As a consequence, the Hough transform is a kind of projection integral essentially.
Firstly, we convert the TF distribution into the grayscale image. Then, by performing Hough transform on the grayscale image, we can obtain the corresponding curves of each TF points in the parameter domain. These curves exist multiple intersections, and each intersection corresponds to a detected line segment in the plane coordinate domain. For N LFM signals in TF domain, 2N peak points (θ_{1}, ρ_{1}), (θ_{2}, ρ_{2}), ⋯, (θ_{2N}, ρ_{2N}) can be searched by setting appropriate threshold value and carrying accumulation statistics; this is because the TF distribution of LFM signal under STFT has somewhat wide. The 2N peak points correspond to 2N line segments of the plane coordinate domain, and the mathematical equation of these lines can be calculated based on the endpoint coordinates, namely,
where k_{1}, k_{2}, ⋯, k_{2N} and b_{1}, b_{2}, ⋯, b_{2N} are slopes and intercepts of lines, respectively.
Since the noise in TF domain has been removed, the energy value in the TF ridgeline is much stronger than other TF points. Therefore, based on the Hough transform, we can precisely extract the line segments where TF distribution boundary of LFM signals is located in TF domain. Assuming that there exist Q (Q ≤ N(N − 1)/2) intersections between N signals, then the detected 2N line segments which corresponds to N signals will have P (P ≥ 4Q) intersections. If N LFM signals only pairwise intersect, the detected line segments have 4Q intersections. However, when there exists the situation that multiple signals intersect at one point, the number of intersections among 2N line segments will be more than 4Q. For any intersection in TF domain, we can calculate all intersections (t_{1}, f_{1}), (t_{2}, f_{2}), ⋯, (t_{P}, f_{P}) according to the line equation in the grayscale image. Then the minimum min(t_{1}, t_{2}, ⋯t_{P}) and maximum max(t_{1}, t_{2}, ⋯t_{P}) of the sampling time axis can be obtained in all intersections. In addition, the corresponding min(f_{1}, f_{2}, ⋯f_{U}) and max(f_{1}, f_{2}, ⋯f_{U}) can also be solved based on the (9), where f_{1}, f_{2}, ⋯f_{U} are all frequency values that min(t_{1}, t_{2}, ⋯t_{P}) and max(t_{1}, t_{2}, ⋯t_{P}) correspond to detected line segments. If multiplesource TF points cannot be completely removed, these TF points are selected to estimate DOAs, which will result in the failure of DOA estimation. So we remove multiplesource TF points corresponding to the region of (min(t_{1}⋯t_{P}), max(t_{1}⋯t_{P})) and (min(f_{1}⋯f_{U}), max(f_{1}⋯f_{U})) and then obtain singlesource TF points set Ω_{S} = ∪ Ω_{n} for n = 1, 2, ⋯, N, where Ω_{n} denotes singlesource TF points set of the nth signal.
Constructing STFD matrix
In this subsection, we firstly calculate the singlesource TF points set belonging to each signal and then construct the STFD matrix for subsequent DOA estimation. According to reference [15], for the signal STFD matrix D_{SS}(t, f), the diagonal elements have a larger value, and the remaining elements are close to zero at the singlesource TF points. As a result, based on the (4), the singlesource TF points of the nth signal in Ω_{n} satisfies
For any two singlesource TF points (t_{i}, f_{i}) and (t_{j}, f_{j}) in Ω_{n}, (10) can be rewritten as
It can be seen from (10–12) that the STFD matrix D_{XX} of different TF points in Ω_{n} has same eigenvector a_{n}(t).
For any TF point (t_{p}, f_{p}) in Ω_{S}, we can compute the main eigenvalue λ(t_{p}, f_{p}) and the corresponding principal eigenvector e(t_{p}, f_{p}) from the STFD matrix. Without loss of generality, we can make the first element of e(t_{p}, f_{p}) real and positive, namely,
where e_{m}(t_{p}, f_{p}) denotes the mth element of eigenvector e(t_{p}, f_{p}).
For singlesource TF points set Ω_{S}, mathematically, (t_{i}, f_{i}) and (t_{j}, f_{j}) belong to the same signal if they have following relationship
where d(·) is the Euclidean distance operator between \( \tilde{\mathbf{e}}\left({t}_i,{f}_i\right) \) and \( \tilde{\mathbf{e}}\left({t}_j,{f}_j\right) \). ε_{2} is an empirical threshold value, which is greater than 0. Typically, ε_{2} = 0.05 when SNR = 10 dB [15]. By traversing entire set Ω_{S}, we can obtain singlesource TF points set Ω_{n} of the nth signal.
In order to reduce the calculation error and make full use of singlesource TF points, we can compute the averaged STFD matrix of all TF points. For the nth signal, the averaged STFD matrix is expressed as
where \( {\overline{\mathbf{D}}}_n\left(t,f\right) \) is similar to the covariance matrix in the subspacebased algorithm and ΣΩ_{n} denotes the number of singlesource TF points.
Therefore, based on the MUSIC algorithm, we can carry eigendecomposition on the matrix \( {\overline{\mathbf{D}}}_n\left(t,f\right) \) and obtain the signal subspace and noise subspace. Finally, we can construct the spectrum peak searching function and perform DOA estimation
where b(θ) is the steering vector and U_{n} is the noise subspace.
For the readers’ convenience, the procedure of singlesource TF points selection algorithm based on Hough transform and STFT is summarized as follows:
 (1)
Filter out noise according to empirical threshold value ε_{1} in TF domain.
 (2)
Based on Hough transform, remove multiplesource TF points, and obtain all singlesource TF points set Ω_{S}.
 (3)
Classify signals according to the Euclidean distance operator, and get singlesource TF points set Ω_{n} of each signal.
 (4)
For every signal, calculate the averaged STFD matrix \( {\overline{\mathbf{D}}}_n\left(t,f\right) \) in the set Ω_{n}, and then construct the spectrum peak search function P(θ) = 1/b^{H}(θ)U_{n}U_{n}^{H}b(θ) based on the MUSIC algorithm.
Results and discussion
In order to verify the performance of the proposed algorithm, the following simulations are carried. Here, we set some general simulation parameters. The ULA has M = 8 sensors, and the array element spacing is the half of the wavelength which corresponds to the lowest frequency of LFM signals. The sampling rate is 360 MHz and snapshots are 1024. Since the STFT distribution can obviously gather the signal energy in TF domain and the SNR of signals is improved in TF domain by filtering the noise, we select SNR = 10 dB in the following analyses. Meanwhile, this paper involves the selection of two empirical threshold values. The threshold value ε_{1} eliminates the effect of noise in TF domain, which is related to the SNR. We take ε_{1} = 0.05 when SNR = 10 dB, and ε_{1} has been verified by [18]. Similarly, ε_{2} = 0.05 when SNR = 10 dB, which is also verified by [15]. In addition, we proceed with the comparison simulations among STFTMUSIC algorithm [20], STFTFBSSMUSIC algorithm [25], the reference [24], and the proposed algorithm. Firstly, the general performance of the proposed algorithm is verified, which includes the DOA estimation precision, the spatial resolution, and underdetermined DOA estimation, as is shown in Sections 3.1 and 3.2. Then, we verify the new property of the proposed algorithm, i.e., the DOA estimation performance when multiple LFM signals intersect at one point, as is shown in Section 3.3. In Section 3.4, the computational complexity of different algorithms is analyzed.
DOA estimation accuracy analysis
We consider that three spectrally overlapped LFM signals in TF domain come from− 30 °, 0, ° and 30 °. The frequency distribution of three signals are [10, 80] MHz, [170, 10] MHz, and [60, 170] MHz, respectively, and all signals have same power. Figure 3 illustrates the calculation process of the singlesource TF points selection method. Figure 3a displays the TF distribution of three LFM signals after filtering noise, whose grayscale image is showed in Fig. 3b. Based on Hough transform, we can search six peak points which is indicated by white in Fig. 3c, and Fig. 3d shows the detected six line segments. After removing multiplesource TF points, the singlesource TF points set is shown in Fig. 3e. According to the Euclidean distance operator, we can classify signals and get singlesource TF points set of each signal, as is shown in Fig. 3f.
Based on the abovementioned analysis, Fig. 4a plots ten typical independent estimated spatial spectra of the proposed algorithm, where the dashed lines represent the true DOAs. It can be seen that the proposed algorithm can accurately achieve DOA estimation. In order to quantitatively analyze the DOA estimation accuracy of different algorithms, the rootmeansquare error (RMSE) is introduced, which is denoted as
where M is the number of Monte Carlo trials, N is the number of signals, and \( {\hat{\theta}}_{mn} \) is the estimated value of θ_{n} in the mth simulation.
By carrying 1000 times independent Monte Carlo trials, Fig. 4b shows that the DOA estimation RMSE of 4 algorithms vary with the SNR. From Fig. 4b, we can see that DOA estimation RMSE of four algorithms decreases rapidly with the increase of SNR. In general, the STFTFBSSMUSIC algorithm has the worst DOA estimation accuracy, and the estimation precision is 0.7594 ° when SNR is 10 dB. Because of performing the spatial smoothing operation, the array aperture of the STFTFBSSMUSIC algorithm is reduced. As a result, the estimation precision is reduced. Then, it is followed by the STFTMUSIC algorithm, and the estimation precision is 0.3970 ° when SNR is 10 dB. The STFTMUSIC algorithm performs DOA estimation by using the TF points at the TF ridgeline, and only few TF points are used to estimate DOAs, which results in low estimation precision. In addition, it is not realistic to assume that the TF points are available at the TF ridgeline. By removing multiplesource TF points, the reference [24] and the proposed algorithm can obtain singlesource TF points set belonging to each signal. Then, by computing the averaged STFD matrix of all TF points for every signal, the DOA estimation precision of these two methods is improved. Compared with the [24], the proposed algorithm removes multiplesource TF points based on Hough transform and reserves as many singlesource TF points as possible. Therefore, the proposed algorithm possesses the best estimation precision, which can achieve 0.1474 ° when SNR is 10 dB.
The angular resolution analysis and underdetermined DOA estimation
In order to verify the angular resolution of the proposed algorithm, we consider two spectrally overlapped LFM signals coming from−0.5 ° and 0.5 °. The frequency distribution of two signals are [170, 10] MHz and [10, 170] MHz, respectively. Figure 5a plots ten typical independent estimated spatial spectra of the proposed algorithm, which shows that the proposed algorithm can achieve DOA estimation when the spatial spacing of signals is 1 °. By performing 1000 times Monte Carlo trials, Fig. 5b displays that the DOA estimation RMSE of LFM signals at different SNR, which can be seen that the proposed algorithm has better DOA estimation accuracy when the DOA spacing is 1 °. Although the spatial spacing is very close, they are still distinctly distributed in TF domain. Therefore, the proposed algorithm has better angular resolution.
It is known that conventional subspacebased algorithms will fail in the underdetermined cases. However, the TF analysis can solve this problem. Based on the abovementioned simulation condition, we assume that four spectrally overlapped LFM signals come from− 15 °,− 5 °, 5 °, and 15 °. Figure 6a plots ten typical independent estimated spatial spectra of this case. Then, we reduce the number of sensors to M = 3, and the incident signals remain unchanged; ten independent estimated spatial spectra is plotted in Fig. 6b. It is seen that the proposed algorithm can correctly estimate DOAs in the underdetermined cases. The reason is that the proposed algorithm separately handles each signal in TF domain. Although the number of signals is more than that of sensors, the proposed algorithm can still achieve DOA estimation.
According to Sections 3.1 and 3.2, we know that the STFTMUSIC algorithm is not practical in the engineering applications and the DOA estimation precision of the STFTFBSSMUSIC algorithm is poor. Although the reference [24] has good DOA estimation accuracy, it is only applicable to the case that two LFM signals intersect at one point. When multiple signals intersect at one point, the algorithm will fail. However, the proposed algorithm is not restricted by this case. Based on the Hough transform, the proposed algorithm can achieve DOA estimation when multiple signals intersect at one point. The following experiment will illustrate this property.
The DOA estimation analysis when multiple LFM signals intersect at one point
We consider that three spectrally overlapped LFM signals come from− 30 °, 0 °, and 30 °, and these signals intersect at one point in TF domain. The frequency distribution of three signals are [170,10] MHz, [90,90] MHz , and [10,170] MHz, respectively. Other settings remain unchanged; the TF distribution of three signals is shown in Fig. 7a when SNR = 10 dB, and Fig. 7b shows the detected results by using the Hough transform. Based on the proposed algorithm, the DOA estimation results are shown in Fig. 7c and d. It can be seen that the proposed algorithm has higher DOA estimation precision and the estimation precision can reach 0.1153 ° when SNR = 10 dB. Therefore, the proposed algorithm can achieve DOA estimation when multiple LFM signals intersect at one point.
The computational complexity analysis
The computational complexity comparison among the proposed algorithm and three existing methods are presented in Table. 1. We consider a ULA with M sensors and N farfield LFM signals imping on the ULA. L_{s} snapshots are used. In the STFTFBSSMUSIC algorithm, assuming that subarray elements are K(K < M). Considering that the TFMUSIC algorithm is based on the quadratic TF operation, it has the highest computational complexity O(NM^{2}L_{s}^{2}log_{2}L_{s}). Due to performing the FBSS algorithm, the STFTFBSSMUSIC algorithm possesses the lowest computational complexity O(NKL_{s}^{2}log_{2}L_{s}). However, its estimation precision is poor. The reference [24] and the proposed algorithm have similar computational complexities, where Δ_{1} and Δ_{2} denote the increased calculation burden due to extract singlesource TF points, respectively. Δ_{1} and Δ_{2} are similar and are much smaller than O(NML_{s}^{2}log_{2}L_{s}). Therefore, the proposed algorithm has better estimation performance and acceptable computational complexity, which will have broad application prospects in the engineering.
Conclusion
Conventional DOA estimation algorithms will fail when LFM signals are spectrally overlapped in TF domain. In order to solve this problem, this paper proposes singlesource TF points selection algorithm based on Hough transform. For the purpose of reducing computational complexity, we select the STFT to analyze LFM signals. The proposed algorithm not only has higher DOA estimation precision and angular resolution but also can achieve DOA estimation in the underdetermined cases. In addition, the proposed algorithm can achieve DOA estimation when multiple signals intersect at one point compared with existing algorithms. Although the current algorithm is only applicable to LFM signals which satisfy the straight line distribution in TF domain, generalized Hough transform can detect other forms of signals with different shapes in TF domain, such as sinusoidal FM signal and frequencyshiftkeying signal. Therefore, the proposed algorithm has a broad application prospect in the electronic countermeasures.
Availability of data and materials
Data and materials are not possible to share publicly. If you have any interest in our data and materials, please contact the corresponding author by email.
Abbreviations
 DOA:

Directionofarrival
LFM
Linear frequency modulation
STFT
Shorttime Fourier transform
STFD
Spatial timefrequency distribution
TF
Timefrequency
MUSIC
Multiple signal classification
FBSS
Forwardbackward spatial smoothing
ULA
Uniform linear array
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The authors would like to thank all anonymous reviewers and editors for their helpful suggestions for the improvement of this paper.
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WZ and KC proposed the original idea of the full text; WZ and YY designed and implemented the simulation experiments; WZ and TX wrote the manuscript under the guidance of NY; WZ and NY revised the manuscript; XZ revised the manuscript and made great contribution to the final manuscript. All authors read and approved the final manuscript.
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State Key Laboratory of Complex Electromagnetic Environment Effects on Electronics and Information System, National University of Defense Technology. No. 109, Deya Road, Kaifu District, Changsha, 410073, China.
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Correspondence to Weike Zhang.
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Zhang, X., Zhang, W., Yuan, Y. et al. DOA estimation of spectrally overlapped LFM signals based on STFT and Hough transform. EURASIP J. Adv. Signal Process. 2019, 58 (2019) doi:10.1186/s1363401906540
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Keywords
 DOA estimation
 LFM signals
 TF analysis
 STFT
 Hough transform
 Singlesource TF points