https://orcid.org/0000-0003-0795-8206
Figures
Abstract
SONAR signal processing plays an indispensable role when it comes to parameter estimation of Direction of Arrival (DOA) of acoustic plane waves for closely spaced target exclusively under severe noisy environments. Resolution performance of classical MUSIC and ESPRIT algorithms and other subspace-based algorithms decreases under scenarios like low SNR, smaller number of snapshots and closely spaced targets. In this study, optimization strength of swarm intelligence of Cuckoo Search Algorithm (CSA) is accomplished for viable DOA estimation in different scenarios of underwater environment using a Uniform Linear Array (ULA). Higher resolution for closely spaced targets is achieved using smaller number of snapshots viably with CSA by investigating global minima of the highly nonlinear cost function of ULA. Performance analysis of CSA for different number of targets employing estimation accuracy, higher resolution, variance analysis, frequency distribution of RMSE over the monte Carlo runs and robustness against noise in the presence of additive-white Gaussian measurement noise is achieved. Comparative studies of CSA with Root MUSIC and ESPRIT along with Crammer Rao Bound analysis witnesses better results for estimating DOA parameters which are further endorsed from the results of Monte Carlo simulations.
Citation: Ahmed N, Wang H, Ahmad R, Salem AA, Abd Rahman R, Kashif M, et al. (2022) High resolution DOA estimation of acoustic plane waves: An innovative comparison among Cuckoo search heuristics and subspace based algorithms. PLoS ONE 17(6): e0268786. https://doi.org/10.1371/journal.pone.0268786
Editor: Seyedali Mirjalili, Torrens University Australia, AUSTRALIA
Received: January 6, 2022; Accepted: May 8, 2022; Published: June 16, 2022
Copyright: © 2022 Ahmed et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting Information files.
Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Direction of arrival has remained an active topic for researchers in array signal processing for its vast applications in the field of electromagnetic signal processing, seismic exploration, acoustic signal processing, speech signal processing, defense (warfare), and biomedical imaging [1–4]. Acoustics plays a pivotal role in underwater wireless communication due to its nature of robust propagation in water as compared to electromagnetic waves.
DOA estimation for the pool of practical problems restraints the application of conventional subspace based algorithms due to limited number of snapshots and low SNR for closely spaced targets. CBF and spectral search algorithms are reliable for the applications involving greater number of snapshots, higher SNR and incoherent sources hence, it is challenging to apply these algorithms for DOA estimation. Maximum Likelihood (ML) [5] is another approach for parameter estimation of DOA having higher computational complexity which restrains its implementation for such problems [6–9]. Estimation accuracy in DOA is highly dependent on signal power and rate of transmission so, low SNR based DOA estimation is an innovative and challenging task in real time applications [10–13].
Recently, a pool of robust optimization algorithms have been introduced for DOA estimation such as Genetic Algorithm (GA) [14], Differential Evolution (DE) method [15], Particle Swarm Optimization (PSO) [16], Seeker Optimization Algorithm (SOA) [17], Sine Cosine algorithm (SCA) [18], Invasive Weed Optimization (IWO) [19], Squirrel Search Algorithm (SSA) [20], etc. In [14], GA is analyzed showing better results for the DOA estimation against its counterparts i.e. ML and MUSIC algorithm in terms of robustness against noise, computational cost and number of snapshots. A modified version of GA applied to highly nonlinear function estimating DOA parameters is presented in [21]. In [16], the PSO algorithm develops the mechanism to estimate the parameters of a multimodal function. In [22], PSO ML estimator shows very healthy and reliable results as compared to conventional parameter estimation techniques for DOA. Using Ant Colony Optimization (ACO) by extending the pheromone, DOA parameters are estimated in [23] with very good results and low computational complexity. In [24], Artificial Bee Colony (ABC) algorithm is used to achieve higher statistical performance. In 2019, Squirrel Search Algorithm (SSA) was proposed which is a novel numerical optimization algorithm. It focuses on foraging and gliding behavior of flying squirrels to determine their efficient way of locomotion. Gliding is a powerful technique used by small mammals for traveling long distances. Present work mathematically models this behavior to comprehend the process of optimization. These features may be helpful to improve convergence, reduce the number of iterations of the SSA algorithm and in determining ML DOA estimation [20].
All the above Evolutionary Algorithms (EA) perform well in locating single-dimensional optimization problem but fail to provide multiple solutions. Several approaches have been exploited in evolutionary algorithms to achieve multimodal optimization, such as fitness sharing [25, 26], deterministic crowding [27], probabilistic crowding [26, 28], clustering based niching [29], clearing procedure [30], species conserving genetic algorithm [31] and elitist-population strategies [32]. However, problems may still exist while implementing these strategies such as tuning niching parameters, retaining perceived solution in a run, extra computational costs and poor scalability for multidimensional problems. Moreover, these methods are hatched for spanning the search capacities of popular evolutionary algorithms such as GA and PSO, which fail to exploit the balance between exploration and exploitation, exclusively for multimodal objective functions [33]. Furthermore, they do not probe whole search span productively and most of the time do not converge. Recently, a novel nature-inspired algorithm known as Cuckoo Search Algorithm (CSA) [34], has been introduced for multidimensional optimization problems. The CSA is based on constrained brood-parasitic strategy of some cuckoo birds. One of the most significant characteristic of CSA is the employment of Levy flights to construct new solutions. Using this approach, the solutions are modified by adopting many small steps resulting in improvement of the relationship between exploration and exploitation [35]. Recent studies revealed that the CSA is far more efficient than PSO and GA [36]. Such characteristics have prompted the use of CSA to manipulate many engineering problems such as mesh generation [37], embedded systems [38], steel frame design [39], scheduling problems [40], thermodynamics [41] and distribution networks [42].
In this study, the optimization strength of swarm intelligent approach of the Cuckoo Search Algorithm (CSA) is accomplished for viable DOA estimation in different scenarios of underwater environment using Uniform Linear Array (ULA). The high resolution for closely spaced targets is achieved using fewer snapshots viably with CSA by investigating the global minima of the highly non-linear cost function of ULA. Performance analysis is conducted for the different number of targets employing estimation accuracy, resolution ability, frequency distribution of RMSE over the Monte Carlo runs and robustness against noise in the presence of additive white Gaussian measurement noise reveals better results. Comparative studies with Root MUSIC and ESPRIT counterparts along with Crammer Rao Bound analysis witnesses the worth of the scheme for estimating DOA parameters which are further endorsed from Monte Carlo simulation results.
The rest of the paper is organized as: In section II the data model for uniform linear array (ULA) is defined. The theoretical principle of the CSA algorithm for DOA estimation is described in section III. Performance analysis of algorithms in terms of RMSE is illustrated in section IV. Potential use and limitations are discussed in section V. The final section explains the main contributions of the proposed study.
2. Data model
In this work, the DOA model was estimated as shown in Fig 1. a ULA of hydrophones is used for DOA estimation of underwater multiple targets. A ULA having the characteristics of impinged plane waves from the far-field region is phase-shifted versions of neighbor hydrophones. DOA of multiple targets can be expressed as:
(1)
where θD is associated angle to Dth acoustic source.
(2)
(3)
(4)
where,
, while β is wavelength. z(t) is the hydrophone’s output vector with dimension Wx1 and can be called as array response.
The steering matrix S of dimension WxD comprises the time delay entities of signals for each hydrophone. Where n(t) is additive white Gaussian noise of zero mean with the dimension Wx1. The co-variance matrix is defined as:
(6)
When a finite number of snapshots are available, the above equation can be written as:
(7)
Where E[.] and [.]H are ensemble average and Hermitian operators. So the correlation matrix can be written as:
(8)
Where Ry is correlation associated with signal and σ2ID is the noise correlation matrix. The actual signal received on the array would be:
(9)
Whereas, the approximated signal after getting optimum solutions would be:
(10)
So, objective function will become,
(11)
The general goal of DOA estimation is a continuous optimization that is used to find the θbest which satisfies:
(12)
where SϵRD and f(θ) comprises the cost values of corresponding solution θ. Therefore, the actual goal of the optimizer is to compute the associated argument for the minimum cost function. So, population of N individuals will be used to solve the optimization problem having T iterations (trials). The set of D-Dimensional vectors (total N vectors) for i iterations can be denoted as:
(13)
Since Cuckoo Search Algorithm [and Deb2009], can solve the multidimensional optimization problems so, the best solution at iteration i can be found as:
(14)
3. Basic principle of CSA
There are two steps which are performed to get the optimum solution. Both of these steps belong to global and local searches. The global search is associated with levy flights and the local search is associated with random walks. Following are the implementation steps of the Cuckoo Search Algorithm:
- A population of solutions is chosen randomly in the lower and upper bounds.
- Fitness of each solution is computed and the best solution is selected among them (current best).
- Each solution from the population is modified by the equation: (Global Optimization).
(15)
where s is generated by symmetric Levy Distribution as:
and λ = 1.5 for all optimization problems.Where, u and v are calculated from normal distributions (U~N(0, σ2) and V~N(0,1)) and σ2 can be computed as
After modification, both previous and modified solutions are compared and one of them based on minimum fitness value is selected. The fitness value would be calculated by inserting the modified solution θi(n+1) as θe:
(16)
This step is known as greedy selection or elitist selection.
4. Next step is associated with the local search. In this operation, solutions are probabilistically selected and modified with a new value. This operation is performed by generating a random number ε within a range of [0,1]. If the number is less than pa then solution is modified by (below equation) otherwise, the solution remains unchanged (most of the practical problems pa = 0.2)
(17)
Where i and j are chosen randomly from the population. After modification, an elitist selection strategy is performed once. The simulation conditions are listed in Table 1 and the CSA steps was illustrated in flowchart in Fig 2.
4. Results and discussion
In this section, numerous simulations have been presented to assess the performance of CSA versus state of the art counterparts. The performance is evaluated in terms of estimation accuracy, convergence analysis, robustness against noise, and statistical analysis of RMSE over the Monte Carlo runs. The simulation conditions for different algorithms are given in Table 2. The performance metrics signifies the comprehensive analysis of algorithms as explained in the following sections:
4.1. Estimation accuracy
Estimation accuracy of Root MUSIC, ESPRIT and CSA algorithms are examined here by taking spatially separated targets and different levels of additive white Gaussian noise. The best, mean and worst numerical results have been presented for 300 independent Monte Carlo runs. It can be seen from Tables 2–4 that the CSA algorithm outperforms the state of the art schemes against all performance measures.
4.2. Convergence analysis
In this subsection the performance has been assessed in terms of convergence of the CSA towards solution. The performance has been measured for multiple scenarios of noise. It can be seen from the Figs 3–5 shows that the CSA algorithm converged in 50 iterations for two sources. Moreover, the CSA converged at 100 and 200 iterations for three and four sources respectively. The very earlier convergence of the algorithm reveals the strength of the algorithm for the optimization of highly non-linear cost function of DOA of underwater multi-targets.
4.3. Robustness against noise
This subsection describes the trend of RMSE against the linearly varying level of SNR. The convergence analysis has been analyzed in various cases of targets.
Figs 6 and 7 reveals that the convergence trend of RMSE of CSA algorithm against low SNR is exclusively far better than Root MUSIC and ESPRIT algorithms which have been also validated by CRB by taking 300 Monte Carlo trials.
4.4. Frequency distribution of RMSE
Histogram analysis describes the frequent successful happenings (trials with minimum RMSE value) over the Monte Carlo trials. Wider shape depicts lesser frequency distribution. It can be seen from Figs 8–10 that the frequency of the minimum valued RMSE is higher for CSA as compared to the ESPRIT and Root MUSIC algorithm.
4.5. Empirical CDF of RMSE
This subsection describes the probability distribution of RMSE against the monte carlo run for two, three and four targets having different level of noise. The performance metric also depicts the dynamics of RMSE. It can be evaluated that more than 90 percent of the runs are with minimum valued RMSE against the state of the art counterparts i.e. ESPRIT and Root MUSIC as shown in Figs 11–13.
4.6. Variability analysis of RMSE
In this subsection, the sorted RMSE observations are distributed in five quartiles (minimum to first 25 percent observations and from 25 to 75 percent and from 75 percent to above). The first and the last observation is the minimum and maximum value of the RMSE, respectively. This distribution gives an insight into the performance of algorithms. The symbol + denotes the outliers (abnormalities) in the observations. It can be found from Figs 14–16 that the minimum spread out depicts the consistency of algorithms. The dispersion rate of the CSA is less as compared to state of the art algorithms i.e. RMSE is about 0.045 to 0.11 for 5dB of SNR. Moreover, asymmetry is depicted by the outliers in the figure. The RMSE of the CSA is symmetric as compared to ESPRIT and Root MUSIC.
4.7. Resolution ability for closely spaced targets
The resolution ability is one of the important performance parameters to analyze the closely spaced targets. The probability of resolution can be defined as:
where △θ = |θ1−θ2|. The resolution ability of both closely spaced sources is shown in Figs 17 and 18. The performance has been evaluated for two different separations between two sources. The number of successful runs (the runs having RMSE less than the threshold) is higher for the CSA algorithm as compared to ESPRIT and Root MUSIC algorithms.
5. Potential use and limitations
Acoustic plane waves mostly suffer in the low SNR in underwater environment due to the far-field regime. Consequently, the estimation accuracy becomes an indespensible challenge for DOA estimation and target tracking in underwater multi-sources. In the results and discussion section, it can be seen that the state of the art counterparts i.e. MUSIC, ESPRIT and RMUSIC deteriotes in terms of estimation accuracy with high RMSE. Statistical analysis of the RMSE in Monte Carlo trials i.e. ECDF of RMSE, variability analysis of RMSE, frequency distribution of RMSE and the probability of resolution witnesses the strength of CSA in this challenging environment. Moreover, the computation complexity will become a trade off for the large number of snapshots exclusively for swarming intelligent algorithms.
6. Conclusions
Estimation accuracy and resolution ability are the main challenges in DOA estimation for closely spaced targets and CSA outperforms its counterpart algorithms (ESPRIT and Root MUSIC) in these performance measures. To be more specific, the state of the art counterparts i.e. MUSIC, ESPRIT and RMUSIC deteriotes in terms of estimation accuracy with high RMSE. Statistical analysis of the RMSE in Monte Carlo trials i.e. ECDF of RMSE, variability analysis of RMSE, frequency distribution of RMSE and the probability of resolution witnesses the strength of CSA in this challenging environment of low SNR and less number of snapshots. In the future, estimation of 2D-DOA using a uniform circular array is planned using the CSA algorithm and other modern computing paradigms [43–47].
References
- 1. Keyvan Zarifi and Alex B Gershman. “General- ized correlation decomposition-based blind channel estimation in ds-cdma systems with unknown wide-sense stationary noise”. IEEE transactions on signal processing, vol. 56, no. 11, 5605–5617,2008.
- 2. Qian Cheng, Huang Lei, Nicholas D Sidiropou- los and Hing Cheung So. “Enhanced puma for direction-of-arrival estimation and its performance analysis”. IEEE Transactions on Signal Processing, vol. 64, no. 16, 4127 4137, 2016.
- 3. Marija M Nikolic, Arye Nehorai and Antonije R Djordjevic. “Estimation of direction of arrival using multipath on array platforms”. IEEE transactions on antennas and propagation, vol. 60, no. 7, 3444–3454, 2012.
- 4. Ulrich Nickel. “Overview of generalized monopulse estimation”. IEEE Aerospace and Electronic Systems Magazine, vol. 21, no. 6, 27–56, 2006.
- 5. Hamid Krim and Mats Viberg. “Two decades of array signal processing research: the parametric approach”. IEEE signal processing magazine, vol. 13, no. 4, 67–94, 1996.
- 6. Petre Stoica and Arye Nehorai. “Music, maximum likelihood, and cramer-rao bound”. IEEE Transactions on Acoustics, speech, and signal processing, vol. 37, no. 5,720–741,1989.
- 7. Petre Stoica and Arye Nehorai. “Performance study of conditional and unconditional direction of arrival estimation”. IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, no. 10, 1783–1795, 1990.
- 8.
Björn Ottersten, Mats Viberg, Petre Stoica and Arye Nehorai. “Exact and large sample maximum likelihood techniques for parameter estimation anddetection in array processing”. In Radar array processing, pages 99–151. Springer, 1993.
- 9. AB Gershman P Stoica M Pesavento and Erik G Larsson. “Stochastic cramér–rao bound for direction estimation in unknown noise fields”. IEEE Proceedings-Radar, Sonar and Navigation, vol. 149, no. 1, 2–8, 2002.
- 10. Howland PF. “Target tracking using television based bistatic radar”. IEEE Proceedings-Radar, Sonar and Navigation, vol. xxx, no. 3, 166–174, 1999.
- 11. Colone F, Carlo Bongioanni and Lombardo P. “Multifrequency integration in fm radio-based passive bistatic radar. part i: Target detection”. IEEE Aerospace and Electronic Systems Magazine, vol. 28, no. 4, 28–39, 2013.
- 12. Francesca Filippini, Tatiana Martelli, Fabiola Colone and Roberta Cardinali. “Target doa estimation in passive radar using non-uniform linear arrays and multiple frequency channels”. In 2018 IEEE Radar Conference (RadarConf18), pages 1290–1295. IEEE, 2018.
- 13. Pasupathy S and Alford WJ. “Range and bearing estimation in passive sonar”. IEEE Transactions on Aerospace and Electronic Systems, vol. xxx, no. 2, 244–249, 1980.
- 14. Li M and Lu Y. “Genetic algorithm based maximum likelihood doa estimation”. xxx, 2002.
- 15. Ali Sheikh Y, Zaman F, Qureshi IM and Atique- ur Rehman M. “Amplitude and direction of arrival estimation using differential evolution”. In 2012 International Conference on Emerging Technologies, pages 1–4. IEEE, 2012.
- 16. Abhinav Sharma and Sanjay Mathur. “Comparative analysis of ml-pso doa estimation with conventional techniques in varied multipath channel environment”. Wireless Personal Communications, vol. 100, no. 3, 803–817, 2018.
- 17.
Chaohua Dai, Yunfang Zhu and Weirong Chen. “Seeker optimization algorithm”. In International Conference on Computational and Information Science, pages 167–176. Springer, 2006.
- 18. Seyedali Mirjalili. “Sca: a sine cosine algorithm for solving optimization problems”. Knowledge based systems, vol. 96, 120–133, 2016.
- 19. Ali Reza Mehrabian and Caro Lucas. “A novel numerical optimization algorithm inspired from weed colonization”. Ecological informatics, vol. 1, no. 4, 355–366, 2006.
- 20. Mohit Jain, Vijander Singh and Asha Rani. “A novel nature-inspired algorithm for optimization: Squirrel search algorithm”. Swarm and evolutionary computation, vol. 44, 148–175, 2019.
- 21. Minghui Li and Yilong Lu. “A refined genetic al- gorithm for accurate and reliable doa estimation with a sensor array”. Wireless Personal Communications, vol. 43, no. 2, 533–547, 2007.
- 22. Chih-Chang Shen. “Maximum likelihood doa estimation using particle swarm optimization under sensor perturbation conditions”. xxx, vol. 16, no.5,847–855,2015.
- 23. Wentao Shi, Jianguo Huang and Yunshan Hou. “Fast doa estimation algorithm for mimo sonar based on ant colony optimization”. Journal of Systems Engineering and Electronics, vol. 23, no. 2, 173–178,2012.
- 24. Zhicheng Zhang, Jun Lin and Yaowu Shi. “Application of artificial bee colony algorithm to maximum likelihood doa estimation”. Journal of Bionic Engineering, vol. 10, no. 1, 100–109, 2013.
- 25. David Beasley , David R Bull and Ralph R Martin. “A sequential niche technique for multimodal function optimization”. Evolutionary computation, vol. 1, no. 2, 101–125, 1993.
- 26. Rene Thomsen. “Multimodal optimization using crowding-based differential evolution”. InProceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No. 04TH8753), volume 2, pages 1382–1389. IEEE, 2004.
- 27. Wagner F Sacco, Cláudio MNA Pereira et al. “Two stochastic optimization algorithms applied to nuclear reactor core design”. Progress in Nuclear Energy, vol. 48, no. 6, 525–539, 2006.
- 28.
Ole J Mengshoel, David E Goldberg et al. “Probabilistic crowding: Deterministic crowding with probabilistic replacement”. In Proceedings of the Genetic and Evolutionary Computation Conference, volume 1, pages 409–416. Morgan Kauffman, 1999.
- 29.
Xiaodong Yin and Noel Germay. “A fast genetic algorithm with sharing scheme using cluster analysis methods in multimodal function optimization”. In Artificial neural nets and genetic algorithms, pages 450–457.Springer,1993.
- 30. Alain Pétrowski. “A clearing procedure as a niching method for genetic algorithms”. In Proceed- ings of IEEE international conference on evolutionary computation, pages 798–803. IEEE, 1996.
- 31. Li Jian-Ping, Marton E Balazs, Geoffrey T Parks and John Clarkson P. “A species conserving genetic algorithm for multimodal function optimization”. Evolutionary computation, vol. 10, no. 3, 207–234, 2002. pmid:12227994
- 32. Yong Liang and Kwong-Sak Leung. “Genetic algorithm with adaptive elitist-population strategies for multimodal function optimization”. Applied Soft Computing, vol.11,no.2,2017–2034,2011.
- 33. Gang Chen, Chor Ping Low and Zhonghua Yang“Preserving and exploiting genetic diversity in evolutionary programming algorithms”. IEEE Transactions on Evolutionary Computation, vol.13,no.3,661–673,2009.
- 34. Xin-She Yang and Suash Deb. “Cuckoo search via lévy flights”. In 2009 World congress on nature & biologically inspired computing (NaBIC), pages210–214.IEEE,2009.
- 35.
Sean Walton, Oubay Hassan, Kenneth Morgan and Rowan Brown M. “A review of the development and applications of the cuckoo search algorithm”. In Swarm intelligence and bio-inspired computation, pages 257–271.Elsevier,2013.
- 36. Yang Xin-She and Deb Suash. “Engineering optimisation by cuckoo search”. International Journal of Mathematical Modelling and Numerical Optimisation, vol. 1, no. 4, 330–343, 2010.
- 37. Walton S, Hassan O, Morgan K and Brown MR. “Modified cuckoo search: a new gradient free optimisation algorithm”. Chaos, Solitons & Fractals, vol. 44, no. 9, 710–718,2011.
- 38.
Anil Kumar and Shampa Chakarverty. “Design optimization for reliable embedded system using cuckoo search”. In 2011 3rd International Conference on Electronics Computer Technology, volume 1, pages 264 268.IEEE,2011.
- 39. Kaveh A and Bakhshpoori T. “Optimum design of steel frames using cuckoo search algorithm with\ lévy flights”. The Structural Design of Tall and\ Special Buildings,vol.22,no.13,1023–1036,2013.
- 40.
Lim Huai Tein and Razamin Ramli. “Recent advancements of nurse scheduling models and a potential path”. In Proc. 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA 2010), pages 395–409,2010.
- 41. Vaibhav Bhargava, Seif-Eddeen K Fateen and Adrian Bonilla-Petriciolet. “Cuckoo search: a new nature-inspired optimization method for phase equilibrium calculations”. Fluid Phase Equilibria, vol. 337, 191–200, 2013.
- 42. Zahra Moravej and Amir Akhlaghi. “A novel ap- proach based on cuckoo search for dg allocationin distribution network”. International Journal of Electrical Power & Energy Systems, vol. 44, no. 1, 672–679, 2013.
- 43. Muhammad Asif Zahoor Raja, Muhammad Saeed Aslam, Naveed Ishtiaq Chaudhary and Wasim Ul- lah Khan. “Bio-inspired heuristics hybrid with interior-point method for active noise control systems without identification of secondary path”. Frontiers of Information Technology & Electronic Engineering, vol. 19, no. 2, 246–259,2018.
- 44. Muhammad Asif Zahoor Raja, Iftikhar Ahmad, Imtiaz Khan, Muhammed Ibrahem Syam and Abdul Majid Wazwaz. “Neuro-heuristic computational intelligence for solving nonlinear pantograph systems”. Frontiers of Information Technology & Electronic Engineering, vol. 18, no. 4,464–484,2017.
- 45. Jing-ming Kuang, Yuan Zhou and Ze-song Fei. “Joint doa and channel estimation with data detection based on 2d unitary esprit in massive mimo systems”. Frontiers of Information Technology & Electronic Engineering, vol.18,no.6,841–849,2017.
- 46. Xin Guan, Li-hua Zhong, Dong-hui Hu and Chi- biao Ding. “An extended processing scheme for coherent integration and parameter estimationbased on matched filtering in passive radar”. Journal of Zhejiang University SCIENCE C, vol. 15,no.11,1071–1085,2014.
- 47. Shi Wen-tao, Zhang Qun-fei, He Cheng-bing and Han Jing. “Taylor expansion music method for joint dod and doa estimation in a bistatic mimo array”. Frontiers of Information Technology & Electronic Engineering, vol. 20, no. 6, 842–848, 2019.