Fast FOCUSS method based on bi-conjugate gradient and its application to space-time clutter spectrum estimation

Research Paper


The focal underdetermined system solver (FOCUSS) is a powerful tool for sparse representation in complex underdetermined systems. This paper presents the fast FOCUSS method based on the bi-conjugate gradient (BICG), termed BICG-FOCUSS, to speed up the convergence rate of the original FOCUSS. BICGFOCUSS was specifically designed to reduce the computational complexity of FOCUSS by solving a complex linear equation using the BICG method according to the rank of the weight matrix in FOCUSS. Experimental results show that BICG-FOCUSS is more efficient in terms of computational time than FOCUSS without losing accuracy. Since FOCUSS is an efficient tool for estimating the space-time clutter spectrum in sparse recoverybased space-time adaptive processing (SR-STAP), we propose BICG-FOCUSS to achieve a fast estimation of the space-time clutter spectrum in mono-static array radar and in the mountaintop system. The high performance of the proposed BICG-FOCUSS in the application is demonstrated with both simulated and real data.


focal underdetermined system solver (FOCUSS) sparse recovery (SR) bi-conjugate gradient (BICG) space-time adaptive processing (STAP) space-time clutter spectrum 



This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61421001, 61331021, 61671060).


  1. 1.
    Gorodnitsky I F, George J S, Rao B D. Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm. Electroencephalogr Clin Neurophysiol, 1995, 95: 231–251CrossRefGoogle Scholar
  2. 2.
    Gorodnitsky I F, Rao B D. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Signal Process, 1997, 45: 600–616CrossRefGoogle Scholar
  3. 3.
    Yang X Y, Chen B X, Chen Y H. An eigenstructure-based 2D DOA estimation method using dual-size spatial invariance array. Sci China Inf Sci, 2011, 54: 163–171CrossRefMATHGoogle Scholar
  4. 4.
    Sun K, Meng H, Wang Y, et al. Direct data domain STAP using sparse representation of clutter spectrum. Signal Process, 2011, 91: 2222–2236CrossRefMATHGoogle Scholar
  5. 5.
    Alonso M T, Lopez-Dekker P, Mallorqui J J. A novel strategy for radar imaging based on compressive sensing. IEEE Trans Geosci Remote Sens, 2010, 48: 4285–4295CrossRefGoogle Scholar
  6. 6.
    Bu H X, Bai X, Tao R. Compressed sensing SAR imaging based on sparse representation in fractional Fourier domain. Sci China Inf Sci, 2012, 55: 1789–1800MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Yang J Y, Peng Y G, Xu W L, et al. Ways to sparse representation: an overview. Sci China Ser F-Inf Sci, 2009, 52: 695–703MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Tropp J. Greed is good: algorithmic results for sparse approximation. IEEE Trans Inf Theory, 2004, 50: 2231–2242MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Wu R, Huang W, Chen D R. The exact support recovery of sparse signals with noise via orthogonal matching pursuit. IEEE Signal Process Lett, 2013, 20: 403–406CrossRefGoogle Scholar
  10. 10.
    Chen S S, Donoho D L, Saunders M A. Atomic decomposition by basis pursuit. SIAM J Sci Comput, 1998, 20: 33–161MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Selesnick I W, Bayram I. Sparse signal estimation by maximally sparse convex optimization. IEEE Trans Signal Process, 2014, 62: 1078–1092MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rao B D, Kreutz-Delgado K. An affine scaling methodology for best basis selection. IEEE Trans Signal Process, 1999, 47: 187–200MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Xie K, He Z, Cichocki A. Convergence analysis of the FOCUSS algorithm. IEEE Trans Neural Netw Learn Syst, 2015, 26: 601–613MathSciNetCrossRefGoogle Scholar
  14. 14.
    He Z, Cichocki A, Zdunek R, et al. Improved FOCUSS method with conjugate gradient iteration. IEEE Trans Signal Process, 2009, 57: 399–404MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hu C X, Liu Y M, Li G, et al. Improved FOCUSS method for reconstruction of cluster structured sparse signals in radar imaging. Sci China Inf Sci, 2012, 55: 1776–1788MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sun K, Zhang H, Li G, et al. Airborne radar STAP using sparse recovery of clutter spectrum. arXiv:1008.4185Google Scholar
  17. 17.
    Yang Z, de Lamare R C, Li X. Sparsity-aware space-time adaptive processing algorithms with L1-norm regularization for airborne radar. IET Signal Process, 2012, 6: 413–423MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yang Z, Li X, Wang H, et al. On clutter sparsity analysis in space-time adaptive processing airborne radar. IEEE Geosci Remote Sens Lett, 2013, 10: 1214–1218CrossRefGoogle Scholar
  19. 19.
    Wang L, Liu Y, Ma Z, et al. A novel STAP method based on structured sparse recovery of clutter spectrum. In: Proceedings of IEEE Radar Conference (RadarCon), Arlington, 2015. 561–565Google Scholar
  20. 20.
    Yang Z, Liu Z, Li X, et al. Performance analysis of STAP algorithms based on fast sparse recovery techniques. Prog Electromagn Res B, 2012, 41: 251–268CrossRefGoogle Scholar
  21. 21.
    Sen S. Low-rank matrix decomposition and spatio-temporal sparse recovery for STAP radar. IEEE J Sel Top Signal Process, 2015, 9: 1510–1523CrossRefGoogle Scholar
  22. 22.
    Fletcher R. Conjugate gradient methods for indefinite systems. In: Numerical Analysis. Berlin: Springer, 1976. 73–89Google Scholar
  23. 23.
    Joly P, Meurant G. Complex conjugate gradient methods. Numer Math, 1993, 4: 379–406MathSciNetMATHGoogle Scholar
  24. 24.
    Mihalyffy L. An alternative representation of the generalized inverse of partitioned matrices. Linear Algebra Appl, 1971, 4: 95–100MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Saad Y. Iterative Methods for Sparse Linear Systems. Boston: PWS-Kent, 1995Google Scholar
  26. 26.
    Bank R E, Chan T F. An analysis of the composite step biconjugate gradient algorithm for solving nonsymmetric systems. Numer Math, 1993, 66: 295–319MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rao B D, Engan K, Cotter S F, et al. Subset selection in noise based on diversity measure minimization. IEEE Trans Signal Process, 2003, 51: 760–770CrossRefGoogle Scholar
  28. 28.
    Peng Y, Fei Y, Feng Y. Sparse array synthesis with regularized FOCUSS algorithm. In: Prceedings of International Symposium of the IEEE Antennas and Propagation Society, 2013. 1406–1407Google Scholar
  29. 29.
    Golub G H, Van-Loan C F. Matrix Computations. 3rd ed. Boltimore and London: The Johns Hopkins University Press, 1996MATHGoogle Scholar
  30. 30.
    Duan K Q, Xie W C, Wang Y L. Nonstationary clutter suppression for airborne conformal array radar. Sci China Inf Sci, 2011, 54: 2170–2177MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wu R B, Jia Q Q, Li H. A novel STAP method for the detection of fast air moving targets from high speed platform. Sci China Inf Sci, 2012, 55: 1259–1269MathSciNetCrossRefGoogle Scholar
  32. 32.
    Peckham C D, Haimovich A M, Ayoub T F, et al. Reduced-rank STAP performance analysis. IEEE Trans Aerosp Electron Syst, 2000, 36: 664–676CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Gatai Bai
    • 1
    • 3
  • Ran Tao
    • 1
    • 2
    • 3
  • Juan Zhao
    • 2
    • 3
  • Xia Bai
    • 2
    • 3
  • Yue Wang
    • 1
    • 2
    • 3
  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  2. 2.School of Information and ElectronicsBeijing Institute of TechnologyBeijingChina
  3. 3.Beijing Key Laboratory of Fractional Signals and SystemsBeijingChina

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