Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

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


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.

This is a preview of subscription content, log in to check access.


  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: 231–251

  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: 600–616

  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: 163–171

  4. 4

    Sun K, Meng H, Wang Y, et al. Direct data domain STAP using sparse representation of clutter spectrum. Signal Process, 2011, 91: 2222–2236

  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: 4285–4295

  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: 1789–1800

  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: 695–703

  8. 8

    Tropp J. Greed is good: algorithmic results for sparse approximation. IEEE Trans Inf Theory, 2004, 50: 2231–2242

  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: 403–406

  10. 10

    Chen S S, Donoho D L, Saunders M A. Atomic decomposition by basis pursuit. SIAM J Sci Comput, 1998, 20: 33–161

  11. 11

    Selesnick I W, Bayram I. Sparse signal estimation by maximally sparse convex optimization. IEEE Trans Signal Process, 2014, 62: 1078–1092

  12. 12

    Rao B D, Kreutz-Delgado K. An affine scaling methodology for best basis selection. IEEE Trans Signal Process, 1999, 47: 187–200

  13. 13

    Xie K, He Z, Cichocki A. Convergence analysis of the FOCUSS algorithm. IEEE Trans Neural Netw Learn Syst, 2015, 26: 601–613

  14. 14

    He Z, Cichocki A, Zdunek R, et al. Improved FOCUSS method with conjugate gradient iteration. IEEE Trans Signal Process, 2009, 57: 399–404

  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: 1776–1788

  16. 16

    Sun K, Zhang H, Li G, et al. Airborne radar STAP using sparse recovery of clutter spectrum. arXiv:1008.4185

  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: 413–423

  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: 1214–1218

  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. 561–565

  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: 251–268

  21. 21

    Sen S. Low-rank matrix decomposition and spatio-temporal sparse recovery for STAP radar. IEEE J Sel Top Signal Process, 2015, 9: 1510–1523

  22. 22

    Fletcher R. Conjugate gradient methods for indefinite systems. In: Numerical Analysis. Berlin: Springer, 1976. 73–89

  23. 23

    Joly P, Meurant G. Complex conjugate gradient methods. Numer Math, 1993, 4: 379–406

  24. 24

    Mihalyffy L. An alternative representation of the generalized inverse of partitioned matrices. Linear Algebra Appl, 1971, 4: 95–100

  25. 25

    Saad Y. Iterative Methods for Sparse Linear Systems. Boston: PWS-Kent, 1995

  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: 295–319

  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: 760–770

  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. 1406–1407

  29. 29

    Golub G H, Van-Loan C F. Matrix Computations. 3rd ed. Boltimore and London: The Johns Hopkins University Press, 1996

  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: 2170–2177

  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: 1259–1269

  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: 664–676

Download references


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

Author information

Correspondence to Ran Tao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bai, G., Tao, R., Zhao, J. et al. Fast FOCUSS method based on bi-conjugate gradient and its application to space-time clutter spectrum estimation. Sci. China Inf. Sci. 60, 082302 (2017).

Download citation


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