Application of RELAX in Line Spectrum Estimation

  • Renbiao WuEmail author
  • Qiongqiong Jia
  • Lei Yang
  • Qing Feng


In the field of signal processing, line spectrum estimation is a classic research topic and is widely applied in the fields of communications, radar, sonar, seismology, etc. [1–14]. In these areas, the signal being processed can often be represented using a sinusoidal signal model. As a result, scholars around the world have attempted to solve the parameter estimation problem of sinusoidal signals, focusing primarily on issues of accuracy and computational complexity. The earliest paper regarding this subject can be traced back to an article published by Prony in 1795 [15].


  1. 1.
    Stoica P, et al. Modern signal spectrum analysis. Trans. RB Wu, Beijing: Publishing House of Electronics Industry; 2012. Electronic Industry Press; 2012.Google Scholar
  2. 2.
    Stoic P, Moses R. Spectral analysis of signals. New Jersey: Prentice-Hall; 2005.Google Scholar
  3. 3.
    Su ZG, Peng YN, Wu RB. Spectral estimation method based on singular covariance matrix. J Tsinghua Univ (Nat Sci). 2008;48(1):55–8.Google Scholar
  4. 4.
    Su ZG, Peng YN, Wu RB, et al. Spectral estimation of dynamic constraints based on sampling covariance matrix structure. J Electron. 2008;36(3):458–62.Google Scholar
  5. 5.
    Su ZG, Peng YN, Wu RB. Minimum norm singular capon spectral estimator. In: IEEE antennas and propagation society international symposium; 2007, p. 1140–3.Google Scholar
  6. 6.
    Su ZG, Zhang KX, Peng YN, et al. Rank-deficient APES filter for complex spectral estimation. In: IEEE radar conference; 2008, p. 1–6.Google Scholar
  7. 7.
    Viti V, Petrucci C, Barone P. Prony method in NMR spectroscopy. Int J Imaging Syst Technol. 1997;8(6):565–71.Google Scholar
  8. 8.
    Leonowicz Z, Lobos T, Rezmer J. Advanced spectrum estimation methods for signal analysis in power electronics. IEEE Trans Industr Electron. 2003;50(3):514–9.Google Scholar
  9. 9.
    Krim H, Viberg M. Two decades of array signal processing research: the parametric approach. IEEE Signal Process Mag. 1996;13(4):67–94.Google Scholar
  10. 10.
    Borcea L, Papanicolaou G, Tsogka C. Theory and applications of time reversal and interferometric imaging. Inverse Prob. 2003;19(6):5139–64.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Maravic I, Kusuma J, Vetterli M. Low-sampling rate UWB channel characterization and synchronization. J Commun Netw. 2004;5(4):319–27.Google Scholar
  12. 12.
    Carriere R, Moses RL. High resolution radar target modeling using a modified Prony estimator. IEEE Trans Antennas Propag. 1992;40(1):13–8.Google Scholar
  13. 13.
    Munson DCJ, O’Brien JD, Jenkins W. A tomographic formulation of spotlight-mode synthetic aperture radar. Proc IEEE. 1983;71(8):917–25.Google Scholar
  14. 14.
    Ausherman DA, Kozma A, Walker JL, et al. Developments in radar imaging. IEEE Trans Aerosp Electron Syst. 1984;20(4):363–400.Google Scholar
  15. 15.
    De Prony BGR. Essai experimental et analytique: sur les lois de la dilatabilite de fluides elastique et sur celles de la force expansive de la vapeur de l’alkool, a differentes temperatures. Journal de lecole polytechnique. 1795;1(22):24–76.Google Scholar
  16. 16.
    Tufts DW, Melissinos C. Simple effective computation of principal eigenvectors and their eigenvalues and application to high-resolution estimation of frequencies. IEEE Trans Acoust Speech Signal Process. 1986;34(5):1046–53.Google Scholar
  17. 17.
    Roy R, Kailath T. ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans Acoust Speech Signal Process. 1990;37(7):984–95.zbMATHGoogle Scholar
  18. 18.
    Li J, Stoica P. Efficient mixed-spectrum estimation with applications to target feature extraction. IEEE Trans Signal Process. 1996;44(2):281–95.Google Scholar
  19. 19.
    Liu ZS, Li J, Stoica P. RELAX-based estimation of damped sinusoidal signal parameters. Sig Process. 1997;62(3):311–21.zbMATHGoogle Scholar
  20. 20.
    Besson O, Stoica P. Nonlinear least-squares approach to frequency estimation and detection for sinusoidal signals with arbitrary envelope. Digit Signal Proc. 1999;9(1):45–56.Google Scholar
  21. 21.
    Chatterjee C, Kashyap R, Boray G. Estimation of close sinusoids in colored noise and model discrimination. IEEE Trans Acoust Speech Signal Process. 1987;35(3):328–37.Google Scholar
  22. 22.
    Kay SM, Nagesha V. Maximum likelihood estimation of signals in autoregressive noise. IEEE Trans Signal Process. 1994;42(1):88–101.Google Scholar
  23. 23.
    Priestley MB. Spectral analysis and time series. J Am Stat Assoc. 1981;79(385).Google Scholar
  24. 24.
    Högbom JA. Aperture synthesis with a non-regular distribution of interferometer baselines. Astron & Astrophys Suppl. 1974;15(15):417.Google Scholar
  25. 25.
    Gough PT. A fast spectral estimation algorithm based on the FFT. IEEE Trans Signal Process. 1994;42(6):1317–22.Google Scholar
  26. 26.
    Ziskind I, Wax M. Maximum likelihood localization of multiple sources by alternating projection. IEEE Trans Acoust Speech Signal Process. 1988;36(10):1553–60.zbMATHGoogle Scholar
  27. 27.
    Hwang JK, Chen YC. Super-resolution frequency estimation by alternating notch periodogram. IEEE Trans Signal Process. 1993;41(2):727–41.zbMATHGoogle Scholar
  28. 28.
    Tsao J, Steinberg BD. Reduction of sidelobe and speckle artifacts in microwave imaging: the CLEAN technique. IEEE Trans Antennas Propag. 1988;36(4):543–56.Google Scholar
  29. 29.
    Kay SM. Modern spectral estimation: theory and application. London: Prentice Hall International; 1988.zbMATHGoogle Scholar
  30. 30.
    Stoica P, Nehorai A. Statistical analysis of two nonlinear least-squares estimators of sine-wave parameters in the colored-noise case. Circuits, Syst, Signal Process. 1989;8(1):3–15.MathSciNetzbMATHGoogle Scholar
  31. 31.
    Strang G. Linear algebra and its applications. Brooks Cole: Cengage Learning; 1976.zbMATHGoogle Scholar
  32. 32.
    Stoica P, Moses RL, Friedlander B, et al. Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements. IEEE Trans Acoust Speech Signal Process. 2010;37(3):378–92.Google Scholar
  33. 33.
    Soderstrom T, Stoica P. System identification. London: Prentice Hall International; 1989.zbMATHGoogle Scholar
  34. 34.
    Graham A. Kronecker products and matrix calculus with applications. Chichester: Ellis Horwood; 1982.zbMATHGoogle Scholar
  35. 35.
    Liu ZS, Li J. Implementation of the RELAX algorithm. IEEE Trans Aerosp Electron Syst. 1998;34(2):657–64.Google Scholar
  36. 36.
    Bandler JW. Optimization methods for computer-aided design. IEEE Trans Microw Theory Tech. 1969;17(8):533–52.Google Scholar
  37. 37.
    Stoica P, Eykhoff P, Janssen P. Model-structure selection by cross-validation. Int J Control. 1986;43(6):1841–78.MathSciNetzbMATHGoogle Scholar
  38. 38.
    Hannan EJ. Estimating the dimension of a linear system. J Multivar Anal. 1981;11(4):459–73.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kay SM, Marple SLJ. Spectrum analysis—a modern perspective. Proc IEEE. 1981;69(11):1380–419.Google Scholar
  40. 40.
    Stoica P, Nehorai A. MUSIC maximum likelihood and Cramer-Rao bound. IEEE Trans Acoust Speech Signal Process. 1989;37(5):720–41.MathSciNetzbMATHGoogle Scholar
  41. 41.
    Yau SF, Bresler Y. A compact Cramer-Rao bound expression for parametric estimation of superimposed signals. IEEE Trans Signal Process. 1992;40(5):1226–30.zbMATHGoogle Scholar
  42. 42.
    Shu JJ. Parameter estimation algorithm for exponential decay sinusoidal signal and its application. Wuhan University of Engineering Master’s thesis, WuHan;2011.Google Scholar
  43. 43.
    Umesh S, Tufts DW. Estimation of parameters of exponentially damped sinusoids using fast maximum likelihood estimation with application to NMR spectroscopy data. IEEE Trans Signal Process. 1996;44(9):2245–59.Google Scholar
  44. 44.
    Tufts DW, Kumaresan R, Kirsteins I. Data adaptive signal estimation by singular value decomposition of a data matrix. Proc IEEE. 1982;70(6):684–5.Google Scholar
  45. 45.
    Cadzow JA. Signal enhancement-a composite property mapping algorithm. IEEE Trans Acoust Speech Signal Process. 1988;36(1):49–62.MathSciNetzbMATHGoogle Scholar
  46. 46.
    van Huffel S. Enhanced resolution based on minimum variance estimation and exponential data modeling. Sig Process. 1993;33(93):333–55.Google Scholar
  47. 47.
    Li J, Stoica P. Angle and waveform estimation via RELAX. IEEE Trans Aerosp & Electron Syst Aes. 1997;33(3):1077–87.Google Scholar
  48. 48.
    Karmanov VG. Programmation mathematique. Moscow: Editions Mir; 1977.zbMATHGoogle Scholar
  49. 49.
    Zangwill WI, Mond B. Nonlinear programming: a unified approach. New Jersey: Prentice-Hall; 1969.zbMATHGoogle Scholar
  50. 50.
    Bunday BD. Basic optimization methods. London: Edward Arnold Ltd; 1984.Google Scholar
  51. 51.
    Bresler Y, Macovski A. Exact maximum likelihood parameter estimation of superimposed exponential signals in noise. IEEE Trans Acoust Speech Signal Process. 1986;34(5):1081–9.Google Scholar
  52. 52.
    Kumaresan R, Scharf L, Shaw A. An algorithm for pole-zero modeling and spectral analysis. IEEE Trans Acoust Speech Signal Process. 1986;34(3):637–40.Google Scholar
  53. 53.
    Li J, Stoica P, Liu ZS. Comparative study of IQML and mode for direction-of-arrival estimation. IEEE Int Conf Acoust, Speech, Signal Process. 1997;5:3509–12.Google Scholar
  54. 54.
    Kumaresan R, Tufts DW. Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise. IEEE Trans Acoust Speech Signal Process. 1982;30(6):833–40.Google Scholar
  55. 55.
    Stoica P, Li J, Söderström T. On the inconsistency of IQML. Sig Process. 1997;56(2):185–90.zbMATHGoogle Scholar
  56. 56.
    Stoica P, Nehorai A. Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Trans Acoust Speech Signal Process. 1990;38(10):1783–95.zbMATHGoogle Scholar
  57. 57.
    Bangs WJ. Array processing with generalized beamformers. Ph.D. dissertation, Yale University, New Haven; 1971.Google Scholar
  58. 58.
    Wu R, Li J, Bi Z, et al. SAR image formation via semiparametric spectral estimation. IEEE Trans Aerosp Electron Syst. 1999;35(4):1318–33.Google Scholar
  59. 59.
    Zhou G. Random amplitude and polynomial phase modeling of non stationary processes using higher-order and cyclic statistics. Ph.D. dissertation, University of Virginia, Charlottesville; 1995.Google Scholar
  60. 60.
    Besson O, Stoica P. Sinusoidal signals with random amplitude: least-squares estimators and their statistical analysis. IEEE Trans Signal Process. 1995;43(11):2733–44.Google Scholar
  61. 61.
    Doviak RJ, Zrnic DS. Doppler radar & weather observations. New York: Dover Publications Inc; 2014.Google Scholar
  62. 62.
    Baggeroer AB, Metzger LS, Moura JMF, et al. Detection, estimation, and modulation theory. A Papoulis Probab Random Var & Stoch Process. 1968;8(10):293–303.Google Scholar
  63. 63.
    Stoica P, Jakobsson A, Li J. Cisoid parameter estimation in the colored noise case: asymptotic Cramer-Rao bound, maximum likelihood, and nonlinear least-squares. IEEE Trans Signal Process. 1997;45(8):2048–59.Google Scholar
  64. 64.
    Rosenlicht M. Introduction to spectral analysis. New York: Dover Publications Inc; 1997.zbMATHGoogle Scholar
  65. 65.
    Proakis J. Digital communications. New York: McGraw-Hill Science; 1995.zbMATHGoogle Scholar
  66. 66.
    Zhou G, Giannakis GB. Harmonics in multiplicative and additive noise: performance analysis of cyclic estimators. IEEE Trans Signal Process. 1995;43(6):1445–60.Google Scholar
  67. 67.
    Giannakis GB, Zhou G. Harmonics in multiplicative and additive noise: parameter estimation using cyclic statistics. IEEE Trans Signal Process. 1995;43(9):2217–21.Google Scholar
  68. 68.
    Francos JM, Friedlander B. Bounds for estimation of multicomponent signals with random amplitude and deterministic phase. IEEE Trans Signal Process. 1995;43(5):1161–72.Google Scholar

Copyright information

© Science Press, Beijing and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Renbiao Wu
    • 1
    Email author
  • Qiongqiong Jia
    • 1
  • Lei Yang
    • 1
  • Qing Feng
    • 1
  1. 1.Tianjin Key Lab for Advanced Signal ProcessingCivil Aviation University of ChinaTianjinChina

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