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Professor C R Rao’s contributions in statistical signal processing and its long-term implications

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Abstract

Professor C R Rao has made significant contributions in different areas of statistics and in related fields particularly in inference, biometrics, design of experiments, linear models, variance components, econometrics, and most of his contributions are well known to the statistical community. But it may not be known to many statisticians that Professor Rao has worked quite extensively, for about six to seven years, on statistical signal processing and has made some fundamental contributions which has generated significant interest along that direction. He along with his collaborators have worked mainly on three classical signal processing problems, and provided theoretical foundations and efficient estimation procedures. In my opinion, the main contribution of Professor Rao is that he has provided new insights in all these problems, which has helped to bring new way of solving these and some related problems. He has guided two Ph.D. students in the area of Statistical Signal Processing. The main aim of this article is two-fold. First, we would like to introduce to the statistical community the contribution of Professor Rao in this area and our second aim is to provide a class of different related open problems which are of interest and which require sophisticated statistical tools to provide efficient solutions.

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References

  1. Abatzoglou T, Fast maximum likelihood joint estimation of frequency and frequency rate, IEEE Trans. Aerosp. Electron. Syst. 22 (1986) 708– 715

    Google Scholar 

  2. Bai Z D, Krishnaiah R and Zhao L C, On the simultaneous estimation of the number of signals and frequencies under a model with multiple sinusoids, Technical Report, 86–37 (1986) (Center for Multivariate Analysis, University of Pittsburgh)

  3. Bai Z D, Miao B Q and Rao C R, Estimation of direction of arrival of signals: asymptotic results, Advances in Spectrum Analysis and Array Processing, edited by S Kaykin (1991) (NJ: Prentice Hall, Englewood, Cliffs) vol. II, Chapter 9

  4. Bai Z D and Rao C R, Spectral analytic methods for the estimation of signals and direction of arrival, Spectral Analysis in One or Two dimensions, edited by S Prasad and R L Kashyap (1989) (New Delhi: Oxford and IBH Publications Co.) pp. 493–507

  5. Bai Z D, Rao C R, Chow M and Kundu D, An efficient algorithm for estimating the parameters of superimposed exponential signal, J. Stat. Plan. Inference (2003) vol. 110, pp. 23–34

    MathSciNet  MATH  Google Scholar 

  6. Bai Z D, Rao C R, Wu Y, Zen M-M and Zhao L, The simultaneous estimation of the number of signals and frequencies of multiple sinusoids when some observations are missing: I. Asymptotic, Proc. Natl. Acad. Sci. 96 (1999) 11106–11110

    MathSciNet  MATH  Google Scholar 

  7. Barbieri M M and Barone P, A two dimensional Prony’s method for spectral estimation, IEEE Trans. Signal Process. 40(11) (1992) 2747–2756

    MATH  Google Scholar 

  8. Bates D M and Watts G W, Nonlinear Regression Analysis and Its Applications (1988) (New York: Wiley)

    MATH  Google Scholar 

  9. Bienvenu G and Kopp L, Adaptivity of background noise spatial coherence for high resolution passive methods, Proceedings of the ICASSP 80 (1980) (CO: Denver) pp. 307–310

  10. Bresler Y and Macovski A, Exact maximum likelihood parameter estimation of superimposed exponential signals in noise, IEEE Trans. Acoust. Speech Signal Process. ASSP-34 (1986) pp. 1081–1089

    Google Scholar 

  11. Bronez T P and Cadzow J A, An algebraic approach to super-resolution array processing, IEEE Trans. Aerosp. Electron. Syst. 19 (1983) 123–133

    Google Scholar 

  12. Cabrera S D and Bose N K, Prony’s method for two-dimensional complex exponential modelling, (Chapter 15), Applied Control Theory, edited by S G Tzafestas (1993) (New York: Marcel and Dekker) pp. 401–411

  13. Chun J and Bose N K, Parameter estimation via signal selectivity of signal subspaces (PESS) and its applications, Digital Signal Process. 5 (1995) 58–76

    Google Scholar 

  14. Djurić P M and Kay S M, Parameter estimation of chirp signals, IEEE Trans. Acoust. Speech Signal Process. 38 (1990) 2118–2126

    Google Scholar 

  15. Froberg C E, Introduction to Numerical Analysis, 2nd edition (1969) (Boston: Addision-Wesley Publ. Co.)

    Google Scholar 

  16. Grover R, Frequency and Frequency Rate Estimation of Some Non-stationary Signal Processing Models, Ph.D. thesis (2020) (Kanpur: Indian Institute of Technology Kanpur)

  17. Guo J, Zou H, Yang X and Liu G, Parameter estimation of multicomponent chirp signals via sparse representation, IEEE Trans. Aerosp. Electron. Syst. 47 (2011) 2261–2268

    Google Scholar 

  18. Hilderband F B, An Introduction to Numerical Analysis (1956) (New York: MCGraw Hill)

  19. Hua Y, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil, IEEE Trans. Signal Process. 40 (1992) 2267–2280

    MATH  Google Scholar 

  20. Jennrich R I, Asymptotic properties of the non-linear least squares estimation, Ann. Math. Stat. 40 (1969) 633–643

    MATH  Google Scholar 

  21. Johnson D H and Degra S R, Improving the resolution of bearing in passive sonar arrays by eigenanalysis, IEEE Trans. Acoust. Speech Signal Process 29 (1982) 401–413

    Google Scholar 

  22. Kannan N, Estimation of Direction of Arrival of Signals in Signal Processing, Ph.D. thesis (1992) (USA: Department of Statistics, The Pennsylvania State University, University Park)

  23. Kannan N and Kundu D, On modified EVLP and ML methods for estimating superimposed exponential signals, Signal Process. 39 (1994) 223–233

    MATH  Google Scholar 

  24. Kay S M and Nekovei R, An efficient two-dimensional frequency estimator, IEEE Trans. Acoust. Speech Signal Process. 38 (1990) 1807–1809

    Google Scholar 

  25. Kumaresan R and Tufts D, Estimation of arrival of multiple plane waves, IEEE Trans. Aerosp. Electron. Syst. 19 (1983) 134–139

    Google Scholar 

  26. Kundu D, Asymptotic properties of the complex valued non-linear regression model, Commun. Stat. Theory Methods 20 (1991) 3793–3803

    MATH  Google Scholar 

  27. Kundu D, Estimating the parameters of undamped exponential signals, Technometrics 35 (1993) 215–218

    MathSciNet  MATH  Google Scholar 

  28. Kundu D, Modified MUSIC algorithm for estimating DOA of signals, Signal Process. 48 (1996) 85–90

    Google Scholar 

  29. Kundu D, Estimating DOA of signals: some asymptotic results Sankhya Ser. A 61 (1999) 120–138

    MathSciNet  MATH  Google Scholar 

  30. Kundu D, Estimating the number of signals in the presence of white noise, J. Stat. Plan. Inference 90 (2000) 57–68

    MathSciNet  MATH  Google Scholar 

  31. Kundu D and Gupta R D, Asymptotic properties of the least squares estimators of a two dimensional texture model, Metrika 48 (1998) 83–97

    MathSciNet  MATH  Google Scholar 

  32. Kundu D and Kundu R, Consistent estimates of super imposed exponential signals when some observations are missing, J. Stat. Plan. Inference 44 (1995) 205–218

    MathSciNet  MATH  Google Scholar 

  33. Kundu D and Mitra A, A note on the consistency of the undamped exponential signals model, Statistics 28 (1996) 25–33

    MathSciNet  MATH  Google Scholar 

  34. Kundu D and Mitra A, On asymptotic properties and confidence intervals for exponential signals, Signal Process. 72 (1999) 129–139

    MATH  Google Scholar 

  35. Kundu D and Nandi S, Determination of discrete spectrum in a stationary random field, Stat. Neerl. 57 (2003) 258–284

    MATH  Google Scholar 

  36. Kundu D and Nandi S, Statistical Signal Processing: Frequency Estimation (2012) (New Delhi: Springer)

  37. Lahiri A, Estimators of Parameters of Chirp Signals and their Properties, Ph.D. thesis (2012) (Kanpur: Indian Institute of Technology Kanpur)

  38. McClellan J H, Multidimensional spectral estimation, Proc. IEEE 70 (1982) 1029–1039

    Google Scholar 

  39. Miao B Q, Wu Y and Zhao L C, On strong consistency of a two-dimensional frequency estimation algorithm, Stat. Sin. 8 (1998) 559–570

    MATH  Google Scholar 

  40. Nandi S, Prasad A and Kundu D, An efficient and fast algorithm for estimating the parameters of two-dimensional sinusoidal signals, J. Stat. Plan. Inference 140 (2010) 153–168

    MathSciNet  MATH  Google Scholar 

  41. Owsley N L, Spectral signal set extraction, Aspects of Signal Processing, Part II, edited by G Eacconi (1977) pp. 469–475

  42. Paulraj A, Otteresten B, Roy R, Swindlehurst A, Xu G and Kailath T, Subspace methods for direction of arrival estimation, Handbook of Statistics, edited by N K Bose and C R Rao (1993) (Amsterdam: Elsevier Science Publishers) vol. 10, pp. 693–739 (Chapter 16)

  43. Pillai S U, Array Signal Processing (1989) (New York: Springer)

  44. Rao C R, Sastry C R and Zhou B, Tracking the direction of arrival of multiple moving targets, IEEE Trans. Signal Process. 42(5) (1994) 1133–1144

    Google Scholar 

  45. Rao C R, Zhang L and Zhao L C, New algorithm for multitarget angle tracking, IEE Proc. F 140(5) (1993) 335–338

    Google Scholar 

  46. Rao C R, Zhang L and Zhao L C, Multiple target angle tracking using sensor array outputs, IEEE Trans. Aerosp. Electron. Syst. 29 (1993) 268–271

    Google Scholar 

  47. Rao C R, Zhang L and Zhao L C, Multitarget angle tracking an algorithm for data association, IEEE Trans. Signal Process. 42(2) (1994) 459–462

    Google Scholar 

  48. Rao C R and Zhao L C, Asymptotic behavior of maximum likelihood estimators of superimposed exponential signals, IEEE Trans. Signal Process. 41(3) (1993) 1461–1464

    MATH  Google Scholar 

  49. Rao C R, Zhao L and Zhou B, Maximum likelihood estimation of 2-D superimposed exponential signals, IEEE Trans. Signal Process. 42(7) (1994) 1795–1802

    Google Scholar 

  50. Roy R, Paulraj A and Kailath T, ESPRIT—a subspace rotation approach to estimation of parameters of cisoids in noise, IEEE Trans. Acoust. Speech Signal Process. 34 (1986) 1340–1342

    Google Scholar 

  51. Schmidt R O, A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation, Ph.D. thesis (1981) (USA: Stanford University)

  52. Seber G A F and Wild C J, Nonlinear Regression (1989) (New York: Wiley)

  53. Stoica P and Moses R, Spectral Analysis of Signals (2005) (New Jersey, USA: Prentice Hall)

  54. Wax M, Schmidt R O and Kailath T, Eigen-structure method for retrieving the poles from the natural response, Proceedings of Second CDC, TX (1983) pp. 1343–1344

  55. Wu C F J, Asymptotic theory of non-linear least squares estimation, Ann. Stat. 9 (1981) 501–513

    MATH  Google Scholar 

  56. Zhang H and Mandrekar V, Estimation of hidden frequencies for 2D stationary processes, J. Time Ser. Anal. 22(5) (2001) 613–629

    MathSciNet  MATH  Google Scholar 

  57. Zhao L C, Krishnaiah P R and Bai Z D, On detection of the number of signals in presence of white noise, J. Multivar. Anal. 20 (1986a) 1–25

    MathSciNet  MATH  Google Scholar 

  58. Zhao L C, Krishnaiah P R and Bai Z D, On detection of the number of signals when the noise covariance matrix is arbitrary, J. Multivar. Anal. 20 (1986b) 26–49

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is thankful to Professor Partha Pratim Mazumdar, the President of Indian Academy of Sciences, for organizing the birth centenary symposium of the legendary scientist Professor Rao on December 11–12, 2019 at Bengaluru and also for inviting the author to give a talk. The paper is based on the presentation given at that symposium. Part of the work has been supported by a grant from SERB, Government of India.

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  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208 016, India

    Debasis Kundu

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  1. Debasis Kundu
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Correspondence to Debasis Kundu.

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This article is part of the “Special Issue in Honour of Professor C R Rao on His Birth Centenary”

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Kundu, D. Professor C R Rao’s contributions in statistical signal processing and its long-term implications. Proc Math Sci 130, 43 (2020). https://doi.org/10.1007/s12044-020-00582-8

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