Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Bounds on SIMO and MIMO Channel Estimation and Equalization with Side Information

  • Published:
Journal of VLSI signal processing systems for signal, image and video technology Aims and scope Submit manuscript

Abstract

Constrained Cramér-Rao bounds are developed for convolutive multi-input multi-output (MIMO) channel and source estimation in additive Gaussian noise. Properties of the MIMO Fisher information matrix (FIM) are studied, and we develop the maximum rank of the unconstrained FIM and provide necessary conditions for the FIM to achieve full rank. Equality constraints on channel and signal parameters provide a means to study the potential value of side information, such as training symbols (semi-blind case), constant modulus (CM) sources, or known channels. Nonredundant constraints may be combined in an arbitrary fashion, so that side information may be different for different sources. The bounds are useful for evaluating the performance of SIMO and MIMO channel estimation and equalization algorithms. We present examples using the constant modulus blind equalization algorithm. The constrained bounds are also useful for evaluating the relative value of different types of side information, and we present examples comparing semi-blind, constant modulus, and known channel constraints. While the examples presented are primarily in the communications context, the CRB framework applies generally to convolutive source separation problems.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B.M. Sadler, R.J. Kozick, and T. Moore, “Constrained CRBs for Channel and Signal Estimation of MIMO Systems,” in Proc. of 2001 Conf. on Info. Sci. and Syst. (CISS'01), Johns Hopkins University, March 2001.

  2. B.M. Sadler, R.J. Kozick, and T. Moore, “Bounds on MIMO Channel Estimation and Equalization with Side Information,” in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), Salt Lake City, Utah, 2001.

  3. L. Tong and S. Perreau, “Multichannel Blind Identification: From Subspace to Maximum Likelihood Methods,” IEEE Proc., vol. 86,no. 10, 1998, pp. 1951-1968.

    Article  Google Scholar 

  4. J.K. Tugnait, L. Tong, and Z. Ding, “Single-User Channel Estimation and Equalization,” IEEE Sig. Proc. Mag., vol. 17,no. 3, 2000, pp. 16-28.

    Article  Google Scholar 

  5. B. Agee, “The Least-Squares CMA: A New Technique for Rapid Correction of Constant Modulus Signals,” in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), 1986, pp. 953-956.

  6. R. Gooch and J. Lundell, “The CM Array: An Adaptive Beamformer for Constant Modulus Signals,” in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), 1986, pp. 2523-2526.

  7. A.-J. van der Veen and A. Paulraj, “An Analytical Constant Modulus Algorithm,” IEEE Trans. Signal Processing, vol. 44,no. 5, 1996, pp. 1136-1155.

    Article  Google Scholar 

  8. P. Comon, “Independent Component Analysis—A New Concept?,” Signal Processing, vol. 36,no. 3, 1994, pp. 287-314.

    Article  MATH  Google Scholar 

  9. A. Swami, G. Giannakis, and S. Shamsunder, “Multichannel ARMA Processes,” IEEE Trans. Signal Processing, vol. 42,no. 4, 1994, pp. 898-913.

    Article  Google Scholar 

  10. B.M. Sadler and R.J. Kozick, “Bounds on Uncalibrated Array Signal Processing,” in 10th IEEE Workshop on Statistical Signal and Array Processing, (SSAP'00), Poconos Manor, PA, Aug. 2000, pp. 73-77.

  11. B.M. Sadler, R.J. Kozick, and T. Moore, “Semi-Blind Array Processing with Constant Modulus Signals,” in Proc. 4th World Multiconf. on Syst., Cyb. and Inform. (SCI 2000), Orlando, FL, July 23–26, 2000, vol. VI, pp. 241-246.

    Google Scholar 

  12. B.M. Sadler, R.J. Kozick, and T. Moore, “Bounds on Bearing and Symbol Estimation with Side Information,” IEEE Trans. Signal Processing, vol. 49,no. 4, 2001, pp. 822-834.

    Article  Google Scholar 

  13. J.D. Gorman and A.O. Hero “Lower Bounds for Parametric Estimation with Constraints,” IEEE Trans. Info. Theory, vol. 26,no. 6, 1990, pp. 1285-1301.

    Article  MathSciNet  Google Scholar 

  14. P. Stoica and B.C. Ng, “On the Cramér-Rao Bound Under Parametric Constraints,” IEEE Sig. Proc. Letters, vol. 5,no. 7, 1998, pp. 177-179.

    Article  Google Scholar 

  15. P. Stoica and B.C. Ng, “Performance Bounds for Blind Channel Estimation,” in Signal Processing Advances in Wireless Communications, Volume 1: Trends in Channel Estimation and Equalization, G.B. Giannakis, Y. Hua, P. Stoica, and L. Tong (Eds.), Prentice-Hall, Upper Saddle River, NJ, USA, 2001.

    Google Scholar 

  16. T.L. Marzetta, “A Simple Derivation of the Constrained Multiple Parameter Cramér-Rao Bound,” IEEE Trans. Signal Processing, vol. 41,no. 6, 1993, pp. 2247-2249.

    Article  MATH  Google Scholar 

  17. S. Barbarossa, A. Scaglione, and G.B. Giannakis, “Performance Analysis of a Deterministic Channel Estimator for Block Transmission Systems with Null Guard Intervals,” IEEE Trans. Signal Processing, to appear.

  18. Z. Liu, G.B. Giannakis, A. Scaglione, and S. Barbarossa, “Decoding and Equalization of Unknown Multipath Channels Based on Block Precoding and Transmit-Antenna Diversity,” in Proc. 33rd Asilomar Conf. on Sigs., Syst., and Comp., 1999, vol. 2, pp. 1557-1561.

    Google Scholar 

  19. A. Dogandzic and A. Nehorai, “Estimating Evoked Dipole Responses in Unknown Spatially Correlated Noise with EEG/MEG Arrays,” IEEE Trans. Signal Processing, vol. 48,no. 1, 2000, pp. 13-25.

    Article  MATH  Google Scholar 

  20. E. De Carvalho and D.T.M. Slock, “Cramér-Rao Bounds for Semi-Blind, Blind and Training Sequence Based Channel Estimation,” in Proc. 1997 IEEE Workshop on Sig. Proc. Adv. in Wireless Comm. (SPAWC'97), Paris, France, April 1997, pp. 129-132.

  21. J.L. Bapat, “Partially Blind Estimation: ML-Based Approaches and Cramér-Rao Bound,” Signal Processing, vol. 71, 1998, pp. 265-277.

    Article  MATH  Google Scholar 

  22. A.O. Hero, J.A. Fessler, and M. Usman, “Exploring Estimator Bias-Variance Tradeoffs Using the Uniform CR Bound,” IEEE Trans. Signal Processing, vol. 44, 1996, pp. 2026-2041.

    Article  Google Scholar 

  23. P. Stoica and T.L. Marzetta, “Parameter Estimation Problems with Singular Information Matrices,” IEEE Trans. Signal Processing, vol. 49,no. 1, 2001, pp. 87-90.

    Article  MathSciNet  Google Scholar 

  24. Y. Hua, “Fast Maximum Likelihood for Blind Identification of Multiple FIR Channels,” IEEE Trans. Signal Processing, vol. 44,no. 3, 1996, pp. 661-672.

    Article  Google Scholar 

  25. S.M. Kay, Fundamentals of Statistical Signal Processing, Estimation Theory, Prentice-Hall, Upper Saddle River, NJ, USA, 1993.

    MATH  Google Scholar 

  26. B. Hochwald and A. Nehorai, “On Identifiability and Information-Regularity in Parameterized Normal Distributions,” Circuits, Systems, Signal Processing, vol. 16,no. 1, 1997, pp. 83-89.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Soderstrom and P. Stoica, System Identification, Prentice-Hall, Cambridge University Press, 1989.

  28. K. Abed-Meraim and Y. Hua, “Strict Identifiability of Multichannel FIR Systems: Further Results and Developments,” in Proc. Int. Conf. on Telecomm., Melbourne, Australia, April 1997, pp. 1029-1032.

  29. K. Abed-Meraim, W. Qiu, and Y. Hua, “Blind System Identification,” IEEE Proc., vol. 85,no. 8, 1997, pp. 1310-1322.

    Article  Google Scholar 

  30. Z. Ding, “Characteristics of Band-Limited Channels Unidentifiable from Second-Order Cyclostationary Statistics,” IEEE Signal Processing Letters, vol. 3,no. 5, 1996, pp. 150-152.

    Article  Google Scholar 

  31. Y. Hua and M. Wax, “Strict Identifiability of Multiple FIR Channels Driven by an Unknown Arbitrary Sequence,” IEEE Trans. Signal Processing, vol. 44,no. 3, 1996, pp. 756-759.

    Article  Google Scholar 

  32. V.U. Reddy, C.B. Papadias, and A.J. Paulraj, “Blind Identifiability of Certain Classes of Multipath Channels From Second-Order Statistics Using Antenna Arrays,” IEEE Signal Processing Letters, vol. 4,no. 5, May 1997, pp. 138-141.

    Article  Google Scholar 

  33. J.K. Tugnait, “On Blind Identifiability of Multipath Channels Using Fractional Sampling and Second-Order Cyclostationary Statistics,” IEEE Trans. Info. Theory, vol. 41,no. 1, 1995, pp. 308-311.

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Serpedin, A. Chevreuil, G.B. Giannakis, and P. Loubaton, “Blind Channel and Carrier Frequency Offset Estimation Using Periodic Modulation Precoders,” IEEE Trans. Signal Processing, vol. 48,no. 8, 2000, pp. 2389-2405.

    Article  Google Scholar 

  35. S. Barbarossa and A. Scaglione, “Blind Equalization Using Cost Functions Matched to the Signal Constellation,” in Proc. 31st Asilomar Conf. Sig. Sys. Comp., Pacific Grove, CA, Nov. 1997, vol. 1, pp. 550-554.

    Google Scholar 

  36. A. Swami, S. Barbarossa, and B.M. Sadler, “Blind Source Separation and Signal Classification,” in Proc. 34th Asilomar Conf. Sig. Sys. Comp., Pacific Grove, CA, Nov. 2000.

  37. S. Barbarossa, A. Swami, B.M. Sadler, and G. Spadafora, “Classification of Digital Constellations Under Unknown Multipath Propagation Conditions,” in Proc. SPIE, Digital Wireless Comm. II, Orlando, FL, April 2000.

  38. D.T.M. Slock, “Blind Fractionally-Spaced Equalization, Perfect Reconstruction Filter Banks, and Multichannel Linear Prediction,” in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), Adelaide, Australia, 1994, pp. 585-588.

  39. M. Dong and L. Tong, “Optimal Design and Placement of Pilot Symbols for Channel Estimation,” in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), Salt Lake City, Utah, 2001.

  40. S. Ohno and G.B. Giannkis, “Optimal Training and Redundant Precoding for Block Transmissions with Application to Wireless OFDM,” in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc. (ICASSP), Salt Lake City, Utah, 2001.

  41. C.R. Johnson, Jr., P. Schniter, T.J. Endres, J.D. Behm, D.R. Brown, and R.A. Casas, “Blind Equalization Using the Constant Modulus Criterion: A Review,” IEEE Proc., vol. 86,no. 10, 1998, pp. 1927-1950.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Army Research Laboratory, AMSRL-CI-CN, 2800 Powder Mill Road, Adelphi, MD, 20783, USA

    Brian M. Sadler, Terrence Moore & Ananthram Swami

  2. Department of Electrical Engineering, Bucknell University, Lewisburg, PA, 17837, USA

    Richard J. Kozick

Authors
  1. Brian M. Sadler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard J. Kozick
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Terrence Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ananthram Swami
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sadler, B.M., Kozick, R.J., Moore, T. et al. Bounds on SIMO and MIMO Channel Estimation and Equalization with Side Information. The Journal of VLSI Signal Processing-Systems for Signal, Image, and Video Technology 30, 107–126 (2002). https://doi.org/10.1023/A:1014046808970

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014046808970

Search

Navigation