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Reconstruction of Nonuniformly Sampled Bandlimited Signals by Means of Time-Varying Discrete-Time FIR Filters

  • Håkan Johansson
  • Per Löwenborg
Open Access
Research Article
Part of the following topical collections:
  1. Frames and Overcomplete Representations in Signal Processing, Communications, and Information Theory

Abstract

This paper deals with reconstruction of nonuniformly sampled bandlimited continuous-time signals using time-varying discrete-time finite-length impulse response (FIR) filters. The main theme of the paper is to show how a slight oversampling should be utilized for designing the reconstruction filters in a proper manner. Based on a time-frequency function, it is shown that the reconstruction problem can be posed as one that resembles an ordinary filter design problem, both for deterministic signals and random processes. From this fact, an analytic least-square design technique is then derived. Furthermore, for an important special case, corresponding to periodic nonuniform sampling, it is shown that the reconstruction problem alternatively can be posed as a filter bank design problem, thus with requirements on a distortion transfer function and a number of aliasing transfer functions. This eases the design and offers alternative practical design methods as discussed in the paper. Several design examples are included that illustrate the benefits of the proposed design techniques over previously existing techniques.

Keywords

Filter Bank Reconstruction Problem Important Special Case Deterministic Signal Reconstruction Filter 

References

  1. 1.
    Marvasti F (Ed): Nonuniform Sampling: Theory and Practice. Kluwer Academic, New York, NY, USA; 2001.zbMATHGoogle Scholar
  2. 2.
    Black WC, Hodges DA: Time interleaved converter arrays. IEEE Journal of Solid-State Circuits 1980, 15(6):1022–1029. 10.1109/JSSC.1980.1051512CrossRefGoogle Scholar
  3. 3.
    Yen JL: On nonuniform sampling of bandwidth-limited signals. IRE Transactions on Circuit Theory 1956, 3(4):251–257.CrossRefGoogle Scholar
  4. 4.
    Jerri AJ: The Shannon sampling theorem—its various extensions and applications: a tutorial review. Proceedings of IEEE 1977, 65(11):1565–1596.CrossRefGoogle Scholar
  5. 5.
    Papoulis A: Generalized sampling expansion. IEEE Transactions on Circuits and Systems 1977, 24(11):652–654. 10.1109/TCS.1977.1084284MathSciNetCrossRefGoogle Scholar
  6. 6.
    Choi H, Munson DC: Analysis and design of minimax optimal interpolators. IEEE Transactions on Signal Processing 1998, 46(6):1571–1579. 10.1109/78.678470CrossRefGoogle Scholar
  7. 7.
    Eldar YC, Oppenheim AV: Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples. IEEE Transactions on Signal Processing 2000, 48(10):2864–2875. 10.1109/78.869037MathSciNetCrossRefGoogle Scholar
  8. 8.
    Saramäki T: Finite impulse response filter design. In Handbook for Digital Signal Processing. Edited by: Mitra SK, Kaiser JF. John Wiley & Sons, New York, NY, USA; 1993:155–277. chapter 4Google Scholar
  9. 9.
    Jin H, Lee EKF: A digital-background calibration technique for minimizing timing-error effects in time-interleaved ADCs. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 2000, 47(7):603–613. 10.1109/82.850419CrossRefGoogle Scholar
  10. 10.
    Higgins J: A sampling theorem for irregularly spaced sample points (Corresp.). IEEE Transactions on Information Theory 1976, 22(5):621–622. 10.1109/TIT.1976.1055596MathSciNetCrossRefGoogle Scholar
  11. 11.
    Namgoong W: Finite-length synthesis filters for non-uniformly time-interleaved analog-to-digital converter. Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '02), May 2002, Phoenix-Scottsdale, Ariz, USA 4: 815–818.Google Scholar
  12. 12.
    Prendergast RS, Levy BC, Hurst PJ: Reconstruction of bandlimited periodic nonuniformly sampled signals through multirate filter banks. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 2004, 51(12):1612–1622.CrossRefGoogle Scholar
  13. 13.
    Johansson H, Löwenborg P: Reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters. IEEE Transactions on Signal Processing 2002, 50(11):2757–2767. 10.1109/TSP.2002.804089CrossRefGoogle Scholar
  14. 14.
    Johansson H, Löwenborg P: Reconstruction of nonuniformly sampled bandlimited signals using time-varying discrete-time FIR filters. Proceedings of 12th European Signal Processing Conference (EUSIPCO '04), September 2004, Vienna, AustriaGoogle Scholar
  15. 15.
    Johansson H, Löwenborg P: Reconstruction of periodically nonuniformly sampled bandlimited signals using time-varying FIR filters. Proceedings of 4th International Workshop Spectral Methods Multirate Signal Processing (SMMSP '04), September 2004, Vienna, AustriaGoogle Scholar
  16. 16.
    Maeda N: Transversal filters with nonuniform tap spacings. IEEE Transactions on Circuits and Systems 1980, 27(1):1–11. 10.1109/TCS.1980.1084712MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jarske P, Saramäki T, Mitra SK, Neuvo Y: On properties and design of nonuniformly spaced linear arrays [antennas]. IEEE Transactions on Acoustics, Speech, Signal Processing 1988, 36(3):372–380. 10.1109/29.1534CrossRefGoogle Scholar
  18. 18.
    Coleman JO: Choosing nonuniform tap spacings for a tapped-delay-line filter. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 1996, 43(4):298–303. 10.1109/82.488284CrossRefGoogle Scholar
  19. 19.
    Nash SG, Sofer A: Linear and Nonlinear Programming. McGraw-Hill, New York, NY, USA; 1996.Google Scholar
  20. 20.
    Laakso TI, Valimäki V, Karjalainen M, Laine UK: Splitting the unit delay [FIR/all pass filters design]. Signal Processing Magazine 1996, 13(1):30–60. 10.1109/79.482137CrossRefGoogle Scholar
  21. 21.
    Vaidyanathan PP: Multirate Systems and Filter Banks. Prentice-Hall, Englewood Cliffs, NJ, USA; 1993.zbMATHGoogle Scholar
  22. 22.
    Vaidyanathan PP, Liu VC: Efficient reconstruction of band-limited sequences from nonuniformly decimated versions by use of polyphase filter banks. IEEE Transactios on Acoustics, Speech, Signal Processing 1990, 38(11):1927–1936. 10.1109/29.103094CrossRefGoogle Scholar
  23. 23.
    Johansson H, Löwenborg P: Reconstruction of a class of nonuniformly sampled and decimated bandlimited signals. Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '02), May 2002, Phoenix-Scottsdale, Ariz, USA 2: 604–607.Google Scholar
  24. 24.
    Parks TW, Burrus CS: Digital Filter Design. John Wiley & Sons, New York, NY, USA; 1987.zbMATHGoogle Scholar
  25. 25.
    Hanzo L, Münster M, Choi BJ, Keller T: OFDM and MC-CDMA for Broadband Multi-user Communications, WLANs and Broadcasting. John Wiley & Sons, West Sussex, UK; 2003.CrossRefGoogle Scholar
  26. 26.
    Vesma J, Saramäki T: Optimization and efficient implementation of FIR filters with adjustable fractional delay. Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '97), June 1997, Hong Kong 4: 2256–2259.Google Scholar
  27. 27.
    Johansson H, Löwenborg P: On the design of adjustable fractional delay FIR filters. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 2003, 50(4):164–169. 10.1109/TCSII.2003.809712CrossRefGoogle Scholar

Copyright information

© Johansson and Löwenborg 2006

Authors and Affiliations

  • Håkan Johansson
    • 1
  • Per Löwenborg
    • 1
  1. 1.Electronics Systems of the Department of Electrical EngineeringLinköping UniversityLinköpingSweden

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