Advertisement

Interpolation

  • Saeed V. Vaseghi
Chapter

Abstract

Interpolation is the estimation of the unknown, or the lost, samples of a signal using a weighted average of a number of known samples at the neighbourhood points. Interpolators are used in various forms in most signal processing and decision making systems. Applications of interpolators include conversion of a discrete-time signal to a continuous-time signal, sampling rate conversion in multi-rate systems, low bit rate speech coding, upsampling of a signal for improved graphical representation, and restoration of a sequence of samples irrevocably distorted by transmission errors, impulsive noise, drop outs etc.

This chapter begins with a study of the ideal interpolation of a band limited signal, a simple model for the effects of a number of missing samples, and the factors that affect interpolation. The classical approach to interpolation is to construct a polynomial that passes through the known samples. In Section 10.2 a general form of polynomial interpolation, and its special forms Lagrange, Newton, Hermite, and cubic spline interpolators are considered. Optimal interpolators utilise predictive and statistical models of the signal process. In Section 10.3 a number of model-based interpolation methods are considered. These methods include maximum a posterior interpolation and least squared error interpolation based on an autoregressive model. Finally we consider time-frequency interpolation, and interpolation through search of an adaptive codebook for the best signal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Bogner R.E., Li T. (1989), Pattern Search Prediction of Speech, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP-89, Pages 180–183, Glasgow.Google Scholar
  2. Cohen L. (1989), Time-Frequency Distributions- A review, Proceedings of the IEEE, Vol. 77(7), Pages 941–81.CrossRefGoogle Scholar
  3. Crochiere R.E., Rabiner L.R. (1981), Interpolation and Decimation of Digital Signals-A Tutorial review, Proc. IEEE, Vol. 69, Pages. 300–331, March.CrossRefGoogle Scholar
  4. Godsill S.J. (1993), The Restoration of Degraded Audio Signals, Ph.D. Thesis, Cambridge University, Cambridge.Google Scholar
  5. Godsill S.J. and P.J.W. Rayner (1993), Frequency domain interpolation of sampled signals, IEEE Int. Conf., Speech and Signal Processing, ICASSP-93, Minneapolis.Google Scholar
  6. Janssen A.J., Veldhuis R., Vries L.B (1984),Adaptive Interpolation of Discrete-Time Signals that can be Modelled as Autoregressive Processes, Proc. IEEE Trans. Acoustics, Speech and Signal Processing, Vol. ASSP-34, No. 2, Pages 317–330 June.Google Scholar
  7. Janssen A.J., Vries L.B (1984), Interpolation of Band-Limited Discrete-Time Signals by Minimising Out-of Band Energy, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP-84.Google Scholar
  8. Kay S.M. (1983), Some Results in Linear Interpolation theory, IEEE Trans. Acoustics Speech and Signal Processing, Vol. ASSP-31, Pages 746–49, June.MathSciNetCrossRefGoogle Scholar
  9. Kay S. M. (1988) Modern Spectral Estimation: Theory and Application. Prentice-Hall, Englewood Cliffs, N.J.zbMATHGoogle Scholar
  10. Kolmogrov A.N. (1939),Sur 1’ Interpolation et Extrapolation des Suites Stationaires, Comptes Rendus de l’Academie des Sciences, Vol. 208, Pages 2043–45.Google Scholar
  11. Kubin G., Kleijin W.B. (1994) Time-Scale Modification of Speech Based on a Nonlinear Oscillator Model, Proc. IEEE Int. Conf., Speech and Signal Processing, ICASSP-94, Pages 1453–56, Adelaide.Google Scholar
  12. Lochart G.B., Goodman D.J. (1986), Reconstruction of missing speech packets by waveform substitution Signal processing 3: Theories and Applications, Pages 357–360.Google Scholar
  13. Marks R.J. (1983), Restoring Lost Samples from an over-sampled band-limited signal, IEEE Trans. Acoustics, Speech and Signal Processing, Vol. ASSP-31, No 2. Pages.752–55, June.CrossRefGoogle Scholar
  14. Marks R.J. (1991), Introduction to Shannon Sampling and Interpolation Theory, Springer Verlag.zbMATHCrossRefGoogle Scholar
  15. Mathews J.H. (1992), Numerical Methods for Mathematics, Science and Engineering, Prentice-Hall, Englewood Cliffs, N.J.zbMATHGoogle Scholar
  16. Musicus B. R. (1982), Iterative Algorithms for Optimal Signal Reconstruction and Parameter Identification Given Noisy and Incomplete Data, Ph.D. Thesis, MIT August.Google Scholar
  17. Platte H.J., Rowedda V. (1985), A Burst Error Concealment Method for Digital Audio Tape Application. AES preprint, 2201:1–16.Google Scholar
  18. Press W.H., Flannery B.P., Teukolsky S.A., Vettereling W.T. (1992), Numerical Recepies in C., Second edition, Cambridge University Press, Cambridge.Google Scholar
  19. Nakamura S. (1991), Applied Numerical Methods with Software, Prentice-Hall, Englewood Cliffs, N.J.zbMATHGoogle Scholar
  20. Schafer, R.W., Rabiner, L.R. (1973), A Digital Signal Processing Approach to Interpolation, Proc. IEEE, Vol. 61, Pages 692–702, June.CrossRefGoogle Scholar
  21. Steele R., Jayant N. S. (1980), Statistical Block Coding for DPCM-AQF Speech, IEEE Trans On Communications, Vol. COM-28, No. 11, Pages 1899–1907, Nov.CrossRefGoogle Scholar
  22. Tong H. (1990), Nonlinear Time Series A Dynamical System Approach, Oxford University Press.Google Scholar
  23. Vaseghi S.V. (1988), Algorithms for Restoration of Gramophone Records, Ph.D. Thesis, Cambridge University, Cambridge.Google Scholar
  24. Veldhuis R. (1990), Restoration of Lost samples in Digital Signals. Prentice-Hall.Google Scholar
  25. Verhelst W., Roelands M. (1993), An Overlap-Add Technique Based on Waveform Similarity (Wsola) for High Quality Time-Scale Modification of Speech, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP-93, Pages II-554–II-557, Adelaide.Google Scholar
  26. Wiener N. (1949), Extrapolation, Interpolation and Smoothing of Stationary Time Series With Engineering Applications, MIT Press, Cambridge, Mass.zbMATHGoogle Scholar

Copyright information

© John Wiley & Sons Ltd. and B.G. Teubner 1996

Authors and Affiliations

  • Saeed V. Vaseghi
    • 1
  1. 1.Queen’s UniversityBelfastUK

Personalised recommendations