Quaternionic Spectral Analysis of Non-Stationary Improper Complex Signals

Part of the Trends in Mathematics book series (TM)


We consider the problem of the spectral content of a complex improper signal and the time-varying behaviour of this spectral content. The signals considered are one-dimensional (1D), complex-valued, with possible correlation between the real and imaginary parts, i.e., improper complex signals. As a consequence, it is well known that the ‘classical’ (complex-valued) Fourier transform does not exhibit Hermitian symmetry and also that it is necessary to consider simultaneously the spectrum and the pseudo-spectrum to completely characterize such signals. Hence, an ‘augmented’ representation is necessary. However, this does not provide a geometric analysis of the complex improper signal. We propose another approach for the analysis of improper complex signals based on the use of a 1D Quaternion Fourier Transform (QFT). In the case where complex signals are non-stationary, we investigate the extension of the well-known ‘analytic signal’ and introduce the quaternion-valued ‘hyperanalytic signal’. As with the hypercomplex two-dimensional (2D) extension of the analytic signal proposed by Bülow in 2001, our extension of analytic signals for complex-valued signals can be obtained by the inverse QFT of the quaternion-valued spectrum after suppressing negative frequencies. Analysis of the hyperanalytic signal reveals the time-varying frequency content of the corresponding complex signal. Using two different representations of quaternions, we show how modulus and quaternion angles of the hyperanalytic signal are linked to geometric features of the complex signal. This allows the definition of the angular velocity and the complex envelope of a complex signal. These concepts are illustrated on synthetic signal examples. The hyperanalytic signal can be seen as the exact counterpart of the classical analytic signal, and should be thought of as the very first and simplest quaternionic time-frequency representation for improper non-stationary complex-valued signals.


Quaternions complex signals Fourier transform analytic signal 


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  1. [1]
    T. Bülow. Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD thesis, University of Kiel, Germany, Institut für Informatik und Praktische Mathematik, Aug. 1999.Google Scholar
  2. [2]
    T. Bülow and G. Sommer. Hypercomplex signals – a novel extension of the analytic signal to the multidimensional case. IEEE Transactions on Signal Processing, 49(11):2844–2852, Nov. 2001.MathSciNetCrossRefGoogle Scholar
  3. [3]
    M.A. Delsuc. Spectral representation of 2D NMR spectra by hypercomplex numbers. Journal of magnetic resonance, 77(1):119–124, Mar. 1988.Google Scholar
  4. [4]
    T.A. Ell and S.J. Sangwine. Hypercomplex Wiener-Khintchine theorem with application to color image correlation. In IEEE International Conference on Image Processing (ICIP 2000), volume II, pages 792–795, Vancouver, Canada, 11–14 Sept. 2000. IEEE.Google Scholar
  5. [5]
    R.R. Ernst, G. Bodenhausen, and A. Wokaun. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. International Series of Monographs on Chemistry. Oxford University Press, 1987.Google Scholar
  6. [6]
    M. Felsberg and G. Sommer. The monogenic signal. IEEE Transactions on Signal Processing, 49(12):3136–3144, Dec. 2001.MathSciNetCrossRefGoogle Scholar
  7. [7]
    D. Gabor. Theory of communication. Journal of the Institution of Electrical Engineers, 93(26):429–457, 1946. Part III.Google Scholar
  8. [8]
    S.L. Hahn. Multidimensional complex signals with single-orthant spectra. Proceedings of the IEEE, 80(8):1287–1300, Aug. 1992.CrossRefGoogle Scholar
  9. [9]
    S.L. Hahn. Hilbert transforms. In A.D. Poularikas, editor, The transforms and applications handbook, chapter 7, pages 463–629. CRC Press, Boca Raton, 1996. A CRC handbook published in cooperation with IEEE press.Google Scholar
  10. [10]
    S.L. Hahn. Hilbert transforms in signal processing. Artech House signal processing library. Artech House, Boston, London, 1996.Google Scholar
  11. [11]
    W.R. Hamilton. Lectures on Quaternions. Hodges and Smith, Dublin, 1853. Available online at Cornell University Library:
  12. [12]
    A.J. Hanson. Visualizing Quaternions. The Morgan Kaufmann Series in Interactive 3D Technology. Elsevier/Morgan Kaufmann, San Francisco, 2006.Google Scholar
  13. [13]
    J.B. Kuipers. Quaternions and Rotation Sequences. Princeton University Press, Princeton, New Jersey, 1999.Google Scholar
  14. [14]
    N. Le Bihan and S.J. Sangwine. About the extension of the 1D analytic signal to improper complex valued signals. In Eighth International Conference on Mathematics in Signal Processing, page 45, The Royal Agricultural College, Cirencester, UK, 16–18 December 2008.Google Scholar
  15. [15]
    N. Le Bihan and S.J. Sangwine. The H-analytic signal. In Proceedings of EUSIPCO 2008, 16th European Signal Processing Conference, page 5, Lausanne, Switzerland, 25–29 Aug. 2008. European Association for Signal Processing.Google Scholar
  16. [16]
    J.M. Lilly and S.C. Olhede. Bivariate instantaneous frequency and bandwidth. IEEE Transactions on Signal Processing, 58(2):591–603, Feb. 2010.MathSciNetCrossRefGoogle Scholar
  17. [17]
    S.J. Sangwine and T.A. Ell. The discrete Fourier transform of a colour image. In J.M. Blackledge and M.J. Turner, editors, Image Processing II Mathematical Methods, Algorithms and Applications, pages 430–441, Chichester, 2000. Horwood Publishing for Institute of Mathematics and its Applications. Proceedings Second IMA Conference on Image Processing, De Montfort University, Leicester, UK, September 1998.Google Scholar
  18. [18]
    S.J. Sangwine, T.A. Ell, and N. Le Bihan. Hypercomplex models and processing of vector images. In C. Collet, J. Chanussot, and K. Chehdi, editors, Multivariate Image Processing, Digital Signal and Image Processing Series, chapter 13, pages 407–436. ISTE Ltd, and John Wiley, London, and Hoboken, NJ, 2010.Google Scholar
  19. [19]
    S.J. Sangwine and N. Le Bihan. Hypercomplex analytic signals: Extension of the analytic signal concept to complex signals. In Proceedings of EUSIPCO 2007, 15th European Signal Processing Conference, pages 621–4, Poznan, Poland, 3–7 Sept. 2007. European Association for Signal Processing.Google Scholar
  20. [20]
    S.J. Sangwine and N. Le Bihan. Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley–Dickson form. Advances in Applied Clifford Algebras, 20(1):111–120, Mar. 2010. Published online 22 August 2008.Google Scholar
  21. [21]
    J. Ville. Théorie et applications de la notion de signal analytique. Cables et Transmission, 2A:61–74, 1948.Google Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.GIPSA-Lab/CNRSSaint Martin d’HèresFrance
  2. 2.School of Computer Science and Electronic EngineeringUniversity of EssexColchesterUK

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