Adaptive Generalised Fractional Spectrogram and Its Applications


The generalised time–frequency transform (GTFT) is a powerful tool to analyse a large variety of frequency-modulated signals. However, it is not adequate to represent the variation of frequency over time for non-stationary signals. To solve this problem, short-time GTFT and short-time GTFT-based adaptive generalised fractional spectrogram (AGFS) are proposed. The AGFS is capable of providing a high concentration, high resolution, cross-term-free time–frequency distribution for analysing multicomponent frequency-modulated signals. It is also a generalisation of the short-time Fourier transform-based spectrogram and the short-time fractional Fourier transform-based spectrogram. The uncertainty principle for short-time GTFT is derived, and its time-bandwidth product is compared with other time–frequency distributions. With the help of simulated data examples, the effectiveness of AGFS is demonstrated in comparison with other time–frequency distributions for resolving and extracting individual components of multicomponent quadratic chirps. Robustness of AGFS is demonstrated under different input signal-to-noise ratio conditions. A local spectrogram optimisation technique is adopted for AGFS to represent simulated and real chirp signals. Finally, an application of the AGFS is presented to resolve multiple ground moving targets in synthetic aperture radar data and obtain its focused synthetic aperture radar image.

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  1. 1.

    L.B. Almeida, The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)

    Google Scholar 

  2. 2.

    F. Auger, P. Flandrin, Y.T. Lin, S. McLaughlin, S. Meignen, T. Oberlin, H.T. Wu, Time-frequency reassignment and synchrosqueezing: an overview. IEEE Signal Process. Mag. 30(6), 32–41 (2013)

    Google Scholar 

  3. 3.

    M. Awal, Design and optimisation of time-frequency analysis for multichannel neonatal eeg background features in term neonates with hypoxic ischaemic encephalopathy: characterisation, classification and neurodevelopmental outcome prediction. Ph.D. thesis, University of Queensland, Brisbane, Australia (2018)

  4. 4.

    M.A. Awal, S. Ouelha, S. Dong, B. Boashash, A robust high-resolution time-frequency representation based on the local optimization of the short-time fractional Fourier transform. Digital Signal Process. 70, 125–144 (2017)

    Google Scholar 

  5. 5.

    X. Bai, R. Tao, L.J. Liu, J. Zhao, Autofocusing of SAR images using STFRFT-based preprocessing. Electron. Lett. 48(25), 1622–1624 (2012)

    Google Scholar 

  6. 6.

    B. Barkat, K. Abed-Meraim, Algorithms for blind components separation and extraction from the time-frequency distribution of their mixture. EURASIP J. Adv. Signal Process. 2004(13), 2025–2033 (2004)

    Google Scholar 

  7. 7.

    B. Boashash, Time–Frequency Signal Analysis and Processing: A Comprehensive Reference (Academic Press, Orlando, 2015)

    Google Scholar 

  8. 8.

    B. Boashash, N.A. Khan, T. Ben-Jabeur, Time-frequency features for pattern recognition using high-resolution TFDs: a tutorial review. Digital Signal Process. 40, 1–30 (2015)

    MathSciNet  Google Scholar 

  9. 9.

    B. Boashash, S. Ouelha, Designing high-resolution time-frequency and time-scale distributions for the analysis and classification of non-stationary signals: a tutorial review with a comparison of features performance. Digital Signal Process. 77, 120–152 (2018)

    Google Scholar 

  10. 10.

    C. Capus, K. Brown, Fractional Fourier transform of the gaussian and fractional domain signal support. IEE Proc. Vis. Image Signal Process. 150(2), 99–106 (2003)

    Google Scholar 

  11. 11.

    C. Capus, K. Brown, Short-time fractional Fourier methods for the time-frequency representation of chirp signals. J. Acoust. Soc. Am. 113(6), 3253–3263 (2003)

    Google Scholar 

  12. 12.

    C. Tian-Wen, L. Bing-Zhao, X. Tian-Zhou, The ambiguity function associated with the linear canonical transform. EURASIP J. Adv. Signal Process. 2012(1), 1–14 (2012)

    Google Scholar 

  13. 13.

    V.C. Chen, The Micro-Doppler Effect in Radar (Artech House, Norwood, 2019)

    Google Scholar 

  14. 14.

    V.C. Chen, D. Tahmoush, W.J. Miceli, Radar Micro-Doppler Signatures: Processing and Applications (Institution of Engineering and Technology, Herts, 2014)

    Google Scholar 

  15. 15.

    L. Cohen, Time–Frequency Analysis: Theory and Applications (Prentice Hall, Englewood Cliffs, 1995)

    Google Scholar 

  16. 16.

    I.G. Cumming, F.H. Wong, Digital Processing of Synthetic Aperture Radar Data (Artech House, Norwood, 2005)

    Google Scholar 

  17. 17.

    I. Djurović, C. Ioana, T. Thayaparan, L. Stanković, P. Wang, V. Popović, M. Simeunović, Cubic-phase function evaluation for multicomponent signals with application to SAR imaging. IET Signal Process. 4(4), 371–381 (2010)

    Google Scholar 

  18. 18.

    I. Djurovic, T. Thayaparan, L. Stankovic, SAR imaging of moving targets using polynomial Fourier transform. IET Signal Process. 2(3), 237–246 (2008)

    Google Scholar 

  19. 19.

    M. El-Mashed, O. Zahran, M.I. Dessouky, M. El-Kordy, F.A. El-Samie, Synthetic aperture radar imaging with fractional Fourier transform and channel equalization. Digital Signal Process. 23(1), 151–175 (2013)

    MathSciNet  Google Scholar 

  20. 20.

    P. Huang, G. Liao, Z. Yang, X.G. Xia, J. Ma, J. Zheng, Ground maneuvering target imaging and high-order motion parameter estimation based on second-order keystone and generalized Hough-HAF transform. IEEE Trans. Geosci. Remote Sens. 55(1), 320–335 (2017)

    Google Scholar 

  21. 21.

    D. Iatsenko, P.V. McClintock, A. Stefanovska, Linear and synchrosqueezed time-frequency representations revisited: Overview, standards of use, resolution, reconstruction, concentration, and algorithms. Digital Signal Process. 42, 1–26 (2015)

    MathSciNet  Google Scholar 

  22. 22.

    N.A. Khan, B. Boashash, Instantaneous frequency estimation of multicomponent nonstationary signals using multiview time-frequency distributions based on the adaptive fractional spectrogram. IEEE Signal Process. Lett. 20(2), 157–160 (2013)

    Google Scholar 

  23. 23.

    X. Li, G. Bi, Y. Ju, Quantitative SNR analysis for ISAR imaging using LPFT. IEEE Trans. Aerosp. Electron. Syst. 45(3), 1241–1248 (2009)

    Google Scholar 

  24. 24.

    J.G. Liu, B.C. Yuan, The analysis and simulation of the detectors based on FRFT statistic performance, in 2008 Asia Simulation Conference-7th International Conference on System Simulation and Scientific Computing, pp. 1543–1548 (2008)

  25. 25.

    H.M. Ozaktas, O. Arikan, M.A. Kutay, G. Bozdagt, Digital computation of the fractional Fourier transform. IEEE Trans. Signal Process. 44(9), 2141–2150 (1996)

    Google Scholar 

  26. 26.

    S.C. Pei, S.G. Huang, STFT with adaptive window width based on the chirp rate. IEEE Trans. Signal Process. 60(8), 4065–4080 (2012)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    R. Pelich, N. Longépé, G. Mercier, G. Hajduch, R. Garello, Vessel refocusing and velocity estimation on SAR imagery using the fractional Fourier transform. IEEE Trans. Geosci. Remote Sens. 54(3), 1670–1684 (2016)

    Google Scholar 

  28. 28.

    B. Peng, X. Wei, B. Deng, H. Chen, Z. Liu, X. Li, A sinusoidal frequency modulation Fourier transform for radar-based vehicle vibration estimation. IEEE Trans. Instrum. Meas. 63(9), 2188–2199 (2014)

    Google Scholar 

  29. 29.

    R. Perry, R. Dipietro, R. Fante, SAR imaging of moving targets. IEEE Trans. Aerosp. Electron. Syst. 35(1), 188–200 (1999)

    Google Scholar 

  30. 30.

    G. Pooniwala, The generalised time frequency transform: Properties and application. thesis, Indian Institute of Technology, Bombay, Mumbai, India (2014)

  31. 31.

    V. Popović, I. Djurović, L. Stanković, T. Thayaparan, M. Daković, Autofocusing of SAR images based on parameters estimated from the PHAF. Signal Process. 90(5), 1382–1391 (2010)

    MATH  Google Scholar 

  32. 32.

    B.Z.L. Rui-Feng Bai, Q.Y. Cheng, Wigner-Ville distribution associated with the linear canonical transform. J. Appl. Math. 1–14 (2012)

  33. 33.

    P. Sahay, A. Anjarlekar, S.A. Jain, P. Radhakrishna, V.M. Gadre, Generalized fractional matched filtering and its applications, in 2020 National Conference on Communications (NCC), pp. 1–6 (2020)

  34. 34.

    P. Sahay, I.A. Shaik Rasheed, S.A. Jain, A. Anjarlekar, P. Radhakrishna, V.M. Gadre, Generalized fractional ambiguity function and its applications. Circuits, Systems, and Signal Processing (2020). (in press)

  35. 35.

    S. Sahay, D. Pande, N. Wing, V. Gadre, P. Sohani, A novel generalized time-frequency transform inspired by the fractional Fourier transform for higher order chirps, in International Conference on Signal Processing and Communications (SPCOM), pp. 1–5 (2012)

  36. 36.

    S.B. Sahay, Parameter estimation of chirp signals. Ph.D. thesis, Indian Institute of Technology, Bombay, Mumbai, India (2015)

  37. 37.

    S.B. Sahay, T. Meghasyam, R.K. Roy, G. Pooniwala, S. Chilamkurthy, V. Gadre, Parameter estimation of linear and quadratic chirps by employing the fractional fourier transform and a generalized time frequency transform. Sadhana 40(4), 1049–1075 (2015)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    E. Sejdić, I. Djurović, J. Jiang, Time-frequency feature representation using energy concentration: an overview of recent advances. Digital Signal Process. 19(1), 153–183 (2009)

    Google Scholar 

  39. 39.

    E. Sejdić, I. Djurović, L. Stanković, Fractional Fourier transform as a signal processing tool: an overview of recent developments. Signal Process. 91(6), 1351–1369 (2011)

    MATH  Google Scholar 

  40. 40.

    S. Shinde, V.M. Gadre, An uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 49(11), 2545–2548 (2001)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Y.E. Song, X.Y. Zhang, C.H. Shang, H.X. Bu, X.Y. Wang, The Wigner-Ville distribution based on the linear canonical transform and its applications for QFM signal parameters estimation. J. Appl. Math. 1–8 (2014)

  42. 42.

    L. Stankovic, M. Dakovic, T. Thayaparan, Time–Frequency Signal Analysis with Applications (Artech House, Norwood, 2014)

    MATH  Google Scholar 

  43. 43.

    S. Starosielec, D. Hägele, Discrete-time windows with minimal rms bandwidth for given rms temporal width. Signal Process. 102, 240–246 (2014)

    Google Scholar 

  44. 44.

    R. Tao, B. Deng, W.Q. Zhang, Y. Wang, Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 56(1), 158–171 (2008)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    R. Tao, X.M. Li, Y.L. Li, Y. Wang, Time-delay estimation of chirp signals in the fractional Fourier domain. IEEE Trans. Signal Process. 57(7), 2852–2855 (2009)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    R. Tao, Y.L. Li, Y. Wang, Short-time fractional Fourier transform and its applications. IEEE Trans. Signal Process. 58(5), 2568–2580 (2010)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    R. Tao, Y.E. Song, Z.J. Wang, Y. Wang, Ambiguity function based on the linear canonical transform. IET Signal Process. 6(6), 568–576 (2012)

    MathSciNet  Google Scholar 

  48. 48.

    T. Thayaparan, L. Stankovic, C. Wernik, M. Dakovic, Real-time motion compensation, image formation and image enhancement of moving targets in ISAR and SAR using S-method-based approach. IET Signal Process. 2(3), 247–264 (2008)

    MathSciNet  Google Scholar 

  49. 49.

    X.G. Xia, G. Wang, V.C. Chen, Quantitative snr analysis for ISAR imaging using joint time-frequency analysis-short time Fourier transform. IEEE Trans. Aerosp. Electron. Syst. 38(2), 649–659 (2002)

    Google Scholar 

  50. 50.

    Z. Xinghao, T. Ran, D. Bing, Practical normalization methods in the digital computation of the fractional Fourier transform, in Proceedings of 7th International Conference on Signal Processing, pp. 105–108 (2004)

  51. 51.

    M. Xu, W. Tang, Multi-component LFM signal filtering based on the short-time fractional Fourier transform, in Proceedings of the 32nd Chinese Control Conference, pp. 4507–4512 (2013)

  52. 52.

    J. Yang, C. Liu, Y. Wang, Detection and imaging of ground moving targets with real SAR data. IEEE Trans. Geosci. Remote Sens. 53(2), 920–932 (2015)

    Google Scholar 

  53. 53.

    J. Yang, Y. Zhang, An airborne SAR moving target imaging and motion parameters estimation algorithm with azimuth-dechirping and the second-order keystone transform applied. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens. 8(8), 3967–3976 (2015)

    MathSciNet  Google Scholar 

  54. 54.

    Y. Yang, Z. Peng, X. Dong, W. Zhang, G. Meng, Application of parameterized time-frequency analysis on multicomponent frequency modulated signals. IEEE Trans. Instrum. Meas. 63(12), 3169–3180 (2014)

    Google Scholar 

  55. 55.

    Y. Yang, Z. Peng, X. Dong, W. Zhang, G. Meng, General parameterized time-frequency transform. IEEE Trans. Signal Process. 62(11), 2751–2764 (2014)

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Y. Yang, W. Zhang, Z. Peng, G. Meng, Multicomponent signal analysis based on polynomial chirplet transform. IEEE Trans. Ind. Electron. 60(9), 3948–3956 (2013)

    Google Scholar 

  57. 57.

    A.I. Zayed, A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998)

    Google Scholar 

  58. 58.

    J.D. Zhu, J.I. Li, X.D. Gao, L.B. Ye, H.Y. Dai, Adaptive threshold detection and estimation of linear frequency-modulated continuous-wave signals based on periodic fractional Fourier transform. Circuits Syst. Signal Process. 35(7), 2502–2517 (2016)

    MATH  Google Scholar 

  59. 59.

    L. Zuo, M. Li, Z. Liu, L. Ma, A high-resolution time-frequency rate representation and the cross-term suppression. IEEE Trans. Signal Process. 64(10), 2463–2474 (2016)

    MathSciNet  MATH  Google Scholar 

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The authors would like to thank the DRDO, Ministry of Defence, Govt. of India for sponsorship of Peeyush Sahay, Sc ‘E’ (Ph.D. student), and B. S. Teza, Sc ‘C’ (Master student) under the R&D scheme, at IIT Bombay. The authors would like to thank Mr. Shubham Anand Jain (B.Tech, IIT Bombay), Mr. Ameya Anjarlekar (B.Tech, IIT Bombay), Mr. Adway Girish (B.Tech, IIT Bombay), Mr. Shubham Kar (B.Tech, IIT Bombay), Mr. Ayush Agarwal (B.Tech, IIT Dharwad), Mr. Shaan Ul Haque (B.Tech, IIT Bombay), Mr. Izaz Ahamed Shaik Rasheed (M.Tech, IIT Bombay), and Mr. Gaurav Pooniwala (B.Tech, IIT Bombay) for improving the quality of the paper. The authors wish to thank Curtis Condon, Ken White and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this paper.

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Sahay, P., Teza, B.S., Kulkarni, P. et al. Adaptive Generalised Fractional Spectrogram and Its Applications. Circuits Syst Signal Process 39, 5982–6033 (2020).

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  • Higher-order chirps
  • Synthetic aperture radar
  • Short-time fractional Fourier transform
  • Short-time generalised time–frequency transform
  • Time–frequency distribution