Skip to main content
SpringerLink
Log in

Bridging the gap between the short-time Fourier transform (STFT), wavelets, the constant-Q transform and multi-resolution STFT

  • Original Paper
  • Published:
Signal, Image and Video Processing Aims and scope Submit manuscript

Abstract

The short-time Fourier transform (STFT) is extensively used to convert signals from the time-domain into the time–frequency domain. However, the standard STFT has the drawback of having a fixed window size. Recently, we proposed a variant of that transform which fixes the window size in the frequency domain (STFT-FD). In this paper, we revisit that formulation, showing its similarity to existing techniques. Firstly, the formulation is revisited from the point of view of the STFT and some improvements are proposed. Secondly, the continuous wavelet transform (CWT) equation is used to formulate the transform in the continuous time using wavelet theory and to discretize it. Thirdly, the constant-Q transform (CQT) is analyzed showing the similarities in the equations of both transforms, and the differences in terms of how the sweep is carried out are discussed. Fourthly, the analogies with multi-resolution STFT are analyzed. Finally, the representations of a period chirp and an electrocardiogram signal in the time–frequency domain and the time-scale domain are obtained and used to compare the different techniques. The analysis in this paper shows that the proposed transform can be expressed as a variant of STFT, and as an alternative discretization of the CWT. It could also be considered a variant of the CQT and a special case of multi-resolution STFT.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. \( p \) is the number of samples per cycle in the STFT-FD methodology.

  2. The ECG signal corresponds to 10 s of a signal downloaded from http://eleceng.dit.ie/dorran/matlab/ecg.txt, with a sample frequency of 100 Hz (original source: https://physionet.org).

References

  1. Xing, F., Chen, H., Xie, S., Yao, J.: Ultrafast three-dimensional surface imaging based on short-time Fourier transform. IEEE Photon. Teh. Lett. 27, 2264–2267 (2015)

    Article  Google Scholar 

  2. Zhang, H., Hua, G., Yu, L., Cai, Y., Bi, G.: Underdetermined blind separation of overlapped speech mixtures in time-frequency domain with estimated number of sources. Speech Commun. 89, 1–16 (2017). https://doi.org/10.1016/j.specom.2017.02.003

    Article  Google Scholar 

  3. Liu, H., Li, L., Ma, J.: Rolling Bearing Fault Diagnosis Based on STFT-Deep Learning and Sound Signals. Shock Vib. 2016, 12 pp (2016)

  4. Zhang, W.Y., Hao, T., Chang, Y., Zhao, Y.H.: Time-frequency analysis of enhanced GPR detection of RF tagged buried plastic pipes. NDT E Int. 92, 88–96 (2017). https://doi.org/10.1016/j.ndteint.2017.07.013

    Article  Google Scholar 

  5. Kara, S., Içer, S., Erdogan, N.: Spectral broadening of lower extremity venous Doppler signals using STFT and AR modeling. Digit. Signal Process. 18, 669–676 (2008)

    Article  Google Scholar 

  6. Gabor, D.: Theory of communication. J. Inst. Electr. Eng. London. 93, 429–457 (1946). https://doi.org/10.1049/ji-3-2.1946.0074

    Article  Google Scholar 

  7. Xie, H., Lin, J., Lei, Y., Liao, Y.: Fast-varying AM–FM components extraction based on an adaptive STFT. Digit. Signal Process. 22, 664–670 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Gnann, V., Becker, J.: Signal reconstruction from multiresolution STFT magnitudes with mutual initialization. In: 45th International Conference: Applications of Time-Frequency Processing in Audio. pp. 2–3, (2012)

  10. Pihlajamäki, T.: Multi-resolution short-time Fourier transform implementation of directional audio coding. pp 56–73 (2009)

  11. Bonada, J.: Automatic technique in frequency domain for near-lossless time-scale modification of audio. Int. Comput. Music Conf. 396–399 (2000)

  12. Juillerat, N., Müller Arisona, S., Schubiger-banz, S.: Enhancing the quality of audio transformations using the multi-scale short time Fourier transform. Proc. 10th IASTED Int. Conf. Signal Image Process. 623, 379–387 (2008)

    Google Scholar 

  13. Miao, H., Zhang, F., Tao, R.: Fractional Fourier analysis using the Möbius inversion formula. IEEE Trans. Signal Process. 67, 3181–3196 (2019). https://doi.org/10.1109/TSP.2019.2912878

    Article  MathSciNet  MATH  Google Scholar 

  14. Wei, D., Li, Y.M.: Generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 64, 4861–4874 (2016). https://doi.org/10.1109/TSP.2016.2560148

    Article  MathSciNet  MATH  Google Scholar 

  15. Tao, R., Li, Y.L., Wang, Y.: Short-time fractional fourier transform and its applications. IEEE Trans. Signal Process. 58, 2568–2580 (2010). https://doi.org/10.1109/TSP.2009.2028095

    Article  MathSciNet  MATH  Google Scholar 

  16. Capus, C., Brown, K.: Short-time fractional Fourier methods for the time-frequency representation of chirp signals. J. Acoust. Soc. Am. 113, 3253 (2003). https://doi.org/10.1121/1.1570434

    Article  Google Scholar 

  17. Oktem, F., Ozaktas, H.: Exact relation between continuous and discrete linear canonical transforms. IEEE Signal Process. Lett. 16, 727–730 (2009). https://doi.org/10.1109/LSP.2009.2023940

    Article  Google Scholar 

  18. Wei, D., Yang, W., Li, Y.M.: Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain. J. Franklin Inst. 356, 7571–7607 (2019). https://doi.org/10.1016/j.jfranklin.2019.06.031

    Article  MathSciNet  MATH  Google Scholar 

  19. Wei, D., Li, Y.M.: Convolution and multichannel sampling for the offset linear canonical transform and their applications. IEEE Trans. Signal Process. 67, 6009–6024 (2019). https://doi.org/10.1109/TSP.2019.2951191

    Article  MathSciNet  MATH  Google Scholar 

  20. Bao, Y.P., Zhang, Y.N., Song, Y.E., Li, B.Z., Dang, P.: Nonuniform sampling theorems for random signals in the offset linear canonical transform domain. Proc. - 9th Asia-Pacific Signal Inf. Process. Assoc. Annu. Summit Conf. APSIPA ASC 2017. 2018-Febru, 94–99 (2018). https://doi.org/10.1109/APSIPA.2017.8282008

  21. Rioul, O., Vetterli, M.: Wavelets and signal processing. IEEE Signal Process. Mag. 8, 14–38 (1991). https://doi.org/10.1109/79.91217

    Article  Google Scholar 

  22. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, Cambridge (2009)

    MATH  Google Scholar 

  23. Gargour, C., Gabrea, M., Ramachandran, V., Lina, J.M.: A short introduction to wavelets and their applications. IEEE Circuits Syst. Mag. 9, 57–68 (2009). https://doi.org/10.1109/MCAS.2009.932556

    Article  Google Scholar 

  24. Fanaee, F., Yazdi, M., Faghihi, M.: Face image super-resolution via sparse representation and wavelet transform. Signal Image Video Process. 13, 79–86 (2019). https://doi.org/10.1007/s11760-018-1330-9

    Article  Google Scholar 

  25. Chen, J., Phung, T., Blackburn, T., Ambikairajah, E., Zhang, D.: Detection of high impedance faults using current transformers for sensing and identification based on features extracted using wavelet transform. IET Gener. Transm. Distrib. 10, 2990–2998 (2016). https://doi.org/10.1049/iet-gtd.2016.0021

    Article  Google Scholar 

  26. Tantawi, M.M., Revett, K., Salem, A.B., Tolba, M.F.: A wavelet feature extraction method for electrocardiogram (ECG)-based biometric recognition. Signal Image Video Process. 9, 1271–1280 (2015). https://doi.org/10.1007/s11760-013-0568-5

    Article  Google Scholar 

  27. Alajlan, N., Bazi, Y., Melgani, F., Malek, S., Bencherif, M.A.: Detection of premature ventricular contraction arrhythmias in electrocardiogram signals with kernel methods. Signal Image Video Process. 8, 931–942 (2014). https://doi.org/10.1007/s11760-012-0339-8

    Article  Google Scholar 

  28. Sheng, Y.: Wavelet transform. In: The Transforms and Applications Handbook, Second Edition (2000)

  29. Hlawatsch, F., Boudreaux-Bartels, G.F.: Linear and quadratic time-frequency signal representations. IEEE Signal Process. Mag. 9, 21–67 (1992)

    Article  Google Scholar 

  30. Gu, Y.H., Bollen, M.H.J.: Time-frequency and time-scale domain analysis of voltage disturbances. IEEE Trans. Power Deliv. 15, 1279–1284 (2000). https://doi.org/10.1109/61.891515

    Article  Google Scholar 

  31. Wang, J., Ding, Y., Ren, S., Wang, W.: Sampling and reconstruction of multiband signals in multiresolution subspaces associated with the fractional wavelet transform. IEEE Signal Process. Lett. 26, 174–178 (2019). https://doi.org/10.1109/LSP.2018.2883832

    Article  Google Scholar 

  32. Brown, J.C.: Calculation of a constant Q spectral transform. J. Acoust. Soc. Am. 89, 425–434 (1991). https://doi.org/10.1121/1.400476

    Article  Google Scholar 

  33. Bachhav, P.B., Todisco, M., Mossi, M., Beaugeant, C., Evans, N.: Artificial bandwidth extension using the constant Q transform. ICASSP IEEE Int. Conf. Acoust. Speech Signal Process. Proc. 5550, 5554 (2017). https://doi.org/10.1109/ICASSP.2017.7953218

    Article  Google Scholar 

  34. Dorran, D.: Audio Time-Scale Modification, http://arrow.dit.ie/engdoc, (2005)

  35. Bo, H., Li, H., Ma, L., Yu, B.: A Constant Q Transform based approach for robust EEG spectral analysis. In: 2014 International Conference on Audio, Language and Image Processing. pp. 58–63. IEEE (2014)

  36. Mateo, C., Talavera, J.A.: Short time Fourier transform with the window size fixed in the frequency domain. Digit. Signal Process. 77, 13–21 (2018)

    Article  MathSciNet  Google Scholar 

  37. Mateo, C., Talavera, J.A.: Short time Fourier transform with the window size fixed in the frequency domain: implementation. SoftwareX. 8, 5–8 (2018)

    Article  Google Scholar 

  38. Bravo, J.P., Roque, S., Estrela, R., Leão, I.C., De Medeiros, J.R.: Wavelets: a powerful tool for studying rotation, activity, and pulsation in Kepler and CoRoT stellar light curves. Astron. Astrophys. 568, A34 (2014). https://doi.org/10.1051/0004-6361/201323032

    Article  Google Scholar 

  39. Brown, J.C., Puckette, M.S.: An efficient algorithm for the calculation of a constant Q transform. J. Acoust. Soc. Am. 92, 2698–2701 (1992). https://doi.org/10.1121/1.404385

    Article  Google Scholar 

  40. Dobre, R.A., Negrescu, C.: Automatic music transcription software based on constant Q transform. Proc. 8th Int. Conf. Electron. Comput. Artif. Intell. ECAI 2016. (2017). https://doi.org/10.1109/ECAI.2016.7861193

  41. Chang, C.C., Hsu, H.Y., Hsiao, T.C.: The interpretation of very high frequency band of instantaneous pulse rate variability during paced respiration. Biomed. Eng. Online. (2014). https://doi.org/10.1186/1475-925X-13-46

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute for Research in Technology, School of Engineering (ICAI), Universidad Pontificia Comillas, Madrid, 28015, Spain

    Carlos Mateo

  2. Esdras A., S.L., Las Rozas de Madrid, Spain

    Juan Antonio Talavera

Authors
  1. Carlos Mateo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Antonio Talavera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Mateo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mateo, C., Talavera, J.A. Bridging the gap between the short-time Fourier transform (STFT), wavelets, the constant-Q transform and multi-resolution STFT. SIViP 14, 1535–1543 (2020). https://doi.org/10.1007/s11760-020-01701-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11760-020-01701-8

Keywords

Search

Navigation