Many domains, including communication, signal processing, and image processing, use the Fourier Transform as a mathematical tool for signal analysis. Although it can analyze signals with steady and transitory properties, it has limits. The Wavelet Packet Decomposition (WPD) is a novel technique that we suggest in this study as a way to improve the Fourier Transform and get beyond these drawbacks. In this experiment, we specifically considered the utilization of Daubechies level 4 for the wavelet transformation. The choice of Daubechies level 4 was motivated by several reasons. Daubechies wavelets are known for their compact support, orthogonality, and good time-frequency localization. By choosing Daubechies level 4, we aimed to strike a balance between preserving important transient information and avoiding excessive noise or oversmoothing in the transformed signal. Then we compared the outcomes of our suggested approach to the conventional Fourier Transform using a non-stationary signal. The findings demonstrated that the suggested method offered a more accurate representation of non-stationary and transient signals in the frequency domain. Our method precisely showed a 12% reduction in MSE and a 3% rise in PSNR for the standard Fourier transform, as well as a 35% decrease in MSE and an 8% increase in PSNR for voice signals when compared to the traditional wavelet packet decomposition method.
References
[1]
Guo, H., Member, S., Sitton, G.A., Member, S. and Sydney Burrus, C. (1998) The Quick Fourier Transform: An FFT Based on Symmetries. IEEE Transactions on Signal Processing, 46, 335-341. https://doi.org/10.1109/78.655419
[2]
Escabí, M.A. (2005) Biosignal Processing. In: Enderle, J.D., Blanchard, S.M. and Bronzino, J.D., Eds., Introduction to Biomedical Engineering, Academic Press, Cambridge, MA, 549-625. https://doi.org/10.1016/B978-0-12-238662-6.50012-4
[3]
Lai, E. (2003) Frequency-Domain Representation of Discrete-Time Signals. In: Practical Digital Signal Processing, Newnes, London, 61-78. https://doi.org/10.1016/B978-075065798-3/50004-7
[4]
Sugianto, S. and Suyanto, S. (2019) Voting-Based Music Genre Classification Using Melspectogram and Convolutional Neural Network. 2019 2nd International Seminar on Research of Information Technology and Intelligent Systems, Yogyakarta, 5-6 December 2019, 330-333. https://doi.org/10.1109/ISRITI48646.2019.9034644
[5]
Gokhale, M.Y. and Khanduja, D.K. (2010) Time Domain Signal Analysis Using Wavelet Packet Decomposition Approach. Network and System Sciences, 3, 321-329. https://doi.org/10.4236/ijcns.2010.33041
[6]
González-Velasco, E.A. (1995) Fourier Series. In: Fourier Analysis and Boundary Value Problems, Academic Press, Cambridge, MA, 23-83. https://doi.org/10.1016/B978-012289640-8/50002-2
[7]
Jia, M., Li, F., Wu, J., Chen, Z. and Pu, Y. (2020) Robust QRS Detection Using High-Resolution Wavelet Packet Decomposition and Time-Attention Convolutional Neural Network. IEEE Access, 8, 16979-16988. https://doi.org/10.1109/ACCESS.2020.2967775
[8]
Huang, D., Zhang, W.A., Guo, F., Liu, W. and Shi, X. (2023) Wavelet Packet Decomposition-Based Multiscale CNN for Fault Diagnosis of Wind Turbine Gearbox. IEEE Transactions on Cybernetics, 53, 443-453. https://doi.org/10.1109/TCYB.2021.3123667
[9]
Min, H., Wei, G., Xu, Y. and Zhang, Y. (2022) Wavelet Packet Sub-Band Cepstral Coefficient for Speaker Verification. 2022 IEEE 6th Advanced Information Technology, Electronic and Automation Control Conference, Beijing, 3-5 October 2022, 1713-1717. https://doi.org/10.1109/IAEAC54830.2022.9929836
[10]
Özkurt, N. and Savacı, F.A. (2005) Determination of Wavelet Ridges of Nonstationary Signals by Singular Value Decomposition. IEEE Transactions on Circuits and Systems-II: Express Briefs, 52, 480-485. https://doi.org/10.1109/TCSII.2005.849041
[11]
Zhao, M., Zhou, S., Yu, Q., Hu, X. and Sun, X. (2021) Research on Guided Wave Signal Processing Method for Wing Icing Quantitative Detection Based on Wavelet Packet Decomposition-Singular Value. IEEE International Ultrasonics Symposium (IUS), Xi’an, 11-16 September 2021, 1-4. https://doi.org/10.1109/IUS52206.2021.9593825
[12]
Ponomareva, O., Ponomarev, A. and Smirnova, N. (2021) Two-Dimensional Discrete Fourier Transform with Variable Parameter in the Spatial-Frequency Domain. 2021 23rd International Conference on Digital Signal Processing and Its Applications, DSPA 2021, Moscow, 24-26 March 2021, 1-4. https://doi.org/10.1109/DSPA51283.2021.9535997
[13]
Das, A. (2012) Fourier Transform. In: Signal Conditioning. Signals and Communication Technology, Springer, Berlin, Heidelberg, 51-76. https://doi.org/10.1007/978-3-642-28818-0_3
[14]
Akhtar, J. (2021) Discrete Fourier Transform with Neural Networks. IEEE Vehicular Technology Conference, Norman, 27-30 September 2021, 1-5. https://doi.org/10.1109/VTC2021-Fall52928.2021.9625247
[15]
Serov, V. (2017) Connection Between the Discrete Fourier Transform and the Fourier Transform. Applied Mathematical Sciences (Switzerland), 197, 85-92. https://doi.org/10.1007/978-3-319-65262-7_12
[16]
Gao, R.X. and Yan, R. (2011) Wavelet Packet Transform. In: Wavelets, Springer, Boston, MA, 69-81. https://doi.org/10.1007/978-1-4419-1545-0_5
[17]
Thyagarajan, K.S. (2019) Discrete Fourier Transform. In: Introduction to Digital Signal Processing Using MATLAB with Application to Digital Communications, Springer, Cham, 151-188. https://doi.org/10.1007/978-3-319-76029-2_5
[18]
Das, A. (2012) Discrete Fourier Transform. In: Signal Conditioning. Signals and Communication Technology, Springer, Berlin, Heidelberg, 159-192. https://doi.org/10.1007/978-3-642-28818-0_7
[19]
Pal, P., Banerjee, S., Ghosh, A., Vago, D.R. and Brewer, J. (2021) DFT21: Discrete Fourier Transform in the 21st Century. TechRxiv. https://doi.org/10.36227/techrxiv.16543521
[20]
Han, J., Wang, Q. and Qin, K. (2014) The Non-Stationary Signal of Time-Frequency Analysis Based on fractional Fourier Transform and Wigner-Hough Transform. In: Wang, W., Ed., Mechatronics and Automatic Control Systems. Lecture Notes in Electrical Engineering, Vol. 237, Springer, Cham, 1047-1054. https://doi.org/10.1007/978-3-319-01273-5_118
[21]
Alessio, S.M. (2016) Non-stationary Spectral Analysis. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology, Springer, Cham, 573-642. https://doi.org/10.1007/978-3-319-25468-5_13
[22]
Gao, R.X. and Yan, R. (2011) Discrete Wavelet Transform. In: Wavelets, Springer, Boston, MA, 49-68. https://doi.org/10.1007/978-1-4419-1545-0_4
[23]
Guntoro, A. and Glesner, M. (2010) A Lifting-Based Discrete Wavelet Transform and Discrete Wavelet Packet Processor with Support for Higher Order Wavelet Filters. IFIP Advances in Information and Communication Technology, 313, 154-173. https://doi.org/10.1007/978-3-642-12267-5_9
[24]
Li, M., He, Y. and Long, Y. (2011) Analogue Implementation of Wavelet Transform Using Discrete Time Switched-Current Filters. In: Zhu, M., Ed., Electrical Engineering and Control. Lecture Notes in Electrical Engineering, Vol. 98, Springer, Berlin, Heidelberg, 677-682. https://doi.org/10.1007/978-3-642-21765-4_84
[25]
Serbes, G., Aydin, N. and Ozcan Gulcur, H. (2013) Directional Dual-Tree Complex Wavelet Packet Transform. 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Osaka, 3-7 July 2013, 3046-3049. https://doi.org/10.1109/EMBC.2013.6610183
[26]
Eren, L., Unal, M. and Devaney, M.J. (2007) Harmonic Analysis via Wavelet Packet Decomposition Using Special Elliptic Half-Band Filters. IEEE Transactions on Instrumentation and Measurement, 56, 2289-2293. https://doi.org/10.1109/TIM.2007.908244
[27]
Song, R., Chen, W., Wang, Y., Du, L. and Wang, P. (2023) Transformer Aging Diagnosis Method Based on Raman Spectroscopy Wavelet Packet-SPCA Feature Extraction. IEEE Transactions on Instrumentation and Measurement, 72, Article No. 2503008. https://doi.org/10.1109/TIM.2022.3225005
[28]
Jagadeesh, B.K. and Siva Kumar, B. (2014) Psychoacoustic Model-1 Implementation for MPEG Audio Encoder Using Wavelet Packet Decomposition. In: Sridhar, V., Sheshadri, H. and Padma, M., Eds., Emerging Research in Electronics, Computer Science and Technology. Lecture Notes in Electrical Engineering, Vol. 248, Springer, New Delhi, 495-505. https://doi.org/10.1007/978-81-322-1157-0_51
[29]
Sun, L., Zhang, J., Wang, S. and Li, X. (2011) EEG Signal Decomposition Using Wavelet Packet on DSP. Proceedings of 2011 International Conference on Electronic and Mechanical Engineering and Information Technology, 2, 750-753. https://doi.org/10.1109/EMEIT.2011.6023204
[30]
Arslan, Ö. (2020) Classification of Pathological and Healthy Voice Using Perceptual Wavelet Packet Decomposition and Support Vector Machine. 2020 Medical Technologies Congress, Antalya, 19-20 November 2020, 1-4. https://doi.org/10.1109/TIPTEKNO50054.2020.9299290
[31]
Wang, X.-Y. and Yuan, H.-M. (2018) Wavelet Decomposition Transform System Based on FPGA. 2018 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), Wuhan, 31 May-2 June 2018, 2698-2703. https://doi.org/10.1109/ICIEA.2018.8398167
