The fractional Fourier transform as a biomedical signal and image processing tool: A review

• Autorzy: Gómez-Echavarría Alejandro , Ugarte Juan P. , Tobón Catalina

Identyfikatory

DOI

10.1016/j.bbe.2020.05.004

• Języki publikacji: EN

Abstrakty

EN

This work presents a literature review of the fractional Fourier transform (FrFT) investiga-tions and applications in the biomedical field. The FrFT is a time-frequency analysis tool that has been used for signal and image processing due to its capability in capturing the nonstationary characteristics of real signals. Most biomedical signals are an example of such non-stationarity. Thus, the FrFT-based solutions can be formulated, aiming to enhance the health technology. As the literature review indicates, common applications of the FrFT involves signal detection, filtering and features extraction. Establishing adequate solutions for these tasks requires a proper fractional order estimation and implementing the suitable numeric approach for the discrete FrFT calculation. Since most of the reports barely describe the methodology on this regard, it is important that future works include detailed information about the implementation criteria of the FrFT. Although the applications in biomedical sciences are not yet among the most frequent FrFT fields of action, the growing interest of the scientific community in the FrFT, supports its practical usefulness for developing new biomedical tools.

Słowa kluczowe

EN

biomedical signal processing; fractional Fourier transform; time-frequency analysis; nonstationary signals

PL

przetwarzanie sygnałów biomedycznych; transformata Fouriera; analiza częstotliwości; sygnał niestacjonarny

• Wydawca: Nałęcz Institute of Biocybernetics and Biomedical Engineering of the Polish Academy of Sciences ; Elsevier
• Czasopismo: Biocybernetics and Biomedical Engineering
• Rocznik: 2020
• Tom: Vol. 40, no. 3
• Strony: 1081--1093
• Opis fizyczny: Bibliogr. 124 poz., rys., tab., wykr.

Twórcy

autor

Gómez-Echavarría Alejandro

• Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia

autor

Ugarte Juan P.

• GIMSC, Universidad de San Buenaventura, Medellín, Colombia

autor

Tobón Catalina

• Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia

Bibliografia

• [1] Escabí M. Chapter 11 – biosignal processing.. In: Enderle JD, Bronzino JD, editors. Introduction to biomedical engineering. Third ed. Biomedical Engineering; Academic Press; 2012. p. 667–746.
• [2] Tabar YR, Halici U. A novel deep learning approach for classification of EEG motor imagery signals. J Neural Eng 2016;14(1):16003.
• [3] Huang J, Chen B, Yao B, He W. ECG arrhythmia classification using STFT-based spectrogram and convolutional neural network. IEEE Access 2019;7:92871–80.
• [4] Tsipouras MG. Spectral information of EEG signals with respect to epilepsy classification. Eurasip J Adv Signal Process 2019;2019(1).
• [5] Boashash B. Time-frequency signal analysis and processing: a comprehensive reference. Academic Press; 2015.
• [6] Healy JJ, Kutay MA, Ozaktas HM, Sheridan JT. Linear canonical transforms: theory and applications, vol. 198. Springer; 2015.
• [7] Almeida LB. The fractional fourier transform and time-frequency representations. IEEE Trans Signal Process 1994;42(11):3084–91.
• [8] Man'ko MA, Man'ko VI, Mendes RV. Tomograms and other transforms: a unified view. J Phys A: Math Gen 2001;34 (40):8321–32.
• [9] Mendlovic D, Ozaktas HM. Fractional Fourier transforms and their optical implementation I. J Opt Soc Am A 1993;10 (12):1875–81.
• [10] Ozaktas HM, Mendlovic D. Fractional Fourier transform and their optical implementation. II. J Opt Soc Am A 1993;10(12):2522–31.
• [11] Bernardo LM, Soares OD. Fractional Fourier transforms and optical systems. Optics Commun 1994;110(5–6):517–22.
• [12] Sejdic´ E, Djurovic´ I, Stankovic´ L. Fractional Fourier transform as a signal processing tool: an overview of recent developments. Signal Process 2011;91(6):1351–69.
• [13] Ozaktas HM, Kutay MA. The fractional Fourier transform. 2001 European Control Conference (ECC). 2001. pp. 1477–83.
• [14] Ozaktas HM, Kutay MA, Mendlovic D. Introduction to the fractional Fourier transform and its applications. Adv Imaging Electron Phys 1999;106:239–91.
• [15] Moody GB, Mark RG. The impact of the MIT-BIH arrhythmia database. IEEE Eng Med Biol Mag 2001;20 (3):45–50.
• [16] Santhanam B, McClellan J. The discrete rotational Fourier transform. IEEE Trans Signal Process 1996;44(4):983–7.
• [17] Dickinson BW, Steiglitz K. Eigenvectors and functions of the discrete Fourier transform. IEEE Trans Acoust Speech Signal Process 1982;30(1):25–31.
• [18] Pei SC, Yeh MH. Improved discrete fractional Fourier transform. Optics Lett 1997;22(14):1047–9.
• [19] Candan Ç, Kutay MA, Ozaktas HM. The discrete fractional Fourier transform. Water Resour Manag 2000;32(12):3887–902.
• [20] Ozaktas HM, Ankan O, Kutay MA, Bozdagt G. Digital computation of the fractional Fourier transform. IEEE Trans Signal Process 1996;44(9):2141–50.
• [21] Zhang YD, Wang SH, Yang JF, Zhang Z, Phillips P, Sun P, et al. A comprehensive survey on fractional Fourier transform. Fundam Inform 2017;151(1–4):1–48.
• [22] Su X, Tao R, Kang X. Analysis and comparison of discrete fractional Fourier transforms. Signal Process 2019;160:284–98.
• [23] Pei SC, Ding JJ. Closed-form discrete fractional and affine Fourier transforms. IEEE Trans Signal Process 2000;48 (5):1338–53.
• [24] Pei SC, Tseng CC, Yeh MH. A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by largrange multiplier method. IEEE Trans Circuits Syst II: Analog Digit Signal Process 1999;46(9):1240–5.
• [25] Pei SC, Yeh MH, Tseng CC. Discrete fractional Fourier transform based on orthogonal projections. IEEE Trans Signal Process 1999;47(5):1335–48.
• [26] Hanna MT, Seif NPA, Ahmed WAEM. Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value decomposition of its orthogonal projection matrices. IEEE Trans Circuits Syst I: Regular Pap 2004;51(11):2245–54.
• [27] Pei SC, Hsue WL, Ding JJ. Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices. IEEE Trans Signal Process 2006;54(10):3815–28.
• [28] Candan Ç. On higher order approximations for Hermite- Gaussian functions and discrete fractional Fourier transforms. IEEE Signal Process Lett 2007;14(10):699–702.
• [29] Hanna MT. Direct batch evaluation of optimal orthonormal eigenvectors of the DFT matrix. IEEE Trans Signal Process 2008;56(5):2138–43.
• [30] Pei SC, Hsue WL, Ding JJ. DFT-commuting matrix with arbitrary or infinite order second derivative approximation. IEEE Trans Signal Process 2009;57(1):390–4.
• [31] Serbes A, Durak-Ata L. Efficient computation of DFT commuting matrices by a closed-form infinite order approximation to the second differentiation matrix. Signal Process 2011;91(3):582–9.
• [32] Serbes A, Durak-Ata L. The discrete fractional Fourier transform based on the DFT matrix. Signal Process 2011;91 (3):571–81.
• [33] Candan Ç. On the eigenstructure of DFT matrices. IEEE Signal Process Mag 2011;28(2):105–8.
• [34] Hanna MT. Direct sequential evaluation of optimal orthonormal eigenvectors of the discrete Fourier transform matrix by constrained optimization. Digit Signal Process: Rev J 2012;22(4):681–9.
• [35] Hanna MT. The direct batch generation of Hermite- Gaussian-like eigenvectors of the DFT matrix using the notion of matrix pseudoinverse. ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing – Proceedings; 2013. p. 6063–7.
• [36] Kuznetsov A. Explicit hermite-type eigenvectors of the discrete Fourier transform. SIAM J Matrix Anal Appl 2015;36(4):1443–64.
• [37] De Oliveira Neto JR, Lima JB. Discrete fractional fourier transforms based on closed-form Hermite-Gaussian-like DFT eigenvectors. IEEE Trans Signal Process 2017;65 (23):6171–84.
• [38] De Oliveira Neto JR, Lima JB, da Silva GJ, Campello de Souza RM. Computation of an eigendecomposition-based discrete fractional Fourier transform with reduced arithmetic complexity. Signal Process 2019;165:72–82.
• [39] Santhanam B, McClellan J. The DRFT-a rotation in time-frequency space. 1995 International Conference on Acoustics, Speech, and Signal Processing; 1995. pp. 921–4.
• [40] Cariolaro G, Erseghe T, Kraniauskas P, Laurenti N. A unified framework for the fractional Fourier transform. IEEE Trans Signal Process 1998;46(12):3206–19.
• [41] Richman MS, Parks TW. Understanding discrete rotations. ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing – Proceedings, vol. 3; 1997. p. 2057–60.
• [42] Deng X, Li Y, Fan D, Qiu Y. A fast algorithm for fractional Fourier transforms. Optics Commun 1997;138(4–6):270–4.
• [43] Ikram MZ, Abed-Meraim K, Hua Y. Fast quadratic phase transform for estimating the parameters of multicomponent chrip signals. Digit Signal Process: Rev J 1997;7(2):127–35.
• [44] Bi G, Wei Y, Li G, Wan C, Si B. Radix-2 DIF fast algorithms for polynomial time-frequency transforms. IEEE Trans Aerosp Electron Syst 2006;42(4):1540–6.
• [45] Ju Y, Bi G. Generalized fast algorithms for the polynomial time-frequency transform. IEEE Trans Signal Process 2007;55(10):4907–15.
• [46] Bi G, Ju Y, Li X. Fast algorithms for polynomial time-frequency transforms of real-valued sequences. IEEE Trans Signal Process 2008;56(5):1905–15.
• [47] Ozaktas HM, Mendlovic D, Onural L, Barshan B. Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms. J Opt Soc Am A 1994;11(2):547–59.
• [48] Ozaktas HM, Barshan B, Mendlovic D. Convolution and filtering in fractional Fourier domains. Opt Rev 1994;1(1):15–6.
• [49] Kutay MA, Ozaktas HM, Ankan O. Optimal filtering in fractional Fourier domains. IEEE Trans Signal Process 1997;45(5):1129–43.
• [50] Erden MF, Kutay MA, Ozaktas HM. Applications of the fractional Fourier transform to filtering, estimation and restoration. NSIP 1999;481–5.
• [51] Durak L, Aldirmaz S. Adaptive fractional Fourier domain filtering. Signal Process 2010;90(4):1188–96.
• [52] Kumar S, Saxena R. fFrMF: fractional Fourier matched filter. Circuits Syst Signal Process 2018;37(1):49–80.
• [53] Zhang XZ, Ling BWK, Dam HH, Teo KL, Wu C. Optimal joint design of discrete fractional Fourier transform matrices and mask coefficients for multichannel filtering in fractional Fourier domains. IEEE Trans Signal Process 2018;66(22):6016–30.
• [54] Zhao Y, Yu H, Wei G, Ji F, Chen F. Parameter estimation of wideband underwater acoustic multipath channels based on fractional Fourier transform. IEEE Trans Signal Process 2016;64(20):5396–408.
• [55] Lu Y, Kasaeifard A, Oruklu E, Saniie J. Fractional Fourier transform for ultrasonic chirplet signal decomposition. Adv Acoust Vib 2012. 2012.
• [56] Bhalke DG, Rao CB, Bormane DS. Automatic musical instrument classification using fractional Fourier transform based- MFCC features and counter propagation neural network. J Intell Inf Syst 2016;46(3):425–46.
• [57] Shi Q, Li W, Tao R. Classification in remote sensing imagery. 2018 10th IAPR Workshop on Pattern Recognition in Remote Sensing (PRRS); 2018. pp. 1–5.
• [58] Gu FC, Chen HC, Chen BY. A fractional Fourier transform- based approach for gas-insulated switchgear partial discharge recognition. J Electr Eng Technol 2019;14 (5):2073–84.
• [59] Kumar S, Saxena R, Singh K. Fractional Fourier transform and fractional-order calculus-based image edge detection. Circuits Syst Signal Process 2017;36(4):1493–513.
• [60] Saxena N, Sharma KK. Pansharpening scheme using filtering in twodimensional discrete fractional Fourier transform. IET Image Process 2018;12(6):1013–9.
• [61] Qiu F, Liu Z, Liu R, Quan X, Tao C, Wang Y. Fluid flow signals processing based on fractional Fourier transform in a stirred tank reactor. ISA Trans 2019;90:268–77.
• [62] Chen H, Liu Z, Chen Q, Blondel W, Varis P. Color image cryptosystem using Fresnel diffraction and phase modulation in an expanded fractional Fourier transform domain. Laser Phys 2018;28(5).
• [63] Liu Z, Chen H, Blondel W, Shen Z, Liu S. Image security based on iterative random phase encoding in expanded fractional Fourier transform domains. Optics Lasers Eng 2018;105(December 2017):1–5.
• [64] Chen H, Liu Z, Zhu L, Tanougast C, Blondel W. Asymmetric color cryptosystem using chaotic Ushiki map and equal modulus decomposition in fractional Fourier transform domains. Optics Lasers Eng 2019;112(August 2018):7–15.
• [65] Liansheng S, Xiao Z, Chongtian H, Ailing T, Krishna Asundi A. Silhouette-free interference-based multiple-image encryption using cascaded fractional Fourier transforms. Optics Lasers Eng 2019;113(September 2018):29–37.
• [66] Yu SS, Zhou NR, Gong LH, Nie Z. Optical image encryption algorithm based on phase-truncated short-time fractional Fourier transform and hyper-chaotic system. Optics Lasers Eng 2020;124(July 2019).
• [67] Farah MA, Guesmi R, Kachouri A, Samet M. A novel chaos based optical image encryption using fractional Fourier transform and DNA sequence operation. Optics Laser Technol 2020;121(April 2019):105777.
• [68] Pedersen AF, Simons H, Detlefs C, Poulsen HF. The fractional Fourier transform as a simulation tool for lens-based X-ray microscopy. J Synchrotron Radiat 2018;125 (3):717–28.
• [69] Yang L, Guo P, Yang A, Qiao Y. Blind third-order dispersion estimation based on fractional Fourier transformation for coherent optical communication. Optics Laser Technol 2018;99:86–90.
• [70] Habibi F, Moradi M. Propagation of an airy beam through atmospheric turbulence with optical vortex under fractional Fourier transforms. Optics Laser Technol 2018;107:313–24.
• [71] Habibi F, Moradi M, Ansari A. Study on the Mainardi beam through the fractional Fourier transforms system. Comput Optics 2018;42(5):751–7.
• [72] Saad F, Ebrahim AA, Khouilid M, Belafhal A. Fractional Fourier transform of double-half inverse Gaussian hollow beams. Opt Quantum Electron 2018;50(2):1–12.
• [73] Sreekumar G, Mary L, Unnikrishnan A. Performance analysis of fractional Fourier domain beam-forming methods for sensor arrays. Smart Sci 2018;7(1):28–38.
• [74] Hanbali SBS, Kastantin R. Fractional Fourier transform- based chirp radars for countering self-protection frequency-shifting jammers. Int J Microw Wirel Technol 2017;9(8):1687–93.
• [75] Wang F, Wang Y, Liu J, Wang Y. Optical excitation fractional Fourier transform (FrFT) based enhanced thermal-wave radar imaging (TWRI). Optics Express 2018;26(17):21403–17.
• [76] Tang Z, Bao Q, Chen Z, Lin C, Wang S. A new target detection method for noncooperative bistatic radar based on fractional Fourier transform and wavelet transform. Proceedings of 2018 2nd IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference, IMCEC 2018; 2018. p. 834–8.
• [77] Fang X, Cao Z, Min R, Pi Y. Radar maneuvering target detection based on two steps scaling and fractional Fourier transform. Signal Process 2019;155:1–13.
• [78] Gaglione D, Clemente C, Ilioudis CV, Persico AR, Proudler IK, Soraghan JJ, et al. Waveform design for communicating radar systems using fractional Fourier transform. Digit Signal Process: Rev J 2018;80:57–69.
• [79] Ali M, Ahn CW, Pant M. An efficient lossless robust watermarking scheme by integrating redistributed invariant wavelet and fractional Fourier transforms. Multimed Tools Appl 2018;77(10):11751–73.
• [80] Abdelhakim AM, Saad MH, Sayed M, Saleh HI. Optimized SVD-based robust watermarking in the fractional Fourier domain. Multimed Tools Appl 2018;77(21):27895–917.
• [81] Zhang XZ, Li Y, Ling BWK, Song C, Teo KL. Spread spectrum compressed sensing magnetic resonance imaging via fractional Fourier transform. 2017 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC). 2017. pp. 90–3.
• [82] Naveen Kumar R, Jagadale BN, Bhat JS. A lossless image compression algorithm using wavelets and fractional Fourier transform. SN Appl Sci 2019;1(3):1–8.
• [83] Gong L, Deng C, Pan S, Zhou N. Image compression- encryption algorithms by combining hyper-chaotic system with discrete fractional random transform. Optics Laser Technol 2018;103:48–58.
• [84] Zhang D, Liao X, Yang B, Zhang Y. A fast and efficient approach to color-image encryption based on compressive sensing and fractional Fourier transform. Multimed Tools Appl 2018;77(2):2191–208.
• [85] Chen B, Yu M, Tian Y, Li L, Wang D, Sun X. Multiple-parameter fractional quaternion Fourier transform and its application in colour image encryption. IET Image Process 2018;12(12):2238–49.
• [86] Li Y, Song Z, Sha X. The multi-weighted type fractional fourier transform scheme and its application over wireless communications. Eurasip J Wirel Commun Netw 2018 2018;1.
• [87] Li J, Sha X, Fang X, Mei L, Dxwkru F, Vkd H, et al. 8- Weighted-type fractional Fourier transform based three-branch transmission method. China Commun 2018;15 (9):147–59.
• [88] Ni L, Da X, Hu H, Liang Y, Xu R. PHY-aided secure communication via weighted fractional Fourier transform. Wirel Commun Mob Comput 2018 2018.
• [89] Zhou L, Zhao Q, Chi S, Li Y, Liu L, Yu Q. A fractional Fourier transform-based channel estimation algorithm in single- carrier direct sequence code division multiple access underwater acoustic communication system. Int J Distrib Sens Netw 2019;15(1).
• [90] Zhang Y, Zhang Q, Wu S. Biomedical signal detection based on fractional Fourier transform.5th Int. Conference on Information Technology and Applications in Biomedicine, ITAB 2008 in Conjunction With 2nd Int. Symposium and Summer School on Biomedical and Health Engineering, IS3BHE. 2008. 2008. p. 349–52.
• [91] Iwai R, Yoshimura H. A new method for improving robustness of registered fingerprint data using the fractional Fourier transform. Int J Commun Netw Syst Sci 2010;03(09):722–9.
• [92] Iwai R, Yoshimura H. New method for increasing matching accuracy and reducing process time of fingerprint data by the fractional Fourier transform. Proceedings – International Conference on Image Processing, ICIP; 2010. p. 3061–4.
• [93] Iwai R, Yoshimura H. Matching accuracy analysis of fingerprint templates generated by data processing method using the fractional Fourier transform. Int J Commun Netw Syst Sci 2011;04(01):24–32.
• [94] Iwai R, Yoshimura H. Accuracy analysis in fingerprint authentication using the fractional Fourier transform without misalignment correction of scanned images. Int J Commun Netw Syst Sci 2012;05(03):178–86.
• [95] Guerrero-Mosquera C, Verleysen M, Navia Vazquez A. EEG feature selection using mutual information and support vector machine: a comparative analysis. 2010 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBC'10; 2010. pp. 4946–9.
• [96] Subramaniam SR, Hon TK, Georgakis A, Papadakis G. Fractional fourier-based filter for denoising elastograms. 2010 Annual International Conference of the IEEE Engineering in Medicine and Biology. 2010. pp. 4028–31.
• [97] Madhu A, Jayasree VK, Thomas V. Seizure detection in epileptic EEGs using short time fractional Fourier transform. Int J Adv Eng Emerg Technol (IJAEET) 2011;2 (2):9–16.
• [98] Rizwan-I-Haque I, Khan MF, Saleem M, Rao NI. Network weight adjustment in a fractional fourier transform based multi-channel brain computer interface for person authentication. 2012 11th International Conference on Information Science, Signal Processing and Their Applications (ISSPA); 2012. pp. 900–5.
• [99] Zheng L, Shi D. Maximum amplitude method for estimating compact fractional Fourier domain. IEEE Signal Process Lett 2010;17(3):293–6.
• [100] Gencer M, Bilgin G, Aydin N. Embolic Doppler ultrasound signal detection via fractional Fourier transform. 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC). 2013. pp. 3050–3.
• [101] Wang S, Zhang Y, Yang X, Sun P, Dong Z, Liu A, et al. Pathological brain detection by a novel image feature-fractional Fourier entropy. Entropy 2015;17(12):8278–96.
• [102] Zhang YD, Chen S, Wang SH, Yang JF, Phillips P. Magnetic resonance brain image classification based on weighted- type fractional Fourier transform and nonparallel support vector machine. Int J Imaging Syst Technol 2015;25(4):317–27.
• [103] Zhang YD, Sun Y, Phillips P, Liu G, Zhou X, Wang S. A multilayer perceptron based smart pathological brain detection system by fractional Fourier entropy. J Med Syst 2016;40(7).
• [104] Zhang YD, Wang SH, Liu G, Yang J. Computer-aided diagnosis of abnormal breasts in mammogram images by weighted-type fractional Fourier transform. Adv Mech Eng 2016;8(2):1–11.
• [105] Wang S, Yang M, Zhang Y, Li J, Zou L, Lu S, et al. Detection of left-sided and right-sided hearing loss via fractional Fourier transform. Entropy 2016;18(5):1–10.
• [106] Sud S. Blind separation of twin fetal heartbeats in an electrocardiogram using the fractional Fourier transform. Int J Eng Res Appl 2016;6(4):14–8.
• [107] Kumar P, Kansal S. Noise removal in speech signal using fractional Fourier transform. 2017 International Conference on Information, Communication, Instrumentation and Control (ICICIC). 2017. pp. 1–4.
• [108] Fei K, Wang W, Yang Q, Tang S. Chaos feature study in fractional Fourier domain for preictal prediction of epileptic seizure. Neurocomputing 2017;249:290–8.
• [109] Awal MA, Ouelha S, Dong S, Boashash B. A robust high-resolution time-frequency representation based on the local optimization of the short-time fractional Fourier transform. Digit Signal Process: Rev J 2017;70:125–44.
• [110] Belousov YM, Elkin NN, Man'ko VI, Popov EG, Revenko SV. Tomographic representation of electrocardiogram signals. J Russ Laser Res 2018;39(3):302–13.
• [111] Zhang Y, Hu Q, Guo Z, Xu J, Xiong K. Multi-class brain images classification based on reality-preserving fractional Fourier transform and adaboost. 2018 3rd IEEE International Conference on Image, Vision and Computing, ICIVC 2018, vol. 2; 2018. pp. 444–7.
• [112] Keshishzadeh S, Fallah A, Rashidi S. Electroencephalogram based biometrics: a fractional Fourier transform approach. Proceedings of the 2018 2nd International Conference on Biometric Engineering and Applications; 2018. p. 1–5.
• [113] Guo ZP, Xin Y, Zhao YZ. Cancer classification using entropy analysis in fractional Fourier domain of gene expression profile. Biotechnol Biotechnol Equip 2018;32 (4):1042–6.
• [114] Gupta V, Mittal M. A comparison of ECG signal pre-processing using FrFT. FrWT IPCA Improv Anal Irbm 2019;40(3):145–56.
• [115] Mendlovic D, Zalevsky Z, Mas D, García J, Ferreira C. Fractional wavelet transform. Appl Optics 1997;36 (20):4801–6.
• [116] Yang Q, Wu Y, Cao R. Brain–computer interface system of steady-state visual evoked potentials based on fractional domain features. Proceedings of 2019 IEEE 8th Data Driven Control and Learning Systems Conference, DDCLS 2019; 2019. p. 1128–31.
• [117] Bajaj A, Kumar S. QRS complex detection using fractional Stockwell transform and fractional Stockwell Shannon energy. Biomed Signal Process Control 2019;54:101628.
• [118] Wang T, Liu N, Su Z, Li C. A new time-frequency feature extraction method for action detection on artificial knee by fractional Fourier transform. Micromachines 2019;10(5).
• [119] Abduh Z, Nehary EA, Wahed MA, Kadah YM. Classification of heart sounds using fractional Fourier transform based mel-frequency spectral coefficients and stacked autoencoder deep neural network. J Med Imaging Health Inform 2019;9(1):1–8.
• [120] Abduh Z, Nehary EA, Abdel Wahed M, Kadah YM. Classification of heart sounds using fractional Fourier transform based mel-frequency spectral coefficients and traditional classifiers. Biomed Signal Process Control 2020;57:101788.
• [121] Bultheel A, Martínez Sulbaran HE. Computation of the fractional Fourier transform. Appl Comput Harmonic Anal 2004;16(3):182–202.
• [122] Sreekumar G, Mary L, Unnikrishnan A. Beam-forming of broadband QFM signals using generalized time-frequency transform. Int J Electron 2019;0(0):1.
• [123] Xu X, Wang Y, Chen S. Medical image fusion using discrete fractional wavelet transform. Biomed Signal Process Control 2016;27:103–11.
• [124] Abdelliche F, Charef A, Ladaci S. Complex fractional and complex morlet wavelets for QRS complex detection. 2014 International Conference on Fractional Differentiation and Its Applications, ICFDA 2014; 2014. pp. 1–5.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).