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Discrete cosine transform interpolation based design of two-dimensional FIR fractional order digital differentiator

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Abstract

In this paper a two-dimensional (2-D) DCT interpolation based method for the designing of a 2-D fractional order digital differentiator (FODD) is presented. The modeling of the FODD is achieved in the form of a finite impulse response (FIR) filter. Here, Grun-wald Letnikov partial fractional derivative of two variable function with discrete cosine transform (DCT) interpolation is used to estimate the impulse response of an ideal 2-D FODD. Here, 2-D DCT-II and DCT-III methods are employed to evaluate the optimal values of coefficients of the 2-D fractional order differentiator. Simulation results demonstrate that the proposed method surpasses the existing method in terms of integral square magnitude error (ISME). The simulated results reflect that the improved response gives a much reduced error of 0.0404 and 0.0165 using 2-D DCT-II and DCT-III methods respectively.The proposed 2-D FODD is applied on an image for edge detection to demonstrate the effectiveness of the method.

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References

  • Adams, J. L. (2013). Approximate realization of fractional-order 2-D IIR frequency-planar filters. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 338–345.

    Article  Google Scholar 

  • Aggarwal, A., Kumar, M., & Rawat, T. K. (2016a). Optimal design of 2-D FIR digital differentiator using L1-norm based cuckoo-search algorithm. Multi Dimensional System Signal Process, 28, 1569–1587.

  • Aggarwal, A., Rawat, T. K., & Kumar, M. (2016b). Design of optimal 2-D FIR differentiators with quadrantally symmetric properties using the L1-method. In International conference signal processing and communication systems (ICSPCS) (pp. 1–6).

  • Aggarwal, A., Rawat, T. K., Kumar, M., & Upadhyay, D. K. (2016c). Optimal design of FIR band stop filter using L1-norm based RCGA. Elsevier’s Ain Shams Engineering Journal, 9(2), 277–289.

  • Aggarwal, A., Rawat, T. K., & Upadhyay, D. K. (2015d). Optimal design of FIR high pass filter based on L1 error approximation using real coded genetic algorithm. Engineering Science and Technology International Journal, 18, 594–602.

  • Aggarwal, A., Rawat, T. K., & Upadhyay, D. K. (2016e). Optimal design of L1-norm based IIR digital differentiators and integrators using the bat algorithm. IET Signal Process, 11, 26–35.

  • Ali, T. A. A., Xiao, Z., Mirjalili, S., & Havyarimana, V. (2020). Efficient design of wideband digital fractional order differentiators and integrators using multi-verse optimizer. Applied Soft Computing Journal, 93, 1–10.

    Article  Google Scholar 

  • Benmalek, M., & Charef, A. (2008). Digital fractional order operators for R-wave detection in the electrocardiogram signal. IET Signal Processing, 3(5), 381–391.

    Article  Google Scholar 

  • Bensouici, T., Charef, A., & Assadi, I. (2017) New approximation to design fractional order digital FIR differentiators. In IET 3rd international IEEE conference on intelligent signal processing (ISP 2017) (pp. 1–5).

  • Chen, D., Chen, Y. Q., & Xue, D. (2011). Digital fractional order Savitzky–Golay differentiator. IEEE Transactions on Circuits an Systems-II: Express Briefs, 58(11), 758–762.

    Article  Google Scholar 

  • Chen, H., & Zeng, B. (2012). New transforms tightly bounded by DCT and KLT. IEEE Signal Processing Letters, 19(6), 344–347.

    Article  Google Scholar 

  • Chen, Y. Q., & Vinagre, B. M. (2003). A new IIR-type digital fractional order differentiator. Signal Processing, 83(11), 2359–2365.

    Article  MATH  Google Scholar 

  • Dabbaghchian, S., Ghaemmaghami, M. P., & Aghagolzadeh, A. (2010). Feature extraction using discrete cosine transform and discrimination power analysis with a face recognition technology. Pattern Recognition, 43, 1431–1440.

    Article  MATH  Google Scholar 

  • Dwivedi, A. K., Ghosh, S., & Londhe, N. D. (2016). Low power FIR filter design using modified multi-objective artificial bee colony algorithm. Engineering Applications of Artificial Intelligence, 55, 58–69.

    Article  Google Scholar 

  • Ernawan, F., Ariatmanto, D., & Firdaus, A. (2021). An improved image watermarking by modifying selected DWT-DCT coefficients. IEEE Access, 9, 45474–45485.

    Article  Google Scholar 

  • Ferdi, Y., Herbeuval, J. P., Charef, A., & Boucheham, B. (2000). R wave detection using fractional digital differentiation. ITBM-RBM, 24(5–6), 273–280.

    Google Scholar 

  • Goswami, O. P., Rawat, T. K., & Upadhyay, D. K. (2020). A novel approach for the design of optimum IIR differentiators using fractional interpolation. Circuits, Systems, and Signal Processing, 39, 1688–1698.

    Article  Google Scholar 

  • Gupta, M., & Garg, A. K. (2012). Analysis of image compression algorithm using DCT. International Journal of Engineering Research and Applications, 2(1), 515–521.

    Google Scholar 

  • Gupta, M., & Yadav, R. (2014). New improved fractional order differentiator models based on optimized digital differentiators. Science World Journal, 2014, 1–11.

    Google Scholar 

  • Jiang, C. X., Carletta, J. E., & Hartley, T. T. (2007). Implementation of fractional-order operators on field programmable gate arrays. In Advances in fractional calculus (pp. 333–346). Springer.

  • Kafai, M., Eshghi, K., & Bhanu, B. (2014). Discrete cosine transform locality-sensitive hashes for face retrieval. IEEE Transaction on Multimedia, 16(4), 1090–1103.

    Article  Google Scholar 

  • Koshita, S. (2014). A simple ladder realization of maximally flat allpass fractional delay filters. IEEE Transaction, 61(3), 203–207.

    Google Scholar 

  • Krishna, B. T. (2011). Studies on fractional order differentiators and integrators: A survey. Signal Processing, 91(3), 386–426.

    Article  MATH  Google Scholar 

  • Kumar, A., Komaragiri, R., & Kumar, M. (2019). Design of efficient fractional operator for ECG signal detection in implantable cardiac pacemaker systems. International Journal of Circuit Theory and Applications, 47(9), 1459–1476.

    Article  Google Scholar 

  • Kumar, M. (2019a). Fractional order FIR differentiator design using particle swarm optimization algorithm. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(2), e2514.

  • Kumar, M. (2019b). Fractional order FIR differentiator design using particle swarm optimization algorithm. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(2), e2514.

  • Kumar, M., Mittal, A., & Rawat, T. K. (2016). Fractional constraints based designing of 2-dimensional FIR filters. In 3rd International IEEE conference on signal processing and integrated networks (SPIN) (pp. 481–485).

  • Kumar, M., & Rawat, T. K. (2014). On the designing of fractional order FIR differentiator using radial basis function and window. WSEAS Transactions on Signal Processing, 10, 538–543.

    Google Scholar 

  • Kumar, M., & Rawat, T. K. (2015a). Optimal fractional delay-IIR filter design using cuckoo search algorithm. Journal of Elsevier ISA Transaction, 59(3), 39–54.

  • Kumar, M., & Rawat, T. K. (2015b). Optimal design FIR fractional order differentiator using cuckoo search algorithm. Expert System Application, 42(7), 3433–3449.

  • Kumar, M., & Rawat, T. K. (2016). Fractional order digital differentiator design based on power function and least squares. International Journal of Electronics, 103, 1639–1653.

    Article  Google Scholar 

  • Kumar, M., & Rawat, T. K. (2017). Design of fractional order differentiator using type-III and type-IV discrete cosine transform. Engineering Science Technical International Journal, 20(1), 51–58.

    Google Scholar 

  • Kumar, M., Rawat, T. K., Jain, A., Singh, A. A., & Mittal, A. (2015). Design of digital differentiators using interior search algorithm. Procedia Computer Science, 57, 368–376.

    Article  Google Scholar 

  • Kumar, M., Rawat, T. K., Singh, A. A., Mittal, A., & Jain, A. (2015). Optimal design of wideband digital integrators using gravitational search algorithm. In International conference on international conference on computing, communication and automation (ICCCA) (pp. 1314–1319).

  • Kumar, N., Kumar, A., & Bajaj, V. (2020). A new design approach for nearly linear phase stable IIR filter using fractional derivative. IEEE/CAA Journal of Automatica Sinica, 7(2), 527–538.

    Article  MathSciNet  Google Scholar 

  • Lukin, V. V., Fevralev, D. V., Nikolay, N., & Abramov, S. (2010). Discrete cosine transform-based local adaptive filtering of images corrupted by non stationary noise. Journal of Electronic Imaging, 19(2), 1–15.

    Article  Google Scholar 

  • Mahata, S., Saha, S. K., Kar, R., & Mandal, D. (2017). Optimal and accurate design of fractional order digital differentiator—An evolutionary approach. IET Signal Processing, 11(2), 181–196.

    Article  Google Scholar 

  • Matusu, R. (2011). Application of fractional order calculus to control theory. International Journal of Mathematical Models and Methods in Applied Sciences, 5, 1162–1169.

    Google Scholar 

  • Mirjalili, S., Mirjalili, S. M., & Hatamlou, A. (2016). Multi-verse optimizer: A nature inspired algorithm for global optimization. Neural Computer Applications, 27, 495–513.

    Article  Google Scholar 

  • Nayak, C., Saha, S. K., Kar, R., & Mandal, D. (2019a). An optimally designed digital differentiator based preprocessor for R-peak detection in electrocardiogram signal. Biomedical Signal Processing and Control, 49, 440–464.

  • Nayak, C., Saha, S. K., Kar, R., & Mandal, D. (2019b). An efficient and robust digital fractional order differentiator based ECG pre-processor design for QRS detection. IEEE Transactions on Biomedical Circuits and Systems, 13(4), 682–696.

  • Nayak, C., Saha, S. K., Kar, R., & Mandal, D. (2020). Efficient design of zero-phase Riesz fractional order digital differentiator using manta-ray foraging optimisation for precise electrocardiogram QRS detection. IEEE Open Journal of Circuits and Systems, 1, 280–292.

    Article  Google Scholar 

  • Petras, I., & Vinagre, B. M. (2002). Practical application of digital fractional-order controller to temperature control. Acta Montanistica Slovca, 2, 131–137.

    Google Scholar 

  • Rabie, T., & Kamel, I. (2016). High-capacity steganography: A global-adaptive-region discrete cosine transform approach. Multimedia Tools and Applications, 76, 6473–6493.

    Article  Google Scholar 

  • Raid, A. M., Khedr, W. M., El-dosuky, M. A., & Ahmed, W. (2014). Jpeg image compression using discrete cosine transform—A survey. International Journal of Computer Science & Engineering Survey, 5(2), 39–47.

    Article  Google Scholar 

  • Rajeswari, S. S. (2019). Implementation of 2D-DCT as an efficient accelerator for HEVC video CODEC. International Journal of Engineering and Advanced Technology, 9(2), 4320–4325.

    Article  MathSciNet  Google Scholar 

  • Sharma, A., & Rawat, T. K. (2019). Design and FPGA implementation of lattice wave fractional order digital differentiator. Microelectronics Journal, 88, 67–78.

    Article  Google Scholar 

  • Sheng, H., Chen, Y. Q., & Qiu, T. S. (2012). Fractional processes and fractional-order signal processing (pp. 31–46). Springer-Verlag. https://doi.org/10.1007/978-1-4471-2233-3

  • Tseng, C. C. (2007). Design of FIR and IIR fractional order Simpson digital integrator. Signal Processing, 87(5), 1045–1057.

    Article  MATH  Google Scholar 

  • Tseng, C. C. (2013). Design of 2-D variable fractional delay FIR filter using 2-D differentiators. In Proceedings IEEE international symposium circuits systems (Vol. 4, pp. 189–192).

  • Tzeng, S. T. (2014). Genetic algorithm approach for designing 2-D FIR digital filters with 2-D symmetric properties. Signal Processing, 84(10), 1883–1893.

    Article  MATH  Google Scholar 

  • Tseng, C. C., & Lee, S. L. (2012). Design of linear phase FIR filters using fractional derivative constraints. Signal Processing, 92, 1317–1327.

    Article  Google Scholar 

  • Tseng, C. C. & Lee, S. L. (2013a). Closed-form design of fractional order differentiator using discrete cosine transform. In IEEE international symposium on circuits and systems (ISCAS) (pp. 2609–2612).

  • Tseng, C. C., & Lee, S. L. (2013b). Designs of two-dimensional linear phase FIR filters using fractional derivative constraints. Elsevier Journal of Signal Processing, 93, 1141–1151.

  • Tseng, C. C., & Lee, S. L. (2014). Designs of fractional derivative constrained 1-D and 2-D FIR filters in the complex domain. Signal Processing, 95, 111–125.

    Article  Google Scholar 

  • Yadav, S., Kumar, M., Yadav, R., & Kumar, A. (2020). A novel approach for optimal design of digital FIR filter using grasshopper optimization algorithm. ISA Transactions, 108, 196–206.

    Article  Google Scholar 

  • Yadav, S., Yadav, R., Kumar, A., & Kumar, M. (2020). Design of optimal two-dimensional FIR filters with quadrantally symmetric properties using vortex search algorithm. Journal of Circuits, Systems and Computers, 29(10), 1–23.

    Article  Google Scholar 

  • Yadav, S., Yadav, R., Kumar, A., & Kumar, M. (2021). A novel approach to design optimal 2-D digital differentiator using vortex search optimization algorithm. Multimedia Tools and Applications, 80, 5901–5916.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics and Communication Engineering, IGDTUW, New Delhi, New Delhi, 110006, India

    Shilpa garg & Richa Yadav

  2. Department of Electronics and Communication Engineering, JIMSEMTC, Greater Noida, U.P., 201308, India

    Shilpa garg

  3. Department of Electronics and Communication Engineering, DTU, New Delhi, New Delhi, 110042, India

    Manjeet Kumar

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  1. Shilpa garg
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garg, S., Yadav, R. & Kumar, M. Discrete cosine transform interpolation based design of two-dimensional FIR fractional order digital differentiator. Multidim Syst Sign Process 33, 1367–1386 (2022). https://doi.org/10.1007/s11045-022-00846-8

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