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Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II)

Yıl 2024, Cilt: 16 Sayı: 2, 419 - 425, 31.12.2024

Hesham Sharkas

https://doi.org/10.47000/tjmcs.1424850

Öz

This work derives an identity that maps between the $2$-norm of two multiplied $2\pi$-periodic functions in $L^2$ space (i.e., $||f.g||^2_{L^2 (-\pi,\pi)}$) to the individual Fourier coefficients of $f$ and $g$. Alternately, it maps between the $2$-norm of two multiplied discrete-time Fourier transforms (i.e., $||\mathscr{F}\{f\}.\mathscr{F}\{g\}||^2_{L^2 (-\pi,\pi)}$) to the discrete-time samples of $f$ and $g$. The results are equality to Cauchy–Schwarz inequality, and extend the results of our previous paper that map between $||f||^4_{L^4 (-\pi,\pi)}$ to the Fourier coefficients of $f$, alternately $||\mathscr{F}\{f\}||^4_{L^4 (-\pi,\pi)}$ to the discrete-time samples of $f$.

Anahtar Kelimeler

$L^2$-norm of two multiplied exponential Fourier series $L^4$-norm of exponential Fourier series $L^4$-norm of a DFT/IDFT sequence Equality of Cauchy–Schwarz inequality

Teşekkür

Thanks to the God.

Kaynakça

  • Aldaz, J.M., Barza, S., Fujii, M., Moslehian, M.S., Advances in operator cauchy–schwarz inequalities and their reverses, Annals of Functional Analysis, 6(3)(2015), 275–295.
  • Barkat, M., Signal Detection and Estimation, Artech House radar library, Artech House, 2005.
  • Bastianello, N., Carli, R., Simonetto, A., Extrapolation-based prediction-correction methods for time-varying convex optimization, Signal Processing, 210(2023), 109089.
  • Bouniakowsky, V., Sur quelques inegalit´es concernant les int´egrales aux diff´erences finies, Mem. Acad. Sci. St. Petersbourg, 7(1)(1859), 9.
  • Chen, S., Tan, S., Xu, L., Hanzo, L., Adaptive minimum error-rate filtering design: A review, Signal Processing, 88(7)(2008), 1671–1697.
  • Diaz, M., Kairouz, P., Liao, J., Sankar, L., Neural network-based estimation of the MMSE, In IEEE International Symposium on Information Theory (ISIT), (2021), 1023–1028.
  • Dragomir, S.S., A survey on cauchy-bunyakovsky-schwarz type discrete inequalities, Journal of inequalities in pure and applied mathematics, 4(3)(2003).
  • Hosseini, S.M.A.T., Amindavar, H., Ritcey, J.A., Robust detection in ultra-wideband impulse radar using DPSS-MMSE estimator, EURASIP Journal on Advances in Signal Processing, 2016(1)(2016), 60.
  • Hu, X.-M., Tian, J.-F., Chu, Y.-M., Lu, Y.-X., On cauchy–schwarz inequality for n-tuple diamond-alpha integral, Journal of Inequalities and Applications, 2020(1)(2020), 8.
  • Huang, J.C., Ko, K.M., Shu, M.H., Hsu, B.M., Application and comparison of several machine learning algorithms and their integration models in regression problems, Neural Computing and Applications, 32(10)(2020), 5461–5469.
  • Jiang, K., Shi, Y., Yang, Z., Zhang, X., Vehicle self-positioning via Kalman filter using multi-station non-circular signals, Signal Processing, 226(2025), 109658.
  • Lei, W., Zhang, Y., Chen, Z., Minimum peak sidelobe ratio filter for MIMO radar via convex optimization, Signal Processing, 226(2025), 109626.
  • Mecklenbr¨auker, C.F., Gerstoft, P., Ollila, E., Park, Y., Robust and sparse M-estimation of DOA, Signal Processing, 220(2024), 109461.
  • Nekrasov, A., Khachaturian, A., Veremyev, V., Bogachev, M. Doppler navigation system with a non-stabilized antenna as a sea-surface wind sensor, Sensors, 17(6)(2017).
  • Neumann, D., Wiese, T., Utschick, W., Learning the MMSE channel estimator, IEEE Transactions on Signal Processing, 66(11)(2018), 2905–2917.
  • Oh, J., Kwak, N., Discriminative subspace learning using generalized mean, Signal Processing, 219(2024), 109421.
  • Plotnitskaya, E., Heister, S., Veremyev, V., Mathematical model for a radar signal reflected from drone propellers as applied to the method of inverse synthetic aperture radar in bi-static radar, Journal of the Russian Universities. Radio-electronics, 26(6)(2023), 41–53.
  • Rao, Y., He, H., Wan, X., Yi, J., Range-angle dependent beam-pattern synthesis method for OFDM-based passive radar, Wuhan Univ. J. Nat. Sci., 27(3)(2022), 255–260.
  • Ren, J., Zhang, M., Yu, C., Liu, Z., Balanced MSE for imbalanced visual regression, In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2022), 7926–7935.
  • Sharkas, H., Solution of integral of the fourth power of a finite-length exponential Fourier series, Turkish Journal of Mathematics and Computer Science, 15(2)(2023), 403–406.
  • Sharkas, H., Echigo, H., Beamforming solutions for efficient link setup and link maintenance in mmwave communication systems, Master’s thesis, Lund University, Sweden, 2019.
  • Shevchenko, M., Malyshev, V., Gorovoy, A., Cherepanov, A., Spatial filtering of signals under imprecise calibration of antenna arrays, Journal of the Russian Universities. Radio-electronics, 26(6)(2023), 27–40.
  • Song, J., Choi, J., Love, D.J., Common codebook millimeter wave beam design: Designing beams for both sounding and communication with uniform planar arrays, IEEE Transactions on Communications, 65(4)(2017), 1859–1872.
  • Stoica, P., Moses, R.L., Spectral Analysis of Signals, Pearson Prentice Hall, 2005.
  • Takano, Y., Juntti, M., Matsumoto, T., ℓ1 LS and ℓ2 MMSE-based hybrid channel estimation for intermittent wireless connections, IEEE Transactions on Wireless Communications, 15(1)(2016), 314–328.
  • Tang, B., Zhang, N., Zhang, S., Huang, Z., MMSE-based waveform design for the distributed MIMO radar in spectrally crowded environments, The Journal of Engineering, 2019(20)(2019), 6603–6607.
  • Tohidi, E., Coutino, M., Gesbert, D., Revisiting matching pursuit: Beyond approximate submodularity, Signal Processing, 226(2025), 109638.
  • Vorobev, E., Veremyev, V., Kholodnyak, D., Recognition of propeller-driven aircraft in a passive bi-static radar, Journal of the Russian Universities. Radio-electronics, (6)(2018), 75–90.
  • Wiener, N., Generalized harmonic analysis, Acta Mathematica, 55(1930), 117–258.
  • Xu, J., Zhang, X., Li, Y., Kernel MSE algorithm: a unified framework for KFD, LS-SVM and KRR, In IJCNN’01. International Joint Conference on Neural Networks Proceedings (Cat. No.01CH37222), 2(2001), 1486–1491.
  • Yi, J., Wan, X., Leung, H., ℓ0-regularized least squares versus matched filtering, Signal Processing, 192(2021), 108398.
  • Yoshikawa, E., Takizawa, N., Kikuchi, H., Mega, T., Ushio, T., An estimator for weather radar doppler power spectrum via minimum mean square error, IEEE Transactions on Geoscience and Remote Sensing, 60(2022), 1–16.
  • Yoshikawa, E., Ushio, T., Kawasaki, Z., Yoshida, S., Morimoto, T. et al., MMSE beam forming on fast-scanning phased array weather radar, IEEE Transactions on Geoscience and Remote Sensing, 51(5)(2013), 3077–3088.
  • Yu, Y., Dou, W., Pseudo-Bessel beams in millimeter and sub-millimeter range, In Advanced Microwave and Millimeter Wave Technologies, IntechOpen, (2010), chapter 24.

Yıl 2024, Cilt: 16 Sayı: 2, 419 - 425, 31.12.2024

Hesham Sharkas

https://doi.org/10.47000/tjmcs.1424850

Öz

Kaynakça

  • Aldaz, J.M., Barza, S., Fujii, M., Moslehian, M.S., Advances in operator cauchy–schwarz inequalities and their reverses, Annals of Functional Analysis, 6(3)(2015), 275–295.
  • Barkat, M., Signal Detection and Estimation, Artech House radar library, Artech House, 2005.
  • Bastianello, N., Carli, R., Simonetto, A., Extrapolation-based prediction-correction methods for time-varying convex optimization, Signal Processing, 210(2023), 109089.
  • Bouniakowsky, V., Sur quelques inegalit´es concernant les int´egrales aux diff´erences finies, Mem. Acad. Sci. St. Petersbourg, 7(1)(1859), 9.
  • Chen, S., Tan, S., Xu, L., Hanzo, L., Adaptive minimum error-rate filtering design: A review, Signal Processing, 88(7)(2008), 1671–1697.
  • Diaz, M., Kairouz, P., Liao, J., Sankar, L., Neural network-based estimation of the MMSE, In IEEE International Symposium on Information Theory (ISIT), (2021), 1023–1028.
  • Dragomir, S.S., A survey on cauchy-bunyakovsky-schwarz type discrete inequalities, Journal of inequalities in pure and applied mathematics, 4(3)(2003).
  • Hosseini, S.M.A.T., Amindavar, H., Ritcey, J.A., Robust detection in ultra-wideband impulse radar using DPSS-MMSE estimator, EURASIP Journal on Advances in Signal Processing, 2016(1)(2016), 60.
  • Hu, X.-M., Tian, J.-F., Chu, Y.-M., Lu, Y.-X., On cauchy–schwarz inequality for n-tuple diamond-alpha integral, Journal of Inequalities and Applications, 2020(1)(2020), 8.
  • Huang, J.C., Ko, K.M., Shu, M.H., Hsu, B.M., Application and comparison of several machine learning algorithms and their integration models in regression problems, Neural Computing and Applications, 32(10)(2020), 5461–5469.
  • Jiang, K., Shi, Y., Yang, Z., Zhang, X., Vehicle self-positioning via Kalman filter using multi-station non-circular signals, Signal Processing, 226(2025), 109658.
  • Lei, W., Zhang, Y., Chen, Z., Minimum peak sidelobe ratio filter for MIMO radar via convex optimization, Signal Processing, 226(2025), 109626.
  • Mecklenbr¨auker, C.F., Gerstoft, P., Ollila, E., Park, Y., Robust and sparse M-estimation of DOA, Signal Processing, 220(2024), 109461.
  • Nekrasov, A., Khachaturian, A., Veremyev, V., Bogachev, M. Doppler navigation system with a non-stabilized antenna as a sea-surface wind sensor, Sensors, 17(6)(2017).
  • Neumann, D., Wiese, T., Utschick, W., Learning the MMSE channel estimator, IEEE Transactions on Signal Processing, 66(11)(2018), 2905–2917.
  • Oh, J., Kwak, N., Discriminative subspace learning using generalized mean, Signal Processing, 219(2024), 109421.
  • Plotnitskaya, E., Heister, S., Veremyev, V., Mathematical model for a radar signal reflected from drone propellers as applied to the method of inverse synthetic aperture radar in bi-static radar, Journal of the Russian Universities. Radio-electronics, 26(6)(2023), 41–53.
  • Rao, Y., He, H., Wan, X., Yi, J., Range-angle dependent beam-pattern synthesis method for OFDM-based passive radar, Wuhan Univ. J. Nat. Sci., 27(3)(2022), 255–260.
  • Ren, J., Zhang, M., Yu, C., Liu, Z., Balanced MSE for imbalanced visual regression, In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2022), 7926–7935.
  • Sharkas, H., Solution of integral of the fourth power of a finite-length exponential Fourier series, Turkish Journal of Mathematics and Computer Science, 15(2)(2023), 403–406.
  • Sharkas, H., Echigo, H., Beamforming solutions for efficient link setup and link maintenance in mmwave communication systems, Master’s thesis, Lund University, Sweden, 2019.
  • Shevchenko, M., Malyshev, V., Gorovoy, A., Cherepanov, A., Spatial filtering of signals under imprecise calibration of antenna arrays, Journal of the Russian Universities. Radio-electronics, 26(6)(2023), 27–40.
  • Song, J., Choi, J., Love, D.J., Common codebook millimeter wave beam design: Designing beams for both sounding and communication with uniform planar arrays, IEEE Transactions on Communications, 65(4)(2017), 1859–1872.
  • Stoica, P., Moses, R.L., Spectral Analysis of Signals, Pearson Prentice Hall, 2005.
  • Takano, Y., Juntti, M., Matsumoto, T., ℓ1 LS and ℓ2 MMSE-based hybrid channel estimation for intermittent wireless connections, IEEE Transactions on Wireless Communications, 15(1)(2016), 314–328.
  • Tang, B., Zhang, N., Zhang, S., Huang, Z., MMSE-based waveform design for the distributed MIMO radar in spectrally crowded environments, The Journal of Engineering, 2019(20)(2019), 6603–6607.
  • Tohidi, E., Coutino, M., Gesbert, D., Revisiting matching pursuit: Beyond approximate submodularity, Signal Processing, 226(2025), 109638.
  • Vorobev, E., Veremyev, V., Kholodnyak, D., Recognition of propeller-driven aircraft in a passive bi-static radar, Journal of the Russian Universities. Radio-electronics, (6)(2018), 75–90.
  • Wiener, N., Generalized harmonic analysis, Acta Mathematica, 55(1930), 117–258.
  • Xu, J., Zhang, X., Li, Y., Kernel MSE algorithm: a unified framework for KFD, LS-SVM and KRR, In IJCNN’01. International Joint Conference on Neural Networks Proceedings (Cat. No.01CH37222), 2(2001), 1486–1491.
  • Yi, J., Wan, X., Leung, H., ℓ0-regularized least squares versus matched filtering, Signal Processing, 192(2021), 108398.
  • Yoshikawa, E., Takizawa, N., Kikuchi, H., Mega, T., Ushio, T., An estimator for weather radar doppler power spectrum via minimum mean square error, IEEE Transactions on Geoscience and Remote Sensing, 60(2022), 1–16.
  • Yoshikawa, E., Ushio, T., Kawasaki, Z., Yoshida, S., Morimoto, T. et al., MMSE beam forming on fast-scanning phased array weather radar, IEEE Transactions on Geoscience and Remote Sensing, 51(5)(2013), 3077–3088.
  • Yu, Y., Dou, W., Pseudo-Bessel beams in millimeter and sub-millimeter range, In Advanced Microwave and Millimeter Wave Technologies, IntechOpen, (2010), chapter 24.
Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Reel ve Kompleks Fonksiyonlar, Uygulamalı Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Hesham Sharkas 0000-0001-5359-4744

Yayımlanma Tarihi 31 Aralık 2024
Gönderilme Tarihi 24 Ocak 2024
Kabul Tarihi 5 Aralık 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 16 Sayı: 2

Kaynak Göster

APA Sharkas, H. (2024). Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II). Turkish Journal of Mathematics and Computer Science, 16(2), 419-425. https://doi.org/10.47000/tjmcs.1424850
AMA Sharkas H. Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II). TJMCS. Aralık 2024;16(2):419-425. doi:10.47000/tjmcs.1424850
Chicago Sharkas, Hesham. “Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II)”. Turkish Journal of Mathematics and Computer Science 16, sy. 2 (Aralık 2024): 419-25. https://doi.org/10.47000/tjmcs.1424850.
EndNote Sharkas H (01 Aralık 2024) Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II). Turkish Journal of Mathematics and Computer Science 16 2 419–425.
IEEE H. Sharkas, “Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II)”, TJMCS, c. 16, sy. 2, ss. 419–425, 2024, doi: 10.47000/tjmcs.1424850.
ISNAD Sharkas, Hesham. “Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II)”. Turkish Journal of Mathematics and Computer Science 16/2 (Aralık 2024), 419-425. https://doi.org/10.47000/tjmcs.1424850.
JAMA Sharkas H. Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II). TJMCS. 2024;16:419–425.
MLA Sharkas, Hesham. “Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II)”. Turkish Journal of Mathematics and Computer Science, c. 16, sy. 2, 2024, ss. 419-25, doi:10.47000/tjmcs.1424850.
Vancouver Sharkas H. Mapping of $L^2 -$norm of Two Multiplied $2\pi-$Periodic Functions to Their Fourier Coefficients (Part II). TJMCS. 2024;16(2):419-25.