Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches

Nur ; Andi Tenri Ajeng ; Khotimah ; Husnul ; Cayo ; Putri Nilam
Bilangan : Jurnal Ilmiah Matematika, Kebumian dan... 📅 2025 Vol 3, No 6 🔗 DOI: 10.62383/bilangan.v3i6.849
Abstract
The Hartley transform provides a real-valued alternative to the classical Fourier transform, offering structural advantages for the analysis of real-valued signals. This paper presents a systematic study of the continuous Hartley transform, including its definition, inversion formula, Plancherel identity, and core operational properties such as shifting, modulation, and convolution. The analytical framework is developed in parallel with the classical Fourier theory to highlight structural similarities and distinctions between the two transforms. Furthermore, we establish a Hartley-type Heisenberg uncertainty principle using two complementary approaches: a direct method based on intrinsic properties of the Hartley kernel, and a Fourier-based method that exploits the algebraic relationship between the Hartley and Fourier transforms. These results provide a unified and rigorous foundation for understanding uncertainty relations within real-valued transform frameworks, and they demonstrate the continued relevance of the Hartley transform in harmonic analysis, integral transforms, and modern signal processing.
Keywords
Fourier transform; Hartley transform; Heisenberg inequality; Signal Analysis; Uncertainty principle
How to Cite

Nur, et al. (2025). Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches. Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa, 3(6). https://doi.org/10.62383/bilangan.v3i6.849

Nur, Andi Tenri Ajeng; Khotimah, Husnul; Cayo, Putri Nilam, "Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches," Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa, vol. 3, no. 6, 2025.

Nur, Andi Tenri Ajeng; Khotimah, Husnul; Cayo, Putri Nilam. "Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches." Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa, vol. 3, no. 6, 2025.

Nur, Andi Tenri Ajeng; Khotimah, Husnul; Cayo, Putri Nilam. "Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches." Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa 3, no. 6 (2025).

Nur, et al. (2025) 'Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches', Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa, 3(6). doi: 10.62383/bilangan.v3i6.849.

Nur, Andi Tenri Ajeng; Khotimah, Husnul; Cayo, Putri Nilam. Uncertainty Principle for the Hartley Transform: Direct and Fourier–Based Approaches. Bilangan : Jurnal Ilmiah Matematika, Kebumian dan Angkasa. 2025;3(6).

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