Fourier Analysis on 3D Complex Numbers and Communication Applications Theoretical Framework and Numerical Verification 三维复数傅里叶分析与通信应用 理论框架与数值验证

This paper systematically establishes the Fourier analysis theory on the 3D complex algebra T₃ and verifies its potential applications in signal processing and communication systems through numerical experiments. T₃ is a three-dimensional commutative associative algebra over the real numbers with basis {1,e,i} satisfying e²=1, i²=−1, ei=0. Its Fourier transform is a natural generalization of the standard Fourier transform to three-dimensional signal space. Core Results: • Fourier Transform Unitarity: The 3D complex Fourier transform is numerically exactly unitary, with reconstruction error 2.46×10⁻¹⁷, consistent with the precision of the standard Fourier transform. • Directional Filtering: The 3D complex Fourier transform supports independent frequency-domain filtering along three orthogonal directions, with uniform energy distribution across channels and crosstalk at the 10⁻⁴ level. • Constellation Optimization: After optimization, the minimum distance of the 8-point 3D complex constellation increases from 1.1547 to 1.1685 (1.2% improvement), approaching the geometric limit of the 8-point Tammes problem on a 3D sphere. • High-Order Modulation: The 16-point 3D complex modulation at 4 bits/symbol shows a BER curve comparable to 16QAM; the 32-point modulation at 5 bits/symbol exhibits a clear convergence trend. 摘要 本文系统建立三维复数 T₃ 的傅里叶分析理论,并通过数值实验验证其在信号处理与通信系统中的潜在应用。三维复数是实数域上的三维交换结合代数,基底 {1,e,i} 满足 e²=1, i²=−1, ei=0,其傅里叶变换是标准傅里叶变换在三维信号空间中的自然推广。