基于非圆复数的自适应滤波算法完备均方性能分析及应用

It has been illustrated in both theory and practice that unlike the case in the real domain, there is a special type of noncircular signals in the complex domain. Their existence leads to the situations that the traditional mean square analysis of adaptive filtering algorithms for the circular signals cannot be extended to noncircular signals, and even worse, the real-valued mean square performance evaluation of adaptive filtering algorithms is not able to separate the individual mean square performance for real part (I-channel) and imaginary part (Q-channel) of adaptive filtering algorithms. To this end, enlightened by the full second order statistics of complex signals, we first introduce the noncircular nature of both the estimation error and the weight error, and propose a novel concept called complementary mean square performance to represent their noncircularity. We second explore the convergence analysis and the steady-state evaluation of the proposed complementary mean square performance, which, together with the standard mean square performance, establish the full mean square analysis of adaptive filtering algorithms in both transient and steady-state stages. Meanwhile, by employing the duality between real and complex domains, we decouple the individual mean square performance of real part (I-channel) and imaginary part (Q-channel) of adaptive filtering algorithms. Finally, we explore the applications of the adaptive filtering algorithms and the proposed full mean square performance analysis in a typical application scenario of noncircular complex signals, that is, IQ imbalance compensation of wide-band multi-carrier communication systems built upon Orthogonal Frequency Division Multiplexing (OFDM) and Generalized Frequency Division Multiplexing (GFDM) techniques.

理论与实际均表明了不同于实数域，复数域存在一类特殊的非圆信号。这一方面造成传统基于圆性信号假设的自适应滤波算法均方性能的理论分析结果无法推广至非圆复数，另一方面使得基于实数表征的均方性能标准在复数域具有局限性，无法衡量算法实部（I路）与虚部（Q路）各自的均方性能。基于此，本项目以复数的完备二阶统计理论为指导，引入自适应滤波算法估计误差与权重误差的非圆性，提出补充均方性能的概念。其次，探索补充均方特性的收敛性与稳态性，协同标准的均方性能分析，建立各类自适应滤波算法的完备均方性能从收敛到稳态全程的理论分析。同时，利用复数域与实数域的二元性，从完备均方性能中解耦实现算法实部（I路）与虚部（Q路）各自的均方性能。最后，针对非圆复数的典型应用场景，探索各类自适应滤波算法及其完备均方性能分析在正交频分复用及广义频分复用宽带多载波通信系统IQ不平衡补偿问题中的应用。

本项目“基于非圆复数的自适应滤波算法完备均方性能分析及应用” 以非圆复数的完备二阶统计理论为指导，引入自适应滤波算法估计误差与权重误差的非圆性，提出了补充均方性能的概念。其次，探索了补充均方特性的收敛性与稳态性，协同标准的均方性能分析，建立了各类自适应滤波算法的完备均方性能从收敛到稳态全程的理论分析。同时，利用复数域与实数域的二元性，从完备均方性能中解耦实现算法实部（I路）与虚部（Q路）各自的均方性能。最后，针对非圆复数的典型应用场景，本项目探索各类自适应滤波算法及其完备均方性能分析在正交频分复用（OFDM）及广义频分复用（GFDM）宽带多载波通信系统IQ不平衡补偿问题中的应用。研究意义在于为复数自适应估计领域以及 OFDM 和 GFDM 宽带多载波通信系统中 IQ 不平衡问题的解决提供新理论、新技术及新测量标准。