Statistical Signal Processing of Nonstationary Processes

信号处理通常依赖于广义平稳（WSS）信号的频率解释，以傅立叶基的形式给出。WSS信号的频谱表示是具有正交增量的复杂过程，隐含地假设为圆形分布（旋转不变概率密度函数）。直观地说，这意味着相位在时域中均匀分布，或者换句话说，是i.i.d.。这些假设不足以模拟真实世界的信号，因为它们不能满足两个关键现象：（i）确定性;和（ii）nonstationarity.Deterministic频谱过程是非圆形的，因为它们的相位是确定性分布在时域中。因此，完全描述其二阶统计量所需的足够统计量必须包括：（i）埃尔米特方差（功率谱）;和（ii）互补方差（互补功率谱或全景图）。非平稳性需要丢失傅里叶基或正交增量约束。放松后者导致了时频表示的基础，这自然迎合了非平稳性，例如在循环平稳过程中遇到的非平稳性，循环平稳过程是一类重要的具有周期性变化的二阶矩的过程，并且通常发生在科学和技术中，包括通信，气象学，海洋学，气候学，天文学和经济学。和信号分析是根据频谱表示来措辞的，并且随着最近在频谱非圆性方面建立的基础，有可能更准确地模拟真实世界的信号。尽管如此，仍然存在许多问题和目标，我们的目标是解决：理论基础-遍历估计条件的自卷积和全景的一个单一的实现;-维纳-欣钦定理，分别链接的自相关和自卷积函数的功率谱和全景;-最大似然估计的充分的频谱统计，和相关的克拉美-饶下界;- 非平稳确定性信号中的谱非圆性证明（例如线性啁啾）和非零交叉频谱统计的表现（Hermitian和互补）;使用非傅立叶基的张量变量系统中的确定性-单变量和张量变量系统的确定性非傅立叶基的估计，使用Koopman算子及其相关的动力系统理论的频谱扩展;-非傅立叶基中的“确定性指标”，类似于傅立叶基中非圆形频谱过程的“圆形系数”;应用-一般高斯噪声中正弦曲线的统计有效估计和检测- 考虑自相关和自卷积的最大熵谱估计;- 通过从非圆形频谱过程采样来生成替代数据，以及用于非线性检测的改进的延迟向量方差（DVV）方法; o概率谱分解，假设非圆形谱过程嵌入各向同性圆形高斯噪声中;- 使用张量分解的多通道频谱分析，维纳滤波和MUSIC信号处理。理论发展将应用于真实世界的心电图（ECG），脑电图（EEG），上述研究目标和应用与相关的EPSRC研究领域一致，包括：-数字信号处理;-统计和应用概率;-非线性系统。

Signal processing commonly relies on the frequency interpretation of wide-sense stationary (WSS) signals, given in terms of the Fourier basis. The spectral representation of a WSS signal is a complex process with orthogonal increments, implicitly assumed to be circularly distributed (rotation-invariant probability density function). Intuitively, this implies that the phase is uniformly distributed in the time-domain, or in other words, is i.i.d.. These assumptions are inadequate to model real-world signals, since they fail to cater for two critical phenomena: (i) determinism; and (ii) nonstationarity.Deterministic spectral processes are noncircular, as their phase is deterministically distributed in the time domain. The sufficient statistics required to fully describe its second-order statistics must therefore include: (i) the Hermitian variance (power spectrum); and (ii) the complementary variance (complementary power spectrum or panorama).Nonstationarity requires the loss of either the Fourier basis, or the orthogonal increment constraint. Relaxing the latter leads to a basis for time-frequency representations, which naturally caters for nonstationarity such as that encountered in cyclostationary processes, an important class of processes that have periodically varying second-order moments, and commonly occurs in science and technology, including communications, meteorology, oceanography, climatology, astronomy, and economics.Most problems in detection, estimation, and signal analysis are phrased in terms of the spectral representation, and with the recently established foundations in spectral noncircularity, there is potential to more accurately model real-world signals. Still, there remain numerous issues and objectives which we aim to address:Theoretical foundations- Ergodic estimation conditions for the autoconvolution and panorama of a single realisation;- Wiener-Khinchin theorem which respectively links the autocorrelation and autoconvolution functions to the power spectrum and panorama;- Maximum likelihood estimator of the sufficient spectral statistics, and the associated Cramer-Rao Lower Bounds;- Proof for spectral noncircularity in nonstationary deterministic signals (e.g. linear chirps), and the manifestation of non-zero cross-frequency spectral statistics (Hermitian and complementary);Determinism in tensor-variate systems using non-Fourier bases- Estimation of the deterministic non-Fourier basis of uni- and tensor-variate systems, using the Koopman operator and its associated spectral expansion from dynamical systems theory;- "Determinism indicators" in non-Fourier bases, analogous to the "circularity coefficient" for noncircular spectral processes in the Fourier basis; Applications- Statistically efficient estimation and detection of a sinusoid in general Gaussian noise (colored and white) using the sufficient spectral statistics;- Maximum entropy spectral estimation accounting for the autocorrelation and autoconvolution;- Surrogate data generation by sampling from a noncircular spectral process, and an improved delay vector variance (DVV) methodology for nonlinearity detection;- Applications which utilise cross-frequency spectral statistics:o Wiener filtering;o Generalised likelihood ratio tests for detecting the number of deterministic components in a signal;o Probabilistic spectral decomposition which assumes the noncircular spectral process is embedded in isotropic circular Gaussian noise;- Multichannel spectral analysis, Wiener filtering, and MUSIC signal processing using tensor decompositions.The theoretical developments will be applied to real-world electrocardiogram (ECG), electroencephalogram (EEG), speech and power system signals.The aforementioned research objectives and applications align with relevant EPSRC research areas, including: - Digital Signal Processing;- Statistics and Applied Probability;- Non-Linear Systems.