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，以傅立叶基数给出。WSS信号的谱表示是一个具有正交增量的复杂过程，隐含地假设为圆形分布(旋转不变的概率密度函数)。直观地说，这意味着相位在时间域中均匀分布，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)决定论；(Ii)非平稳性。确定谱过程是非循环的，因为它们的相位在时间域中是确定性分布的。因此，完全描述其二阶统计量所需的充分统计量必须包括：(I)厄米方差(功率谱)；和(Ii)互补方差(互补功率谱或全景)。非平稳性需要失去傅立叶基或正交增量约束。放松后者产生了时频表示的基础，它自然满足非平稳性，例如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，通常发生在通信、气象、海洋学、气候学、天文学和经济等科学技术领域。检测、估计和信号分析中的大多数问题都是根据频谱表示来描述的，随着最近建立的频谱非循环基础的建立，有可能对真实世界的信号进行更准确的建模。尽管如此，我们仍然有许多问题和目标需要解决：理论基础--单个实现的自卷积和全景的遍历估计条件；-维纳-金钦定理，它分别将自相关和自卷积函数与功率谱和全景图联系起来；-充分谱统计的最大似然估计器，以及相关的Cramer-Rao下界；-非平稳确定性信号(例如线性啁啾)中频谱非圆性的证明，以及非零交叉频谱统计的表现(厄米特和互补性)；使用非傅立叶基的张量系统中的确定性--使用库普曼算符及其来自动力系统理论的相关谱展开估计单变量和张量系统的确定性非傅立叶基；-非傅立叶基中的“确定性指标”，类似于傅立叶基中非圆谱过程的“圆度系数”；应用--使用充分的谱统计对一般高斯噪声(有色和白色)中的正弦信号进行统计上有效的估计和检测；-考虑自相关和自卷积的最大熵谱估计；-通过从非循环频谱过程采样生成替代数据，以及用于非线性检测的改进的延迟向量方差(DVV)方法；-利用交叉频谱统计的应用：o维纳滤波；o用于检测信号中确定性分量数量的广义似然比测试；o假设非循环频谱过程嵌入各向同性循环高斯噪声中的概率频谱分解；-多通道频谱分析、维纳滤波和使用张量分解的音乐信号处理。理论发展将应用于真实世界的心电、脑电、语音和电力系统信号。上述研究目标和应用与EPSRC的相关研究领域一致，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.