# Statistical Signal Processing of Nonstationary Processes

Lead Research Organisation:
Imperial College London

Department Name: Electrical and Electronic Engineering

### Abstract

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.

(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.

## People |
## ORCID iD |

Danilo Mandic (Primary Supervisor) | |

Bruno Scalzo Dees (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509486/1 | 01/10/2016 | 30/09/2021 | |||

1859640 | Studentship | EP/N509486/1 | 01/10/2016 | 31/03/2020 | Bruno Scalzo Dees |