Detecting Periodic Signals Based On A Statistical Theory Of Signal Coherence

Nature does not produce perfect periodic signals. There is always some variation in the waveform which is labeled as periodic signal. All periodic signals have some sort of variability from period to period regardless of how stable they appear to be. For example, a true sinusoidal time series is a deterministic function of time, so it has zero bandwidth around the sinusoid's frequency. As all known, a zero bandwidth is impossible in nature. Deterministic sinusoids are used to model cycles as a mathematical convenience, and not accurate. It is time to break away from this simplification in order to model various periodic signals that observed in different sciences fields, so we need set up a new periodic signals model.The traditional signal detection is often assumed that the probability characteristics of the observing sample is known or have some prior knowledge. Because of signal detection in most cases are in strong noise background, traditional time-domain detection have poor performance. And in strong noise environment, when signal is non-stationary, especially when has unknown parameters, the conventional detection method does not applicable. We know that sinusoidal signal in noise when the signal's frequency is known, but amplitude and phase is unknown, if lack of the priori knowledge of unknown parameters, then we use the generalized likelihood ratio test method. However, this method does not take full advantage of the signal priori knowledge, at the actual periodic signal detection circumstances, we probably know the frequency of the signal, even if do not know, we know it random distribution in a limited band. At this point, we put it as completely unknown signal to detect, obviously is wrong. In order to improve signal to noise ratio and adapt the complicated signal, it is essential to carry out such periodic signal detection research. Therefore, we should further research the detection theory, develop new detection theory, introduct new detection methods, and improve detector's detection performance in a complex environment and low SNR environment.This paper set up a new periodic signal model, with changed periodic signal to simulate observed signal in the different science fields. Such random modulation periodic signal model comparing with original periodic signal model which can produce zero-bandwidth harmonic. This signal model more realistic, and this modulation assumption more appropriate.By analysis of the theory of signals in frequency domain coherent, I determine the signal coherence function applied to the periodic signal detection. The principle of signal coherence function is derived and theoretical analysis, and then it conducts a series of statistical analysis.This paper presents a specific signal detection method which is based on the statistical theory of signal coherence, and this method is easy computer implementation, and easy automation. Application of the method and current practical application methods to simulate and to compare, proved that additional noise whether or not, this detection method is superior to the standard spectral waterfall method. Analysis of the false alarm probability, detection probability, as well as the detection influence of parameters, and it draw the additional noise only required to satisfy stationarity and finite dependence, do not need the assumption noise is Gaussian noise. It gives a situation of the signal's existence, we known the signal frequency range, the method of determine the signal basic cycle.The following gives the main work of this paper in detail.1. I comprehensively collect, organize, read and studied a variety of domestic and foreign literature related to the signal detection, sum up the background and application of the signal detection and periodic signal detection. Aiming at the problems that the nature does not produce perfect periodic signals and deterministic sinusoids are used to model cycles as a mathematical convenience, not accurate, we need to set up a new model. Aiming at the problems of the traditional methods did not take full advantage of the prior knowledge of the signals, so we need to study new detection methods.2. It introduces the coherent signal theory, gives the proof of traditional frequency-domain coherence function equal one. It gives detailed description on the relevant signal statistical theory, including the evaluation criteria and some formula. It gives the periodic signal express in time domain and frequency domain and the classification of the periodic signal.3. It introduces the detection classification, points out that according to the noise type of detection this paper belongs to non-parametric testing, according to the type of signal types this paper belongs to unknown parameters detection, according to the type of observing sample values to deal with this paper belongs to non-fixed observation sample value. A detailed analysis of the signal detection rules, we derived calculating likelihood ratio of maximum posteriori probability criterion, Bayesian criterion and the minimum error probability criterion need to know a priori probability. For this paper, obtain a priori probability is very difficult, in order to guarantee a low probability of false alarm, only to restrict the probability of false alarm and to make the probability of detection greatest, so we should use Neyman-Pearson criteria. It introduces the periodic signal detection methods, infers the periodic signal detection problem in strong noise background, an intuitive way is to improve the signal to noise ratio. Under this paper, gives detailed description on time-domain average processing, signal correlation and periodogram at periodic signal detection applications, and gives the simulation results and analysis.4. Aiming at the fact that the nature does not produce perfect periodic signals, I find a random modulated periodic signal model which can descript the changes in the periodic signal, analysis the model and formula for deformation, study the elimination of the linear and nonlinear trend of the signal, and use matlab to simulate them.5. The principle of the signal coherence function is analyzed and formula derivation, it gives the statistical analysis of the signal coherence function, according to the traditional amplitude squared coherence function's conclusions, it draws the signal coherence function's messages, and simulates the coherent function and studies it performance.6. It presents a specific signal detection method which based on a statistical theory of signal coherence, using this method simulation comparing with the periodogram. Analysis of the false alarm probability, detection probability, as well as the detection influence of parameters. It gives a situation of the existence of signal, we known the signal frequency range, the method of determine the basic cycle.