The Study Of A Regionally Optimized Time-frequency Distribution Based On The Gaussian Mixture Model

As an emerging signal processing method, time-frequency analysis has gotten more and more widespread recognition in recent years. The basic thought of the time-frequency analysis is to design the union function of time and frequency, which is used to describe the energy density or the intensity of the signal with different time and frequency. The union function is called time frequency representation (TFR). The TFR is used to analysis the non-steady signal like radar signal, the seismic signal and so on, it can be used to indicate the energy accumulation situation near the instantaneous frequency at each time location and to time-frequency filter or signal systemization.There are two kinds of the TFR. One is the linear TFR,the short-time Fourier transform (STFT),the wavelet transform, and the Gabor expansion are the most common linear TFR. The second kind of TFR is quadratic time-frequency, also called time frequency distribution (TFD),it has the special advantage ,because the quadratic is the representation of energy of the signal. So the quadratic TFR is contact to the concept of energy, and is widely used.Because of the difficulties in suppressing the disturbance of the interference terms, the use of TFD became more limited. And suppressing the interference terms of the TFD has been the key question which the people study。The basic thought of most methods to suppress the interference terms is to reduce the iterations between the signal components, but at the same time, the time-frequency resolution is also reduced, so the suppression of the interference terms and the enhancement of the time-frequency resolution become a pair of contradictory.This thesis study a regionally optimized time-frequency distribution, it cansuppress the interference terms and enhance the TF resolution of the multi-component signal in the same time. This TFD is based on a TF energy model , the model can not only descript the TF energy structure of the signal reasonably but also distinguish the number of the signal components and the region which occupied by each component while descript the TF energy structure . this thesis use a set of Gaussian radial function (or called a finite mixture model of Gaussian function) to construct the energy model. After the structure of the model has been determined, for a given signal, the parameters of it have to be optimized. the whole course is divided into two parts. First, the initial TDF is thought as a sample of the true structure of the signal energy, taking this as the foundation, the probability estimation technology is used to optimize the parameters of the Gauss basis function in fixed number model. And then the approximate numbers of the Gaussian function in the model is determined by the function merging technology. A parameter-determined model can approximately indicate the number of the components of the signal and the TF region occupied by each component. This information is used to decompose the signal and analysis the decomposed components separately. Then after combining the results together, the final distribution is gotten.The generation of the regionally optimized time-frequency distribution consists of the following stages:1. An initial TFD is generated. The TFD which can suppress the interference terms well. The component concentration is recovered in the final distribution. The TFDs like SPWD are all satisfactory. The TFD is chosen depend on the real signal.2. The initial TFD is modeled by a Gaussian mixture model with N Gaussians, where N is more than the number of the components in the signal. Good initializations of the parameters provide a good coverage of the initial TFD.3. The parameters of the N-Gaussian mixture model are determined. The parameter determination is considered as an optimization problem, the aim is to obtain the best representation of the energy distribution of the signal. In this thesis, a variant of the expectation-maximization algorithm is used to optimize the model parameters.4. After the parameters optimization, the model provides a reasonable approximation to the energy distribution, but the number of the Gaussians is more than the number of the signal components, so the functional merging technique is applied to merge the Gaussians which represent the same component of the signal, then the number of the Gaussian can approximately equal to the true number of components in the distribution.5. Using the model, a Bayesian classification approach is used to define the TF region occupied by each component, and effectively divide the TF plan into several regions.6. Based on the defined regions, time-frequency projection filter is designed to extract the component, so the signal is divided into several single component signals.7. using WVD analysis the single component signals, and then sum the results. The final regionally optimized time-frequency distribution is generated. After the experiment, a nicer result is got. The final regionally optimized TFD can not only suppress the interference terms, but also get a well resolution.It is emphasized that, the model is made up with the Gaussians, and one of the limitations of the mixture model is that its components are restricted to occupying an elliptical shape in the TF plan. Highly non-linear components cannot be satisfactorily represented by a single model component. And the basic assumption of the functional merging procedure is that separate Gaussians in the model shouldrepresent separate components, and this cannot be satisfied for a (highly) non-linear signal, so we cannot get an exact number of the signal components. And the TF filter cannot be carried out, the regionally optimized TFD cannot be exactly generated either. So the regionally optimized TFD generated in this thesis is not suitable for the highly non-linear signal.