In recent years, Wigner-Ville distribution has been recognized as a useful tool and applied to various types of mechanical noise and vibration signals. However, its limitations which mainly come from cross-talk have not been well addressed. The cross-talk takes place for a signal with multiple components, simply because the Wigner-Ville distribution is a bilinear transform. The cross-talk can cause negative value in the distribution, and degrade its readability. The cross-talk must be eliminated or reduced. Smoothing the Wigner-Ville distribution by convoluting it with a window, is the most commonly used method to reduce the cross-talk.
The main goal of this thesis is to develop a distribution which can effectively reduce the cross-talk. In order to accomplish this, we have examined the structure of cross-talk in the ambiguity function domain; more specifically, the relative location of signal components and cross-talk. For the quantitative analysis of the structure, Wigner-Ville distribution of arbitrary shape is approximated as a sum of linear segments. From this model, it is found that the signal components appear as the lines crossing the origin by preserving their own angles and lengths, and the cross-talk distributes widely in the ambiguity function plane.
Based on this observation, a new window is designed to maintain only the signal components. The new window is named as the angle weighted window, which is to emphasize the fact that the window is weighted the power of the distribution along the angle. The angle dependent power can be obtained by rotating and multiplying a narrow strip to the ambiguity function. The angle weighted window is examined numerically. It is better in reducing cross-talk than a conventional window, the Gaussian window.
Applying the angle weighted window to the ambiguity function corresponds to smoothing the Wigner-Ville distribution. In order to examine the effects of smoothing with the angle weighted window, the bias error has been examined. The bias error is proportional to the second order derivatives of the Wigner-Ville distribution and the second momenta of the smoothing window. The cross momentum which is zero for the Gaussian window, has an important role in smoothing the Wigner-Ville distribution. The angle weighted window changes not only the Wigner-Ville distribution but also its properties. The properties of the new distribution are examined and is to be same as the distribution, applying the Gaussian window.
Waves, such as sound waves and many structural waves, convey energy by the motion of the media in which the waves propagate. The media itself is not usually transported. One often wants to know the direction of the wave transportation. This is the energy flux, which can be regarded as the intensity. The time-frequency analysis of intensity is possible by means of instantaneous intensity. The instantaneous intensity shows how energy propagates with regard to time and frequency.
Instantaneous intensity can be estimated by using the cross Wigner- Ville distribution. Cross Wigner-Ville distribution transforms the cross correlation function, therefore has phase information of two signals. Its characteristics has not been well addressed, especially the way of smoothing and its properties of cross ambiguity function. In the cross ambiguity function, the signal components do not cross the origin. This can restrict the use of the proposed window in the ambiguity function domain. Windowing leaves only the components close to the origin, and attenuates the components far from the origin. The effects of window on the estimating the instantaneous intensity is examined.
At discontinuities in a structure, such as boundaries or junctions, the waves experience reflections, transmission, and/or conversion to different types of waves. These phenomena can be illustrated by using the instantaneous structural intensity. We examined a T-shaped beam which has a junction. Instantaneous structural intensity gives us a graphical image concerning the wave propagation, reflection at the discontinuities, and transmission at the junction. The cross-talk can be reduced by appropriately smoothing the Wigner-Ville distributions. For this purpose, the angle weighted window is applied to the estimation of the instantaneous structural intensity. It shows good performance in reducing cross-talk.