International Journal of Wavelets, Multiresolution and Information Processing

It is well known that a Hilbert transform pair of wavelet bases improves the lack of translation invariance of the discrete wavelet transform. However, its shapes and improvement are limited by the difficulty in applying the Hilbert transform pair to a discrete signal. In this paper, novel Hilbert transform pairs of wavelet bases, which are based on a Meyer wavelet and have a wide range of shapes, are proposed to create perfect translation invariance, and their calculation method is designed to apply these wavelet bases to any discrete signal. Therefore, perfect translation invariance is achieved with a wide range of shapes of the Hilbert transform pairs of wavelet bases.

This paper is concerned with an input-output relation of linear systems when using the complex wavelet packet transform. In general, the linear relation between input-output is guaranteed by the following two conditions: (1) the mother wavelet has a better frequency resolution, and (2) the real and imaginary parts in the mother wavelet consist of a Hilbert transform pair. The complex wavelet satisfying the above conditions has been used for demonstrating the linear relation. In this paper, a complex wavelet packet transform is applied for improving the frequency resolution. The validity of our approach is shown through a numerical experiment.

The useful theorems for achieving perfect translation invariance have already been proved, and based on these theorems, dual-tree complex discrete wavelet transforms with perfect translation invariance have been proposed. However, due to the complication of frequency divisions with wavelet packets, it is difficult to design complex wavelet packet transforms with perfect translation invariance. In this paper, based on the aforementioned theorems, novel complex wavelet packet transforms are designed to achieve perfect translation invariance. These complex wavelet packet transforms are based on the Meyer wavelet, which has the important characteristic of possessing a wide range of shapes. In this paper, two types of complex wavelet packet transforms are designed with the optimized Meyer wavelet. One of them is based on a single Meyer wavelet and the other is based on a number of different shapes of the Meyer wavelets to create good localization of wavelet packets.

The solutions of convection and diffusion equations are analyzed using Multiresolution Multiwavelet (MRMW) basis functions of multidimension including time. The distribution representing the solution reconstructed in non-standard form of the higher order MRMW reproduces the original solution in the multidimensional space much more efficiently than the lower order one. Smooth movement of the distribution is observed except for the initial area even using the MRMW of zeroth order but the shape of the distribution is not well reproduced for the case. The initial area is better reproduced by the MRMW of higher order and the wavelet component decays much quicker both in low and high levels of resolution.

The cocktail party problem deals with the specialized human listening ability to focus one's listening attention on a single talker among a cacophony of conversations and background noises. The blind source separation problem is how to enable computers to solve the cocktail party problem in a satisfactory manner. The simplest version of spatio-temporal mixture problem, which is a type of blind source separation problem, has been solved by a generalized version of the quotient signal estimation method based on the analytic wavelet transform, under the assumption that the time delays are integer multiples of the sampling period. The analytic wavelet transform is used to represent time-frequency information of observed signals. Without the above assumption, improved algorithms, utilizing phase information of the analytic wavelet transforms of the observed signals, are proposed. A series of numerical simulations is presented.

Independent component analysis (ICA) is a useful method for blind source separation of two or more signals. We have previously proposed a new method combining ICA with the complex discrete wavelet transform (CDWT), in which voice and noise signals were separated using a new method. At that time, we used a simulated signal. In this study, we analyze measured biological signals by using a new method, and discuss its effectiveness. As an experiment, we try to separate an electromyogram (EMG) signal from an electrocardiogram (ECG) signal.

Recent research on time-frequency analysis using wavelet transforms has focused on analyzing wavelets using a mathematical approach. In this study, a measured signal is adopted as the wavelet, and we analyze the correlation between acoustic signals in the car cabin and air intake suction noise signals by applying the proposed system. Since the traditional calculation of correlation repeats the averaging procedure, the original signal must be stationary. Consequently, a technique for separating and identifying noise from each part of the engine is used for analyzing the noise source contribution. To apply the method to time-varying signals, we introduce the concept of an instantaneous correlation factor (ICF), and we prove that the dominant feature of the correlation can be estimated by the ICF. However, the ICF method has not previously used a time-varying analyzing wavelet. In this research, to conduct a practical analysis we proposed Time-Time analysis, which was developed based on the ICF. Furthermore, we also proposed Complex Time-Time analysis, extending into the complex region for improvement in accuracy. First, we verified the effectiveness of the ICF using simulated signals, and then we investigated the contribution of intake noise (which is one of the engine noises) to the car interior noise during acceleration. Using an ICF in which signals relating to each noise source are selected for the analyzing wavelet (AW), we showed that this technique is also useful for contribution analysis. We also conducted a fundamental experiment about audibility impressions.

We have already proposed a new type of wavelet, named the "motion wavelet", whose mother wavelet is varied according to a certain reference signal, and at the same time, transforms another target signal. The wavelet allows variable frequency elements to be picked out from the target signal according to the reference signal. This motion wavelet can also be applied to making entertainment sounds. However, it generates only monaural sound and its effect is simple. In this paper, we propose a new type of motion wavelet having double the modulation performance by the real and imaginary parts of its complex mother wavelet varied according to two different reference signals. Furthermore, we introduce a fast calculation method for it and one of its applications to entertainment sounds.

This paper proposes a new method of road type estimation by means of the running tire sound (the tire inner sound). The sound is collected with a microphone installed in a tire and a wireless communication and power supply system. Characteristic frequency of each road condition is extracted from wavelet transform of the sound signal. An experiment result shows the proposing logic can discriminate the road conditions, including flat, pave, rough and ditch.

In this study, a novel direction selection method using the two-dimensional complex discrete wavelet transform (2D-CDWT) is proposed. In order to achieve arbitrary direction selection, the directional filters are first designed. Calculation procedure of directional selection can be shown as follows: (1) The 16 sub-images are generally generated from the original image by the 2D-CDWT without a down-sampling process and the 12 sub-images that correspond to the high-frequency components are selected. (2) The 12 sub-images are filtered by using the designed directional filter. (3) The down-sampling process is carried out and the resulting images are obtained. Furthermore, this method is applied to the surface analysis of a wafer, and it is confirmed that our method is effective in detecting irregular direction components.

The authorship of this paper should read as S. K. Kaushik and Varinder Kumar.