<title>Q-Gabor wavelet</title>

Abstract

The Gabor wavelet is well-known tool in the various fields, such as computational neuroscience, multi-resolutional analysis and so on. The Gabor wavelet is a kind of the Gaussian modulated sinusoidal wave or a kind of windowed Fourier transformation with the Gaussian kernel window. The Gabor wavelet attains the minimum of the uncertainty relation. However the width and the height of the time-frequency window do not change in their lengths depending on the analyzing frequency. This makes the application area of the Gabor wavelet narrow. On the other hand, instead of using the conventional Gaussian distribution as a kernel of the Gabor wavelet, if the q-normal distribution is used, we can get the q-Gabor wavelet as a possible generalization of the Gabor wavelet. The q-normal distribution, which is given by the author, is one of the generalized Gaussian distribution. In this paper, we give the definitions of the q-Gabor wavelet for continuous version and discrete version. The discrete version consists of two different types of the q-Gabor wavelet. One is similar to the conventional Gabor wavelet with respect to the width and the height of the time-frequency window, the other is similar to the conventional discrete wavelet system such that the width and the height of the time-frequency window change in their lengths depending on the frequency. The mother wavelet is also given for the orthonormal q-Gabor wavelet with some approximation.