Efficient Time-Frequency Localization of a Signal

A new representation of the Fourier transform in terms of time and scale localization is discussed that uses a newly coined A-wavelet transform (Grigoryan 2005). The A-wavelet transform uses cosine- and sine-wavelet type functions, which employ, respectively, cosine and sine signals of length . For a given frequency , the cosine- and sine-wavelet type functions are evaluated at time points separated by on the time-axis. This is a two-parameter representation of a signal in terms of time and scale (frequency), and can find out frequency contents present in the signal at any time point using less computation. In this paper, we extend this work to provide further signal information in a better way and name it as -wavelet transform. In our proposed work, we use cosine and sine signals defined over the time intervals, each of length , , and are nonnegative integers, to develop cosine- and sine-type wavelets. Using smaller time intervals provides sharper frequency localization in the time-frequency plane as the frequency is inversely proportional to the time. It further reduces the computation for evaluating the Fourier transform at a given frequency. The A-wavelet transform can be derived as a special case of the -wavelet transform. 1. Introduction There are two domains for representing a signal: time and frequency domains. Depending upon the information required, either representation can be used. Fourier analysis has been the main technique for transforming a signal from one representation into another. In spite of the fact that the Fourier analysis is an ideal solution for deterministic and stationary signals, it is hardly of any use for time-varying signals or nonstationary signals, because analysis of these types of signals compromises between their transition and long term behaviors. For these types of signals, a transform is desired to represent the signal in a two-parameter form. The very first such transform in literature is the short time Fourier transform (STFT) [1]. This transform uses a time-window function to decompose the signal into segments and then the Fourier analysis is carried out on individual segments. The STFT provides local features that are present in the signal in a limited form because it uses the time-window function of fixed width for all frequency contents and thus it is unable to extract the required information in any given signal. So, a transform that can represent the signal in two-parameter form and uses the time-window function of different lengths is needed. Application of the short time Fourier transform multiple times