Everywhere around us are signals that need to be analysed. Seismic tremors, human speech, engine vibrations, medical images, financial data, music, and many other types of signals have to be efficiently encoded, compressed, cleaned up, reconstructed, described, simplified, modelled, distinguished, or located. Wavelet analysis is a new and promising set of tools and techniques for doing this. FFT analysis gives only frequency domain information it won’t give time domain information so further improvement to obtain time information was by gabber that is short time Fourier transform but it also fails to explain what spectral components exist at what particular time interval to obtain this the ultimate solution is wavelet transform.
The wavelet transform found to be particular useful for analysing signals which can be aperiodic, noisy, intermittent, transient and so on. Its ability to examine the signal simultaneously both in time and frequency in a distinctly different way from the traditional short time Fourier Transform has spawned a number of sophisticated wavelet based methods for signal manipulation and interrogation.
Fourier analysis: Signal analysts already have at their disposal an impressive arsenal of tools. Perhaps the most well-known of these is Fourier analysis, which breaks down a signal into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a mathematical technique for transforming our view of the signal from a time-based one to a frequency-based one.
Fourier transforms and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is also used in Nuclear Magnetic Resonance (NMR) and in other kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier transformed to a Lorentzian line shape in the frequency domain. The Fourier transform is also used in magnetic resonance imaging (MRI) and mass spectrometry.
We propose a new analysis tool for signals, called signature, that is based on complex wavelet signs. The complexvalued signature of a signal at some spatial location is deﬁned as the ﬁne-scale limit of the signs of its complex wavelet coefﬁcients. We show that the signature equals zero at sufﬁciently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. We establish that signature is invariant under fractional differentiation and rotates in the complex plane under fractional Hilbert transforms. We derive an appropriate discretization, which shows that wavelet signatures can be computed explicitly. This allows an immediate application to signal analysis.