13.4 Continuous Wavelet II

This paragraph introduces the topic of wavelets, specifically focusing on the continuous wavelet transform (CWT). It sets the stage for a series of lectures that will explore both continuous and discrete wavelet transformations, drawing parallels with the Fourier transform and its discrete counterpart. The paragraph emphasizes the need for an analysis tool that can handle non-stationary signals with varying frequencies over time, pointing out the limitations of traditional Fourier analysis due to its global and time-invariant nature. The introduction of the short-time Fourier transform (STFT) as a precursor to wavelets is also mentioned, highlighting its ability to perform localized frequency analysis by using a sliding window function.

The second paragraph delves into the specifics of the wavelet transform, explaining the process and parameters involved. It contrasts the wavelet transform with the Fourier transform by describing the integration over a wavelet function rather than an exponential function. The paragraph introduces the scale and time parameters, 's' and 'tor', which are crucial for understanding the wavelet's frequency and time localization properties. It also discusses the concept of mother wavelets and daughter wavelets, illustrating how a single mother wavelet can be scaled and translated to generate a family of wavelets that can analyze different frequencies and time segments of a signal. The summary touches on the practical implications of wavelet transforms, such as their use in image processing and data compression.

This paragraph explains the method of generating wavelets at all possible scales and frequencies to cover the entire time domain of a signal. It describes the use of a mother wavelet and how it can be transformed into daughter wavelets by scaling and translating. The process involves adjusting the scale to change the frequency content and translating the wavelet to different time positions. The explanation includes visual examples of how the wavelet changes with different scale and time parameters, emphasizing the flexibility and adaptability of wavelets in signal analysis. The paragraph also discusses the inverse wavelet transform and its application in reconstructing signals from their wavelet coefficients, drawing an analogy with digital cameras and image processing.

The fourth paragraph provides a visualization of how wavelet transforms work in practice, using a chirp signal as an example. It describes the process of convoluting the signal with wavelets of varying scales and translating them to capture the signal's characteristics at different resolutions. The paragraph also explains the concept of wavelet coefficients and how they form a two-dimensional space that can be plotted as a surface plot. The discussion includes the inversion of the wavelet transform, which is akin to reconstructing the original signal from its wavelet coefficients, and it provides an example of how this can be done using a computer program.

This paragraph discusses the mathematical foundations and properties necessary for wavelets to function effectively in signal analysis. It outlines the criteria that wavelets must meet, such as oscillation, localization, and the vanishing of the first p moments. The paragraph also emphasizes the importance of normalization in wavelet transforms and the need for consistency in mathematical treatment to ensure accurate analysis and inversion. It touches on the practical aspects of implementing wavelet transforms, including the assignment given to the audience to convert a program to perform wavelet transformations using different wavelets and test them on various signals.

The final paragraph concludes the lecture by assigning practical work to the audience, encouraging them to apply their understanding of wavelets by converting a program to perform wavelet transformations. The assignment involves using three different wavelets on specific test signals and checking the invertibility of the wavelet transform. The paragraph also previews the next lecture, which will focus on digitizing wavelets and making the process more discrete, thus building upon the foundation laid in this session.