WaveletTransform

The Wavelet Transform is a mathematical tool used for analyzing and representing signals or functions in both time and frequency domains. Unlike the Fourier Transform, which provides frequency information but lacks time localization, the Wavelet Transform offers a time-frequency representation, making it particularly useful for non-stationary signals. This transform uses wavelets, which are small waves of limited duration, to decompose a signal into different scale components.

The Continuous Wavelet Transform (CWT) is defined as the sum of the signal multiplied by scaled and shifted versions of a mother wavelet. The Discrete Wavelet Transform (DWT), on the other hand, is a sampled version of the CWT, making it computationally efficient. The DWT is widely used in signal processing, image compression, and data analysis due to its ability to provide a multi-resolution representation of a signal.

Wavelet transforms have several advantages over traditional Fourier transforms. They can capture both frequency and location information, making them suitable for analyzing signals with transient features. Additionally, wavelets can be adapted to specific signal characteristics, providing a more flexible and efficient analysis tool.

In summary, the Wavelet Transform is a powerful mathematical technique that combines elements of time and frequency analysis, making it a valuable tool in various fields such as signal processing, image analysis, and data compression. Its ability to provide a time-frequency representation and adapt to different signal characteristics sets it apart from other transform methods.