 Wavelet

A wavelet is a localized analyzing function whose shifted and scaled copies examine a signal at different positions and resolutions. Its usefulness is to separate where a change occurs from how large a region that change spans. A brief feature can contribute to a few nearby wavelet coefficients instead of being described entirely through global oscillations.

The Haar wavelet is the simplest example: compare the first half of an interval with the second. A constant value contributes nothing to this contrast, while a difference between the halves survives. Moving the function changes which part of the signal is examined; shrinking it compares smaller neighbouring regions. In an orthonormal wavelet construction, these comparisons fit together without duplicating squared-norm contributions.

A scaling function retains the coarse information that such contrasts omit. The discrete wavelet transform computes both coarse values and details, and the wavelet series recombines them. This gives signal processing a representation that can expose local structure and support image compression. The transform itself is reversible in the complete orthonormal representation; discarding coefficients introduces approximation error. ^[ref]

A wavelet is a mathematical function used to divide data into different frequency components and study each component with a resolution that matches its scale. It is particularly useful for analyzing non-stationary signals, which can change over time, unlike traditional Fourier transformations that analyze signals in a fixed manner.