Перетворення Вейлетова — Мова Часу і Частоти

Limitations of the Fourier Transform

The Fourier transform reveals the frequency content of a signal, but it doesn't provide information about *when* those frequencies occur. A symphony and a randomly scrambled version of the same notes would have identical Fourier magnitude spectra. The Short-Time Fourier Transform attempts to address this by using a sliding window, however, the size of this window is fixed: a short window offers good time resolution but poor frequency resolution, while a long window provides the opposite at every frequency. This represents a fundamental limitation – for any signal, the time spread Δt and frequency spread Δω are related through the uncertainty principle: Δt·Δω ≥ 1/2.

Wavelets: a window that adapts to frequency

A wavelet ψ(t) is a short oscillating function with zero mean and unit energy. From a single mother wavelet, a whole family of daughter wavelets is generated by scaling and translating: ψ_{a,b}(t) = (1/√|a|)·ψ((t−b)/a). Large scale 'a' stretches the wavelet to analyse low frequencies; small 'a' compresses it to analyse high frequencies — the window automatically narrows exactly where fine time resolution is needed.

The Continuous Wavelet Transform correlates the signal with every daughter wavelet, producing a 2D time-scale map called a scalogram.

Haar wavelet:  ψ(t) = +1 on [0, 0.5), −1 on [0.5, 1), 0 elsewhere
Haar transform of [4, 6, 2, 8]:
  averages    = [5, 5]  →  [5]
  differences = [−1, −3] →  [0]
  Haar coefficients: [5 | 0 | −1, −3]

Haar, Daubechies and multiresolution analysis

The Haar wavelet (Alfred Haar, 1909) is the simplest orthonormal wavelet: it splits a signal into pairwise averages and differences recursively, in O(N) time with exact integer arithmetic — but it has only one vanishing moment, producing blocky "staircase" artefacts. Ingrid Daubechies solved this in 1988 by constructing compactly supported wavelets with N vanishing moments (db1 = Haar, db2, db4 ... dbN): a wavelet with N vanishing moments gives exactly zero coefficients for any polynomial signal of degree below N, so smooth regions compress to almost nothing and only genuine discontinuities produce large coefficients. The theoretical backbone is multiresolution analysis (Mallat & Meyer, ~1989), which recursively splits a signal into a coarse approximation and a detail layer via a lowpass/highpass filter bank — the Mallat algorithm computes a full decomposition in O(N), faster than the FFT's O(N log N).

Decomposition of Signals

The core concept behind wavelet analysis is signal decomposition. This process breaks down a complex waveform into simpler components, much like separating the colors in white light using a prism. Wavelets are mathematical functions that represent these individual components, allowing us to analyze and manipulate them separately.

Discrete Wavelet Transforms (DWTs) are particularly useful for analyzing signals with sharp changes or discontinuities. The DWT decomposes a signal into different frequency bands, providing insights into its underlying structure. This is crucial in applications like image compression and noise reduction.

The choice of wavelet family – Haar, Daubechies, Morlet, etc. – depends on the specific characteristics of the signal being analyzed. Each wavelet has unique properties that make it suitable for certain types of data.

Frequently asked questions

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