Discrete Wavelet Transform

Fourier Analysis · Axiom Academy

LESSON Discrete Wavelet Transform Understanding how continuous wavelets are discretized for efficient computation 1. Discretizing Scale and Translation Instead of using continuous parameters a and b, we restrict them to discrete values that form a dyadic grid : Scale: a = 2 j where j is an integer (scale level) Translation: b = k · 2 j where k is an integer (shift at scale j) This exponential scaling means each level analyzes frequencies at half the resolution of the previous level, while translations become coarser at larger scales. Substituting our discrete parameters into the continuous wavelet transform gives us the dyadic wavelet family : Normalization: The factor 2 -j/2 ensures that wavelets at different scales have the same energy (L 2 norm). Dilation: The term 2 -j t dilates the wavelet by 2 j , making it wider at coarser scales. Translation: Subtracting k shifts the dilated wavelet to position k · 2 j . The DWT can be implemented efficiently using a cascade of digital filters - a structure known as a filter bank : Analysis: Signal is passed through complementary low-pass (h) and high-pass (g) filters, then downsampled by 2. Decomposition: This process repeats on the low-pass output, creating a multiresolution pyramid. Synthesis: Reconstruction reverses the process with upsample-filter-add operations. 4. Perfect Reconstruction Conditions For the DWT to be reversible (allowing perfect signal reconstruction), the filters must satisfy specific mathematical constraints:

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