Wavelet Transform and Multiresolution Analysis

Core Idea

Wavelets are localized oscillatory functions that analyze signals at multiple scales. Continuous wavelet transform (CWT) correlates signal with dilated and translated wavelets, providing time-scale representation. Discrete wavelet transform (DWT) uses dyadic scales and orthonormal wavelets for efficient, non-redundant decomposition into approximation and detail components.

Explainer

The Fourier transform tells you *which* frequencies are present in a signal, but not *when* they occur. The Short-Time Fourier Transform you already know fixes this by chopping the signal into windowed segments and taking a Fourier transform of each. But STFT has a fundamental limitation: you choose one fixed window width and live with the resulting tradeoff — short windows give good time resolution but smear frequency information, while long windows resolve frequency well but blur events in time. You cannot have both at once, at any scale.

Wavelets escape this tradeoff by abandoning the fixed window entirely. A mother wavelet ψ is a short, localized oscillatory pulse with zero mean — not an infinite sinusoid. The continuous wavelet transform (CWT) computes the inner product of the signal with scaled and shifted copies of ψ. Scaling stretches or compresses the wavelet in time: a compressed wavelet oscillates faster and captures high-frequency content with fine time resolution; a stretched wavelet oscillates slowly and captures low-frequency content with fine frequency resolution. This adaptive behavior — fine time resolution at high frequencies, fine frequency resolution at low frequencies — matches the natural structure of many physical signals (a brief mechanical impact needs time resolution; a slow structural resonance needs frequency resolution).

The discrete wavelet transform (DWT) restricts scales to a dyadic grid (2^j for integer j) and uses orthonormal wavelet families such as Haar or Daubechies wavelets. The DWT is implemented as a cascade of highpass and lowpass filter banks. At each level, the signal passes through a highpass filter (producing detail coefficients capturing fast variation) and a lowpass filter (producing approximation coefficients capturing slow variation), each followed by downsampling by 2. The approximation output is then fed back into the same filter pair for the next level. This recursion produces a multi-level decomposition: detail coefficients at level j capture features at scale 2^j, and the final approximation captures the coarsest structure.

Multiresolution analysis (MRA) is the mathematical framework that makes this precise. The signal lives in a Hilbert space that decomposes as a nested sequence of approximation subspaces V_j ⊂ V_{j-1} ⊂ … ⊂ L²(ℝ). Each V_j is generated by scaled versions of a scaling function φ. The orthogonal complement of V_j inside V_{j-1} is the detail subspace W_j, generated by the wavelet ψ at scale 2^j. The DWT is the orthogonal projection of the signal onto each W_j plus the coarsest V_j — a complete, non-redundant decomposition. Unlike the CWT, which is highly redundant, the DWT stores exactly as many coefficients as the original signal, making it ideal for compression and denoising applications.

The practical power of wavelets is that smooth regions of a signal produce small detail coefficients (few coefficients needed), while discontinuities and sharp features produce large detail coefficients localized in time. Compression works by thresholding small coefficients to zero; denoising works by thresholding coefficients below a noise level. JPEG 2000 and many audio codecs use wavelet decomposition precisely because it concentrates signal energy into a small number of large coefficients, leaving most coefficients near zero and highly compressible.