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The continuous wavelet transform (CWT) replaces sinusoidal basis functions with wavelets—finite-duration, oscillatory functions that are scaled and translated. The key innovation lies in the scaling parameter ( a ) and translation parameter ( b ):
[ W(a,b) = \int_-\infty^\infty x(t) \frac1\sqrta \psi^* \left( \fract-ba \right) dt ]
Unlike the STFT’s fixed window, wavelets use short windows at high frequencies (capturing fine temporal details) and long windows at low frequencies (capturing coarse frequency structure). This multi-resolution analysis (MRA) aligns with the natural trade-off of the uncertainty principle but optimizes it for real-world signals. The discrete wavelet transform (DWT) implements this efficiently via filter banks: high-pass and low-pass filters split the signal into detail coefficients (high frequencies) and approximation coefficients (low frequencies), which are then recursively decomposed. This hierarchical decomposition yields a compact, multi-level representation that is sparse for many natural signals. dsj 4 1113 high quality
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The continuous Fourier Transform decomposes a signal ( x(t) ) into complex exponentials: ( X(f) = \int_-\infty^\infty x(t) e^-j2\pi ft dt ). The resulting frequency spectrum ( X(f) ) reveals which frequencies exist but discards all information about when those frequencies occur. For a stationary signal—such as a steady 60 Hz hum—this limitation is irrelevant. However, consider an electrocardiogram (ECG) signal: a sudden QRS complex (a high-frequency transient) followed by a low-frequency T-wave. The FT would smear these events across the entire time axis, rendering it impossible to distinguish the timing of the heartbeat. The Short-Time Fourier Transform (STFT) attempts to remedy this by windowing the signal, yet it suffers from the Heisenberg uncertainty principle: a fixed window size forces a trade-off between time and frequency resolution. Narrow windows capture rapid changes poorly in frequency; wide windows smear transients in time. This rigid trade-off is precisely where wavelets excel. ski jumping simulation
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