Wavelet Transform

The Fourier transform tells you WHAT frequencies are present in a signal, but not WHEN they occur. A musical score shows both the notes (frequencies) AND when they are played, but a Fourier transform gives only the notes without timing. This is the time-frequency uncertainty principle: you cannot simultaneously have perfect resolution in both time and frequency. Wavelets solve this by providing a compromise — they analyze signals at different scales and translations, automatically using short windows for high frequencies (good time resolution) and long windows for low frequencies (good frequency resolution).

The Discrete Wavelet Transform decomposes a signal into approximation coefficients (coarse, low-frequency overview) and detail coefficients (fine, high-frequency details) at multiple levels. Each level is sampled at half the rate of the previous level (dyadic sampling). This creates a multi-resolution analysis where the signal is represented by a hierarchy of details at different scales. The process can be reversed perfectly — the inverse DWT reconstructs the original signal from its coefficients. This is fundamentally different from Fourier where reconstruction requires ALL frequency components.

Wavelets are everywhere in modern signal and image processing. JPEG2000 uses wavelets instead of the discrete cosine transform used in original JPEG, achieving better compression with fewer artifacts. The FBI digitized 30 million fingerprints using wavelet compression (WSQ). Wavelets excel at denoising: threshold the detail coefficients to remove noise while preserving edges. They detect edges in images by identifying where detail coefficients are large. In medical imaging, wavelets analyze EEG and ECG signals. In seismology, they extract information about underground structures from reflected waves.

See how changing the scale of a wavelet captures different features of a signal. Large scales see broad trends; small scales pick up fine details.