Explain wavelet transforms and their advantages over Fourier analysis.

Wavelet transform decomposes signals using scaled and shifted wavelets (localized oscillations) instead of infinite sinusoids. Continuous WT: W(a,b) = integral(x(t)*psi*((t-b)/a)dt). Discrete WT uses dyadic scales: a=2^j, b=k*2^j. Advantages over STFT: Multi-resolution analysis (fine time resolution at high frequencies, fine frequency resolution at low frequencies), Compact support (good for transients), and Efficient computation (fast wavelet transform via filter banks). Applications: Image compression (JPEG2000), denoising, edge detection, and biomedical signals. Wavelet choice (Haar, Daubechies, etc.) affects properties.