Signal Denoising

Fourier Analysis · Axiom Academy

EXAMPLE Signal Denoising with Wavelet Thresholding Walk through a complete wavelet denoising procedure step by step Excellent work! You've completed the wavelet denoising example. Here's what we learned: Wavelet Transform: The discrete wavelet transform decomposes a noisy signal into approximation coefficients (low-frequency content) and detail coefficients (high-frequency content). Noise typically appears in the detail coefficients. Thresholding Methods: Hard thresholding sets coefficients below the threshold to zero, while soft thresholding shrinks all coefficients toward zero. Soft thresholding produces smoother results but may over-smooth the signal. Universal Threshold: The threshold λ = σ√(2 ln N) is based on the noise standard deviation and signal length, providing optimal denoising under certain assumptions. Selective Filtering: Only detail coefficients are thresholded; approximation coefficients preserve the signal's main structure. Quality Metrics: Compare original, noisy, and denoised signals using SNR (Signal-to-Noise Ratio) or RMSE (Root Mean Square Error) to quantify denoising performance. This wavelet thresholding approach is fundamental in signal processing for removing noise while preserving important signal features. Try different wavelets and threshold levels to optimize results for specific applications!

This is the written version of the interactive lesson above. See the full Fourier Analysis course.