Wavelet Transform: Extracting Low-Frequency and High-Frequency Coefficients

First, we need to generate a signal. In Python, this can be implemented using NumPy's signal generation functions like numpy.sin() for sinusoidal waves or numpy.random.randn() for Gaussian noise signals. Next, we apply wavelet transformation to this signal using libraries such as PyWavelets (pywt), which provides functions like pywt.wavedec() for multi-level decomposition. This process yields low-frequency coefficients (approximation coefficients) that capture the signal's overall trends, and high-frequency coefficients (detail coefficients) that represent fine details and transient features. Wavelet transform is a signal processing technique that decomposes signals into different frequency components through convolution with scaled and translated wavelet functions (e.g., Daubechies or Haar wavelets), enabling multiresolution analysis for better understanding of signal characteristics and structures. The decomposition follows a filter bank approach where high-pass and low-pass filters split the signal into frequency subbands at each level.