Introduction to the Wavelet Transform

Overview

• Understanding the mathematics and intuition behind the wavelet transform
• Focuses on its capabilities in signal processing

Basics Before Wavelets

Fourier Transform

• Provides frequency information of a stationary signal
  • Frequencies and their magnitudes
• Ideal for signals that do not change over time
  • Constant frequency throughout
• Limitation: No time localization
  • Lacks capability for non-stationary signals

Short-Time Fourier Transform (STFT)

• Developed to improve time resolution
• Provides time-frequency representation
• Assumes part of the non-stationary signal is stationary
• Method:
  • Divide the signal into stationary parts
  • Use a window function of fixed length moved along the signal
  • Multiply signal and window function
    • Zero-valued outside the interval
• Result: Time and frequency localization
• Limitations:
  • Finite window function reduces frequency resolution
  • Fixed time and frequency resolutions
  • Uncertainty principle
    • Higher time resolution results in lower frequency resolution
    • Bounded by 1 over 4π

Frequency-Time Plane Representation

• STFT produces squares of equal area
  • Narrow window: Good time resolution, bad frequency resolution
  • Wide window: Good frequency resolution, bad time resolution
• Challenges in sound/signal processing
  • Low frequencies: Long duration, need high frequency resolution
  • High frequencies: Short bursts, need high time resolution
• STFT limitations: Fixed time-frequency resolutions

Wavelet Transform

Improvements Over STFT

• Multi-resolution analysis
• Different resolutions for different frequencies
• High frequencies: Good time resolution, poor frequency resolution
• Low frequencies: Good frequency resolution, poor time resolution

Mathematical Formulation

• Uses integral form (Continuous Wavelet Transform)
• Multiplies signal by wavelet's complex conjugate
• Scale parameter (1/frequency)
• Translation parameter (τ)

Wavelets

• Small waves used as basis functions
• Can be scaled (s) to change width and central frequency of the wavelet
• Scaling:
  • Expander wavelet: Resolves low frequencies, bad time resolution (large s)
  • Shrunken wavelet: Resolves high frequencies, good time resolution (small s)
• Wavelet coefficients:
  • Approximation (low frequency)
  • Detail (high frequency)

Visualization

• Wavelet is translated and scaled across the signal
• Produces 3D plot of scale (1/frequency), translation, and amplitude

Discrete Wavelet Transform (DWT)

Computational Efficiency

• Discrete selection of s and τ to reduce data
• Dyadic: Powers of two

Formulation

• Replaces integral with sum
• Uses dyadic values
  • j: Scale index
  • k: Wavelet transformed signal index

Multi-Level Decomposition

• Signal passed into low-pass and high-pass filters
• Low-pass filters: Approximation coefficients (keep low frequencies)
• High-pass filters: Detail coefficients (keep high frequencies)
• Halving of coefficients with each level of decomposition (Decimated DWT)

Future Topics

• Detailed exploration of DWT and multi-level decomposition
• Approximation and detail coefficients
• Introduction to stationary wavelet transforms
• Application in signal denoising (e.g., ECG, MCG signals)