Continuous Wavelet Transform (CWT) From Scratch

An interactive Streamlit web application for performing Continuous Wavelet Transform (CWT) analysis and Fast Fourier Transform (FFT) on signal data, with a focus on Phonocardiogram (PCG) signal analysis.

Features

Signal Input

• Upload Custom Data: Load 2-column .txt files (time and signal values)
• Generate Synthetic Signals: Create sine waves with customizable parameters:
  • Two independent sine wave components
  • Adjustable frequency, amplitude, and noise levels
  • Configurable sampling frequency and duration

Signal Analysis

1. Time-Domain Analysis

   • Normalized signal visualization
   • Shannon Energy envelope computation
   • Interactive plots with zoom capabilities

2. Frequency-Domain Analysis (FFT)

   • Fast Fourier Transform using Radix-2 algorithm (implemented from scratch)
   • Frequency spectrum visualization
   • Magnitude analysis up to Nyquist frequency

3. Time-Frequency Analysis (CWT)

   • Continuous Wavelet Transform using Morlet wavelet
   • Interactive scalogram visualization (contour plot)
   • 3D surface plot of CWT coefficients
   • Configurable CWT parameters:
     • Morlet ω₀ parameter
     • Scale parameters (start, step, count)
     • Number of time translations

PCG Component Detection

• Automated Feature Detection: Identify heart sound components (M1, T1, A2, P2)
• Binary Thresholding: Adjustable amplitude threshold for region detection
• Center of Gravity (CoG): Calculate time, frequency, and amplitude for each detected component
• Component Labeling: Automatic labeling and visualization on scalogram

Installation

Prerequisites

• Python 3.7 or higher
• pip package manager

Dependencies

Install the required packages:

pip install streamlit numpy pandas scipy plotly

Or use the requirements file if available:

pip install -r requirements.txt

Usage

Running the Application

1. Navigate to the project directory:

cd CWT-From-Scratch

2. Launch the Streamlit app:

streamlit run cwt.py

3. The application will open in your default web browser (typically at http://localhost:8501)

Using the Application

Option 1: Upload Signal Data

1. Select "Upload File" in the sidebar
2. Upload a .txt file with two columns: time and signal values
3. The sampling frequency will be automatically calculated from the time data
4. Adjust CWT parameters as needed

Option 2: Generate Synthetic Signal

1. Select "Generate Sine Wave" in the sidebar
2. Configure signal parameters:
   • Sampling frequency (Hz)
   • Duration (seconds)
   • Frequency and amplitude for two sine components
   • Noise level
3. The signal will be generated automatically

Performing CWT Analysis

1. Adjust CWT parameters in the sidebar:

   • ω₀: Morlet wavelet parameter (default: 2π × 0.849)
   • Start Scale (a): Initial scale value
   • Scale Step (da): Increment for scale values
   • Number of Scales: Resolution in frequency dimension
   • Number of Time-Shifts: Resolution in time dimension

2. Click "Compute CWT" to perform the analysis

3. View results:

   • CWT scalogram (contour plot)
   • Binary mask with threshold adjustment
   • Detected PCG components with time, frequency, and amplitude
   • Optional 3D surface plot
   • Scale-to-frequency and translation-to-time mappings

Data Format

Input File Format

The application expects a 2-column space or tab-delimited .txt file:

0.0000    0.1234
0.0010    0.2345
0.0020    0.3456
...

Column 1: Time (seconds)
Column 2: Signal amplitude

Example Data

Sample PCG signal files are included in the data/ directory:

• Print_13_v2_PCG_RV.txt
• Print_14_v2_PCG_RV.txt

Technical Details

Algorithms

CWT Implementation

• Wavelet: Complex Morlet wavelet
• Method: Direct convolution in time domain
• Normalization: Energy-preserving normalization
• Output: Magnitude of complex CWT coefficients

FFT Implementation

• Algorithm: Radix-2 Cooley-Tukey FFT
• Padding: Zero-padding to nearest power of 2
• Scaling: Normalized magnitude spectrum

Shannon Energy

• Formula: E(t) = -1/N × Σ(x²·log(x² + ε))
• Parameters: Frame length and hop size based on sampling rate
• Output: Energy envelope with mean and standard deviation

Performance

• Caching: CWT results are cached using Streamlit's caching mechanism
• Progress Tracking: Real-time progress bar during CWT computation
• Interactive Plots: Plotly-based visualizations with zoom and pan

Applications

This tool is particularly useful for:

• Cardiac Signal Analysis: PCG (phonocardiogram) heart sound analysis
• Signal Processing Education: Understanding time-frequency analysis
• Research: Exploring wavelet transform parameters
• Feature Extraction: Identifying signal components in time-frequency domain

Algorithm Complexity

• CWT Computation: O(N × M × K) where:
  • N = number of scales
  • M = number of translations
  • K = signal length
• FFT Computation: O(N log N) where N is the padded signal length

Limitations

• Large parameter combinations (high scale/translation counts) may result in long computation times
• Memory usage scales with the product of scales and translations
• CWT is computed in the time domain (not optimized with FFT-based convolution)

License

This project is open-source and available for educational and research purposes.

Author

Created as part of signal processing and biomedical engineering research.

Acknowledgments

• Wavelet theory and Morlet wavelet implementation
• PCG signal analysis methodology
• Streamlit framework for interactive web applications