Wavelet Transforms

A Wavelet Transform (WT) is a mathematical technique that transforms a signal into different frequency components, each analyzed with a resolution that matches its scale. Wavelets are small waves with limited duration and they possess both time and frequency localization, which means they can capture both high-frequency and low-frequency information simultaneously.

The basic idea of wavelet analysis is to represent a function or signal in terms of a set of basis functions known as wavelets, which are derived from a single mother wavelet by translation and scaling.

Types of Wavelet Transforms

1. Continuous Wavelet Transform (CWT)

• Provides a continuous mapping of the signal in time and frequency.
• Suitable for detailed analysis and visualization of signals.

2. Discrete Wavelet Transform (DWT)

• Provides a compact representation of signals using wavelet coefficients.
• Used extensively in image and signal compression.

3. Stationary Wavelet Transform (SWT)

• A variant of DWT that is shift-invariant.
• Ideal for feature extraction and denoising applications.

4. Multiresolution Analysis (MRA)

• Decomposes signals into different resolution levels.
• Provides a hierarchical view of signal components.

Continuous Wavelet Transform (CWT)

The Continuous Wavelet Transform (CWT) of a signal x(t) is defined as:

C(a, b) = \int_{-\infty}^{\infty} f(t) \frac{1}{\sqrt{|a|}} \psi\left( \frac{t - b}{a} \right) dt

Where:

• W(a,b) is the wavelet coefficient.
• a is the scale parameter that dilates or compresses the wavelet.
• b is the translation parameter that shifts the wavelet.
• ψ_a,b^∗(t) is the conjugate of the mother wavelet ψ(t).

Discrete Wavelet Transform (DWT)

The Discrete Wavelet Transform (DWT) is a sampled version of the CWT where the scale and translation parameters are discretized. It is defined as:

W(j, k) = \sum_{n=0}^{N-1} x(n) \psi_{j, k}(n)

Where:

• j represents the scale index.
• k is the translation index.
• ψ_j,k[n] is the discrete wavelet obtained by scaling and shifting the mother wavelet.

The DWT decomposes a signal into approximation and detail coefficients, which represent low and high-frequency components, respectively.

Commonly Used Wavelets

1. Haar Wavelet:

• Simplest and most intuitive wavelet.
• Suitable for step-like signals.

2. Daubechies Wavelets (db):

• Provides smooth and compactly supported wavelets.
• Widely used in data compression and denoising.

3. Symlets:

• A modified version of Daubechies wavelets.
• Exhibits better symmetry properties.

4. Coiflets:

• Designed to have better approximation properties.

5. Morlet Wavelet:

• Combines a sinusoidal wave with a Gaussian window.
• Ideal for time-frequency analysis.

Wavelet Transform Implementation in Python

import pywt
import numpy as np
import matplotlib.pyplot as plt

# Generate a sample signal
t = np.linspace(0, 1, 1024)
signal = np.sin(2 * np.pi * 5 * t) + np.sin(2 * np.pi * 20 * t)

# Perform Discrete Wavelet Transform
wavelet = 'db4'
coeffs = pywt.wavedec(signal, wavelet, level=4)

# Plot the original signal and wavelet coefficients
plt.figure(figsize=(10, 8))
plt.subplot(5, 1, 1)
plt.plot(t, signal)
plt.title("Original Signal")

for i, coeff in enumerate(coeffs):
    plt.subplot(5, 1, i + 2)
    plt.plot(coeff)
    plt.title(f"Wavelet Coefficients - Level {i}")

plt.tight_layout()
plt.show()

Output:

• Original Signal: The original time-domain signal that contains multiple frequency components.
• Wavelet Coefficients: Decomposed coefficients at different levels representing details and approximations.
• Higher levels correspond to lower frequency components, capturing smooth variations.
• Lower levels capture finer details and high-frequency components.

Applications of Wavelet Transforms

1. Denoising and Signal Processing: DWT is used to remove noise from signals while preserving essential details.
2. Medical Signal Analysis: EEG, ECG and MRI data analysis benefit from wavelet techniques.
3. Feature Extraction: Wavelets provide features useful for classification in machine learning models.
4. Anomaly Detection: Time-series anomaly detection uses wavelet decomposition to capture patterns.