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Noise Module

The Noise Module provides methods for generating and adding noise to signals, enabling the simulation of realistic noisy environments for signal processing applications. Noise is an inherent part of real-world signals, originating from various sources such as electronic interference, environmental fluctuations, or stochastic variations in a system. In many cases, noise is considered an undesirable artifact that must be reduced or removed. However, noise can also carry valuable information about the system’s state or environment.

The module supports stationary and non-stationary noise models, including:

  • Colored noise synthesis (white, pink, brown, blue, violet, and custom profile).
  • Time-varying noise envelopes (linear, sinusoidal, random walk, and blockwise modulation).
  • Controlled noise injection using signal-to-noise ratio (SNR) scaling.

These methods allow users to introduce realistic noise patterns into their signals, useful for testing denoising algorithms, training machine learning models, and modeling real-world scenarios.

Conceptual Overview

Understanding Noise Characteristics

Noise can be broadly classified into stationary and non-stationary types:

  • Stationary Noise: Statistical properties (e.g., variance, mean, and spectral density) remain constant over time.
    Examples: Thermal noise in electrical circuits, Gaussian noise in images.

  • Non-Stationary Noise: Statistical properties vary over time, often due to changing environmental factors or system dynamics.
    Examples: Wind noise in microphones, motion artifacts in biomedical signals.

Many natural noise sources follow a power-law spectral density, where spectral power decreases as a function of frequency. This principle, observed in fractal and chaotic systems, is the basis for colored noise synthesis.

Colored Noise Synthesis

The Noise Module implements a Fourier-based noise synthesis method, introduced by Poul Bourke (1997), which follows these steps:

  1. Generate a white noise sequence in the time domain.
  2. Transform the noise into the frequency domain using the Fourier transform.
  3. Apply a spectral filter to shape the noise power based on the desired frequency-dependent function \( 1/f^p \).
  4. Perform an inverse Fourier transform to reconstruct the time-domain noise.

This approach allows the generation of different noise types:

  • White noise (\( p=0 \)) → Equal power at all frequencies.
  • Pink noise (\( p=1 \)) → Power decreases as \( 1/f \), common in biological signals.
  • Brown noise (\( p=2 \)) → Power decreases as \( 1/f^2 \), with stronger low-frequency components.

$$ S(f) \propto \frac{1}{f^p}, $$ where \( S(f) \) represents the spectral power density and \( p \) determines the noise color.

Envelope-Based Noise Modulation

To introduce non-stationary noise effects, the module supports four amplitude modulation envelopes:

  1. Linear Envelope → Gradually increases or decreases noise intensity over time.
  2. Sine Envelope → Modulates noise amplitude periodically, introducing cyclic variations.
  3. Random Walk Envelope → Introduces stochastic amplitude changes, simulating unpredictable real-world fluctuations.
  4. Blockwise Envelope → Produces abrupt, step-like noise intensity variations.

These envelopes enable controlled noise dynamics, making the generated noise adaptable to real-world non-stationary environments.

References

  1. Celka, P. (2006). Smoothly Time-Varying AR Models and Applications in Biomedical Signal Processing.
  2. Vaseghi, S. (2008). Advanced Digital Signal Processing and Noise Reduction. Wiley.
  3. Bourke, P. (1997). Frequency Synthesis of Landscapes (and Clouds).
  4. Mitrovic, D. et al. (2010). Analysis and Generation of Synthetic Colored Noise for Machine Learning Applications.