Waveform Functions with Laplace Distribution and Probability Density Functions

This documentation presents a suite of waveform functions with Laplace distribution implementations and multiple probability density functions. These mathematical tools are fundamental in signal processing, communications, and image processing applications. The waveform functions include implementations for generating standard signals (sine, square, triangle waves) along with Laplace-distributed waveforms using the probability density function f(x) = (1/2b) * e^(-|x-μ|/b). Code implementations typically involve parameter configuration for mean (μ) and diversity (b), with efficient random number generation for real-time signal simulation. Probability density functions cover key distributions including Gaussian, Uniform, Exponential, and Poisson, implemented through mathematical computing libraries. Each PDF includes configurable parameters and normalization handling, with algorithms optimized for numerical stability and computational efficiency. Understanding these functions' mathematical foundations is essential for developing advanced algorithms in signal analysis, noise modeling, and data pattern recognition. The Laplace distribution specifically offers robust modeling for systems with heavier tails than Gaussian distributions. These techniques enable extraction of hidden patterns in complex datasets across financial analytics, biological signal processing, and social network analysis. Implementation examples typically utilize vectorized operations and probability sampling techniques for optimal performance.