Generation of Time-Domain Plots for Gaussian Pulses and Their First and Second Derivatives

Generating time-domain plots for Gaussian pulses and their first and second derivatives represents a fundamental signal processing technique. A Gaussian pulse constitutes a transient, high-peak signal that can be derived through differentiation of the normal distribution function. The first and second derivatives of Gaussian signals correspond to their temporal rate-of-change and curvature respectively, serving critical roles in signal processing applications such as filtering and edge detection. These waveforms are extensively employed in radar signal processing and communication systems to enhance signal quality and reliability through improved pulse shaping and interference rejection.

Implementation typically involves defining a Gaussian function using mathematical libraries like NumPy or MATLAB, followed by calculating derivatives through finite difference methods or analytical differentiation. The Gaussian function can be expressed as G(t) = exp(-t²/2σ²), with its first derivative computed as -t/σ²·G(t) and second derivative as (t²/σ⁴ - 1/σ²)·G(t). Key programming considerations include proper time-axis discretization, sigma parameter optimization for desired bandwidth, and visualization using plot() functions with appropriate axis labeling for time-domain analysis.