arXiv:2406.16548v3 Announce Type: replace Abstract: Understanding the probability of error is paramount in the design and analysis of digital communication systems, particularly in Rayleigh fading channels where signal impairments are prevalent. This article presents a unified approach for deriving the probability of error formulations applicable to Binary Phase Shift Keying (BPSK), 16-Quadrature Amplitude Modulation (16-QAM), and 64-QAM in Rayleigh fading channels. This article presents a general approach to derive the probability of error formulations applicable to Binary Phase Shift Keying (BPSK), 16-Quadruple Amplitude Modulation (16-QAM) and 64-QAM in Rayleigh fading channels. The derivation process encompasses the unique characteristics of each modulation scheme and the statistical properties of Rayleigh fading providing a comprehensive framework to analyze error performance. By establishing a unified formulation, this approach simplifies the analysis and facilitates a deeper understanding of error probability behavior across different modulation schemes. The derived formulations offer insights into the impact of fading-induced impairments on system performance, allowing for accurate prediction and optimization of communication systems in real-world fading environments. The insights and techniques presented herein serve as a valuable resource for researchers, engineers, and practitioners engaged in the design and optimization of communication systems operating in challenging fading environments.