  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.
