On The Quotient of a Centralized and a Non-centralized Complex Gaussian Random Variable

Area of Science:

• Probability Theory
• Complex Analysis
• Signal Processing

Background:

• Investigates the statistical properties of the quotient of two independent complex random variables.
• Addresses scenarios where the numerator has zero mean and the denominator has a non-zero mean.

Purpose of the Study:

• To derive the statistics of the modulus, phase angle, real, and imaginary parts of the quotient.
• To determine the probability density function (PDF) of the quotient after normalization.
• To analyze the behavior of the quotient for large means and its relation to practical applications.

Main Methods:

• Normalization of complex random variables to simplify formulation.
• Indirect derivation of statistics for modulus and phase angle.
• Extension of statistical results to real and imaginary parts.
• Asymptotic analysis for large means.

Main Results:

• The PDF of the quotient is expressed as a function of the denominator's mean.
• The quotient approximates a normally-distributed complex random variable for large means.
• Moments of the clipped random variable are derived, relevant for signal processing.
• Tolerance intervals for the ratio of complex random variables are presented.

Conclusions:

• The derived statistical properties offer a comprehensive understanding of complex random variable quotients.
• The findings have direct implications for complex-signal processing and microwave metrology.
• The study provides a foundation for further research in related statistical signal processing areas.