Probability Density Function of Envelopes of Narrow Band Noise with Gaussian Quadrature Components

Introduction:
Narrow band noise is a type of noise where the noise energy is concentrated within a narrow frequency range. Gaussian quadrature components refer to the real and imaginary parts of a complex Gaussian random variable that are orthogonal to each other. The probability density function (PDF) of the envelopes of a narrow band noise with Gaussian quadrature components is of interest in communication systems.

Rayleigh PDF:
The PDF of the envelopes of a narrow band noise with Gaussian quadrature components is known as the Rayleigh PDF. The Rayleigh PDF is a probability distribution function that describes the amplitude of a sinusoidal signal that is corrupted by additive Gaussian noise. The Rayleigh PDF has the following form:

f(x) = (x/σ^2) * exp(-x^2/(2σ^2))

where x is the envelope of the narrow band noise and σ is the standard deviation of the Gaussian quadrature components.

Explanation:
The Rayleigh PDF is the appropriate distribution for the envelopes of a narrow band noise with Gaussian quadrature components because the real and imaginary parts of a complex Gaussian random variable are independent and identically distributed (i.i.d.) Gaussian random variables. The envelope of a complex Gaussian random variable is the magnitude of the complex number, which is the square root of the sum of the squares of the real and imaginary parts. The PDF of the envelope of a complex Gaussian random variable is the Rayleigh PDF.

Conclusion:
In conclusion, the probability density function of the envelopes of a narrow band noise with Gaussian quadrature components is the Rayleigh PDF. The Rayleigh PDF is a probability distribution function that describes the amplitude of a sinusoidal signal that is corrupted by additive Gaussian noise. The Rayleigh PDF has the form f(x) = (x/σ^2) * exp(-x^2/(2σ^2)), where x is the envelope of the narrow band noise and σ is the standard deviation of the Gaussian quadrature components.