91Ó°ÊÓ

Gaussian random variables, also known as normally distributed variables, are fundamental in statistics and communication theory. A Gaussian random variable has a bell-shaped probability distribution characterized by its mean (µ) and variance (σ²). In the context of communication theory, such variables are often encountered as they model noise and signal variations effectively. The Gaussian distribution's probability density function (pdf) is given by:
\[ f_X(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} \]
In our given problem, X and Y are independent Gaussian random variables with a mean of 0 and variance σ², symbolized as N(0, σ²). This implies their joint distribution can be compactly written as: \[ f_{X,Y}(x,y) = \frac{1}{2 \pi \sigma^2} e^{-\frac{x^2 + y^2}{2 \sigma^2}} \]

When dealing with two or more random variables, the joint distribution describes the probability of different outcomes that each variable can take simultaneously.
For our problem, the joint distribution of X and Y is given by:
\[ f_{X,Y}(x,y) = \frac{1}{2 \pi \sigma^2} e^{-\frac{x^2 + y^2}{2 \sigma^2}} \]
This function tells us the likelihood of X taking a specific value x and Y taking a value y concurrently. Because X and Y are independent, their joint pdf is simply the product of their individual pdfs. Additionally, the joint distribution is important to derive the properties of the envelope Z = √(X² + Y²).

The Cumulative Distribution Function (CDF) transformation is a powerful tool for understanding the distribution of a random variable derived from other variables. The CDF of a random variable Z, denoted F_Z(z), is the probability that Z takes values less than or equal to z:
\[ F_Z(z) = P(Z \le z) \]
In our problem, we start with the joint distribution of X and Y and aim to find the CDF of Z. We transform from Cartesian to polar coordinates and integrate the probability density function (pdf) to get the CDF. By integrating the pdf from 0 to z, we obtain the CDF:
\[ F_Z(z) = \int_0^z \frac{t}{\sigma^2} e^{-\frac{t^2}{2 \sigma^2}} dt \]
Using appropriate substitutions, we derive:
\[ F_Z(z) = 1 - e^{-\frac{z^2}{2 \sigma^2}} , \quad z > 0 \]

In many mathematical and engineering problems, polar coordinates simplify transformations and integrations. Instead of using Cartesian coordinates (x, y), we use (r, θ) in polar coordinates, where r is the radius (distance from the origin) and θ is the angle.
In this problem, transforming to polar coordinates is crucial. We transform the Gaussian variables X and Y into polar coordinates Z and θ using:
\[ X = Z \cos(\theta) \] \[ Y = Z \sin(\theta) \]
The Jacobian determinant for this coordinate transformation is Z, which modifies the integration measure to dX dY = Z dZ dθ. Applying this transformation simplifies our integration of the joint distribution significantly, as seen in transforming the pdf f_{X,Y}(x,y).

A probability density function (pdf) gives the likelihood of a random variable to take on specific values. It is integral in understanding distributions. For our variable Z, derived from X and Y, we find its pdf f_Z(z) by transforming the joint pdf in polar coordinates and integrating out irrelevant variables.
The pdf for Z is given by:
\[ f_Z(z) = \frac{z}{\sigma^2} e^{-\frac{z^2}{2 \sigma^2}} , \quad z > 0 \]
This pdf describes the Rayleigh distribution, common in communication theory for signal amplitude modeling. The function peaks at z = σ and decreases exponentially for larger z values, indicating less likelihood of very high amplitudes.