Question

Let x and y denote the amplitudes of noise signals at two antennas. Their joint probability density function is given by:

a) f(x, y) = k * e^(-x² - y²)
b) f(x, y) = k * e^(xy)
c) f(x, y) = k * (1/2π) * e^(-(x² + y²)/2)
d) f(x, y) = k * (1/π) * e^(-(x² + y²))

Answer 1

The correct option is Option c, f(x, y) = k * (1/2π) * e^(-(x² + y²)/2)

Option c represents the standard two-dimensional Gaussian probability density function, likely the correct choice, as it aligns with characteristics of a valid probability distribution, accounts for noise amplitudes approaching infinity, and includes a normalization factor.

Step-by-step explanation:

The question pertains to the determination of the correct joint probability density function (pdf) for noise signals at two antennas, given as options a through d. The joint pdf must satisfy the criteria of being a legitimate probability distribution, meaning it must integrate to 1 over the entire space and yield probabilities that are non-negative.

Option c, f(x, y) = k * (1/2π) * e^(-(x² + y²)/2), resembles the standard two-dimensional Gaussian probability density function which is used for describing independent noise signals in two antennas. In this expression, the term e^(-(x² + y²)/2) ensures that the probability density approaches 0 as the amplitude goes to infinity, and the multiplication by (1/2π) and the normalization constant k is chosen such that the integral of the pdf over all x and y equals 1.