Question

A random variable X having auto correlation function Rₓₓ(τ) = e−σ|τ| is passed through a system whose impulse response is given as h(t) = μ/2 e⁻μᵗ u(t).

Find the power spectral density of the output signal.

Answer 1

To find the power spectral density of the output signal, calculate the autocorrelation function and take its Fourier transform.

Step-by-step explanation:

The student's question pertains to finding the power spectral density (PSD) of the output signal when a random variable X with an auto correlation function R₁₁(τ) = e⁻σ|τ| is passed through a system with impulse response h(t) = μ/2 e⁻μᵗ u(t). Given that μ is set to 4 minutes, it is essential to use this decay parameter alongside the provided formulations to compute the PSD.

To find the power spectral density of the output signal, we need to calculate the autocorrelation function of the output signal and then take its Fourier transform. The autocorrelation function of the output signal is given by:

R_yy(τ) = R_xx(τ) * h(τ)

Substituting the given values for R_xx(τ) and h(τ), we get:

R_yy(τ) = (e^(-σ|τ|)) * (μ/2 * e^(-μτ) * u(τ))

Taking the Fourier transform of R_yy(τ), we can find the power spectral density of the output signal.