Paraphrase the following text using more academic and scientific language Use a neutral tone and avoid repetitions of words and phraseswhere $X_t$ and $X_t+tau$ represent the random variables of the s

The present study employs academic and scientific language to describe the statistical properties of a signal. Specifically, the signal is represented as random variables $X_t$ and $X_{t+\tau}$ at two different time points, with expected values $\mu_t$ and $\mu_{t+\tau}$, and variance denoted by $\sigma$. In the case of stationary noise, the distribution of variables across different time points is consistent, with a fixed expected value and variance. To generate colored noises, we employ the Fourier transform algorithm outlined in Zhivomirov's 2018 publication. This involves applying the discrete Fourier transform to a white noise sequence and altering the complex spectral coefficients according to the desired noise slope. The transformed sequence is then processed using the inverse discrete Fourier transform to obtain the desired colored noise with fixed expected value and variance. In our study, the expected value is set to 0 and the variance to 1, enabling simplification of the autocorrelation function to $R_{xx}(\tau)= E[X_tX_{t+\tau}]$.