Generating Colored Noise using Fourier Transform for Signal Processing

This study examines the generation of colored noise within the framework of signal processing. The signal at distinct time points, represented by $X_t$ and $X_{t+/tau}$, is defined by their respective expected values, $/mu_t$ and $/mu_{t+/tau}$, and variance, $/sigma$. For stationary noise, the probability distributions at different time points are identical, with constant expected values and variances. To generate colored noise, the Fourier transform algorithm, as outlined by Zhivomirov et al. (2018), is employed. This involves applying the discrete Fourier transform (DFT) to a random white noise sequence in the time domain and subsequently modifying the spectrum by multiplying it with the desired noise slope. The resultant sequence undergoes inverse discrete Fourier transform (IDFT) to yield noise with the desired mean and variance. The expected value of the generated colored noise is set to zero, and its variance is set to one. Consequently, the autocorrelation function of the colored noise simplifies to $R_{xx}(/tau)= E[X_tX_{t+/tau}]$.