Gaussian noise has zero mean because it follows a Gaussian distribution, which is symmetric around its mean value. In the case of zero-mean Gaussian noise, the distribution of noise values is centered at zero, meaning that on average the noise does not add any systematic bias or offset to the signal it affects. This characteristic is fundamental in signal processing and communications because it simplifies mathematical analysis and modeling.

Having a zero mean ensures that noise does not introduce any bias into data or signal processing algorithms, allowing researchers and engineers to focus on the statistical properties and variability of the noise itself.

Noise in general is often assumed to have zero mean because it represents random fluctuations or disturbances that do not favor positive or negative deviations over time or between samples. In statistical terms, noise is generally modeled as a stochastic process where each sample or realization of the noise is independent and identically distributed around zero.

This assumption simplifies the analysis and interpretation of noisy signals or data by ensuring that noise does not systematically bias or affect the accuracy of statistical estimates or predictions. Zero noise allows researchers to focus on the variance and other statistical properties that characterize the randomness and unpredictability of the noise process.

A white mean white Gaussian process refers to a stochastic process where each variable or random sample has a Gaussian distribution with a mean and constant variance, and successive samples are statistically independent of each other.

The term “white” indicates that the process has a flat power spectral density across all frequencies, implying that the process exhibits equal intensity at all frequencies within a given bandwidth. These processes are common in signal processing and telecommunications, where they model random variations or background noise that are additive and independent in time or space.

Similarly, a zero-mean white noise process refers to a stochastic process characterized by independent and distributed random variables at identification with zero mean.

Each sample or realization of white noise is uncorrelated with previous or subsequent samples, and the noise exhibits constant variance at all time points or spatial locations.

White noise processes are fundamental in signal processing and statistics because they represent random fluctuations that are equally likely to occur at any time or space, making them useful for modeling uncertainty, chance or background disturbances in various applications.

When the autocorrelation of a signal or stochastic process is zero, it means that there is no correlation between signal values at different times or spatial locations. Autocorrelation measures the degree of similarity between a signal and a delayed version of itself over varying time lags or spatial separations.

Zero autocorrelation indicates that the signal or process exhibits no systematic relationship or predictable pattern between its past and future values. In practical terms, zero autocorrelation implies that successive samples of the signal are statistically independent, which is a desirable property in many applications where randomness or independence of observations is assumed or required