Both signal and noise in communication can be regarded as random processes that vary with time.
The random process has the characteristics of a random variable and a time function, and can be described from two different but closely related perspectives: ① the random process is a collection of infinite sample functions; ② A random process is a set of random variables.
The statistical characteristics of a random process are described by its distribution function or probability density function. If the statistical characteristics of a random process are independent of the time starting point, it is called a strictly stable process.
Digital features are another concise way to describe random processes. If the mean value of the process is constant and the autocorrelation function R (T1, T1+ τ)= R (T), the process is called a generalized stationary process.
If a process is strictly stable, it must be broadly stable; otherwise, it may not be true.If the time average of a process is equal to the corresponding statistical average, the process is ergodic. If a process is ergodic, it is also stable; otherwise, it may not be true.
The autocorrelation function R (T) of the generalized stationary process is an even function of the time difference R, and R (0) is equal to the total average power, which is R（ τ) maximum value. Power spectral density (P) ξ (f) is the Fourier transform’s autocorrelation function R() (Wiener Minchin theorem). This pair of transformations determine the conversion relationship between the time and frequency domains. The Gaussian process’s probability distribution follows the normal distribution, and its complete statistical description requires only its numerical characteristics. The one-dimensional probability distribution only depends on the mean and variance, and the two-dimensional probability distribution mainly depends on the correlation function. The Gaussian process is still a Gaussian process after linear transformation. The relationship between the normal distribution function and the Q (x) or ERF (x) function is very useful in analyzing the anti-noise performance of digital communication systems. A stochastic process that is stationary After I (T) passes through the linear system, its output process ξ 0 (T) is also stable.
The statistical characteristics of narrow-band random processes and sine waves plus narrow-band Gaussian noise is more suitable for the analysis of modulation systems, band-pass systems, and wireless communication fading multipath channels. The three common distributions in communication are the Rayleigh distribution, the rice distribution, and the normal distribution: the envelope of a sinusoidal carrier signal plus narrowband. Gaussian noise is generally a rice distribution. When the signal amplitude is large, it tends to normal distribution; when the amplitude is small, it is approximately Rayleigh distribution.
Gaussian white noise is an ideal model to analyze the additive noise of the channel, and the main noise source in communication thermal noise belongs to this kind of noise. Its values at any two different times are uncorrelated and statistically independent. After the white noise passes through the band-limited system, the result is band-limited noise. Low pass white noise and bandpass white noise are common in theoretical analysis.
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