Random Processes in Communication systems

Both signal and noise in communication can be regarded as random processes that change with time.

Random process has the characteristics of random variable and time function, which can be described from two different but closely related perspectives: (1) Random process is the set of infinite sample functions; (2) A random process is a set of random variables.

The statistical properties of random processes are described by their distribution function or probability density function. If the statistical properties of a random process are independent of the time starting point, it is called a strictly stationary process.

Numerical features are another neat way of describing random processes. If the mean of the process is constant and the autocorrelation function R(t1,t1+τ)=R(T), the process is said to be generalized stationary.
If a process is strictly stationary, then it must be broadly stationary, and vice versa is not necessarily true.

A process is ergodic if its time average is equal to the corresponding statistical average.

If a process is ergodic, then it is also stationary, and vice versa is not necessarily true.

The autocorrelation function R(T) of a generalized stationary process is an even function of the time difference r, and R(0) is equal to the total average power and is the maximum value of R(τ). Power spectral density Pξ(f) is the Fourier transform of the autocorrelation function R(ξ) (Wiener – Sinchin theorem). This pair of transformations determines the conversion relationship between the time domain and the frequency domain. The probability distribution of a Gaussian process obeys a normal distribution, and its complete statistical description requires only its numerical characteristics. The one-dimensional probability distribution depends only on the mean and variance, while the two-dimensional probability distribution depends mainly on the correlation function. A 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. After a stationary random process ξi(t) passes through a linear system, its output process ξ0(t) is also stable.

The statistical characteristics of narrow-band random process and sine-wave plus narrow-band Gaussian noise are more suitable for the analysis of fading multipath channels in modulation system/bandpass system/wireless communication. Rayleigh distribution, Rice distribution and normal distribution are three common distributions in communication: the envelope of sinusoidal carrier signal plus narrow-band Gaussian noise is generally 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 the communication, thermal noise, belongs to this kind of noise. Its values at any two different times are uncorrelated and statistically independent. After white noise passes through a band-limited system, the result is band-limited noise. Low-pass white noise and band-pass white noise are common in theoretical analysis.
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