hjr-Communication Principle (V): Stochastic process

First, a concept.

Stochastic process : First imagine a 2-dimensional coordinate system, X represents time, Y is a number, and when x takes a value, its corresponding Y value is a random value (called a random variable )

The Y value of each moment of the x-axis is connected to a line, and this set is called a random process.

distribution function and probability density :,

The statistical properties of the Y-values corresponding to each X-value can be described by distribution function or probability density, and the probability density function after the derivation of the distribution function

A stochastic process y (t), the probability that the random variable is less than or equal to X1 at the moment of T1 is called one-dimensional distribution function of the stochastic process, and the derivative is one-dimensional probability density

At the same time satisfying the probability that the random variable is less than or equal to X2 at the moment of T1 time is called the 2-dimensional distribution function of the stochastic process, n-dimensional and so on, the more the dimension, the more accurate the stochastic process is described.

mean (mathematical expectation): Describes the center point of a sample collection

Standard deviation : Sample collection of each sample point to the mean distance of the average, the smaller the standard deviation indicates the more concentrated sample (stable), remember the high school that archery example, the mean, the standard deviation small archery good, because the stability

Variance : The square of the standard deviation, indicating the random process at the moment T, the degree of deviation from the center point, and the standard deviation function is the same

Standard deviation differs from variance: the standard deviation and the variable are calculated in the same unit, for example, the difference is clear, so many times when we analyze more use of the standard deviation

Correlation function : Describes the degree of correlation between the values of random signal X (t) t1,t2 at any two different times

covariance : Describes how random signals X (t) t1,t2 at any two different times to describe the degree of fluctuation (relative to mean) of X (t) at two-hour values

In the stochastic process of communication system, the variance and correlation function are used mainly

A concept for stochastic processes is a stationary stochastic process, similar to a linear time-invariant system in the study of signals, and there must always be a precondition, otherwise it is impossible to study

And it is divided into a wide stable and strict stability, strict stability must be broadly stable, the contrary is not established

Strict stability : the statistical characteristics of stationary stochastic systems do not change over time

Generalized Stationary

(1) mean value is not independent of T is constant

(2) autocorrelation function is only related to time interval

Gaussian (normal) stochastic process : Random process N-dimensional distributions are subject to normal distribution, such as thermal noise

In real life, most communication systems are narrow band-pass, and the signal or noise through narrowband systems is necessarily a narrow-band stochastic process.

Narrow-Band Stochastic process : The spectral density of the stochastic process is concentrated in a narrow band near the center frequency, and the center frequency is away from 0 frequency

Four frequency characteristics of the signal

1. spectral density : frequency-amplitude curve, time-domain signal Fourier transform, Unit (v/hz) amplitude/Hz

2, the power spectrum density is multiplied by a factor, the Unit (w/hz) w/Hz

Power spectral density P (w): Stationary stochastic process, P (w) <=> R (t) power spectral density is the Fourier transform of autocorrelation function

Signal Power S: Stationary stochastic process, signal power equal to R (0), that is, autocorrelation function time is 0 moment value