第二章 随机过程的一般概念2.1 随机过程的基本概念和例子定义2.1.1：设(P ,,F )Ω为概率空间，T 是某参数集，若对每一个，是该概率空间上的随机变量，则称为随机过程(Stochastic Process)。
T t ∈),(w t X ),w t (X 随机过程就是定义在同一概率空间上的一族随机变量。
随机过程可以看成定义在),(w t X Ω×T 上的二元函数，固定Ω∈0w ，即对于一个特定的随机试验，称为样本路径(Sample Path)，或实现(realization)，这是通常所观测到的过程；另一方面，固定，是一个随机变量，按某个概率分布随机取值。
),(0w t X T t ∈0),(0w t X抽象一点：令，即∏∈=Tt T R R T R 中的元素为),(T t x X t t ∈=,为其Borel域(插乘)(T R B σ域)，随机过程实质上是()F ,Ω到())(,T T R R B 上的一个可测映射，在())(,T TR RB 上诱导出一个概率测度：T P ()B X P B P R B T T T ∈=∈∀)(),(B 。
一般代表的是时间。
根据参数集T 的性质，随机过程可以分为两大类： t 1)为可数集，如T {}L ,2,1,0=T 或{}L L ,1,0,1,−=T ，称为离散参数随机过程，也称为随机序列；2)为不可数集，如T {}0≥=t t T 或{}∞<<∞−=t t T ，称为连续参数随机过程。
随机过程的取值称为过程所处的状态(State)，所有状态的全体称为状态空间(State Space)。
通常以表示随机过程的状态空间。
根据状态空间的特征，一般把随机过程分为两大类：T t t X ∈),(S 1) 离散状态，即取一些离散的值； )(t X 2)连续状态，即的取值范围是连续的。
)(t X离散参数离散状态随机过程： Markov 链 连续参数离散状态随机过程： Poisson 过程 离散参数连续状态随机过程： *Markov 序列连续参数连续状态随机过程： Gauss 过程，Brown 运动例2.1.1：一醉汉在路上行走，以的概率向前迈一步，以q 的概率向后迈一步，以p r 的概率在原地不动，1=++r q p ，选定某个初始时刻，若以记它在时刻的位置，则就是直线上的随机游动(Random Walk)。
Lecture One Random Process13. Probabilty distribution of narrow band process envoloping squareTips: We have talked about the probabilty distribution of envoloping and phase of narrow-band process in both cases of pure noise and sinusoidal signal plus noise and understood the distribution form of the signal, which can determine the detection threshold specifically and find signal from the noise background. Detection method can be used for phase or envelope, but the envelope detection was used commonly because it is relatively easy to implement. Initial phase of incoherent signals which we usually used can ’t use phase detection because of its random change. Therefore, the envelope detection is divided into linear detection and square-law detection. The former has more complex theoretical analysis, but practices more and it is based on the probability distribution of the envelope of narrowband process. In contrast to the former,the latter uses multi-square-law detection in the theoretical analysis and it is based on the probability distribution of the square of the envelope of narrowband process. In fact, these two detectors have little difference in performance. When the information processing needs to know the probability distribution density after detection, the results discussed above is very practical.1. (Pure noise) probability distribution of the envelope square of narrowband normal noisenarrowband normal noise can be represented as:()()()[]()()t Sin t A t Cos t A t t Cos t A t n S C 000ωωϕω-=+=narrowband normal noise envelope:()()()t A t A t A S C22+=The probability density of narrowband normal noise envelope ()t A is222)(σσA e A A p -=——Rayleigh distributionIf envelope detector has half-wave square-law detection characteristics, that is, the output voltage()()t A t u 2= 0,≥A uAccording to the transform of random variable function, the envelope square u of the probability density is()()222221σσA e A A A p du dA u p -⋅=⋅= 2221σσu e -= 0u ≥——exponential distributionMake the normalized random variable is2σu v =, the probability density is()()221v e u p dv du v p -=⋅= 0v ≥——exponential distribution standard expression2. Narrowband normal noise + probability distribution of envelope square in the sinusoidal signal synthesis processsynthesis process is ：()()()()()[]t t Cos t R t n t s t Y Φ+=+=0ωThe probability density of the envelope ()t R in synthesis process:()⎪⎭⎫ ⎝⎛⋅⎥⎦⎤⎢⎣⎡+-=2022222exp σσσA R I A R R R p 0R ≥——Obey thegeneralized Rayleigh distributionIf envelope detector has half-wave square-law detection characteristics, that is, the output voltage()()t R t u 2= 0R ,u ≥According to the of transform random variable function, the probability density of the envelope square u is()()R p du dR u p ⋅=⎪⎪⎭⎫ ⎝⎛⋅⎥⎥⎦⎤⎢⎢⎣⎡+-=202222exp 21σσσA u I A u 0u ≥ 令2σu v =，()()⎪⎪⎭⎫ ⎝⎛⋅⎥⎥⎦⎤⎢⎢⎣⎡+-=⋅=σσA v I v u p dv du v p A 02exp 2122 0v ≥ 3. 2χ distribution and non-Center 2χ distribution(Greek alphabet [kai])（1）Accumulation of the video signal improves the detection performanceTips: When we use envelope detection method to detect the periodic signal noise in the background, we usually choose video accumulation inorder to improve detection performance.Tips: we need to determine whether the output of the receiver has the information which is desired to check in signal detection theory. However, signal is always accompanied by noise and they are mixed together to form a random process. In order to improve the detection performance, we often add up sample values which are outputted by square law detector (accumulation) and then compare with the threshold level before judging whether the signal is appeared or not. Because there is more than one signal the receiver receiving in many applications, for example radar is about to launch many pulse in sequence at work, and the receiver need to observe many echo pulses which are received at different times before judging. These echo pulses usually carrie the same information or all of the reflection information which containing object, or only have noise. That is to say, every pulse signal contains the information whether targets would appear and each pulse signal noise is generally statistically independent to each other. We can combine these signals before judging these "yes" or "no" signals and the detection performance can be significantly improved.Tips: Add up the signal (accumulation) in two situations: one is the addition before detector, another is the addition after detector. We are going to discuss the latter. Amplitude detector has two kinds: one kind is linear detector, the other is square law detector. The distribution of linear detector output process is Rayleigh distribution, the unified exact expressions can not be get by the sum of multiple generalized Rayleigh distribution, so we are not research itThe output of square law detector is different. when its input completely has Rayleigh envelope statistical properties (narrowband gaussian noise), the sum of (non-coherent accumulation of n pulses) its output statistics independent sampling values (condition: sampling time is long enough, sothat the sampling values are independent)will produce the probability density function distribution containing n degrees of freedom 2χ.)()()(222t A t A t R +=)]( cos[)()(t t t R t Y Φω+= ()()()()[]()[]t S i n t n A Si n t C o s t n A C o s t n t s t Y S C 00ωθωθ+-+=+=()t A C ()t A STips: In the diagram, signal plus noise is through the narrow band system and ()t Y becomes signal with narrowband normal process.It is through the square law detector and does envelope detection and getssquare envelope process ()()()t A t A t R S C 222+=.。
