检测技术与信号处理（十三）

　　检测技术与信号处理（十三）
　　根据上次的学习，我们学习了信号数字化出现的问题，接下来我们接着上次的讲解，和大家学习一下信号的自相关函数。
　　According to the previous study, we learned the problems of signal digitization. Next, we will continue with the last time to learn about the autocorrelation function of the signal.
　　什么是自相关函数
　　首先什么是自相关函数，x(t)是各态历经随机过程的一个样本函数，观测时间为T。而x(t+τ)是时移之后的样本函数。这两个样本函数具有相同的均值ux和标准差ox.信号的自相关是描述信号在一个时刻的取值与另一个时刻取值的依赖关系，可以用下式表示（式子中，T为样本记录长度即是观测时间）：
　　First of all, what is the autocorrelation function, x(t) is a sample function of each state undergoing a random process, and the observation time is T. And x(t+τ) is the sample function after time shift. These two sample functions have the same mean value ux and standard deviation ox. The autocorrelation of a signal describes the dependence of the value of the signal at one moment on the value of another moment, which can be expressed by the following formula (where T is The sample record length is the observation time):
　　以下是自相关函数的图像：
　　The following is an image of the autocorrelation function:
　　接下来我们了解一下自相关函数的具体性质:
　　Next we understand the specific nature of the autocorrelation function:
　　1.
　　当数据点分布愈来愈接近一条直线时，pxy的绝对值接近1，x与y的线性相关程度愈好。pxy的正负号表示一变量随另一变量的增加而增加或减少。x↑ y↑，x↓ y↓ 正相关。x↑ y↓，x↓ y↑ 负相关。而pxy→0， x 、y之间无关。但可能存在非线性相关或函数关系。
　　When the distribution of data points is getting closer and closer to a straight line, the absolute value of pxy is close to 1, and the linear correlation between x and y is better. The sign of pxy means that one variable increases or decreases with the increase of another variable. x↑ y↑, x↓ y↓ are positively correlated. x↑ y↓, x↓ y↑ negative correlation. And pxy→0, there is nothing to do between x and y. But there may be non-linear correlation or functional relationship.
　　2.自相关函数在时T=0为最大值，并等于该随机信号的均方值
　　The autocorrelation function is the maximum value at time T=0 and is equal to the mean square value of the random signal
　　3.当T趋近于无穷大时，分为两种情况；
　　（1）随机变量x(t)不含周期成分时，x(t)和x(t+T)不相关，故
　　3. When T approaches infinity, there are two situations;
　　(1) When the random variable x(t) does not contain periodic components, x(t) and x(t+T) are not correlated, so
　　（2）随机变量x(t)含有周期成分时，x(t)和x(t+T)的周期成分相关，保留周期成分的频率、幅值信息
　　(2) When the random variable x(t) contains periodic components, the periodic components of x(t) and x(t+T) are related, and the frequency and amplitude information of the periodic components are retained
　　4.自相关函数为偶函数
　　The autocorrelation function is an even function,
　　5.周期函数的自相关函数仍为同频率的周期函数，其幅值与原周期信号的幅值有关，而丢失了原信号的相位信息
　　举一个比较特别的例子，正弦函数的自相关函数是一个余弦函数，在T=0时具有最大值，但他不随T的增加而衰减至零。它保留了原正弦信号的幅值和频率信息，而丢失了初始相位信息。
　　The autocorrelation function of the periodic function is still a periodic function of the same frequency, and its amplitude is related to the amplitude of the original periodic signal, and the phase information of the original signal is lost.
　　To give a special example, the autocorrelation function of the sine function is a cosine function, which has a maximum value when T=0, but it does not decay to zero as T increases. It retains the amplitude and frequency information of the original sinusoidal signal, but loses the initial phase information.
　　参考资料：
　　文字：重庆邮电大学郑佳老师检测技术与信号处理PPT；
　　图片：百度；
　　翻译：Google翻译.
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