[digital signal processing] correlation function (finite signal | autocorrelation function of finite signal)

One 、 Autocorrelation function of finite signal

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about " Limited signal " $x(n)$ , $n$ The range of phi is zero $[0,N-1]$ Closed interval ;

be Limited signal $x(n)$ Of Autocorrelation function yes :

$$r_x(m)=\frac{1}{N}\sum _{n=0}^{N-1-m}{x}^{\ast }(n)x(n+m)$$

${\sum }_{n=0}^{N-1-m}{x}^{\ast }(n)x(n+m)$ Divide $N$ Equivalent to the $m$ by $0$ when , The value of the autocorrelation function is signal power ;

$$r_x(0)=LetterNumberworkrate$$

seek mean value or variance , Both need to work with $N$ be divided by , $N$ Is time , That is, the number of finite signals , Here is the Yes Time Averaging ;

Limited signal yes Energy signal , " Autocorrelation function " Of " The Fourier transform " yes " Power spectral density function " ,

Limited signal Of Time $n$ Value range of yes $[0,N-1]$ Closed interval , But in the formula Additive formula yes

$$\sum _{n=0}^{N-1-m}$$

No

$$\sum _{n=0}^{N-1}$$

Because it's for $x(n)$ And $x(n+m)$ The correlation function of ;

If The signal displacement $m$ , Beyond the $n$ Value range of $[0,N-1]$ Closed interval , This signal is not the original signal , The autocorrelation function is meaningless ;

So here's $n$ Value , Must be $[0,N-1-m]$ Closed interval ;