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

Signal basis " energy " Can be divided into " Energy signal " and " Power signal " ;

• Signal energy definition : The energy on the whole axis is squared first , And then integral ; If energy Less than infinite , Then the signal yes Energy signal ; A signal in a finite interval is called an energy signal ;

• Signal power definition : In one signal cycle , Perform integral summation operation ; If power Less than infinite , Then the signal yes Power signal ; Periodic signal , Random signals Power signal ;

In this blog Cross correlation function and Autocorrelation function , All are " Power signal " Of Correlation function ;

A power signal is a signal with infinite energy , The energy value cannot be calculated , Here we only calculate the energy value in one cycle ;

One 、 Cross correlation function of power signal

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Power signal Cross correlation function It means Two different signals The correlation between ;

$x(n)$ And $y(n)$ Of " Cross correlation function " as follows ,

$$r_{xy}(m)=\underset{N\to \infty }{\lim }\frac{1}{2N+1}\sum _{n=-N}^{+N}{x}^{\ast }(n)y(n+m)$$

Take the sequence elements in a cycle , seek Correlation function value , Then take the average ;

among $y(n)$ Shifted , Moved left $m$ Company ,

The " Cross correlation function " What I'm looking for is $y(n)$ displacement $m$ The sequence after And $x(n)$ The relationship between sequences ;

Notice the $n$ It means the moment , $m$ It represents the interval of signal movement ;

The " Cross correlation function " It means $x(n)$ The signal , And Separated $m$ After time $y(n)$ The relationship between signals ;

this $2$ A signal ( Sequence ) Between " Relationship " It's a function , The argument to the function is $m$ interval , No $n$ ;

Two 、 Autocorrelation function of power signal

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Power signal Autocorrelation function ( Autocorrelation Function ) :

$$r_x(m)=\underset{N\to \infty }{\lim }\frac{1}{2N+1}\sum _{n=-N}^{+N}{x}^{\ast }(n)x(n+m)$$

Take the sequence elements in a cycle , seek Correlation function value , Then take the average ;

" Autocorrelation function " yes " Own signal " And " After a period of time Own signal " Between The correlation ;

If $m=0$ when , " Own signal " And " After a while $m$ Your own signal after " Completely equal , The value is The energy of the signal ;

$$r_x(0)=\sum _{n=-\infty }^{+\infty }|x(n){|}^2=E$$