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

List of articles

• ​ ​ One 、 Cross correlation function ​​
• ​ ​ Two 、 Autocorrelation function ​​

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 " Energy signal " Of Correlation function ;

One 、 Cross correlation function

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

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

r x y ( m ) = ∑ n = − ∞ + ∞ x ∗ ( n ) y ( n + m ) r_{xy}(m) = \sum_{n=-\infty}^{+\infty} x^*(n) y(n + m) rxy(m)=n=−∞∑+∞x∗(n)y(n+m)

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

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

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

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

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

Two 、 Autocorrelation function

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Autocorrelation function ( Autocorrelation Function ) :

r x x ( m ) = ∑ n = − ∞ + ∞ x ∗ ( n ) x ( n + m ) = r x ( m ) r_{xx}(m) = \sum_{n=-\infty}^{+\infty} x^*(n) x(n + m) = r_x(m) rxx(m)=n=−∞∑+∞x∗(n)x(n+m)=rx(m)

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

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

r x ( 0 ) = ∑ n = − ∞ + ∞ ∣ x ( n ) ∣ 2 = E r_{x}(0) = \sum_{n=-\infty}^{+\infty} |x(n)|^2= E rx(0)=n=−∞∑+∞∣x(n)∣2=E