[digital signal processing] correlation function (property of correlation function | maximum value of correlation function | maximum value of autocorrelation function | maximum value of cross correlat

One 、 Maximum value of correlation function

---

1、 Maximum value of autocorrelation function

Autocorrelation function stay The independent variables $m=0$ when , Always greater than others $m\ne 0$ Value ;

$$r_x(0)\ge r_x(m)$$

in other words , Autocorrelation function Of Maximum , Namely $m=0$ Time value ;

2、 Maximum value of cross correlation function

Cross correlation function Of The maximum is $\sqrt{r_x(0)r_y(0)}$,

$r_x(0)$ yes $x(n)$ The signal energy ;

$r_y(0)$ yes $y(n)$ The signal energy ;

$$|r_{xy}(m)|\le \sqrt{r_x(0)r_y(0)}=\sqrt{E_x{E}_y}$$

Two 、 The correlation function of the energy limited signal is m When approaching infinity 0

---

If The signal $x(n)$ and The signal $y(n)$ All are Energy signal ,

Energy signal refer to Limited energy Of The signal , The energy is Absolute summability Of ,

And Relative yes Power signal , Power signal Energy is infinite ,

Energy signal A certain No Periodic signal ,

$$\underset{m\to \infty }{\lim }{r}_x(m)=0$$

$$\underset{m\to \infty }{\lim }{r}_{xy}(m)=0$$

Physical meaning : When $m=0$ when , The two sequences are similar The most relevant , however With $m$ Add to infinity $\infty$ , Then the correlation becomes $0$ ,

Finite sequence , Once you pan , There is always When staggered , Once staggered , There is no correlation whatsoever , The correlation is $0$ ;