[digital signal processing] linear time invariant system LTI (judge whether a system is a "non time varying" system | case 1 | transform before shift | shift before transform)

One 、 Judge whether the system is " Non time varying "

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1、 Case a

$y(n)=x(-n)$ Whether it is " Time does not change " Of ;

$x(n)$ It's the input sequence , $x(-n)$ Is the output sequence ;

① Time invariant system

Time invariant system ( time-invariant ) : System features , Does not change with time ;

$$y(n-m)=T[x(n-m)]$$

After input delay , The output is also delayed ;

And " Time does not change " The system corresponds to " time varying " System ;

② First transform and then shift

take " Output sequence " Shift , First " Transformation " after " displacement " ;

First the " Input sequence " Conduct " Transformation " operation , obtain " Output sequence " , Then on Output sequence Conduct " displacement " operation ;

among " Transformation " refer to , Discrete time systems , take " Input sequence " Transformation by " Output sequence " , Input sequence To Output sequence Operation between , yes " Transformation " ;

Change operation : First the Input sequence $x(n)$ Conduct Transformation operation , obtain Output sequence $x(-n)$ ,

Shift operation : then Yes $x(-n)$ Output sequence Shift $n-n_0$ obtain $x(-(n-n_0))=x(-n+n_0)$ ,

The complete operation process is as follows :

$$y(n-n_0)=x[-(n-n_0)]=x(-n+n_0)$$

③ Shift first and then transform

yes Shift first , take " Input sequence " to " displacement " operation , obtain new " Input sequence " by $x(n-n_0)$ , then The new input sequence is " Transformation " operation , obtain " Output sequence " ;

The transformation process is $y(n)=x(-n)$ , Transformation time , Just to $n$ Negative value ;

$x(n-n_0)$ Transformation time , Only will $n$ Take the negative , $n_0$ unchanged , The transformation result is as follows $x(-n-n_0)$ ;

The whole process is as follows :

$$T(x(n-n_0))=x(-n-n_0)$$

④ Conclusion

First " Transformation " after " displacement " , The result is $x(-n+n_0)$ ,

First " displacement " after " Transformation " , The result is $x(-n-n_0)$ ,

The system is " Time varying system " ;