Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System?

Question

Linearity: $ \alpha \cdot T\{x_1[n] \} + \beta \cdot T\{x_2[n] \} = T\{\alpha \cdot x_1[n]\ + \beta \cdot x_2[n] \} \\\alpha \cdot (x_1[n]+2) + \beta \cdot(x_2[n]+2)= \alpha \cdot (x_1[n]+2) + \beta \cdot(x_2[n]+2)$

Time-Invariance: $y_1[n] = y[n-n_0] = x[n-n_0]+2 \\ x_2[n]= x_1[n-n_0]\\ y_2 [n]=T\{x_2[n ]\}= x[n-n_0]+2 \\y_1[n]=y_2[n]$

So I would say it is an LTI-System, is that right?

Your linearity calculation is wrong. Assume $\beta=0$. Notice that $T \lbrace \alpha x(t) \rbrace$ is not equal to $\alpha T \lbrace x(t) \rbrace$. — MBaz, Feb 25 '19 at 18:34
it's the difference between the concepts of "linear" and "affine". — robert bristow-johnson, Feb 25 '19 at 21:48

score 0 · Accepted Answer · answered Feb 25 '19 at 20:26

0

A system that adds a constant is rarely linear. If you have two inputs $x_1$ and $x_2$, each one gets the constant, so their combination get the constant two times. While the input $x_1+x_2$ gets the constant only once.

answered Feb 25 '19 at 20:26

Laurent Duval

31,850
3
33
101

Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System?

1 Answers1