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

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

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?

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.