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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.
