# 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?

• 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, 2019 at 18:34
• it's the difference between the concepts of "linear" and "affine". Feb 25, 2019 at 21:48
• it's TI, but not L. Feb 25, 2019 at 21:50

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.