I'm struggling to find any youtube videos for piece-wise examples. Apologies for not knowing the math syntax yet.
Given $y[n]= \begin{cases} x[n] & n\geq 1 \\ 1 & n=0 \\ -x[n] & n\leq 1 \end{cases} $
Determine if the system is linear.
My approach is:
- Set up scalar and additive properties
$x_{1}[n] \to y_{1}[n] = \begin{cases} x_{1}[n] & n\geq 1 \\ 1 & n=0 \\ -x_{1}[n] & n\leq 1 \end{cases} $
$x_{2}[n] \to y_{2}[n] = \begin{cases} x_{2}[n] & n\geq 1 \\ 1 & n=0 \\ -x_{2}[n] & n\leq 1 \end{cases} $
$x_{3}[n] = a_{1}x_{1}[n] + a_{2}x_{2}[n]$
- Compare
$y_{3}[n] = \text{Sys}(x_{3}[n]) = a_{1}y_{1}[n] + a_{2}y_{2}[n]$
- Evaluate (issues)
$$a_{1}y_{1}[n] + a_{2}y_{2}[n] =a_{1} \begin{cases} x_{1}[n] & n\geq 1 \\ 1 & n=0 \\ -x_{1}[n] & n\leq 1 \end{cases} + a_{2} \begin{cases} x_{2}[n] & n\geq 1 \\ 1 & n=0 \\ -x_{2}[n] & n\leq 1 \end{cases} $$ Issue here, can I distribute a into the piecewise?
against $$S(x_{3}[n]) = S(a_{1}x_{1}[n] + a_{2}x_{2}[n]) = \begin{cases} a_{1}x_{1}[n]+a_{2}x_{2}[n] & n\geq 1 \\ a_{1}(1)+a_{2}(1) & n=0 \\ -(a_{1}x_{1}[n]+a_{2}x_{2}[n]) & n\leq 1 \end{cases} $$
Last question, would $x[n]=1$ for $n=0$ regardless of $a_{n}$?