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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:

  1. 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} $

$x_{3}[n] = a_{1}x_{1}[n] + a_{2}x_{2}[n]$

  1. Compare

$y_{3}[n] = \text{Sys}(x_{3}[n]) = a_{1}y_{1}[n] + a_{2}y_{2}[n]$

  1. 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 $a_{n}$?

Garrett Gutierrez
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  • Welcome to MSE! It would be good if you precise in which linear space you're working. I imagine in sequences from $\mathbb Z$ to $\mathbb R$. But better to say it. Also it should be $y(n) = 1$ for $n=1$, as for $x=0$ seems not to make sense. – mathcounterexamples.net Oct 21 '20 at 13:58
  • Welcome to MSE! I edited your question to add proper formatting. If I have changed the information you intended to convey, then please feel free to edit the question. You will find the 'edit' button on the bottom of your question. Just in case you have not found it, here is a good reference on MathJax. –  Oct 21 '20 at 14:16

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