Part E: Linearity and Time Invariance
Part A
For the output signal to be time invariant, the response to the shifted input signal $ x[n-N] $ should be the shifted output ($ y[n-N] $). Basically, this means that if the input signal is shifted along the x-axis by any amount of time, the output signal should produce the same value at $ n + N $ that it used to produce at $ n $ before the shift.
For the system given, let's use the system corresponding to $ X $2 and then select a time $ n $ of 4, and a shifted time $ N $ of 1. If this system was time invariant, $ Y $2$ [n] $ at time $ n = 5 $ should equal $ Y $1$ [n] $ at $ n = 4 $.
$ \,\ X $2$ \,\ [4-2] = $&delta$ \,\ (2) $
$ \,\ X $2$ \,\ [4-1-2] = $&delta$ \,\ (1) $
