Question 9.11: Find the even and odd components of the following discrete-t......

The even component of x[n] is given by

x_e[n] = 0.5(x[n] + x[−n]).

When n = 0 then x_e[0] = 0.5 × 2x[0] = 4, when n > 0 then x_e[n] = 0.5x[n], and when n < 0, then x_e[n] = 0.5x[−n] giving

x_{e}[n]={\left\{\begin{array}{l l}{2+0.5n}&{-4\leq n\leq-1,}\\ {4}&{n=0,}\\ {2-0.5n}&{1\leq n\leq4,}\\ {0}&{\mathrm{otherwise},}\end{array}\right.}

while the odd component

x_{o}[n]=0.5(x[n]-x[-n])={\left\{\begin{array}{l l}{-2-0.5n}&{-4\leq n \leq-1,}\\ {0}&{n=0,}\\ {2-0.5n}&{1\leq n\leq4,}\\ {0}&{{\mathrm{otherwise}}.}\end{array}\right.}

The sum of these two components gives x[n].