$\{$Optional$\}$ Let $[]=11110\mathrm{.}50\mathrm{.}5$

$.$ Sketch: $[][2].$

b$.$ Let $[]=11110\mathrm{.}50\mathrm{.}5$

$.$ Sketch: $[1][3]$

c$.$ In this problem, you will show that any sequence can be formed as the sum of an even $($symmetric$)$

and an odd $($anti$-$symmetric$)$ component.

Let $[]=$

$[]+[]$

$2$

and $[]=$

$[][]$

$2.$

i$)$ Show$/$derive that x$[$n$]=$ xe$[$n$]+$ xo$[$n$].$

ii$)$ Show$/$derive that xe$[$n$]$ is even $($symmetric$).$

iii$)$ Show$/$derive that xo$[$n$]$ is odd $($anti$-$symmetric$).$

iv$)$ Find xe$[$n$]$ and xo$[$n$]$ if $[]=23456$

$.($With this notation, the n$=0$ index is

denoted by the arrow. All out$-$of$-$range sequence values equal zero.$)$

d$.$ Write a general purpose MATLAB function that is provided a sequence $"$x$"$ as its input, and returns

an odd $"$x$\_$o$"$ and an even $"$x$\_$e$"$ sequence as its output. The sum of the odd and even sequences must

equal the input sequence. Note that MATLAB does not have a standard scheme for indexing

$3$ of $3$

sequences that include both negative$-$valued and positive$-$valued indexes. You will want to permit

such sequences. Thus, you must devise such a scheme with this function. Be sure to describe your

scheme in the header comments of your function $($for both the inputs and the outputs to your

function.$)$ Your function must work properly for all general sequences. Test this function on the

sequence from the previous problem. Supply a hard$-$copy of the m$-$file and a printout of the result

from the sequence in the previous problem. $($Your MATLAB result should match the result that you

computed by hand in the previous problem!$)$