Answer it by using Python.

$7\mathrm{.}4$ Down$-$sampling and up$-$conversion. We consider $n-$vectors $x$ that represent signals, with

$x_k$ the value of the signal at time $k$ for $k=1,$dots,$n.$ Below we describe two functions of

$x$ that produce new signals $f(x).$ For each function, give a matrix A such that $f(x)=Ax$

for all $x.$

$($a$)2$ downsampling. We assume $n$ is even and define $f(x)$ as the $\frac{n}{2}-$vector $y$ with

elements $y_k=x_{2k}.$ To simplify your notation you can assume that $n=8,$ i$.$e$.,$

$f(x)=(x_2,x_4,x_6,x_8).$

$($On page $131$ we describe a different type of down$-$sampling, that uses the average

of pairs of original values.$)$

$($b$)2$ up$-$conversion with linear interpolation. We define $f(x)$ as the $(2n-1)-$vector $y$

with elements $y_k=x_{\frac{k+1}{2}}$ if $k$ is odd and $y_k=\frac{x_{\frac{k}{2}}+x_{\frac{k}{2}+1}}{2}$ if $k$ is even. To

simplify your notation you can assume that $n=5,$ i$.$e$.,$

$f(x)=(x_1,\frac{x_1+x_2}{2},x_2,\frac{x_2+x_3}{2},x_3,\frac{x_3+x_4}{2},x_4,\frac{x_4+x_5}{2},x_5).$