Label circle in and out with TikZ

Question

I want to label a circle in and out, like this sketch:

I have also tried to form the circle in my own very simple way as follows:

\documentclass{article}
\usepackage{tikz}
\begin{document}
    \begin{tikzpicture} 
        \draw[|-|] (90.0:3.0) arc(90.0:67.5:3.0); 
        \draw[|-|] (67.5:3.0) arc(67.5:45.5:3.0);
        \draw[|-|] (45.0:3.0) arc(45.0:22.5:3.0);
        \draw[|-|] (22.5:3.0) arc(22.5:00.0:3.0);
        \draw[|-|] (00.0:3.0) arc(00.0:-22.5:3.0); 
        \draw[|-|] (-22.5:3.0) arc(-22.5:-45.0:3.0);
        \draw[|-|] (-45.0:3.0) arc(-45.0:-67.5:3.0);
        \draw[|-|] (-67.5:3.0) arc(-67.5:-90.0:3.0);
        \draw[|-|] (-90.0:3.0) arc(-90.0:-112.5:3.0); 
        \draw[|-|] (-112.5:3.0) arc(-112.5:-135.0:3.0);
        \draw[|-|] (-135.0:3.0) arc(-135.0:-157.5:3.0);
        \draw[|-|] (-157.5:3.0) arc(-157.5:-180.0:3.0);
        \draw[|-|] (-180.0:3.0) arc(-180.0:-202.5:3.0); 
        \draw[|-|] (-202.5:3.0) arc(-202.5:-225.0:3.0);
        \draw[|-|] (-225.0:3.0) arc(-225.0:-247.5:3.0);
        \draw[|-|] (-247.5:3.0) arc(-247.5:-270.0:3.0);
    \end{tikzpicture}
    
\end{document}

What I want

I want my already-drawn circle (through the TikZ code) to be labelled as the image I provided and not you coming up with different code for a different circle as I may not be able to understand your style.

Note

I want an improvement on this my simple style of code that will simply add block of codes to it to make the labels visible and not the one you will come up with a different style. If you come up with different style I will not be able to improve on your answer to use it for what I want.

Answer 1

If I compile this:

\documentclass{article}
\usepackage{xcolor,tikz}
\begin{document}
\begin{tikzpicture}
    \draw[|-|] (90.0:3.0) arc(90.0:67.5:3.0); 
    \draw[|-|] (67.5:3.0) arc(67.5:45.0:3.0);
    \draw[|-|] (45.0:3.0) arc(45.0:22.5:3.0);
    \draw[|-|] (22.5:3.0) arc(22.5:00.0:3.0);
    \draw[|-|] (00.0:3.0) arc(00.0:-22.5:3.0); 
    \draw[|-|] (-22.5:3.0) arc(-22.5:-45.0:3.0);
    \draw[|-|] (-45.0:3.0) arc(-45.0:-67.5:3.0);
    \draw[|-|] (-67.5:3.0) arc(-67.5:-90.0:3.0);
    \draw[|-|] (-90.0:3.0) arc(-90.0:-112.5:3.0); 
    \draw[|-|] (-112.5:3.0) arc(-112.5:-135.0:3.0);
    \draw[|-|] (-135.0:3.0) arc(-135.0:-157.5:3.0);
    \draw[|-|] (-157.5:3.0) arc(-157.5:-180.0:3.0);
    \draw[|-|] (-180.0:3.0) arc(-180.0:-202.5:3.0); 
    \draw[|-|] (-202.5:3.0) arc(-202.5:-225.0:3.0);
    \draw[|-|] (-225.0:3.0) arc(-225.0:-247.5:3.0);
    \draw[|-|] (-247.5:3.0) arc(-247.5:-270.0:3.0);
    \foreach \i in {1,...,16} \node at ({112.5-22.5*\i}:3.45) {\color{blue}{$x_{\i}$}};
    \foreach \i in {17,...,21} \node at ({112.5-22.5*\i}:2.55) {\color{red}{$x_{\i}$}};
\end{tikzpicture}
\end{document}

then I get this:

A quite shorter code that will give you the same output is:

\documentclass{article}
\usepackage{xcolor,tikz}
\begin{document}
\begin{tikzpicture}
  \foreach \i in {1,...,16} {\draw[|-|] ({22.5*\i}:3) arc({22.5*\i}:{22.5*(\i+1)}:3);
    \node at ({112.5-22.5*\i}:3.45) {\color{blue}{$x_{\i}$}};}
  \foreach \i in {17,...,21} \node at ({112.5-22.5*\i}:2.55) {\color{red}{$x_{\i}$}};
\end{tikzpicture}
\end{document}

Answer 2

You may not like an even quicker way to do this, just for fun:

\documentclass[tikz,border=3.14mm]{standalone}
\begin{document}
    \begin{tikzpicture}
        \newcommand{\R}{3}
        \newcommand{\p}{3 pt}
        \draw (0,0) circle (\R cm);
        \foreach \i [count=\j from 17] in {1,...,16}
            {
            \draw (112.5-22.5*\i:\R cm-\p)--(112.5-22.5*\i:\R cm+\p) node[blue,pos=2.5] {$x_{\i}$} \ifnum\j<22 node[red,pos=-1.5] {$x_{\j}$} \fi;
            }
    \end{tikzpicture}
\end{document}