  Motivated by the idea that our access to the spacetime is limited by the
resolution of our measuring device, we give a new description of $K$-homology
with a finite resolution. G. Yu introduced a $C^*$-algebra called the
localization algebra $C^*_L(X)$ which consists of functions from $[1,\infty)$
to the Roe algebra $C^*(X)$ whose propagations converge to $0$ and he showed
that for any finite dimensional simplicial complex $X$ endowed with the
spherical metric, the $K$-theory of the localization algebra is isomorphic to
the $K$-homology of $X$. We give a coarse graining version of this theorem
using controlled $K$-theory (also known as quantitative $K$-theory). Namely,
instead of considering families of operators whose propagations converge to
$0$, we prove that for each dimension $n$, there exists a threshold $r_n>0$
such that the $K$-homology of $n$-dimensional finite simplicial complex $X$ is
isomorphic to a certain group of equivalence classes of operators whose
propagation is less than $r_n$. This picture also enables us to represent any
element in the $K$-homology group $K_*(X)$ by a finite matrix for a finite
simplicial complex $X$.
