  In this article, we give a new method for proving Howe correspondence in the
case of dual pairs of type $({\rm GL}_n, {\rm GL}_m)$ over a non-Archimedean
locally compact field $F$. The proof consists in combining a study on Kudla's
filtration with the results of a previous article of the autor about the
irreducibility of a parabolically induced representation. The proof is valid
for $F$ of any characteristic and allows us to make the correspondence explicit
in terms of Langlands parameters.
