  Let $E/F$ be a quadratic extension of $p$-adic fields and $\textrm{U}_{2r+1}$
be the unitary group associated with $E/F$. We prove the following local
converse theorem for $\textrm{U}_{2r+1}$: given two irreducible generic
supercuspidal representations $\pi,\pi_0$ of $\textrm{U}_{2r+1}$ with the same
central character, if $\gamma(s,\pi\times \tau,\psi)=\gamma(s,\pi_0\times
\tau,\psi)$ for all irreducible generic representation $\tau$ of
$\textrm{GL}_n(E)$ and for all $n$ with $1\le n\le r$, then $\pi\cong \pi_0$.
The proof depends on analysis of the local integrals which define local gamma
factors and uses certain properties of partial Bessel functions developed by
Cogdell-Shahidi-Tsai recently.
