  We provide an explicit construction of representations in the discrete
spectrum of two $p$-adic symmetric spaces. We consider $\mathbf{GL}_n(F) \times
\mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$ and $\mathbf{GL}_n(F)
\backslash \mathbf{GL}_n(E)$, where $E$ is a quadratic Galois extension of a
nonarchimedean local field $F$ of characteristic zero and odd residual
characteristic. The proof of the main result involves an application of a
symmetric space version of Casselman's Criterion for square integrability due
to Kato and Takano.
