Let $\Lambda$ be an integral lattice (i.e. $\Lambda \subset \mathbb{Z}^n$) in $\mathbb{R}^n$. If $\Lambda$ is of full rank, i.e. the rank of $\Lambda$ is $n$, and $\phi:\mathbb{R}^n \to \mathbb{R}$ is a `nice' function, then the Poisson summation formula states that $$ \sum_{\mathbf{x} \in \Lambda} \phi(\mathbf{x}) = \frac{1}{\det \Lambda}\sum_{\mathbf{x} \in \Lambda'}\widehat{\phi}(\mathbf{x}), $$ where $\det \Lambda$ is the volume of the fundamental parallelepiped spanned by any basis of $\Lambda$, $\widehat{\phi}(\mathbf{x})$ is the Fourier transform on $\mathbb{R}^n$ and $$ \Lambda' = \left\{\mathbf{x} \in Span_{\mathbb{R}}(\Lambda) : \langle \mathbf{x}, \mathbf{y}\rangle \in \mathbb{Z} \text{ for every } \mathbf{y} \in \Lambda\right\} $$ is the lattice dual to $\Lambda$.
My question is as follows: is it possible to write down a similar formula if the integral lattice $\Lambda \subset \mathbb{R}^n$ is not of full rank? More precisely, suppose that the rank of $\Lambda$ is $m < n$, and we have $\phi:\mathbb{R}^n \to \mathbb{R}$ a `nice' function, can we relate $\sum_{\mathbf{x} \in \Lambda}\phi(\mathbf{x})$ to a sum over the dual lattice $\Lambda'$, as in the full rank case? If so, can you give me a reference for such a formula?