$\left\{T_k\left(f\right)\right\}_{k\in\mathbb Z}$ ON $\Longleftrightarrow$ $\sum_{k\in\mathbb Z}\left|\widehat f\left(\xi-k\right)\right|^2=1$ a.e.

Question

For every $k\in\mathbb Z$, define $T_k:L^2\left(\mathbb R\right)\to L^2\left(\mathbb R\right)$ by $T_k\left(f\right)\left(x\right)=f\left(x-k\right)$. Let $f\in L^2\left(\mathbb R\right)$. Then

$\left\{T_k\left(f\right)\right\}_{k\in\mathbb Z}$ is orthonormal if and only if $\sum_{k\in\mathbb Z}\left|\widehat f\left(\xi-k\right)\right|^2=1$ almost everywhere.

Could someone please provide a hint?