What is the effect of a sawtooth phase mask in the frequency domain?
let

$$ \phi_0 \triangleq \phi - 2 \pi \left \lfloor \frac{\phi}{2\pi} \right \rfloor $$

where $\lfloor \cdot \rfloor$ is the floor() function, the largest integer not exceeding the argument.

it shouldn't be too hard to show that

$$ e^{j \phi_0} = e^{j \phi} $$

for all real $\phi$.