What is the effect of a sawtooth phase mask in the frequency domain? We know that a linear phase ramp in the frequency domain corresponds to a time shift in the time domain. Qualititavely, it seems a sawtooth phase mask in the frequency domain could produce the same time shift effect as the linear phase ramp, but I need help with the mathematical derivation to see what the actual effect is.
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 $\phi$.