Effect of time constant of integrator on delta modulator operation

[promach]

For delta modulator with low pass RC filter as integrator , why it simply generates a periodic output at half of the clock frequency if the input peaks do not exceed the peaks of Vp ?

Besides, how to derive sensitivity = Vdd * {1-exp[-Tck / (2RdCd ) ] } ?





Note: the screenshot is taken from https://cloudfront.escholarship.org/dist/prd/content/qt1pg6p6nb/qt1pg6p6nb.pdf#page=59