In a previous post, the variance of the in-band quantization noise for a first order sigma delta modulator was derived. Taking it one step furhter, let us find the variance of the quantization noise filtered by a second order filter.
With a first order filter, the quantization noise passes through a system with transfer function (Refer Eq. 9.2.17 in [1]). The frequency response of
is
(Refer Eq. 9.2.18 in [1]).