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]).

For a sigma delta modulator with a second order filter, transfer function of the noise shaping filter is .

The frequency response of is

.

The quantization noise in the desired signal bandwith can be computed as,

.

Applying limits of the integration,

.

Assuming , the three term Taylor series expansion for sine wave is .

Expanding,

.

Note that, the above equation is different from the answer provided in Problem 9.11(c) of [1]. Per my understanding factor of is missing in Problem 9.11(c).

Thus for a second order sigma delta modulator, doubling of the sampling frequency results in 15dB reduction in quantization noise.

In general, for order sigma delta modulator, the transfer function of the noise shaping filter is .

The variance the in-band quantization noise for an order sigma delta modulator can be shown as

.

I hope to able to understand the proof for nth order sigma delta modulator and maybe have a post.

References:

[1] Digital Signal Processing – Principles, Algorithms and Applications, John G. Proakis, Dimitris G. Manolakis