sigma-delta modulator issues

sigma delta stability

Hi, I have two questions about sigma-delta modulator's structure:
1) I think the closed loop transfer functions of single loop and MASH structure modulator are the same. However, there are many papers say that the high order (≥2) MASH structure modulator is stable since each sub-loop(generally 2nd order modulator) and the high order (≥2) single loop modulator may be not stable. My puzzle is:why the TF is the same, the stability is different?
2) Refer to the attached pic(I have not more text explanation about this pic at hand),I am not sure whether the word CANDY is a name. Why is these 2nd single loop structure's stability different?

Would you please give me any hints about above questions?
Thank you very much.

mash 2-2 sigma-delta

The difference between picture a and b is that in a there is not feedback path bewteen the two integrators. Since each integrator as a 90 degrees phase shift the total will be 180 degrees and the system is unstable. For a) the closed loop tf is 1/(s^2+1) and for b) it is 1/(s^2+s+1).

The principle of cascaded sigma-delta modulation is based on the use of multiple sigma-delta modulator stages in a cascade configuration. In cascaded modulators the quantization error of the first path is the input signal to the second path. By using digital filters to combine two outputs of each path, higher order quantization noise shaping can be achieved. Since cascaded modulators are made from loworder modulators, they have a stability advantage. Conventional
second order modulators are stable but often do not provide sufficient resolution. Higher-order modulators have much better signal-to-noise (SNR) performance, but the corner frequency of the loop-filters have to move to a lower frequency
for a fixed sampling frequency to maintain a stable loop. Cascaded modulators combine the better of the two extremes. They have high-order noise shaping characteristics, while the corner frequency is placed at a position for a low-order
modulator.

Hope that helps.
svensl

The answer is that it is not the same TF.
In the Candy structure (Candy is indeed a name see book "Oversampling delta-sigma data converters" from Candy and Temes), the output is fed back into the modulator not only at the input but also after the first integrator. It adds a zero to the loop filter transfer function and improves the stability.

Concerning MASH architectures, a 2-2 MASH or a 4 single-loop do provide the same ideal noise shaping but the loop filter transfer function is again different and it is the loop filter transfer function which matters concerning stability.

For sigma-delta stuff:
http://www.imse.cnm.es/esd-msd/deliverables.html
and look for the courses given in the frame of the mixmodest project.