Here are a few quick notes on digital noise shapers, primarily for my own reference.
Here we have a traditional noise shaper setup, consisting of an Nth-order loop filter and a quantizer:
When a linear model is assumed, the noise transfer function (NTF) can be expresses as a function of the loop filter:
The loop filter also changes the frequency response of the input signal:
Assuming the loop gain is very high in the lower part of the spectrum, where the desired signal is present, this frequency response can be approximated as:
In effect, given enough in-band loop gain, the frequency domain distortion is negligible.
Causal loop filters
Given a desired noise transfer function , the loop filter can be derived as follows:
Now, splitting the NTF into its poles A(z) and zeroes B(z), the loop filter can be expressed as:
For the loop to be causal, the loop filter must have at least one pure delay. Given that , if we force the first coefficient of B(z), i.e. , to be equal to 1, the first coefficient of the resulting loop filter numerator will be zero and the loop filter is guaranteed to have at least one delay.