## Introduction

So, we want to break down our continuous-time sigma-delta feedback into two paths:

- A low-precision tight loop that delivers the first sample to the quantizer
- A higher-precision loop that goes through a clock delay to minimize “metastability” (indecision)

## DAC Feedback

As a small aside, recall that the DAC feedback looks similar to the following:

The input to the quantizer is the convolution of this signal with the impulse response *h*(*t*) of the noise-shaping filter *H*(*s*). As we did with the impulse-invariant transform, we break down the noise-shaping filter into partial-fraction expansion:

`H(s) = \displaystyle \sum_{k} F_{k}(s) = \sum_{k} \frac{g_k}{s - p_k}`

Each of these `F_{k}`

is a single-pole filter, with an impulse response that looks something like:

## Dual-Filter Analysis

The first sample at the quantizer is the convolution of a DAC symbol with the impulse response. This amounts to the integral over one sample period:

The rest of the samples can be passed through a filter that looks like this:

The main difference between this remaining filter and the original is that it is scaled by `A_{1} = e^{-p_{k}T_s}`

. Note that the delay in the filter is already included in the system due to the extra clock delay by the second DAC.

## Summary

So, in the end, we have replaced `H(s)`

with:

`H'(s) = \displaystyle \sum_k e^{-p_{k}T_s}F_{k}(s)`