## 4 The Ordering of the Stages
### The Round-Off Output Noise Variance  
The round-off output noise variance is the summation of the noise sources inside the filter cascade, with suitable weightings and filtering. This variance is affected by the ordering of the stages of the filter structure. The variance of the noise of each rounding operation equals \(\sigma_{e}^{2}\). The input signal of the filter has an amplitude in the interval (\(-1\), \(+1\)). In case \(b+1\) bits (\(b\) bits \(+\) a sign bit) are used to represent the signal in two's complement, the variance of the noise generated by one round-off noise source is (under the assumptions of Section 2.2) given by:  
\[\sigma_{e}^{2}=\frac{q^{2}}{12}. \tag{17}\]  
Here, \(q=2^{-b}\). In general, the variance of the round-off noise source is also given by:  
\[\sigma_{e}^{2}=\frac{1}{2\pi}\int\limits_{-\pi}^{+\pi}P_{\infty}\left(\omega_ {n}\right)d\omega_{n}-\left|m_{e}\right|^{2} \tag{18}\]  
where: \(P_{\infty}(\omega_{n})\) is the Power Spectral Density (PSD) of the noise source. \(m_{e}\) is the mean value of the noise source. Under the assumptions given in Section 2.2, the mean \(m_{e}=0\) and the PSD of the round-off noise source is independent of the frequency, implying:  
\[\sigma_{e}^{2}=P_{ee}\left(\omega_{n}\right). \tag{19}\]  
The PSD of the noise generated at the output of the total filter structure, by the noise source of a (scaled and rounded) stage \(k\) is given by:  
\[P_{vv}\left(\omega_{n}\right)=\left|\overline{G_{k}\left(e^{j\omega_{n}}\right) }\right|^{2}\sigma_{e}^{2} \tag{20}\]  
The variance of this noise is given by:  
\[\sigma_{v}^{2}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}P_{vv}\left(\omega_{n} \right)d\omega_{n}-\left|m_{v}\right|^{2}. \tag{21}\]  
As the noise source has a zero mean value and the stages are linear, \(m_{v}=0\). The PSD of the noise generated at the output of the filter structure, by all round-off noise sources is given by:  
\[P_{m}\left(\omega_{n}\right)=\sigma_{e}^{2}\left[1+\sum_{k=0}^{P-2}\overline{ G_{k}\left(e^{j\omega_{n}}\right)}\right|^{2}\right]. \tag{22}\]  
The output noise variance due to the round-off noise sources of all stages is thus given by (using Equation (11)):  
\[\sigma_{n}^{2}=\sigma_{e}^{2}\left[1+\sum_{k=0}^{P-2}\overline{G_{k}\left(e^{ j\omega_{n}}\right)}\right|^{2}_{2} \tag{23}\]  
As can be seen in Equation (23), the contribution of a noise source of any stage \(k\) to the output noise variance only, depends on the transfer function from this noise source to the output. As this is valid for a general stage \(k\), it is valid for all stages. From this reasoning, it is obvious to order the stages from high power amplification to low power amplification in order to keep all \(\overline{G_{k}\left(e^{j\omega_{n}}\right)}\)\(\left|\overline{G_{k}\left(e^{j\omega_{n}}\right)}\right|^{2}_{2}\) (for any \(k\)) as small as possible. This approach minimizes the output round-off noise variance. However, in Equation (23), the transfer function from a noise source \(k\) to the output is scaled, implying that the optimal ordering must be derived from the scaled stage equations, which is rather inconvenient. Therefore, Equation (23) will be further worked out. It is clear from Figure 3 and Equation (7) that Equation (22) can be written as:  
\[P_{m}\left(\omega_{n}\right)=\sigma_{e}^{2}\left[1+\sum_{k=0}^{P-2}\prod_{m=k+ 1}^{P-1}\left(S_{m}H_{m}\left(e^{j\omega_{n}}\right)\right)\right|^{2}\right]. \tag{24}\]  
Independent of the stage ordering, the following holds for infinity bound and L\({}_{2}\) bound scaling:  
\[\prod_{m=k+1}^{P-1}S_{m}=\frac{\prod_{n=0}^{P-1}S_{m}}{\prod_{n=0}^{k}S_{m}}= \frac{\left|F_{k}\left(e^{j\omega_{n}}\right)\right|_{\alpha}}{\left|H\left(e^ {j\omega_{n}}\right)\right|_{\alpha}} \tag{25}\]  
where, \(\alpha=\infty\) or \(\alpha=2\) respectively.