## 5 The SNR of MFIR Filters
### The SNR and Optimal Ordering for MFIR Filters  
#### 5.3.1 General MFIR SNR Expressions  
In case of forward ordering, Equations (39-41) become:  
\[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[1+\frac{\sum\limits_{i=0}^{p-2}\left(\left\|F_{i}\left(e^{\,/a_{n}} \right)\right\|_{2}^{2}\left\|G_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2} \right)}{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}\right]}, \tag{55}\]  
\[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]  
\[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{abs}=\frac{ 1}{\left[\frac{\left(\sum\limits_{\epsilon=0}^{p}\left|m\left(n\right)\right| \right)^{2}}{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum \limits_{i=0}^{p-2}\left(\left(\sum\limits_{\epsilon=0}^{p_{i}}\left|f_{i} \left(n\right)\right|\right)^{2}\left\|G_{i}\left(e^{\,/a_{n}}\right)\right\|_{2 }^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}\right]}. \tag{57}\]  
Here, \(G_{i}\left(e^{\,/a_{n}}\right)\) and \(F_{i}\left(e^{\,/a_{n}}\right)\) are given by Equations (47) and (49) respectively. The value of \(p_{i}\) is given in Table 2 for forward ordering. \(m(n)\) are the impulse response samples of the total MFIR filter. The value \(p\) in \(\left(\sum\limits_{\epsilon=0}^{p}\left|m\left(n\right)\right|\right)^{2}\) in Equation (57) is the length of the total MFIR filter impulse response and can be found by setting \(i=P-1\) in Table 2 for forward ordering. In case of reverse ordering, Equations (39-41) become:  
\[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[1+\frac{\sum\limits_{i=1}^{P-1}\left(\left\|F_{i}\left(e^{\,/a_{n}} \right)\right\|_{2}^{2}\left\|G_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2} \right)}{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}\right]}, \tag{58}\]  
\[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=1}^{P-1}\left( \left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e^{\,/a_ {n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2 }^{2}}\right]}, \tag{59}\]\[\left(\frac{\sigma_{\varepsilon}^{2}}{\sigma_{in}^{3}}.SNR\right)_{abs}=\frac{1}{ \left[\frac{\left(\sum\limits_{n=0}^{p}\left|m\left(n\right)\right|\right)^{2}} {\left|M\left(e^{i\alpha_{n}}\right)\right|^{2}}+\frac{\sum\limits_{n=0}^{p-1} \left[\left(\sum\limits_{n=0}^{p}\left|f_{i}\left(n\right)\right|\right)^{2} \left|G_{i}\left(e^{i\alpha_{n}}\right)\right|^{2}_{2}\right)}{\left|M\left(e^{ i\alpha_{n}}\right)\right|^{2}_{2}}\right]}. \tag{60}\]  
Here, \(G_{i}\left(e^{i\alpha_{n}}\right)\) and \(F_{i}\left(e^{i\alpha_{n}}\right)\) are given by Equations (50) and (53) respectively. The value of \(p_{i}\) is given in Table 2 for reverse ordering. The value \(p\) in \(\left(\sum\limits_{n=0}^{p}\left|m\left(n\right)\right|\right)^{2}\) in Equation (60) is the length of the total MFIR filter impulse response and can be found by setting \(i=0\) in Table 2 for reverse ordering. In order to keep the text more readable, the figures of the several calculation results of Equations (55-60) are grouped together at the end of the section. Table 3 gives an overview of the results. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Approximation of** & **Scaling type** & **Ordering** & **Range** & **Figure** \\ \hline Real pole & L\({}_{2}\) & Forward and Reverse & \(\left|\lambda\right|\in\left[0.1,\ 1\right)\) & Figure 6 \\ Real Pole & Abs and Inf.