## 5 The SNR of MFIR Filters
### The SNR and Optimal Ordering for MFIR Filters
#### 5.3.1 General MFIR SNR Expressions
\[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]2 \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)SNR\) of an MFIR filter approximating a complex-conjugate pole pair with magnitude 0.9, and \(P=7\). \[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]3 \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)SNR\) of a linear phase MFIR filter approximating the squared magnitude response of a complex-conjugate pole pair with magnitude 0.9 and \(P=7\) in forward ordering. \[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]4 \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)\mathit{SNR}\) of a linear phase MFIR filter approximating the squared magnitude response of a complex-conjugate pole pair with magnitude 0.9 and \(P\) = 7 in reverse ordering. \[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]5 \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)\mathit{SNR}\) of a linear phase MFIR filter in function of the pole angle and magnitude in case of infinity bound scaling and reverse ordering. As explained in Section 4.2, the ordering has no impact on the noise performance when L\({}_{2}\) bound scaling is used. Absolute bound scaling and infinity bound scaling yield the same noise performance and reverse ordering is better (max. 4 dB) than forward ordering. L\({}_{2}\) bound scaling always yields a better noise performance than absolute and infinity bound scaling. In case of absolute or infinity bound scaling, the MFIR filter has a better signal to noise performance than the approximated real pole filter for \(|\lambda|\geq 0.861\). For L\({}_{2}\) bound scaling, only for \(|\lambda|\geq 0.925\) the MFIR filter has a better performance than the approximated IIR filter. In case \(|\lambda|<0.9\), the worst-case differences between the MFIR filter and the IIR filter are 3.6 dB for absolute (and infinity) bound scaling and 5 dB for L\({}_{2}\) bound scaling. This is, however, not dramatic because in the range of interest (\(|\lambda|>0.9\)), in general the MFIR filter has a better signal to noise performance than its corresponding IIR filter (except for the small region between \(|\lambda|=0.9\) and \(|\lambda|=0.925\) for L\({}_{2}\) bound scaling where the difference is maximum 1.2 dB in favor of the IIR filter). The more \(|\lambda|\) approaches the unit circle, the better the noise performance of the MFIR filter compared to the IIR filter. For example, for \(|\lambda|=0.999\), the MFIR approximation is 27 dB better for absolute (and infinity) bound scaling, for L\({}_{2}\) bound scaling the MFIR filter is 17 dB better than the approximated IIR filter.