## 5 The SNR of MFIR Filters
### The SNR and Optimal Ordering for MFIR Filters
#### 5.3.5 SNR Performance of a Linear Phase MFIR filter Approximating the Squared Magnitude  
Response of a Complex-Conjugate Pole Pair Filter  
Figure 15 shows the \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)\)_SNR_  values for pole pairs with magnitude \(r=0.9\) and angles \(\theta\) in the interval [0, \(\pi\)) for forward ordering. Figure 16 shows the values for reverse ordering. In Figure 17, the \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)\)_SNR_  in case of reverse ordering and infinity bound scaling for the pole magnitudes 0.8, 0.85, 0.9 and 0.95 approximated with \(P=5\), 6, 7 and 7 stages respectively, is shown.  
Compared with the non-linear phase approximation, the \(\left(\sigma_{\epsilon}^{\ 2}\right/\sigma_{\ in}^{\ 2})SNR\) is more pole angle dependent. The dips at the pole angles \(\pi/4\), \(\pi/2\) and \(3\pi/4\) are also deeper. The figures show that the ordering of the stages has no effect when L\({}_{2}\) bound scaling is used. In case of absolute or infinity bound scaling, reverse ordering performs better than forward ordering except in the regions where \(\theta=\pi/4,\ \pi/2,\ \text{and}\ 3\pi/4\). However, the width of these regions is pole magnitude dependent. Consequently, in practice both orderings will have to be considered when poles with angles in these regions are to be approximated.  
Unfortunately in the neighborhood of the pole angles \(\theta=\pi/4\), \(\pi/2\) and \(3\pi/4\) the linear phase approximation clearly performs worse than the non-linear phase approximation. The non-optimal performance for pole angles \(\theta=\pi/4\), \(\pi/2\) and \(3\pi/4\) can be explained by using an example. In case \(r=0.9\), \(\theta=\pi/2\) and \(P=9\), \(M_{0}(z)\) is given by (using Equation (5)):  
\[M_{0}\left(z\right)=1+2.045z^{-2}+z^{-4} \tag{62}\]  
and \(M_{8}(z)\) is given by:  
\[M_{8}\left(z\right)=1+1.035*10^{12}z^{-256}+2.678*10^{23}z^{-512}+1.035*10^{12 }z^{-768}+z^{-1024}. \tag{63}\]  
In Section 4.1 it is shown that in case of infinity bound scaling, the stages with the largest peak gains (in the frequency domain), should fall most often in \(G_{k}(z)\) to reduce the output round-off noise variance. In case of absolute bound scaling, the stages with the largest coefficients should fall most often in \(G_{k}(z)\) to reduce the output round-off noise variance.  
It is clear from Equations (62) and (63) that forward ordering (\(M_{8}(z)\) most often in \(G_{k}(z)\) ) is in this case much better than reverse ordering. There is a combined effect involved in the linear phase approximation of complex-conjugate pole pairs with angles in the neighborhood of \(\pi/4\), \(\pi/2\) and \(3\pi/4\):  
* \(1/\,r^{z^{i}}\) factors in Equation (5) can have very large values when \(i\) is large,
* \(\cos\left(2^{i}\theta\right)\) factors in Equation (5) are for most stages close to unity implying the coefficients are not reduced by the cosine functions.  
Even in case of forward ordering, the SNR performance for these angles is not good. Indeed the stages with larger \(i\) values, still have very large coefficients implying that Equation (45) or Equation (46) will never be very small.