## 5 The SNR of MFIR Filters
### The SNR and Optimal Ordering for MFIR Filters
#### 5.3.1 General MFIR SNR Expressions
& Forward and Reverse & \(\left|\lambda\right|\in\left[0.1,\ 1\right)\) & Figure 7 \\ Real Pole & L\({}_{2}\) & Forward and Reverse & \(\left|\lambda\right|\in\left[0.1,\ 1\right)\) & Figure 8 \\ Linear Phase & Abs and Inf. & Forward and Reverse & \(\left|\lambda\right|\in\left[0.1,\ 1\right)\) & Figure 9 \\ Linear Phase & Abs and Inf. & Forward and Reverse & \(\left|\lambda\right|\in\left[0.1,\ 1\right)\) & Figure 9 \\ Compl. Conj. & L\({}_{2}\) & Forward and Reverse & \(r=0.9\) & Figure 10 \\ Pole pair & Inf & Forward and Reverse & \(r=0.9\) & Figure 11 \\ Compl. Conj. & Abs & Forward and Reverse & \(r=0.9\) & Figure 12 \\ Compl. Conj. & & & \(\theta\in\left[0,\ \pi\right)\) & Figure 13 \\ Compl. Conj. & & & \(r=0.9\) & Figure 14 \\ Pole pair & Abs & Forward and Reverse & \(\theta\in\left[0,\ \pi\right)\) & Figure 15 \\ Compl. Conj. & & & \(r=0.9\) & Figure 16 \\ Compl. Conj. & & & \(r=0.8\); \(0.85\); \(0.9\); \(0.95\) & Figure 17 \\ Compl. Conj. & & & \(r=0.9\) & Figure 18 \\ Compl. Conj. & & & \(r=0.9\) & Figure 19 \\ Pole pair & L\({}_{2}\), Inf, Abs & Forward and Reverse & \(\theta\in\left[0,\ \pi\right)\) & Figure 14 \\ Compl. Conj. Pole & & & \(r=0.9\) & Figure 15 \\ pair, Lin Phase & L\({}_{2}\), Inf, Abs & Forward & \(\theta\in\left[0,\ \pi\right)\) & Figure 16 \\ Compl. Conj. Pole & & & \(r=0.8\); \(0.85\); \(0.9\); \(0.95\) & Figure 17 \\ \(\theta\in\left[0,\ \pi\right)\) & & & \(r=0.9\) & Figure 18 \\ Compl. Conj. Pole & & & \(r=0.9\) & Figure 19 \\ Compl. Conj. Pole & & & \(r=0. **Figure 8.**\(\left(\sigma_{e}^{2}\big{/}\sigma_{in}^{2}\right)SNR\) of MFIR filters approximating the squared magnitude response of real poles \(\left|\lambda\right|\) in the interval [0.1, 1) in case of L\({}_{2}\) bound scaling. (The MFIR forward and MFIR reverse results are superimposed). **Figure 9.**\(\left(\sigma_{e}^{2}\big{/}\sigma_{in}^{2}\right)SNR\) of MFIR filters approximating the squared magnitude response of real poles \(\left|\lambda\right|\) in the interval [0.1, 1) in case of absolute and/ or infinity bound scaling. Figure 10: \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)SNR\) for complex-conjugate pole pairs with magnitude 0.9 and L\({}_{2}\) bound scaling; The MFIR approximation uses \(P=7\) stages. (The MFIR forward and MFIR reverse results are superimposed). Figure 11: \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)SNR\) for complex-conjugate pole pairs with magnitude 0.9 and infinity bound scaling; The MFIR approximation uses \(P=7\) stages. \[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]0 \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)\mathit{SNR}\) for complex-conjugate pole pairs with magnitude 0.9 and absolute bound scaling; The MFIR approximation uses \(P=7\) stages. \[\left(\frac{\sigma_{\epsilon}^{2}}{\sigma_{in}^{2}}.\,SNR\,\right)_{L=}=\frac{ 1}{\left[\frac{\left\|M\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}{\left\|M \left(e^{\,/a_{n}}\right)\right\|_{2}^{2}}+\frac{\sum\limits_{i=0}^{p-2} \left(\left\|F_{i}\left(e^{\,/a_{n}}\right)\right\|_{2}^{2}\left\|G_{i}\left(e ^{\,/a_{n}}\right)\right\|_{2}^{2}\right)}{\left\|M\left(e^{\,/a_{n}}\right) \right\|_{2}^{2}}\right]}, \tag{56}\]1 \(\left(\sigma_{e}^{2}\middle/\sigma_{in}^{2}\right)\mathit{SNR}\) in function of pole magnitude \(r\) and pole angle \(\theta\), in case of infinity bound scaling and reverse ordering. (surface = MFIR; discrete curves = IIR filter).