## 4 The Ordering of the Stages
### The Signal to Round-off Noise Ratio  
The round-off noise performance of a filter may not be correctly evaluated by only analyzing the round-off noise. A more reliable result is obtained by calculating the Signal to round-off noise ratio (SNR). The signal to noise ratios (when using the previously discussed scaling methods) are investigated in this section.  
The discrete input signal of the filter is indicated by \(x(n)\) and the discrete output signal by \(y(n).\) The variance of the discrete input signal \(x(n)\) is indicated by \(\sigma_{\text{\tiny{$\mathit{in}$}}}^{2}.\) The variance of the discrete output signal \(y(n)\) of the filter will be indicated by \(\sigma_{\text{\tiny{$\mathit{out}$}}}^{2}.\) All calculations presented in this section are based on the conditions that the input signal \(x(n)\) is zero mean and has a constant, frequency independent, Probability Density Function (PDF).  
In case the input signal is a wide sense stationary random signal with uniform PDF and variance \(\sigma_{in}^{2}\), the output signal variance of the filter is given by:  
\[\sigma_{out}^{2}=\frac{\sigma_{in}^{2}}{2\pi}\int\limits_{-\pi}^{+\pi}\left| \overline{H\left(e^{i\omega_{n}}\right)}\right|^{2}d\omega_{n} \tag{33}\]  
\[\sigma_{out}^{2}=\sigma_{in}^{2}\left\|\overline{H\left(e^{i\omega_{n}}\right) }\right\|_{2}^{2} \tag{34}\]  
\[\sigma_{out}^{2}=\sigma_{in}^{2}\sum\limits_{n}\left|\overline{h\left(n\right) }\right|^{2} \tag{35}\]  
where \(\sum\limits_{n}\left|\overline{h\left(n\right)}\right|^{2}\) is the sum of the squared impulse response samples of the (scaled and rounded) filter.  
* In case of L\({}_{2}\) or L\({}_{\infty}\) bound scaling, Equation (34) can be written as (using Equation (10) or Equation (13) as appropriate): \[\sigma_{out}^{2}=\sigma_{in}^{2}\frac{\left|\overline{H\left(e^{i\omega_{n}} \right)}\right|_{2}^{2}}{\left|\overline{H\left(e^{i\omega_{n}}\right)} \right|_{\alpha}^{2}},\] (36) where \(\alpha=2\) or \(\alpha=\infty\), respectively. Combining Equations (36) and (27) yields. \[SNR=\frac{\sigma_{out}^{2}}{\sigma_{n}^{2}}=\frac{\sigma_{in}^{2}}{\sigma_{e}^{ 2}}\frac{\left|\overline{H\left(e^{i\omega_{n}}\right)}\right|_{2}^{2}}{\left| \overline{H\left(e^{i\omega_{n}}\right)}\right|_{\alpha}^{2}+\sum\limits_{k=0 }^{P-2}\left(\left|\overline{F_{k}\left(e^{i\omega_{n}}\right)}\right|_{\alpha }^{2}\left|\overline{G_{k}\left(e^{i\omega_{n}}\right)}\right|_{2}^{2}\right) \right|}.\] (37)
* In case of absolute bound scaling, the SNR is given by: \[SNR=\frac{\sigma_{out}^{2}}{\sigma_{n}^{2}}=\frac{\sigma_{in}^{2}}{\sigma_{e} ^{2}}\frac{\left|\overline{H\left(e^{i\omega_{n}}\right)}\right|_{2}^{2}}{ \left|\overline{\left(\sum\limits_{n=0}^{P-2}\left|\overline{H\left(n\right) }\right|_{2}^{2}\right)+\sum\limits_{k=0}^{P-2}\left(\left(\sum\limits_{n=0}^ {P_{k}}\left|\overline{f_{k}\left(n\right)}\right|\right)^{2}\left|\overline{G _{k}\left(e^{i\omega_{n}}\right)}\right|_{2}^{2}\right)}\right|}.\] (38)  
In Equations (37) and (38) \(\sigma_{e}^{2}\) is dependent on the type of rounding used and the number of bits that are used to quantize the signal in the filter structure. \(\sigma_{in}^{2}\) depends on the input signal. Quantization and input signal independent factors are given by:  
\[\left(\frac{\sigma_{e}^{2}}{\sigma_{in}^{2}}.SNR\right)_{L2}=\frac{1}{\left[1+ \frac{\sum\limits_{k=0}^{P-2}\left(\left|\overline{F_{k}\left(e^{i\omega_{n}} \right)}\right|_{2}^{2}\left|\overline{G_{k}\left(e^{j\omega_{n}}\right)} \right|_{2}^{2}\right)}{\left|\overline{H\left(e^{i\omega_{n}}\right)}\right|_ {2}^{2}}\right]} \tag{39}\]  
Notice that the equations derived in this section are generally valid for any filter cascade (with one noise source at the output of each filter stage) and not only for MFIR structures.