# Analysis of TDMP Algorithm of LDPC Codes Based on Density Evolution and Gaussian Approximation
## 4 Calculation of TDMP Normalization Factor Based on Density Evolution
### Normalization Factor Calculation  
A TDMP simplification algorithm without non-linear functions is given in Reference [30], which can be expressed as  
\[u=\prod_{j=1}^{dc-1}\text{sgn}(v_{j})\times(\min_{j=1,2-dc-1}|v_{j}|+r(|v_{j}|)), \tag{21}\]  
where \(r|v_{j}|=\ln(1+e^{-|v_{j}|})\approx\ln 2-0.5|v_{j}|\). The update of the check node information can be simplified to  
\[u=\prod_{j=1}^{dc-1}\text{sgn}(v_{j})\times(\min_{j=1,2-dc-1}|v_{j}|+\ln 2-0.5 \cdot|v_{j}|), \tag{22}\]  
In order to improve the decoding performance of the TDMP simplification algorithm, the normalization factor \(\alpha\) is used for improving the calculation of the check node information. Its calculation formula is as shown in Equation (23). The following is the calculation of the normalization factor \(\alpha\) based on density evolution [24]. \[u=\prod_{j=1}^{dc-1}\text{sgn}(v_{j})\times\alpha(\min_{j=1,2-dc-1}|v_{j}|+\ln 2 -0.5\cdot|v_{j}|), \tag{23}\]  
\[ \begin{array}{l}L=\prod_{j=1}^{dc-1}\text{sgn}(v_{j})\times(\min_{j=1,2-dc-1 }|v_{j}|+(\ln 2-0.5\cdot|v_{j}|))\\ E(L)=E(\min_{j=1,2-dc-1}|v_{j}|+(\ln 2-0.5|v_{j}|))=E(L_{1})+E(L_{2})\end{array}  \tag{24}\]  
To facilitate calculation and illustration, variables \(L_{1}\) and \(L_{2}\) are introduced to represent \(\min_{j=1,2,..dc-1}|v_{j}|\) and \(\ln 2-0.5|v_{j}|\), respectively. \[E(L_{1})=E(\min_{j=1,2-dc-1}|v_{j}|), \tag{25}\]  
\[E(L_{2})=E(\ln 2-0.5\cdot|v_{j}|), \tag{26}\]  
\[E(L_{3})=E(|u|)=2\text{tanh}^{-1}(\prod_{j=1}^{dc-1}\text{tanh}\frac{|v_{j}|}{ 2}), \tag{27}\]  
\[\alpha=\frac{E(L_{3})}{E(L_{1})+E(L_{2})}, \tag{28}\]  
According to the knowledge of probability theory, the probability distribution of \(|L_{1}|\) is known [24]. \[Pr(L_{1}>v)=Pr(\min(V_{1},V_{2},\cdots,V_{dc-1})>v)=Pr(V_{1}>v,V_{2}>v,\cdots, V_{dc-1}>v)=[Pr(V_{1}>v)]^{dc-1}, \tag{29}\]  
Thus, its expectation is \[ \begin{array}{l}E(L_{1})=\int_{0}^{\infty}Pr(L_{>}>v)dv=\int_{0}^{\infty}\left[ Pr(V_{1}>v)\right]^{dc-1}dv\\ =\int_{0}^{\infty}\left[1-F(v)\right]^{dc-1}dv=\int_{0}^{\infty}\left[1-(1-Q( \frac{\mu+v}{\sigma})-Q(\frac{\mu-v}{\sigma}))\right]^{dc-1}dv\\ =\int_{0}^{\mu}\left[1-Q(\frac{\mu-v}{\sigma})+Q(\frac{\mu+v}{\sigma})\right]^ {dc-1}dv+\int_{\mu}^{\infty}\left[Q(\frac{v-\mu}{\sigma})+Q(\frac{\mu+v}{ \sigma})\right]^{dc-1}dv\\ \approx\int_{0}^{\mu}\left[1-Q(\frac{\mu-v}{\sigma})+Q(\frac{\mu+v}{\sigma}) \right]^{dc-1}dv\end{array}  \tag{30}\]  
Since \(E(\ln 2-0.5|v_{j}|)=\ln 2-0.5E(|v_{j}|)\), the PDF is \(f(v)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(v-\mu)^{2}}{2\sigma^{2}}}\), where \(\sigma^{2}=2\mu\). By definition of the mean we have  
\[E(|v_{j}|)=\int_{-\infty}^{+\infty}|v_{j}|f(v_{j})dv_{j}=\int_{-\infty}^{+ \infty}|v_{j}|\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(v_{j}-\mu)^{2}}{2\sigma^{2 }}}dv_{j}=2\int_{0}^{+\infty}v_{j}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(v_{j} -\mu)^{2}}{2\sigma^{2}}}dv_{j}\]  
which using integral properties can be written as  
\[ \begin{array}{l}E(|v_{j}|)=2\int_{0}^{+\infty}\left(v_{j}+\mu\right)\frac{1 }{\sigma\sqrt{2\pi}}e^{-\frac{v_{j}^{2}}{2\sigma^{2}}}dv_{j}=2(\int_{0}^{+ \infty}v_{j}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{v_{j}^{2}}{2\sigma^{2}}}dv_{ j}+\mu\int_{0}^{+\infty}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{v_{j}^{2}}{2 \sigma^{2}}}dv_{j})\\ =2(-\sigma\int_{0}^{+\infty}\frac{1}{\sqrt{2\pi}}(-\frac{v_{j}^{2}}{\sigma^{2 }})e^{-\frac{v_{j}^{2}}{2\sigma^{2}}}dv_{j}+\mu\int_{0}^{+\infty}\frac{1}{ \sigma\sqrt{2\pi}}e^{-\frac{v_{j}^{2}}{2\sigma^{2}}}dv_{j})\\ =2(-\frac{d}{\sqrt{2\pi}}\cdot e^{-\frac{v_{j}^{2}}{2\sigma^{2}}}|_{0}^{+ \infty}+\frac{\mu}{2})=2(\frac{d}{\sqrt{2\pi}}+\frac{\mu}{2})=\mu+\sigma\sqrt {\frac{2}{\pi}}\end{array}  \tag{31}\]  
And then the calculation of \(E(L_{3})\).