## 3 The Analysis  
### Estimating the Error Probability  
First, we consider any transmission link among several duplications. It has been proven that, if the generated information bits (with length of \(k\) bits) is first, encoded into a codeword with length of \(n\) symbols and then transmitted through a wireless channel at SNR of \(P\), the channel transmission rate is upper bounded by Shannon capacity as  
\[C(P)=\log_{2}(1+P). \tag{2}\]  
Note that, the upper bound given by Equation (2) requires Gaussian coding and infinite coding length (i.e., \(n\rightarrow\infty\)). Therefore, when the coding length is finite, it is not possible to realize reliable transmission with arbitrarily small error probability. In [12; 13], the maximal rate achievable with error probability \(\epsilon\) and finite coding length \(n\) is approximated. For the wireless channel, its equivalent baseband model is a complex Gaussian channel, which is composed of two real Additive White Gaussian Noise (AWGN) channels. Then, based on Theorem 4 in [12], the transmission rate under given error probability \(\epsilon\), and the finite coding length \(n\) is upper bounded by  
\[R_{1}(n,P,\epsilon)=C(P)-\sqrt{\frac{V(P)}{n}}Q^{-1}(\epsilon)+\frac{\log_{2} (n)}{n}, \tag{3}\]  
where  
\[\begin{split} Q(x)&=\int_{x}^{\infty}\frac{1}{ \sqrt{2\pi}}e^{-t^{2}/2}dt,\\ V(P)&=\frac{P(P+2)}{(P+1)^{2}\ln^{2}2}.\end{split} \tag{4}\]  
One important assumption in Equation (3) is Gaussian coding. However, in practical systems, a constellation diagram with limited points is usually used, such as the widely  
Figure 1: System Model.  
used M-QAM modulation. Then, the maximal rate under infinite coding length M-QAM modulation is no longer \(C(P)\) defined in Equation (2), the following upper bound should be used instead  
\[I(P,M)=\log_{2}M-\frac{1}{M\pi}\sum_{i=1}^{M}\int e^{-\|y-\sqrt{P}_{X_{i}}\|^{2 }}\times\left(\sum_{k=1}^{M}e^{-|y-\sqrt{P}_{X_{i}}|^{2}-|y-\sqrt{P}_{X_{k}}|^{ 2}}\right)\mathrm{d}y. \tag{5}\]  
However, there is no closed form for Equation (5). Fortunately, the authors of [16] found an approximation based on multi-exponential delay curve fitting (M-EDCF) as  
\[I^{\prime}(P,M)\approx\log_{2}M\times\left(1-\sum_{j=1}^{k_{M}}a_{j}^{(M)}e^{- b_{j}^{(M)}P}\right), \tag{6}\]  
where the fitting coefficients for M-QAM are summarized in Table 1.  
Then, for finite coding length and finite constellation, the maximal achievable transmission rate can be approximated as  
\[R_{2}(n,P,\epsilon,M)=I^{\prime}(P,M)-\sqrt{\frac{V(P)}{n}}Q^{-1}(\epsilon)+ \frac{\log_{2}(n)}{n}. \tag{7}\]  
Based on the analysis above, the packet error probability for a single link can be derived by the following lemma.  
**Lemma 1**.: _Consider an information packet with length of \(k\) bits transmitted via a wireless link under channel condition \(P\) and transmission rate \(R_{2}\) using coding and modulation scheme with \(n\) and \(M\), the packet error probability can be obtained by_  
\[\epsilon(n,P,M)=Q\bigg{(}\sqrt{\frac{n}{V(P)}}\bigg{(}I^{\prime}(P,M)-R_{2}+ \frac{\log_{2}(n)}{n}\bigg{)}\bigg{)}. \tag{8}\]  
Note that, in the analysis of this section, the wireless channel is supposed to be a block fading channel, which means the fading in different time-frequency resource is the same for all \(n\) symbols in a transmission block. When the fadings over different symbols are also different, we only need to update the expressions for channel mutual information, i.e., \(C(P)\) and \(I^{\prime}(P,M)\), the analysis in the remaining part of this paper still holds.