In this section, we will study the information-theoretic results of each source-destination channel (i.e. achievable rate and outage probability). Following the reference [5], we first formulate the mutual information of each terminal, then provide the corresponding outage probability and the achievable rate as functions of the network size *m, l* and *s*. Assuming *l* and *s* are linear functions of *m*, at last we provide their numerical analysis when *m* < ∞ as well as their limit evaluation when *m* → ∞. We also assume that perfect channel coding has been performed in each packet, thereby Shannon limitation can be used to denote the information that each terminal can convey per second. For simplicity, we focus on continuous-input, continuous-output channels with Gaussian sources.

### A. Preparations

We begin with introducing some signs for convenient description. We use subscript (*i, d*) to mark "from terminal *i* to the destination".

Since **R**(*i*) = 1 denotes the CSI of terminal *i* is smaller than \sqrt{ln\left(\frac{m}{m-l}\right)} and **R**(*i*) = 1 represents the CSI of terminal *i* is larger than \sqrt{ln\left(\frac{m}{s}\right)}, using the pdf

{p}_{|{h}_{i,d}{|}^{2}}\left(x\right)={e}^{-x}

(5)

we can compute

P\left(R\left(i\right)=1\right)=P\left(|{h}_{i,d}{|}^{2}\le ln\left(\frac{m}{m-l}\right)\right)=\frac{1}{m}

(6)

and

P\left(T\left(i\right)=1\right)=P\left(|{h}_{i,d}{|}^{2}\ge ln\left(\frac{m}{s}\right)\right)=\frac{s}{m}

(7)

where 1 ≤ *i* ≤ *m*.

We use *k*_{1} to denote the number of nonzero elements in **T** and *k*_{2} to mark the number of element 1 in **R**. Since all the channels from terminals to the destination are independent and identically distributed, *k*_{1} and *k*_{2} satisfy Bernoulli distribution and their probability density functions are

P\left({k}_{1}=p\right)=\left(\begin{array}{c}\hfill m\hfill \\ \hfill p\hfill \end{array}\right){\left(\frac{s}{m}\right)}^{p}{\left(1-\frac{s}{m}\right)}^{m-p}

(8)

and

P\left({k}_{2}=q\right)=\left(\begin{array}{c}\hfill m\hfill \\ \hfill q\hfill \end{array}\right){\left(\frac{l}{m}\right)}^{q}{\left(1-\frac{l}{m}\right)}^{m-q}

(9)

respectively, 0 ≤ *p, q* ≤ *m*.

Their expectations are

E\left({k}_{1}\right)=m\times \frac{s}{m}=s

(10)

and

E\left({k}_{2}\right)=m\times \frac{l}{m}=l

(11)

### B. Information-theoretic analysis of FANCC

After making above preparations, we successively analyze mutual information, achievable information rate and outage probability of FANCC.

**1) Mutual Information**: For terminal *i* which gets no error protection in the relay phase, that is, **R**(*i*) = 0. Using the Shannon formula with the instantaneous SNR, the mutual information between terminal *i* and the destination can be directly written as^{b}

{I}_{{R}_{\left(i\right)=0}}=\frac{{k}_{1}}{m+{k}_{1}}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right)

(12)

where the factor \frac{{k}_{1}}{m+{k}_{1}} accounts for: *m* + *k*_{1} time slots are used in FANCC constitutes of *m* slots in the broadcast phase and *k*_{1} slots in the relay phase. Thereby the contribution in broadcast phase is normalized by \frac{{k}_{1}}{m+{k}_{1}} and that in relay phase is \frac{m}{m+{k}_{1}}.

In the relay phase of FANCC, the broadcast-packets from the terminals (**R**(*i*) = 1) are encoded further into relay-packets of the distributed LDGM code and the terminals (**T**(*i*) = 1) convey the different check-sum to the destination. Since the distributed codeword is transmitted by the terminals *i* (**T**(*i*) = 1) through independent channels, the total system mutual information can be written as the sum of the Shannon formula with all the instantaneous SNRs. For fairness, the energy saved in the relay phase of FANCC is added uniformly to the terminals *i* (**T**(*i*) = 1), which renders the total energy consumption the same as that in ANCC. We derive the following expression for the mutual information for terminal *i* (**R**(*i*) = 1):

\begin{array}{ll}\hfill {I}_{{R}_{\left(i\right)=1}}& =\frac{{k}_{1}}{{k}_{1}+m}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right)\phantom{\rule{2em}{0ex}}\\ +\frac{m}{m+{k}_{1}}\times \frac{{k}_{1}}{{k}_{2}}\times \frac{1}{{k}_{1}}\sum _{r=1}^{{k}_{1}}log\left(1+\gamma \frac{m}{{k}_{1}}|{h}_{r,d}{|}^{2}\right)\phantom{\rule{2em}{0ex}}\end{array}

(13)

{I}_{FANCC}\approx \left\{\begin{array}{c}\hfill \frac{s}{m+s}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right),\phantom{\rule{1em}{0ex}}|{h}_{i,d}{|}^{2}>ln\left(\frac{m}{m-l}\right)\hfill \\ \hfill \frac{s}{m+s}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right)+\frac{m}{m+s}\times \frac{1}{l}\sum _{r=1}^{s}log\left(1+\gamma \frac{m}{s}|{h}_{r,d}{|}^{2}\right),\phantom{\rule{1em}{0ex}}0<|{h}_{i,d}{|}^{2}<ln\frac{m}{m-l},|{h}_{r,d}{|}^{2}>ln\frac{m}{s}\hfill \end{array}\right.

(14)

where \frac{{k}_{1}}{{k}_{2}}\times \frac{1}{{k}_{1}}\sum _{r=1}^{{k}_{1}}log\left(1+\gamma \frac{m}{{k}_{1}}|{h}_{r,d}{|}^{2}\right) can be explained like this: the *k*_{1} parity check packets of the network code protect *k*_{2} broadcast packets of terminals *i*(**R**(i) = 1) equally, and the mutual information transmitted by *k*_{1} terminals (**T**(i) = 1) provides uniform contribution to *k*_{2} broadcast packets.

From (12) and (13), we observe that the mutual information in FANCC is not a function of retrieval-set \Re, which is similar to ANCC. For reasonable *R*_{
th
}and *T*_{
th
}, it is always true that the number of broadcast packets which can be allowed to be selected is larger than the fixed number *D*, which guarantees that the resulting LDGM codes are excellent.

From equation (8) and (9), we can know that *k*_{1} and *k*_{2} approach their expectation *s* and *l* when the network size *m* approaches infinity. Therefore, for large networks, *k*_{1} and *k*_{2} could be considered as *s* and *l* approximately. Gathering (6), (7), (10), (11), (12) and (13), the instantaneous mutual information of each terminal can be written as (14).

**2) Achievable Information Rate**: When *m* is large enough, according to its definition, the achievable information rate can be derived as the expectation of the mutual information in (14).

{C}_{FANCC}={E}_{h}\left[{I}_{FANCC}\right]\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}bps\u2215Hz

(15)

Considering

E\left(\sum _{i=1}^{n}f\left({x}_{n}\right)\right)=\sum _{i=1}^{n}E\left(f\left({x}_{n}\right)\right)

(16)

(14) can be simplified as

\begin{array}{ll}\hfill {C}_{FANCC}& \approx {E}_{{h}_{i,d}}\left(\frac{s}{m+s}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right)\right)\phantom{\rule{2em}{0ex}}\\ +{E}_{{h}_{r,d}}\left(\frac{m}{\left(m+s\right)l}\sum _{r=1}^{s}log\left(1+\gamma \frac{m}{s}|{h}_{s,d}{|}^{2}\right)\right)\phantom{\rule{2em}{0ex}}\\ =\frac{s}{m+s}{\int}_{0}^{\infty}log\left(1+\gamma y\right){e}^{-y}dy\phantom{\rule{2em}{0ex}}\\ +\frac{m}{s\left(m+s\right)}\sum _{r=1}^{s}{\int}_{ln\left(\frac{m}{s}\right)}^{\infty}log\left(1+\gamma \frac{m}{s}y\right)\frac{{e}^{-y}}{{\int}_{ln\left(\frac{m}{s}\right)}^{\infty}{e}^{-z}dz}dy\phantom{\rule{2em}{0ex}}\\ =\frac{s}{\left(s+m\right)ln\left(2\right)}Ei\left(\frac{1}{\gamma}\right)exp\left(\frac{1}{\gamma}\right)\phantom{\rule{2em}{0ex}}\\ +\frac{m}{\left(s+m\right)ln\left(2\right)}ln\left(1+\frac{\gamma mln\left(\frac{m}{s}\right)}{s}\right)\phantom{\rule{2em}{0ex}}\\ +\frac{m}{\left(s+m\right)ln\left(2\right)}exp\left(\frac{s}{\gamma m}\right)Ei\left(\frac{s}{\gamma m}+ln\left(\frac{m}{s}\right)\right)\phantom{\rule{2em}{0ex}}\end{array}

(17)

where *Ei* (.) is exponential-integral function defined as:

Ei\left(x\right)=\underset{x}{\overset{\infty}{\int}}\frac{{e}^{-t}}{t}dt,\phantom{\rule{1em}{0ex}}\left(x>0\right)

(18)

The true amazing result comes that the achievable rate of FANCC is not a function of the parameter *l*.

**3) Outage probability**: From equation (14) we can directly derive the outage probability from its definition, as can be seen in(19).

From equation (19), we directly simplify Γ_{1} as

\Gamma \left(R\right)=\left\{\begin{array}{c}\hfill {\Gamma}_{1}=Pr\left[\frac{s}{m+s}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right)<R\right],\phantom{\rule{1em}{0ex}}|{h}_{i,d}{|}^{2}>ln\left(\frac{m}{m-l}\right)\hfill \\ \hfill {\Gamma}_{2}=Pr\left[\frac{s}{m+s}log\left(1+\gamma |{h}_{i,d}{|}^{2}\right)+\frac{m}{\left(m+s\right)l}\sum _{r=1}^{s}log\left(1+\gamma \frac{m}{s}|{h}_{r,d}{|}^{2}\right)<R\right],\phantom{\rule{1em}{0ex}}|{h}_{i,d}{|}^{2}<ln\frac{m}{\left(m-l\right)},|{h}_{r,d}{|}^{2}>ln\frac{m}{s}\hfill \end{array}\right.

(19)

{\Gamma}_{1}=\left\{\begin{array}{c}\hfill 1-exp\left[-\frac{1}{\gamma}\left({2}^{\frac{m+s}{s}R}-1\right)\right],\hfill \\ \hfill 0,\hfill \end{array}\right.\begin{array}{c}\hfill \gamma <\frac{\left({2}^{\left(m+s\right)R\u2215s}-1\right)}{ln\left(m\u2215m-l\right)}\hfill \\ \hfill otherwise\hfill \end{array}

(20)

To compute Γ_{2}, first let us define

{f}_{1}=\frac{s}{m+s}log\left(1+\gamma u\right)

(21)

{f}_{2}=\frac{m}{(m+s)l}\mathrm{log}(1+\gamma \frac{m}{s}{u}^{\prime})]

(22)

Where the pdf of *u* and *u'* is *p*_{
u
}(*x*) = *e*^{-x}, *x* ≥ 0 and {p}_{{u}^{\prime}}\left(x\right)=\frac{m}{s}{e}^{-x},x\le ln\left(\frac{m}{s}\right), respectively. Using the Jacobi law, we can get the pdf of *f*_{1} and *f*_{2}:

\begin{array}{ll}\hfill {p}_{{f}_{1}}\left(y\right)& ={p}_{y}\left({f}_{1}^{-1}\left(y\right)\right)\frac{\partial {f}_{1}^{-1}\left(y\right)}{\partial y}\phantom{\rule{2em}{0ex}}\\ =\frac{\left(s+m\right)ln\left(2\right)}{s\gamma}{2}^{\frac{m+s}{s}y}{e}^{\left(1-{2}^{\frac{m+s}{s}y}\right)\u2215\gamma}\phantom{\rule{2em}{0ex}}\end{array}

(23)

\begin{array}{ll}\hfill {p}_{{f}_{2}}\left(y\right)& ={p}_{{y}^{\prime}}\left({f}_{2}^{-1}\left(y\right)\right)\frac{\partial {f}_{2}^{-1}\left(y\right)}{\partial y}\phantom{\rule{2em}{0ex}}\\ =\frac{\left(s+m\right)lln\left(2\right)}{m\gamma}{2}^{\frac{\left(m+s\right)l}{m}y}{e}^{\frac{s}{m\gamma}\left(1-{2}^{\frac{\left(m+s\right)l}{m}y}\right)}\phantom{\rule{2em}{0ex}}\end{array}

(24)

We then obtain

{\Gamma}_{2}={\int}_{0}^{R}{p}_{{f}_{1}}\left(y\right)\otimes \stackrel{s}{\stackrel{\u23de}{{p}_{{f}_{2}}\left(y\right)\otimes {p}_{{f}_{2}}\left(y\right)\otimes \cdots \otimes {p}_{{f}_{2}}\left(y\right)}}dy

(25)

where ⊗ denotes the convolution operation. From equation (20) and (25), the outage probability of each terminal in FANCC can be written as:

{\Gamma}_{FANCC}\left(R\right)=\frac{l}{m}{\Gamma}_{1}+\frac{m-l}{m}{\Gamma}_{2}

(26)

(25) is hard to simplify. Hence we will numerically analyze it for different parameters, which is presented in Section IV-C.

Next, we analyze its outage probability when the size of network *m* approaches infinity. From the law of large numbers, we get

\begin{array}{ll}\hfill \Omega & =\underset{m\to \infty}{lim}\frac{m}{m+s}\times \frac{1}{l}\sum _{r=1}^{s}log\left(1+\gamma \frac{m}{s}|{h}_{r,d}{|}^{2}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{ms}{l\left(m+s\right)}E\left(log\left(1+\gamma \frac{m}{s}|{h}_{r,d}{|}^{2}\right)\right)\phantom{\rule{2em}{0ex}}\\ =\frac{ms}{l\left(m+s\right)}{\int}_{ln\left(\frac{m}{s}\right)}^{\infty}log\left(1+\gamma \frac{m}{s}y\right){e}^{-y}\frac{m}{s}dy\phantom{\rule{2em}{0ex}}\\ =\frac{{m}^{2}}{l\left(s+m\right)}log\left(1+\frac{\gamma m}{s}ln\left(\frac{m}{s}\right)\right)\frac{s}{m}\phantom{\rule{2em}{0ex}}\\ +\frac{{m}^{2}}{l\left(s+m\right)ln\left(2\right)}exp\left(\frac{s}{\gamma m}\right)Ei\left(\frac{s}{\gamma m}+ln\left(\frac{m}{s}\right)\right)\phantom{\rule{2em}{0ex}}\end{array}

(27)

From (26), we can derive

\begin{array}{ll}\hfill \underset{m\to \infty}{lim}{\Gamma}_{2}& =Pr\left[\frac{s}{s+m}log\left(1+\gamma |{h}_{s,d}{|}^{2}\right)+\Omega <R\right]\phantom{\rule{2em}{0ex}}\\ =1-exp\left[\frac{1}{\gamma}\left(1-{2}^{\vartheta}\right)\right],\phantom{\rule{2em}{0ex}}\end{array}

(28)

where

\begin{array}{ll}\hfill \vartheta & =\frac{s+m}{s}R-\frac{m}{l}log\left(1+\frac{\gamma m}{s}ln\left(\frac{m}{s}\right)\right)\phantom{\rule{2em}{0ex}}\\ -\frac{{m}^{2}}{lsln\left(2\right)}exp\left(\frac{s}{\gamma m}\right)Ei\left(\frac{s}{\gamma m}+ln\left(\frac{m}{s}\right)\right)\phantom{\rule{2em}{0ex}}\end{array}

(29)

Therefore, we can get

\underset{m\to \infty}{lim}{\Gamma}_{FANCC}\left(R\right)=\frac{l}{m}{\Gamma}_{1}+\frac{m-l}{m}\underset{m\to \infty}{lim}{\Gamma}_{2}

(30)

### C. Numerical results

To demonstrate the superiority of FANCC, in this subsection we numerically evaluate the information-theoretic results of the two protocols.

The achievable rates of FANCC (red,solid) and ANCC (black,solid) are demonstrated in Figure 5. Since *C*_{
FANCC
}is irrelevant to *l*, we provide the achievable rates of FANCC with different *s*. From the figure, FANCC outperforms ANCC and *C*_{
FANCC
}decreases with the increase of parameter *s*. When *s* approaches *m, C*_{
FANCC
}gradually gets close to *C*_{
ANCC
}. This is in accord with our prediction: the terminals which have better terminal-destination channel can transmit more information to improve total system capacity. To verify our theoretical results, we present the capacity simulations with corresponding parameters *s, l*, which is plotted by dotted line. From the simulation results, we can see that the simulations is very close to the theoretical results. Furthermore, we also give the simulations with different *m* (m = 30,60,120) to analyze the impact of network size on capacity. Even the network size *m* is equal to 30, the simulated capacity is close to the theoretical capacity. Therefore, our theoretical derivation is inconsistent with the actual networks.

Since the outage of FANCC is a function of *m, l* and *s*, we analyze the their contributions to its outage probability in Figure 6 and Figure 7. Figure 6 shows the outage probabilities with different *l* and *s* = *m* (l=\frac{7}{8}m: black square; l=\frac{5}{6}m: blue diamonds). The curve of ANCC is plotted by red round line. As expected, the outage improves as *l* decreases, which is attributed to the fact that the broadcast-packets in the worse channels receive more error protection as *l* decreases. We take *m* = 48 as an example, at the outage probability of 10^{-7}, FANCC with l=\frac{7}{8}m outperforms ANCC about 2dB and FANCC with l=\frac{5}{6}m gets 2.5dB improvement.

Figure 7 demonstrates that the outage probabilities with different *s* and l=\frac{3}{4}m (*s* =*m*: blue diamonds; s=\frac{7}{8}m: black square; s=\frac{5}{6}m: cyan round). The corresponding curve of ANCC is plotted by red star line. From the numerical results, we also discover the similar phenomenon that, whether the network size is finite or not, the outage probability of FANCC improves with the increase of *s*. It can be explained that the impact of multiplexing gain takes dominant place for l=\frac{3}{4}m. When *m* = 24, we can see that FANCC with s=\frac{5}{6}m outperforms ANCC about 0.3dB at the outage probability of 10^{-7}. The advantage is expanded to about 1dB for s=\frac{7}{8}m and about 1.7dB for *s* = *m*.

From Figure 5 and Figure 6, the outage probability improves with the increase of network size *m*, which is accordance with the fundamental principle that user collaboration can improve the system performance. Moreover, numerical results show that FANCC noticeably outperforms ANCC whether *m* is finite or not. For FANCC with *s* = *m*, l=\frac{5}{6}m in Figure 6, *m* = 24 needs about 4.5dB, *m* = 48 requires about 3dB and *m* = ∞ only demand about -0.2dB, to reach the outage probability of 10^{-7}.

### D. simulation results

After we discuss the effectiveness of FANCC through theoretical analysis such as the capacity and outage probability, the simulation results are analyzed as follows.

Since each packet can use any practical channel code, different channel codes result in different bit error rate (BER) of the whole system. In order to evaluate the impact of user cooperation, we only focus on the performance after network coding and ignore the channel coding used in each packet. Thereby we assume each terminal transmits one bit for itself and relays one bit for others. For comparison purpose, we consider different terminals communicating with a common destination, that is to say, the network sizes, such as *m* = 100, 200, 300. We also assume that every channel, either between terminals or from terminal to destination, has the same SNR. Belief Propagation (BP) algorithm with iteration time 31 is adopted at the destination. For each network size, we choose three fixed number *d* = 5,6,7 and two assembles of the parameters *s* and *l* are selected for every *d*. Therefore, totally we have performed 18 scenario simulations and the results of ANCC is plotted in red while those of FANCC is drawn in black. In this subsection, circle represents *d* = 5, star means *d* = 6 and diamond indicates *d* = 7; The average *E*_{
b
}*/N*_{0} of channels from terminals to destination is plotted in the **X** axis. For fairness, the total energy consumption of FANCC and ANCC for cooperation round are all the half of that in the no-cooperation scheme. The transmission energy is normalized for each terminal for power reduction in FANCC. The **Y** axis denotes the FER averaged over different packets from different terminals.

In Figure 8, the average FER of FANCC with *m* = 100, *l* = 80, *s* = 100 and ANCC is demonstrated. From the figure, we observe that, FANCC has a little worse performance than ANCC at low *E*_{
b
}/*N*_{0} while it has better performance at a little high *E*_{
b
}/*N*_{0}. The larger the fixed number *d*, The greater the superiority of FANCC is. We can see that,FANCC achieves about 1dB improvement over ANCC for *d* = 5 while it gets about 2dB coding gain for *d* = 7.

In Figure 9, we select an assemble of parameters, *s* = *l* = 80 for *d* = 5, 6, 7 to evaluate the effectiveness of FANCC in 100-terminal network. The excitement comes from that FANCC not only has better performance whether *E*_{
b
}/*N*_{0} is low or high but also it saves 10 percent of communication round time than ANCC. The advantage of FANCC is less than 1dB.

Figure 10 and Figure 11 show a comparison of the simulated FER of FANCC and ANCC when the network size is 200. *s* = 200, *l* = 170 and *s* = *l* = 170 are used for FANCC in Figure 10 and Figure 11, respectively. From Figure 10, it can observed that FANCC outperforms ANCC at high *E*_{
b
}/*N*_{0}. However, the opposite is true at low *E*_{
b
}/*N*_{0} but the gap is small. As *d* increases, the intersection of two FER curves moves to the lower right side and the performance gain increases when *E*_{
b
}/*N*_{0} crosses the intersection. For *d* = 5, the intersection coordinates is (3*dB*, 4 × 10^{-2}) and for *d* = 7, the coordinates is (6*dB*, 7 × 10^{-4}). The coding gain is about 1dB.

From Figure 11, FANCC with *s* = *l* = 170 has achieved much better performance over ANCC for *m* = 200. Compared with ANCC, the coding gain of FANCC is becoming smaller and smaller as FER decreases. For *d* = 7, FANCC achieves about 1.5dB coding gain than ANCC at the FER of 10^{-3} while less than 1dB at the FER of 2 × 10^{-5}.

The average FER of FANCC and ANCC for *m* = 300 is further compared in Figure 12 and Figure 13. In Figure 12, we choose *l* = 300, *s* = 250 for FANCC, where the destination only feedbacks 1bit per terminal, that is to say, 300-bit indication message, in the indication phase. Seen from that, FANCC significantly outperforms ANCC at the same FER, which is different from *m* = 100 and *m* = 200. Furthermore, FANCC gets about 2dB coding gain over ANCC at every *d*.

At last, the parameters *l* = *s* = 250 is selected in FANCC for *m* = 300, which reduces 50 time slots than ANCC in one communication round. Figure 13 compares the proposed FANCC with ANCC. In this simulation, the destination broadcasts 600-bit message in the indication phase. From the simulation results, our proposed FANCC achieves about 2dB coding gain compared with ANCC.

From above results, compared with ANCC, FANCC can improve the system performance. However, the parameters *l* and *s* used in the simulation are selected approximately and there may be better parameters. It is also meaningful and interesting to establish a simple model of searching the best *s* and *l* for FANCC, and we will continue this work in the future research.