Multipleaccess relaying with network coding: iterative network/channel decoding with imperfect CSI
 XuanThang Vu^{1}Email author,
 Marco Di Renzo^{1} and
 Pierre Duhamel^{1}
https://doi.org/10.1186/168761802013170
© Vu et al.; licensee Springer. 2013
Received: 8 April 2013
Accepted: 23 October 2013
Published: 11 November 2013
Abstract
In this paper, we study the performance of the fournode multipleaccess relay channel with binary Network Coding (NC) in various Rayleigh fading scenarios. In particular, two relay protocols, decodeandforward (DF) and demodulateandforward (DMF) are considered. In the first case, channel decoding is performed at the relay before NC and forwarding. In the second case, only demodulation is performed at the relay. The contributions of the paper are as follows: (1) two joint network/channel decoding (JNCD) algorithms, which take into account possible decoding error at the relay, are developed in both DF and DMF relay protocols; (2) both perfect channel state information (CSI) and imperfect CSI at receivers are studied. In addition, we propose a practical method to forward the relays error characterization to the destination (quantization of the BER). This results in a fully practical scheme. (3) We show by simulation that the number of pilot symbols only affects the coding gain but not the diversity order, and that quantization accuracy affects both coding gain and diversity order. Moreover, when compared with the recent results using DMF protocol, our proposed DF protocol algorithm shows an improvement of 4 dB in fully interleaved Rayleigh fading channels and 0.7 dB in block Rayleigh fading channels.
Keywords
1 Introduction
In cooperative communications systems, idle nodes have the capability to relay information from other active nodes. Hence multiple copies of the same signal can reach a given destination through independent fading channels, which result in potential spatial diversity gains. However, diversity gains are usually achieved with some loss in system throughput[1, 2].
Network Coding (NC) has recently been introduced as a capacity–achieving routing scheme where intermediate network nodes are allowed to combine several input packets into one output packet[3]. Recent results have shown that NC can also provide improved performance and energy efficiency compared with conventional network routing techniques[4]. However, besides the many potential advantages and applications of NC over classical routing, the NC principle is not without limitations. A fundamental problem that NC needs to take into account over lossy (e.g., wireless) networks is the socalled error propagation problem: corrupted packets injected by some intermediate nodes may propagate through the network until the destination, and might render impossible to decode the original information[5, 6]. It is shown in[5, 6] that error propagation can dramatically degrade performance and reduce the diversity order of cooperative networks.
Among the solutions that are currently being investigated to counteract the error propagation problem[4], joint network channel decoding (JNCD) is gaining a growing interest[7]. The idea behind JNCD is the exploitation of the inherent redundancy of network and channel codes. In[7, 8], it has been shown that, compared to conventional distributed turbo coding and separate network and channel decoding, JNCD can improve the performance of canonical twoway and multipleaccess relay channels. However, these results assume that only correct packets are forwarded from the relay to the destination. Recently, various relaying protocols have addressed the error propagation problem in cooperative communications. In[9–11], the authors propose soft relaying protocols. In softrelaying, the relay does not take any hard decision of the input signals. Instead, the relay computes loglikelihood ratios (LLR) of networkcoded bits and reencodes them using a soft encoder. The relay then forwards encoded soft bits to the receiver. The disadvantage of this method is that it requires higher computational complexity at the relay, as well as larger bandwidth since soft values are transmitted to the destination instead of binary estimates. Another strategy is the socalled thresholdbased relaying[12, 13] where only decoded bits with reliability above a given threshold are forwarded to the destination. Opportunistic relaying is also useful to combat the error propagation[14]. Opportunistic relaying takes advantage of the many potential relay nodes in the network. The relay with the best endtoend link is chosen to forward the received data to the destination. It is wellknown that erroraware relaying provides better performance than errorunaware relaying protocols. Other solutions foresee that the destination takes care of error propagation. The idea is that, if the destination has access to the channel state information (CSI) of the sourcerelay links, it can exploit it to counteract the error propagation problem. In[15], the authors show that channelaware receivers can significantly improve the performance of NC. However, no channel coding is considered in[15]. In[16], a turbolike decoding is proposed. In[17], the authors propose a cooperative communication scheme for multipleinput multipleoutput (MIMO) systems. A similar approach is available in[18] without performing channel decoding at the relay.
All these papers assume that CSI and decoding error probability at the relay are available at the destination, which is not always true in practical wireless systems. It is shown in[19, 20] that imperfect CSI can significantly degrade the performance of cooperative systems. In this paper, we study the impact of both CSI and decoding error probability at the relay in the multiple access relay channel. It is assumed that CSI at the receivers is acquired via the transmission of pilot symbols. The decoding error probability at the relay is not assumed to be available for free at the destination but we propose a practical way of transmitting a quantized version of it. We study the performance of two notable relaying protocols: DecodeandForward (DF) relaying and DemodulateandForward (DMF) relaying. Behind, ComputeandForward has been recently introduced as a new relaying protocol which achieves a higher rate than existing relaying techniques and relies on lattice decoding structure[21]. However, this relaying technique is far different from our work in decoding aspects, hence is out of scope of this paper. In DF relaying, channel decoding is performed at the relay before NC and forwarding. On the other hand, in the DMF case, only demodulation is performed at the relay. As such, DMF has less computational complexity than DF but it is more prone to decoding errors at the relay. For each protocol, we develop two new channelaware JNCD algorithms. To summarize, the contributions of the paper are as follows: we show that JNCD provides better performance than separate network channel decoding only if the destination has enough knowledge of the decoding error probability at the relay; in addition, this gain will be larger as the number of fading blocks per codeword increases. Also, it is shown that the number of pilot symbols mostly affects the coding gain of the system with a negligible impact on the diversity order, at least for the signaltonoise ratio (SNR) range of interest. Finally, it is shown that CSI quantization errors affect both coding gain and diversity order. Additionally, it is shown that, in general, 3bit quantization is sufficient for DMF relaying and 6bit quantization is needed for DF relaying.
The remainder of this paper is organized as follows: section 2 describes system model and notation. Section 3 describes demodulation metrics and channel estimation for imperfect CSI. Section 4 describes the proposed JNCD algorithms. Section 5 describes how to compute the decoding error probability at the relay for various fading situations. Sections 6 and 7 show numerical examples with perfect and imperfect CSIs, respectively. Finally, section 8 concludes this paper.
2 System model
The system model under analysis is given by the canonical multipleaccess relay channel, where two sources, MS1 and MS2, communicate to a base station (BS) with the help of a relay R[7]. We study the realistic situation where all the channels are subject to Rayleigh fading and additive white Gaussian noise (AWGN). The relay is located between the sources and the base station. We note that in practice, when the relay is very close to the sources, sourcerelay channels might be subject to different fading models, e.g., rice fading. In this paper, we consider Rayleigh fading assumption for all links for convenience. In order to avoid mutual interference, we consider that transmissions are scheduled in timeorthogonal timeslots[4]. We study both perfect CSI and imperfect CSI at the receiver. Three fading scenarios are investigated: fully interleaved, block fading with F blocks per codeword, and quasistatic fading, i.e., F = 1.
The source node MSj, j ∈ {1,2}, emits a Klength information message u _{ j }, where K is the number of information bits in u _{ j }. At each source, the information message u _{ j } is processed as follows: (1) first, it is encoded using a recursive convolutional code, which produces a length N codeword c _{ j }, with N = K/R being the length of the codeword and R being the code rate; (2) then, c _{ j } is interleaved and mapped into a 2^{ M } constellation point using Gray mapping. This operation provides the modulated signal x _{ j }. The modulated signal x _{ j }, of length N/M is transmitted to the relay and destination over a Rayleigh fading channel[22] with AWGN. Note that this description involves only the data part. In the imperfect CSI case, we consider that channel estimates are obtained via the use of pilot symbols, and the description will be refined accordingly. These details are provided in the next sections.
3 Channel estimation and modulation metric computation
This section describes the computation of modulation metrics for both perfect and imperfect CSI as well as how channel estimation for imperfect CSI is performed. These metrics will be used in the next subsections to implement the proposed decoders.
3.1 Perfect CSI
where Es_{(.)} is the energy at the destination of the signal received from the sources or from the relay.
Lc _{kl} is sent to the JNCD decoder and is processed as described in the next sections.
3.2 Imperfect CSI
As far as the imperfect CSI case is concerned, we restrict our attention to only block fading channels with F blocks and quasistatic fading with F = 1. The reason is that channel estimation is assumed to be obtained via a pilotbased approach for[23], which is clearly not compatible with fully interleaved Rayleigh fading. The channel gain is assumed to be constant over one block and is assumed to change independently from block to block. In our setting, a codeword covers F blocks, and the relay estimates the error probability of the whole codeword based on the knowledge of the channel gains of all blocks (see section 5 below). These channel gains are estimated via a pilot message, which is inserted at the beginning of each block, and transmitted via BPSK modulation.
where R is the rate of the channel code and N is the codeword’s length.
The difference of the channel rates for F = 1 and F > 1 is negligible and can be ignored in practice. For example, for the parameters used in the simulation section, the actual rates for F = 1 and F = 4 are respectively 0.476 and 0.417. In this paper, block fading channels with F > 4 are not considered.
For simplicity, we drop MS and R indexes in our notation. The channel estimation of the generic link works as follows. Each transmission block first consists in the pilot message x _{ p } followed by the data message x _{ d }. The power of pilot symbols and data symbols are equal. The corresponding received signals y _{ p } and y _{ d } are of a form as in (1) with only one difference that the channel coefficient h in this case is a scale instead of a vector as in (1).
Note that the use of pilot message and its placement can be optimized via a crosslayer pilot design[24]. In this paper, since we just focus on the impact of imperfect CSI on performance of iterative decoding algorithms, a random sequence is used for pilot message, which obviously is a suboptimal solution.
where λ is a normalization factor such that P _{PCSI}(c _{kl} = 1) + P _{PCSI}(c _{kl} = 0) = 1. The LLR demodulation output of the coded bit c _{kl}, Lc _{kl} is computed as in (2) with${D}_{\text{PCSI}}({x}_{k},{y}_{k}\widehat{h})$ is used instead of D _{FCSI}(x _{ k },y _{ k }h).
These LLR are then used by the JNCD decoder for further processing as described in the next sections.
4 Proposed joint network/channel decoding
We propose two JNCD algorithms for the noisy MARC channel. The first algorithm works on possible decoding error of the information bits, while the second algorithm works on possible decoding error of coded bits. The error probability of information and coded bits are denoted by Pe_{bit} and Pe_{code}, respectively.
4.1 Proposed JNCD: Algorithm 1
The first algorithm is developed based on turbolike decoding methods. To fully exploit the potential distributed diversity provided by the relay, the destination needs to know the decoding error probability at the relay, which is estimated and transmitted by the relay as described in the previous section. After receiving three channel observations from the two sources and from the relay, along with the decoding error probability at the relay, the destination runs the algorithm as follows:
where P[.] denotes probability and P[ab] denotes probability of a conditioned on b.

Step 0. (Setup) Let Lc _{1},Lc _{2}, and Lc _{ r } be the LLRs of codewords${\widehat{\mathbf{c}}}_{1},{\widehat{\mathbf{c}}}_{2}$, and${\widehat{\mathbf{c}}}_{r}$, respectively, which are the outputs of the demodulators described in section 2. The k th element of Lc _{ j }, j = 1,2,r is computed as in section 3.

Step 1. (Channel decoding) At the n th iteration, the SISO decoder j, j = 1,2, and SISO decoder R run the BCJR algorithm[25], as follows: Input: extrinsic information of coded bits Lc _{ j }, j = 1,2,r and a priori information${\mathbf{IA}}_{j}^{n}$, Output: extrinsic of information bits${\mathbf{Lu}}_{j}^{\text{Dec}(n)}({u}_{j}),\phantom{\rule{1em}{0ex}}j=1,2$, and${\mathbf{Lu}}_{r}^{\text{Dec}(n)}({w}^{r})$. The upper index (n) indicates the index iteration. In the first iteration, there is no a priori information for SISO decoders 1,2, and R.

Step 2. (Decoding errors are taken into account). The decoding check node updates the extrinsic of the estimated networkcoded information bits${\mathbf{Lu}}_{r}^{\text{Dec}(n)}({w}^{r})$ to get the extrinsic of correct networkcoded information bits L u ^{Dec(n)}(w) by taking into account the decoding error probability Pe _{bit}. Let${\text{Lu}}_{r}^{\text{Dec}(\mathrm{n})}({w}_{k}^{r})\phantom{\rule{0.3em}{0ex}}$ and Lu^{Dec(n)}(w _{ k }) be the k th elements of${\mathbf{Lu}}_{r}^{\text{Dec}(n)}({w}^{r})$ and L u ^{Dec(n)}(w), respectively, then$\begin{array}{l}{\text{Lu}}^{\text{Dec}(n)}\left({w}_{k}\right)=log\frac{\left(1{\mathit{\text{Pe}}}_{\text{bit}}\right)exp\left({\text{Lu}}_{r}^{\text{Dec}(n)}({w}_{k}^{r})\right)+{\text{Pe}}_{\text{bit}}}{{\text{Pe}}_{\text{bit}}exp\left({\text{Lu}}_{r}^{\text{Dec}(n)}({w}_{k}^{r})\right)+1{\text{Pe}}_{\text{bit}}}.\end{array}$(7)

Step 3. (Network decoding) The extrinsic of information bits${\mathbf{Lu}}_{1}^{\text{Dec}(n)}({u}_{1})$,${\mathbf{Lu}}_{2}^{\text{Dec}(n)}({u}_{2})$, L u ^{Dec(n)}(w) are input to the network decoder to output${\mathbf{Lu}}_{1}^{\text{Net}(n)}({u}_{1})$,${\mathbf{Lu}}_{2}^{\text{Dec}(n)}({u}_{2})$, L u ^{Net(n)}(w). Let${\text{Lu}}_{1}^{\text{Net}(n)}({u}_{1k})$,${\text{Lu}}_{2}^{\text{Net}(n)}({u}_{2k})$, Lu^{Net(n)}(w _{ k }) be the k th element of${\mathbf{Lu}}_{1}^{\text{Net}(n)}({u}_{1})$,${\mathbf{Lu}}_{2}^{\text{Dec}(n)}({u}_{2})$, L u ^{Net(n)}(w), respectively. The outputs of the network decoder are computed as follows:$\begin{array}{l}{\text{Lu}}_{1}^{\text{Net}(n)}({u}_{1k})=log\frac{1+exp\left({\text{Lu}}_{2}^{\text{Dec}(n)}({u}_{2k})+{\text{Lu}}^{\text{Dec}(n)}({w}_{k})\right)}{exp\left({\text{Lu}}_{2}^{\text{Dec}(n)}({u}_{2k})\right)+exp\left({\text{Lu}}^{\text{Dec}(n)}({w}_{k})\right)},\\ {\text{Lu}}_{2}^{\text{Net}(n)}({u}_{2k})=log\frac{1\phantom{\rule{0.3em}{0ex}}+exp\left({\text{Lu}}_{1}^{\text{Dec}(n)}({u}_{1k})+{\text{Lu}}^{\text{Dec}(n)}({w}_{k})\right)}{exp\left({\text{Lu}}_{1}^{\text{Dec}(n)}({u}_{1k})\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}exp\left({\text{Lu}}^{\text{Dec}(n)}({w}_{k})\right)},\\ {\text{Lu}}^{\text{Net}(n)}({w}_{k})=log\frac{1+exp\left({\text{Lu}}_{1}^{\text{Dec}(n)}({u}_{1k})\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\text{Lu}}_{2}^{\text{Dec}(n)}({u}_{2k})\right)}{exp\left({\text{Lu}}_{1}^{\text{Dec}(n)}({u}_{1k})\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}exp\left({\text{Lu}}_{2}^{\text{Dec}(n)}({u}_{2k})\right)}.\end{array}$(8)

Step 4. (Decoding errors are taken into account) The decoding check node update L u ^{Net(n)}(w) to get${\mathbf{Lu}}_{r}^{\text{Net}(n)}({w}^{r})$ by taking into account the decoding error probability:${\text{Lu}}_{r}^{\text{Net}(n)}({w}_{k}^{r})=log\frac{(1{\text{Pe}}_{\text{bit}})exp\left({\text{Lu}}^{\text{Net}(n)}({w}_{k})\right)+{\text{Pe}}_{\text{bit}}}{{\text{Pe}}_{\text{bit}}exp\left({\text{Lu}}^{\text{Net}(n)}({w}_{k})\right)+1{\text{Pe}}_{\text{bit}}}.$(9)

Step 5. (Feedback) The extrinsic of information bits${\mathbf{Lu}}_{1}^{\text{Net}(n)}({u}_{1})$,${\mathbf{Lu}}_{2}^{\text{Net}(n)}({u}_{2})$, and L u ^{Net(n)}(w) are feedback to SISO decoders 1,2, and R as a priori information for the next iteration, as follows:${\mathbf{IA}}_{1}^{n+1}({u}_{1})={\mathbf{Lu}}_{1}^{\text{Net}(n)}({u}_{1})$;${\mathbf{IA}}_{2}^{n+1}({u}_{2})={\mathbf{Lu}}_{2}^{\text{Net}(n)}({u}_{2})$;${\mathbf{IA}}_{r}^{n+1}({w}^{r})={\mathbf{Lu}}_{r}^{\text{Net}(n)}({w}^{r})$.

Step 6. Repeat from Step 1.
4.2 Proposed JNCD: Algorithm 2
Let Lc _{1},Lc _{2}, and Lc _{ r } be LLR inputs for sources 1, 2, and relay R, respectively. Let${\mathbf{Lc}}_{1}^{\text{Net}}$ and${\mathbf{Lc}}_{2}^{\text{Net}}$ be the extrinsic information outputs of the network decoder, and${\mathbf{Lc}}_{1}^{\text{Dec}}$ and${\mathbf{Lc}}_{2}^{\text{Dec}}$ be extrinsic information outputs of the coded bits of SISO decoder 1 and SISO decoder 2. Finally, let IA _{1} and IA _{2} be the a priori information (on coded bits) of the network decoder.

Step 0. (Setup) The three demodulators process the received signal to output Lc _{1},Lc _{2},Lc _{ r }. The kth element of Lc _{ j }, j = 1,2,r is computed as in Section 3.
A decoding check node updates Lc _{ r } by taking into account the decoding error probability at the relay, Pe_{code}, to get${\stackrel{~}{\mathbf{Lc}}}_{r}$:${\stackrel{~}{\mathit{\text{Lc}}}}_{\mathit{\text{rk}}}=log\frac{(1{\text{Pe}}_{\text{code}})exp\left({\mathit{\text{Lc}}}_{\mathit{\text{rk}}}\right)+{\text{Pe}}_{\text{code}}}{{\text{Pe}}_{\text{code}}\phantom{\rule{1em}{0ex}}exp\left({\mathit{\text{Lc}}}_{\mathit{\text{rk}}}\right)+1{\text{Pe}}_{\text{code}}},$(11)where Lc _{ rk } and${\stackrel{~}{\mathit{\text{Lc}}}}_{\mathit{\text{rk}}}$ are the k th elements of Lc _{ r } and${\stackrel{~}{\mathbf{Lc}}}_{r}$, respectively.
Step 1. (Network decoding) At the n th iteration, the network decoder decodes Lc _{1}, Lc _{2},${\stackrel{~}{\mathbf{Lc}}}_{r}$, with a priori information${\mathbf{IA}}_{1}^{n}$ and${\mathbf{IA}}_{2}^{n}$ to output the extrinsic information of coded bits${\mathbf{Lc}}_{1}^{\text{Net}(n)}$ and${\mathbf{Lc}}_{2}^{\text{Net}(n)}$. Let Lc _{1k }, Lc _{2k }, and Lc _{ rk } be the k th elements of Lc _{1}, Lc _{2}, and${\stackrel{~}{\mathbf{Lc}}}_{r}$, respectively;$L{c}_{1k}^{\text{Net}(n)}$ and$L{c}_{2k}^{\text{Net}(n)}$ be the k th element of${\mathbf{Lc}}_{1}^{\text{Net}(n)}$ and${\mathbf{Lc}}_{2}^{\text{Net}(n)}$, respectively; and${\text{IA}}_{1k}^{n}$ and${\text{IA}}_{2k}^{n}$ be the k th elements of${\mathbf{IA}}_{1}^{n}$ and${\mathbf{IA}}_{2}^{n}$, respectively. Then$\begin{array}{l}L{c}_{1k}^{\text{Net}(n)})={\mathit{\text{Lc}}}_{1k}+log\frac{exp\left({\stackrel{~}{\mathit{\text{Lc}}}}_{\mathit{\text{rk}}}\right)+exp\left({\mathit{\text{Lc}}}_{2k}+{\text{IA}}_{2k}^{n}\right)}{1+exp\left({\stackrel{~}{\mathit{\text{Lc}}}}_{\mathit{\text{rk}}}+{\mathit{\text{Lc}}}_{2k}+{\text{IA}}_{2k}^{n}\right)},\\ L{c}_{2k}^{\text{Net}(n)})={\mathit{\text{Lc}}}_{2k}+log\frac{exp\left({\stackrel{~}{\mathit{\text{Lc}}}}_{\mathit{\text{rk}}}\right)+exp\left({\mathit{\text{Lc}}}_{1k}+{\text{IA}}_{1k}^{n}\right)}{1+exp\left({\stackrel{~}{\mathit{\text{Lc}}}}_{\mathit{\text{rk}}}+{\mathit{\text{Lc}}}_{1k}+{\text{IA}}_{1k}^{n}\right)}.\end{array}$At the first iteration,${\mathbf{IA}}_{1}^{1}={\mathbf{IA}}_{2}^{1}=0$.

Step 2. (Channel decoding) The SISO decoder j, j = 1,2, run the BCJR algorithm[25] as follows: Input: extrinsic information of coded bits${\mathbf{Lc}}_{j}^{\text{Net}(n)}$; the a priori extrinsic of information bits is equal to 0. Output: extrinsic information of coded bits${\mathbf{Lc}}_{j}^{\text{Dec}(n)}$.

Step 3. (Feedback) The extrinsic information of coded bits${\mathbf{Lc}}_{1}^{\text{Dec}},\phantom{\rule{1em}{0ex}}{\mathbf{Lc}}_{2}^{\text{Dec}}$ is feedback to the network decoder as a priori (of coded bits) information for the next iteration:${\mathbf{IA}}_{j}^{n+1}={\mathbf{Lc}}_{j}^{\text{Dec}(n)},j=1,2$.

Step 4. Repeat from Step 1.
5 Error probability estimation and quantization
In order to apply the algorithms described in the previous section, the destination must estimate Pe_{bit} = P[w ^{ r } ≠ u _{1} ⊕ u _{2}] and Pe_{code} = P[c ^{ r } ≠ c _{1} ⊕ c _{2}]. In this section, we compute these probabilities. Let${\text{Pe}}_{\text{bit}}(j)=P[{u}_{j}^{r}\ne {u}_{j}]$ and${\text{Pe}}_{\text{code}}(j)=P[{c}_{j}^{r}\ne {c}_{j}],j=1,2$ be the decoding error probability of information bits and coded bits, respectively, of the link from source MSj to the relay. We assume, for simplicity, that the networkencoded information bits and networkencoded coded bits are independent (a reasonable assumption if interleavers are used at the relay).
where the expression in (13) is given by the definition of XOR network coding.
In the next subsections,${\text{Pe}}_{\text{bit}}(j)=P[{u}_{j}^{r}\ne {u}_{j}]$ and${\text{Pe}}_{\text{code}}(j)=P[{c}_{j}^{r}\ne {c}_{j}],j=1,2$ are computed for different fading scenarios. In our analysis, we assume Gray mapping. Also, the nearest neighbor approximation is used. This corresponds to the assumptions that if an error occurs, then the transmitted symbol can only be one of the symbols closest to the estimated one. Therefore, due to Gray mapping, one symbol error causes a single coded bit error. As illustrative examples, three cases are considered: F = 1, F = 4, and fully interleaved fading.${\text{Pe}}_{\text{bit}}^{F1,F4,\text{Full}}$ (${\text{Pe}}_{\text{code}}^{F1,F4,\text{Full}}$) denote the decoding error probability of information bits (coded bits) for each case study, respectively. For simplicity, we focus our attention on 16QAM modulation, as used in our numerical examples.
5.1 Error estimation with perfect CSI: computation of${\text{Pe}}_{\text{code}}^{F1,F4,\text{Full}}$
5.1.1 Block Rayleigh fading F = 1
where erfc(·) is related to the Qfunction.
5.1.2 Block Rayleigh fading F = 4
5.1.3 Fullyinterleaved Rayleigh fading
5.2 Error estimation with perfect CSI: computation of${\text{Pe}}_{\text{bit}}^{F1,F4,\text{Full}}$
where d _{ f } is the minimum distance, β(d) is the distance spectrum of the convolutional code, and P _{ c }(d) is the probability of choosing a wrong path in the trellis with distance d from the correct path (usually the allzero path). P _{ c }(d) depends on the channel gains and is computed as follows.
5.2.1 Block Rayleigh fading channel with F = 1
In our simulation results, d _{ f } = 6 for the RSC code [1 15/13] and only two values of d are used.
5.2.2 Block Rayleigh fading channel with F = 4
where${\mathcal{C}}_{k}^{n}$ denotes a combination of k elements of a set of n elements; N _{ s } = N/log_{2}(M) is the length of a signal; m = N _{ s }/F is the block’s length.
5.2.3 Fully interleaved Rayleigh fading
where${\mathcal{C}}_{m}^{n}=\frac{n!}{m!(nm)!}$ denotes the binomial coefficients.
5.3 Error estimation with imperfect perfect CSI
In the imperfect CSI case, we consider F = 1 and F = 4. The error probabilities at the relay can be computed as in the perfect CSI case, except that the estimated channel gain${\widehat{h}}_{j},j=1,2$ is used instead of correct one h _{ j }.
5.4 Error quantization
where M _{ v } = max(v) min(v). The quantization error by Q _{ q }(.) is given by ε _{ q } = M _{ v }/2^{ q+1}. The quantized$\stackrel{\u0304}{v}$ is transmitted over fading plus Gaussian noise to the destination. At the end of the channel estimation phase, the destination recovers the decoding error probability at the relay from the noisy version of the transmitted quantized signal.
6 Numerical results: perfect CSI case
In this section, we study the performance of the proposed JNCD algorithms in various fading scenarios. For this perfect CSI case, assume that the receivers have perfect knowledge of the onehop links CSI. In addition, the destination is assumed to have full knowledge of the decoding error probability at the relay, Pe_{bit} and Pe_{code}, which are estimated as described in the previous sections. We assume a symmetric network topology in which the distance from the sources to the destination is the same. The relay is located between sources and destination. The path loss has been chosen equal to 3.5[29]. Unless stated for specific cases, the recursive convolutional code (RSC) [1 15/3] with rate 1/2 is used. 16QAM is used as the modulation scheme. The number of iterations used to obtain our results is four since we have observed that the algorithms converge to the best performance in four iterations.
Both DF and DMF relaying protocols are studied. In particular, three schemes are studied: (1) DF relaying with the proposed algorithm 1, named DF Algo 1 in the figure; (2) DF relaying with the proposed algorithm 2, named DF Algo 2 in the figure; (3) DMF relaying with the proposed algorithm 2, named DMF Algo 2 in the figure. We note that algorithm 1 cannot be used with the DMF protocol because it performs network coding on the information bits. We also compare our algorithms with[18], which is denoted by Ref. [Yune] in the figure. In addition, we denote by Blind the scenario where the receiver has no information about the decoding error probability at the relay (it assumes perfect decoding at the relay) and by Non Cooperation the conventional pointtopoint communication scenario.
6.1 Effects of iterations
Figure5b shows the simulation results for block fading channel with F = 4. In addition, the theoretical curve SNR ^{4} is plotted to provide some information about the achievable diversity order: (1) iterative decoding improves performance for both DF and DMF relaying. With four iterations, DF relaying with the proposed algorithm 1 gains about 3 dB and with the proposed algorithm 2, it gains about 2 dB at a BER equal to 10^{4} compared to the single iteration case. DMF relaying with algorithm 2 gains about 1 dB. (2) Compared to[18], after four iterations, algorithm 1 and algorithm 2 with DF relaying both gain about 3 dB at a BER equal to 10^{4}. With DMF relaying, the algorithm 2 with four iterations gains about 1 dB at a BER equal to 10^{4}. (3) For both DF and DMF relaying, the receiver loses the diversity order if it has no information about the decoding error probability.
where ⌊x⌋ denotes the largest integer no greater than x, R is the code rate in bits/symbol. In our setup, we have R = 2 bits/symbol and 16QAM. Thus, we get d ^{ H } ≤ 3. It is shown from the simulation that the actual diversity order of the code [1 15/13] is 2 in the SNR range of interest. Therefore, it is reasonable that in the MARC, the relay provides a better diversity gain.
Figure5c shows simulation results for the quasistatic fading channel with F = 1. In addition, the theoretical curve SNR^{2} is plotted as a diversity reference. It is shown in the figure that (1) if the receiver is not informed about the decoding error probability at the relay, the performance is dramatically decreased and it loses diversity order; (2) iterative decoding brings a little gain in both algorithms. Algorithm 2 with four iterations gains about 0.8 dB over the oneiteration case, while algorithm 1 with four iterations gains about 0.5 dB over the oneiteration case; and (3) DMF with algorithm 2 is slightly better than DF with both algorithms.
Consider three fading channels, algorithm 2 is recommended because of its low complexity compared with algorithm 1. With quasistatic block fading scenario, DMF relaying is recommended. In this case, channel decoding at the relay does not improve the system performance. In general Fblock fading channels, DF relaying should be used.
6.2 Effects of location of the relay
6.3 Effects of the channel code
7 Numerical results: imperfect CSI case
This section evaluates the impact of imperfect CSI and quantization error on the performance of the proposed algorithms. The two case studies with F = 1 and F = 4 are investigated. The relay is located at the middle between the sources and the destination. The RSC [1 15/13] is chosen as in the previous section and 16QAM is used. The ML estimator is used for channel estimation. Because the performance of algorithm 2 with DF and DMF relaying is almost the same, we only study algorithm 2 with DMF relaying in this section. Then, in this section, algorithm 1 is linked to DF relaying and algorithm 2 is linked to DMF relaying. In the figures, Full CSI denotes the case when the receivers (relay and destination) have perfect channel state information of the onehop links. On the other hand, Full Pe denotes the case when the destination has full knowledge of the decoding error probability at the relay.
Intensive simulations show that iterative decoding only provides coding gain if the receivers have enough accurate CSI and possible decoding errors. However, concerning realistic imperfect CSI systems where the total number of overhead symbols is not too large, we focus on only separate decoding (one iteration) for block fadings (F = 1 and F = 4) in what follows.
7.1 Effects of pilot length
7.2 Effects of quantization
In conclusion, (1) if F = 1, algorithm 2 has the same performance as algorithm 1 and it is less complex than algorithm 1. However, if F = 4, algorithm 1 outperforms algorithm 2. (2) If the destination is aware enough of the decoding error probability at the relay, pilot length does not change that diversity order of the system. (3) The number of quantization bits for reporting decoding errors from the relay affects both coding gain and diversity order. (4) Sixbit quantization is enough in most analyzed scenarios.
8 Conclusion
In this paper, we have studied the performance of the multiple access relay channel with binary Network Coding and JNCD at the destination in practical situations. Decode and Forward and Demodulate and Forward relaying strategies are investigated. Our results show that iterative Joint Network and Channel Decoding provides better performance than separate network channel decoding only if the destination has enough CSI and knowledge of the decoding error probability at the relay. This gain increases with the number of fading blocks per codeword. It is also shown that the number of pilot symbols mostly affects the coding gain of the system with a negligible impact on the diversity order, at least for the SNR range of interest. Finally, it is shown that CSI quantization errors affect both coding gain and diversity order. In general, representing the BER at the relay using 6bit quantization is sufficient for both DMF relaying and DF relaying. The proposed iterative decoding algorithms can be easily extended to frequency selective fading scenarios, e.g., OFDM systems.
Declarations
Acknowledgements
The research work is supported in part by the European Commission under the auspices of the FP7PEOPLE MITNCROSS FIRE project (Grant 317126) and in part by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM# (Grant agreement no. 318306).
Authors’ Affiliations
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