Asymptotic BER EXIT chart analysis for high rate codes based on the parallel concatenation of analog RCM and digital LDGM codes
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Abstract
This paper proposes an extrinsic information transfer (EXIT) chart analysis and an asymptotic bit error rate (BER) prediction method to speed up the design of high rate RCMLDGM hybrid codes over AWGN and fast Rayleigh channels. These codes are based on a parallel concatenation of a rate compatible modulation (RCM) code with a lowdensity generator matrix (LDGM) code. The decoder uses the iterative sumproduct algorithm to exchange information between the variable nodes (VNs) and the two types of constituent check nodes: RCMCN and LDGMCN. The novelty of the proposed EXIT chart procedure lies on the fact that it mixes together the analog RCM check nodes with the digital LDGM check nodes, something not possible in previous multiedge EXIT charts proposed in the literature.
Keywords
Rate compatible modulation (RCM) Lowdensity generation matrix (LDGM) EXIT charts BER prediction Joint sourcechannel coding (JSCC)Abbreviations
 ACM
Adaptive coded modulation
 APSK
Amplitude and phaseshift keying
 AWGN
Additive white Gaussian noise
 BER
Bit error rate
 BICM
Bitinterleaved coded modulation
 CN(D)
Check node (decoder)
 EXIT
Extrinsic information transfer
 LDGM
Lowdensity generator matrix
 LDPC
Lowdensity parity check
 LLR
Loglikelihood ratio
Probability density function
 QAM
Quadrature amplitude modulation
 RCM
Rate compatible modulation
 RV
Random variable
 SNR
Signal to noise ratio
 SP(A)
Sumproduct (algorithm)
 VN(D)
Variable node (decoder)
1 Introduction
We propose an extrinsic information transfer (EXIT) chart analysis and an asymptotic bit error rate (BER) prediction method to speed up the design of high rate (greater than 2 bits per complex channel symbol) RCMLDGM hybrid channel codes for the transmission of memoryless binary sources over additive white Gaussian noise (AWGN) and fast Rayleigh channels. These hybrid codes consist of the parallel concatenation of a rate compatible modulation (RCM) code (see, e.g., [1, 2]) and a lowdensity generator matrix (LDGM) code (see, e.g., [3, 4]). In what follows, we will refer to these schemes as parallel RCMLDGM codes. Both uniform and nonuniform sources are considered. The reason for considering nonuniform sources is that many data sources (e.g., image or speech signals) are nonuniformly distributed, containing substantial amount of natural redundancy [5, 6, 7, 8]. Even when these sources are compressed, they still exhibit a residual redundancy due to the suboptimality of the compression scheme [9].
RCM codes generate random projections (RP) from weighted linear combinations and are able to achieve smooth rate adaptation in a broad dynamic range. However, they present error floors at high signal to noise ratios (SNRs). In order to solve this drawback, [10, 11] suggested the use of an LDGM code in parallel with the RCM code, aiming at reducing the error floor. Simulation results in [10, 11] show that the parallel RCMLDGM code outperforms RCM schemes significantly, achieving a performance close to the Shannon limit if suitable design parameters are chosen.
One of the main advantages of this class of high rate RCMLDGM codes over other high rate codes, such as the widely adopted bitinterleaved coded modulation (BICM) [12], is the easiness of performing adaptive coded modulation (ACM). Conventional ACM is done by selecting the best combination of channel coding and modulation based on the estimated channel condition. Due to the limited number of rate combinations, a stairshaped rate curve is often obtained. Moreover, ACM requires instant and accurate channel estimation. Due to their intrinsic design, RCM codes are well suited to overcome these adaptation challenges (refer to [2] for a comparison with other ACM schemes). The reason is that their coded symbols are generated by a set of weighted linear combinations of the source binary symbols. By varying the number of linear combinations on a per symbol basis, a smooth rate adaptation is possible.
Another advantage of RCMLDGM codes is in the transmission of nonuniform memoryless sources. Existing low rate joint sourcechannel coding schemes [7, 8, 13, 14] present a gap to the Shannon theoretical limit of about 2 dB for sources with low nonuniformity. However, this gap increases when the source becomes more nonuniform, i.e., when its entropy decreases. Unlike these low rate joint sourcechannel codes, it is shown in [11] that RCMLDGM codes are able to maintain the gap to the theoretical limit as the nonuniformity increases, while keeping very large throughputs. Their robustness against channel and source variations, together with the fact that smooth rate adaption is possible, makes RCMLDGM codes excellent candidates in applications where channel and source variations are encountered. However, the proposed RCMLDGM codes found in the literature [10, 11], have been designed by a trialerror procedure, something that requires a large amount of simulation time. Here, to circumvent this design drawback, we propose an EXIT chart analysis that facilitates the selection of suitable code parameters.
 1
Developing an EXIT chart analysis that is able to deal with the check node disparity encountered in parallel RCMLDGM codes when driven by binary memoryless sources (both uniform and nonuniform) transmitted over AWGN and fast fading Rayleigh channels
 2
Assessment of the time savings achieved by using the EXIT chart analysis, rather than Monte Carlo simulations, for BER predictions
The remainder of the paper is organized as follows. Section 2 briefly reviews previous work on the design of RCM and parallel RCMLDGM codes. Section 3 presents the proposed EXIT chart analysis and BER prediction for RCMLDGM codes. Section 4 evaluates the proposed EXIT chartBER prediction method, comparing the predicted BER with simulation results. Finally, Section 5 concludes this paper.
2 Background: RCM and RCMLDGM code design
where \(\{n_{i}\}_{i=1}^{N}\) are realizations of i.i.d real Gaussian random variables (RVs) with zero mean and variance N_{0}/2 (i.e., \({N_{i}}\sim \mathcal {N}\left (0,N_{0}/2\right)\)). At the receiver side, the decoder estimates the source symbols u from y.
For the sake of clarity in the exposition, we begin by providing a succinct overview of the key concepts of RCM and LDGM codes before covering parallel RCMLDGM codes.
2.1 Rate compatible modulation (RCM) codes
 1Construct the K/2×K matrix G_{0} as$$\begin{array}{*{20}l} {}{G}_{0}=\left[\begin{array}{cccc} \Pi \left(D_{d_{3}}\right) & \Pi \left(D_{d_{4}}\right) & \Pi \left(D_{d_{1}}\right) & \Pi \left(D_{d_{2}}\right) \\ \Pi \left(D_{d_{1}}\right) & \Pi \left(D_{d_{2}}\right) & \Pi \left(D_{d_{3}}\right) & \Pi \left(D_{d_{4}}\right) \\ \Pi \left(D_{d_{4}}\right) & \Pi \left(D_{d_{3}}\right) & \Pi \left(D_{d_{2}}\right) & \Pi \left(D_{d_{1}}\right) \\ \Pi \left(D_{d_{2}}\right) & \Pi \left(D_{d_{1}}\right) & \Pi \left(D_{d_{4}}\right) & \Pi \left(D_{d_{3}}\right) \end{array}\right], \end{array} $$where Π(·) denotes random column permutations of a matrix, and \(D_{d_{l}}\) is a K/8×K/4 sparse matrix given by$$\begin{array}{*{20}l} \tiny{D_{d_{l}}=\left[\begin{array}{ccccccccc} d_{l} & d_{l} & 0 & 0 & 0 & 0 & \ldots & 0 & 0 \\ 0 & 0 & d_{l} & d_{l} & 0 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{l} & d_{l} & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{l} & d_{l} \end{array} \right]}, \end{array} $$
with \(d_{l} \in \mathcal {D}\), for \(l \in \left \{1, \ldots, d_{\text {RCM}}^{(c)}/2\right \}\).
 2
Vertically stack as many G_{0}s as needed. Note that we should keep only as many rows as needed in the last stacked G_{0} matrix so that the required M×K G matrix is obtained.
Observe that \(d_{\text {RCM}}^{(c)}\) gives the number of nonzero entries of any row of G. Similarly, we denote by \(d_{\text {RCM}}^{(v_{k})}\geq 2\) the number of nonzero entries of column k of matrix G, and by \(\overline {d}_{\text {RCM}}^{(v)}\) its average value, i.e, \(\overline {d}_{\text {RCM}}^{(v)} = \frac {1}{K}\sum _{k=1}^{K} d_{\text {RCM}}^{(v_{k})}\).
where [ · ]_{j} is the element at row j and g_{j,i} denotes entry (j,i) of matrix G, where these operations are in the real field.
2.2 Lowdensity generator matrix (LDGM) codes
where in this case the operations are in the binary field. Although LDGM codes have the advantage of linear encoding complexity, unlike general LDPC codes, they can only attain an arbitrarily low error probability by reducing the rate to zero [20]: Independently of the block length, they suffer from a high error floor. Therefore, they have been historically disregarded in favor of other LDPC codes. However, as explained in the next section, they can actually perform well as an aid to reduce the error floor of other codes.
2.3 Parallel RCMLDGM code
2.4 Decoder block
 Step 1. \(q_{k,i}^{(t)}\): Message passing from variable nodes,\(\lbrace U_{k}\rbrace _{k=1}^{K}\), to RCM and LDGM check nodes\(\lbrace T_{i}\rbrace _{i=1}^{M+I}\).$$\begin{array}{*{20}l} q_{k,i}^{(t)}&=\sum_{j\in n(U_{k})\setminus T_{i}}r_{j,k}^{(t1)} + \log\left(\frac{p_{1}}{p_{0}}\right), \end{array} $$(1)
where \(r_{j,k}^{(0)}=0\) for k∈{1,…,K}, j∈n(U_{k})∖T_{i} and (p_{1};p_{0}) is the distribution of the memoryless binary source.
 Step 2. \(r_{i,k}^{(t)}\): Message passing from RCMLDGM check nodes,\(\lbrace T_{i}\rbrace _{i=1}^{M+I}\), to variable nodes\(\lbrace U_{k}\rbrace _{k=1}^{K}\).
 Computation at RCM check nodes\(\lbrace T_{i}\rbrace _{i=1}^{M}\): Observe that \(x_{j}= \sum _{i} g_{j,i}u_{i}\) and define \(a_{j,k}= \sum _{i \sim k} g_{j,i}u_{i}\), where \(\sum _{i \sim k}\) means the sum over all i except k. Combining both terms, we get x_{j}=a_{j,k}+g_{j,k}u_{k} for all k∈n(T_{j}), and the received symbol y_{j}=x_{j}+n_{j}. The message \(r_{i,k}^{(t)}\) is calculated as$$ r_{i,k}^{(t)} = \log \left(\frac{\sum_{z} P^{(t)}(a_{j,k}=z)\cdot e^{(y_{j}  z  g_{j,k})^{2}/N_{0}}}{\sum_{z} P^{(t)}(a_{j,k}=z)\cdot e^{(y_{j}  z)^{2}/N_{0}}}\right) $$(2)
where the sum in z is over all possible values that the RCM symbols can take. Notice that P^{(t)}(a_{j,k}=z), the probability of a_{j,k}=z at iteration t, is calculated in a straightforward manner by convolving the probability density functions (PDFs) of the terms in the summation, where the distribution functions of these terms are obtained from the received LLR messages \(q_{k,i}^{(t)}\). An efficient way to implement these convolutions is explained in [1].
 Computation at LDGM check nodes\(\lbrace T_{i}\rbrace _{i=M+1}^{M+I}\): As in standard LDGM codes, the LLR message transmitted from the ith check node to the variable node U_{k} is given by$$\begin{array}{*{20}l} r_{i,k}^{(t)} = 2\text{atanh}\left(\text{tanh}\left(\frac{\gamma_{i}}{2}\right)\prod_{j\in n(T_{i})\setminus U_{k}}\text{tanh}\left(\frac{q_{k,j}^{(t)}}{2}\right)\right), \end{array} $$(3)
where \(\gamma _{i}= \frac {(y_{i}+1)^{2}  (y_{i}1)^{2}}{N_{0}}\).

3 Methods: EXIT chart analysis for the LDGMRCM code
To obtain codes with near Shannon achieving performance, it is crucial to select good code parameters. For instance, when considering the case of a pure RCM code, one has to find a suitable weight set \(\mathcal {D}\). In [10, 11], simulations of the entire communications system were performed for a given set of design parameters. This procedure was repeated until a good combination of parameters was found. The drawback of this procedure is that it takes a large amount of computational time. To overcome this problem, the authors in [17] proposed an EXIT chart analysis, shortening in this way the parameter selection procedure for pure RCM codes. As already mentioned, this EXIT chart analysis cannot be directly applied in our scheme, since two different types of check nodes, RCM and LDGM, have to be considered. This paper extends the analysis to parallel RCMLDGM codes, considering as well nonuniform sources. Furthermore, it also presents a BER prediction analysis based on EXIT charts that was not previously considered in the literature for this type of codes.
where L∈{A,E} and f_{L}(ξu), for u=0,1, is the conditional probability density function of L given U. As indicated before, f_{L}(ξu) depends on the node decoder under consideration, that is, whether such a node is a VND or a CND, and it is calculated as indicated in Sections 3.1 and 3.2. In the sequel, we will denote I(L;U) for a VND or a CND as I_{L,VND}=I(L^{(vn)};U) or I_{L,CND}=I(L^{(cn)};U), respectively.
which denote the average percentage of edge connections arriving to a VN from an RCM check node and the percentage of edge connections arriving to an RCM check node from a VN, respectively.
3.1 VND EXIT curve for RCMLDGM codes
The EXIT curve of the VND is given by the transfer characteristic between I_{E,VND}=I(E^{(vn)};U) and I_{A,VND}=I(A^{(vn)};U). Note that the realizations of RVs E^{(vn)} and A^{(vn)} are the messages exchanged in the sumproduct algorithm, {r_{i,k}} and {q_{k,i}}, respectively. In order to evaluate these mutual informations from (5), the conditional PDF of the a priori A^{(vn)} and the extrinsic E^{(vn)} at a variable node decoder, given U, have to be found.
3.1.1 Calculation of I_{A,VND}
Different from previous work on EXIT charts, in an RCMLDGM code, one has to consider two types of a priori messages arriving at a VND: first, the messages arriving from an edge connected to an RCM check node, \(A^{(vn)}_{\text {RCM}}\), and second, the messages arriving from an edge connected to an LDGM check node, \(A^{(vn)}_{\text {LDGM}}\).
where \(\sigma _{R,A}^{2}\) and \(\sigma _{L,A}^{2}\) represent the inverse of the variance of the two different virtual channels.
The main challenge of having two different types of CNs rather than one, as in the case of a standard EXIT chart, is that the mutual information I_{A,VND} will now depend on two variables, \(\sigma _{R,A}^{2}\) and \(\sigma _{L,A}^{2}\), rather than just on one. Notice, however, that although \(A^{(vn)}_{\text {RCM}}U\) and \(A^{(vn)}_{\text {LDGM}}U\) can be considered independent, their variances are coupled due to the way the SPA generates the messages (refer to Section 2.4). Therefore, if one of the variances can be expressed as a function of the other, then I_{A,VND} becomes a function of only one variable, which simplifies the analysis.
The constant κ scales the variance of the distribution of \(A^{(vn)}_{\text {LDGM}}\) with respect to the variance of \(A^{(vn)}_{\text {RCM}}\). The steps to compute it are explained in the Appendix.
where the PDFs \(f_{A_{\text {RCM}}^{(vn)}}\) and \(f_{A_{\text {LDGM}}^{(vn)}}\) are given in (11) and (12), respectively, and \(p_{\text {RCM}}^{(vn)}\) in (6). Finally, applying (5), I_{A,VND} is calculated from \(f_{A_{\text {RCM}}^{(vn)}}\) as a parametric expression of \(\sigma _{R,A}^{2}\).
3.1.2 Calculation of I_{E,VND}
where \(p_{\text {RCM}}^{(cn)}\) is given in (7). Finally, applying (5), I_{E,VND} is calculated from \(f_{E_{\text {RCM}}^{(vn)}}\) as a parametric expression of \(\sigma _{R,A}^{2}\).
3.2 CND EXIT curve for the RCMLDGM codes
so that I_{A,CND}=I_{E,VND}.
To compute I_{E,CND}, we need to find the conditional PDF \(\phantom {\dot {i}\!}f_{E^{(cn)}}(eu)\) of the extrinsic LLR E^{(cn)} at the CND. This is done by running step 2 of the sumproduct algorithm (see Section 2.4) and setting q_{k,i}=a, where a are realizations of a random variable A^{(cn)} with conditional PDF (17). The empirical conditional PDF \(\phantom {\dot {i}\!}f_{E^{(cn)}}(eu)\) is now found by the histogram of the realizations {r_{i,k}}.
3.3 Trajectories of iterative decoding and decoding threshold
To account for the iterative nature of the decoding process, both the VND and CND transfer characteristics should be plotted into a single diagram. As long as the SNR is large enough so that both transfer curves do not intersect, the iterative process will achieve its maximum mutual information values, (H(p_{0}),H(p_{0})), consequently achieving a low BER. The smallest SNR value for which both curves do not intersect is defined as the decoding threshold and represents the minimum SNR required to decode without errors an infinite length code with the given configuration. Therefore, the code design problem reduces to find a code configuration, i.e., weight sets \(\mathcal {D}\), and parameters I and \(d_{\text {LDGM}}^{(v)}\), such that the decoding threshold is as close as possible to the corresponding SNR Shannon limit.
Remark
Note that the VND EXIT curve only depends on the values of \(\overline {d}_{\text {RCM}}^{(v)}\) and \(d_{\text {LDGM}}^{(v)}\). On the other hand, the CND EXIT curve depends on all the parameters, i.e., \(\{\mathcal {D}\), SNR, \(\overline {d}_{\text {RCM}}^{(v)}\), \(d_{\text {LDGM}}^{(v)}\), \(d_{\text {LDGM}}^{(c)}\), M, I}. □
Remark
The EXIT chart for a pure RCM code can be calculated as a particular case of the parallel LDGMRCM by taking \(p_{\text {RCM}}^{(vn)}=p_{\text {RCM}}^{(cn)}=1\). □
3.4 Predicting the BER from the EXIT chart
Remark
The BER for a pure RCM code can be estimated as a particular case of the parallel LDGMRCM with \(d_{\text {LDGM}}^{(v)}=0\). □
4 Results and discussion
In this section, we evaluate the proposed EXIT chart analysis and BER prediction method of Section 3 for both AWGN and fast fading Rayleigh channels. We begin by considering the AWGN channel. Section 4.1 presents some mutual information trajectories of actual codes on the corresponding EXIT charts. In Section 4.2, we compare the BER predictions obtained using these charts with the BER obtained by Monte Carlo simulations. In Section 4.3, the EXIT analysis is used to obtain codes that approach the Shannon theoretical limit. Finally, the extension to Rayleigh channels is considered in Section 4.4.
4.1 Trajectories
Observe that by introducing these 200 LDGM coded bits, we avoid the previous intersection of the curves at SNR 20.25 dB, improving in this way the BER at 20.25 dB. The corresponding mutual information trajectory at SNR 20.25 is shown in Fig. 5. Since the channel is open, it reaches its maximal value, i.e., (0.72,0.72). It turns out that SNR = 20.25 is the smallest SNR that allows the channel to remain open, and as such, it is the corresponding decoding threshold of the given code. In the same figure, the trajectory at SNR = 19.25 dB is also shown, but in this case the channel is closed and does not reach the maximum value.
4.2 Bit error rate from the EXIT charts
As explained in Section 3.4, an estimated BER can be assigned to each point of the variable node (VN) curve of the EXIT chart. Therefore, the BER of a particular code at a given SNR is obtained from the value of the VN point where the CN and VN curves intersect.
In this section, we will consider two different RCMLDGM configurations with ρ=4 given by K=25000, M=12365, and I=135, with \(d_{\text {LDGM}}^{(v)}\) 1 and 2. Moreover, we will consider three different sources with p_{0}=0.5, p_{0}=0.8, and p_{0}=0.95 and three different weight sets \(\mathcal {D}=\lbrace 1, 1, 1, 1, 2, 2\rbrace, \lbrace 1, 1, 2, 2, 4, 4\rbrace \), and {2,2,3,3,4,4}. Recall that the VN curve of the EXIT chart depends on M, I, \(d_{\text {LDGM}}^{(v)}\), and \(d_{\text {RCM}}^{(c)}\), whereas the CND curve depends also on the actual values of \(\mathcal {D}\) and on the SNR.
The parameters of these codes have not been optimized, and therefore, they present a large gap to the corresponding Shannon limits given by 10· log10(2^{ρ·H(S)}−1), which correspond to 0.91, 8.06, and 11.76 dB for p_{0}=0.95, p_{0}=0.8, and p_{0}=0.5, respectively. In the next section, we will obtain near capacity high spectral efficiency codes using the EXIT chart analysis.
4.3 Code design based on the decoding threshold for AWGN channels
 1
For sources with smaller entropy, larger RCM weight sets, D, tend to work better, since the sumproduct algorithm is aided by the a priori probability.
 2
When designing the LDGM part of the code, there is a tradeoff regarding the number I of LDGM binary symbols. By increasing I, more residual errors are corrected in the waterfall region, making it steeper. However, larger SNRs are required to reach this waterfall region.
 3
The range for parameter \(d_{\text {LDGM}}^{(v)}\) is between 1 and 5. The larger parameter I is, the larger value for \(d_{\text {LDGM}}^{(v)}\) can be selected.
Best configurations obtained by the EXIT chart analysis for AWGN channels
p _{0}  K  M  I  \(d_{\text {LDGM}}^{(v)}\)  D  Decoding threshold (dB) 

0.5  37000  9800  200  1  {2,3,4,8}  24.15 
0.8  37000  9720  280  4  {2,2,3,3,4,8}  18.2 
0.95  37000  9200  800  3  {1, 1, 1, 1, 1, 1, 1, 1, 1, 1 }  7.25 
Computational time required to predict a BER vs SNR point in Fig. 9
BER  EXIT chart  Monte Carlo simulation  Reduction factor  

Average convergence time per block  Blocks for 10 errors  
10^{−3}  10s  229s  1  22.9 
10^{−4}  10s  113s  3  33.9 
10^{−5}  10s  96s  27  259.2 
10^{−6}  10s  85s  270  2295 
4.4 Extension to fast fading Rayleigh channels
We now look at the behavior of the EXIT chart analysis when considering fast fading Rayleigh channels. Note that the only modification that has to be introduced in this case is in step 2 of the SP algorithm (see Section 3.2). Specifically, since we are assuming perfect channel state information (CSI) at the receiver, the fading factor that multiplies the coded RCMLDGM symbols (i.e., realizations of i.i.d. exponential random variables) has to be provided to the decoder.
Best configurations obtained by the EXIT chart analysis for fast fading Rayleigh channels
p _{0}  K  M  I  \(d_{\text {LDGM}}^{(v)}\)  D  Decoding threshold (dB) 

0.5  37000  9600  300  3  {2,3,4,4,8}  27.7 
0.8  37000  9240  760  5  {2,2,3,3,4,8}  21.3 
0.95  37000  9480  520  4  {1,1,1,1,1,1,1,1,1,1,1,1}  9.55 
5 Conclusion
Parallel RCMLDGM codes are very well suited for implementing smooth high rate adaptation when transmitting uniform and nonuniform binary memoryless sources. However, when long block lengths are considered, the search of good design parameters using a brute force approach is time consuming. To speed up the design process, we have successfully developed an EXIT chart analysis for these codes, which presents the challenge of the combination of analog and digital check nodes, something not encountered in other works. By assuming a linear relationship between the variances of the LLR messages in both types of CNs, very precise EXIT charts are obtained for the case of AWGN and fast fading Rayleigh channels. The predicted BER vs SNR curves are very close to the results obtained through simulations.
6 Appendix
6.1 Obtaining κ
 1
Start with an initial value of κ in (12) (say κ=1), and choose a value for \(\sigma _{R,A}^{2}\) so that the corresponding value of the mutual information computed by the PDF in (11) is in the range (0.5,0.9). For the value of \(\sigma _{R,A}^{2}\) under consideration, generate the extrinsic messages passed from the VN to the RCM and LDGM check nodes according to (14) and (15), respectively.
 2
Run one iteration of the sumproduct algorithm to obtain the extrinsic LLR messages passed from each LDGM and RCM check nodes to the VN, and obtain their empirical conditional PDFs.
 3
Define κ_{1} as the ratio between the variances of the empirical conditional distributions of RCM and LDGM check nodes obtained in step 2.
 4
Repeat the previous 3 steps, using κ_{1} as the initial value for κ, until the value of κ_{1} in step 3 is close enough to the value of κ_{1} in the previous iteration.
 5
Set κ=κ_{1} in the distribution (12).
Footnotes
Notes
Funding
This work was supported in part by the Spanish Ministry of Economy and Competitiveness through the CARMEN project (TEC201675067C43R), the COMONSENS network (TEC201569648REDC) and by NSF Award CCF1618653.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Authors’ contributions
JGF and PMC conceived the research question. IG and PMC proved the main results. IG, PMC, and JGF wrote the paper. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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