LDPC Code Design for
Noncoherent Physical Layer Network Coding
Abstract
This work considers optimizing LDPC codes in the physicallayer network coded twoway relay channel using noncoherent FSK modulation. The errorrate performance of channel decoding at the relay node during the multipleaccess phase was improved through EXITbased optimization of Tanner graph variable node degree distributions. Codes drawn from the DVBS2 and WiMAX standards were used as a basis for design and performance comparison. The computational complexity characteristics of the standard codes were preserved in the optimized codes by maintaining the extended irregular repeataccumulate (eIRA). The relay receiver performance was optimized considering two modulation orders using iterative decoding in which the decoder and demodulator refine channel estimates by exchanging information. The code optimization procedure yielded unique optimized codes for each case of modulation order and available channel state information. Performance of the standard and optimized codes were measured using Monte Carlo simulation in the flat Rayleigh fading channel, and error rate improvements up to dB are demonstrated depending on system parameters.
I Introduction
The twoway relay channel (TWRC) models two source nodes outside radio range exchanging information through a single relay node in range of both. Physicallayer network coding (PNC) [1] may applied in the TWRC to increase throughput by reducing the number of time slots required for exchange. The exchange between the sources is broken into two phases: the multipleaccess (MA) phase and broadcast phase. During MA, the sources transmit at the same time to the relay, and the signal received at the relay is the electromagnetic sum of signals transmitted by the sources. During the broadcast phase, the relay transmits the networkcoded sum of signals to the tranceivers, and each performs network decoding to recover the information transmitted by the opposite sources.
Previous work developed a demodulation and decoding scheme for the relay in the PNCcoded TWRC using Mary FSK modulation [2] [3]. This scheme supports iterative decoding between decoder and demodulator to improve errorrate performance of the received networkcoded information bits. The receiver supports poweroftwo modulation orders and improved decoding performance based on available channel state information (CSI). A simulation regime was performed to demonstrate the performance of the scheme using Turbo decoding at the relay.
The errorrate performance of LDPC codes depends on the structure of the parity check matrix and thus the properties of the Tanner Graph and degree distribution. In this work we apply an optimization technique which identifies variable node degree distributions likely to yield good codes using extrinsic information tranfer characteristic (EXIT) curve fits [4]. LDPC codes taken from the DVBS2 and WiMAX standards are used as a basis for design and performance comparison, and their variable node degree distributions are optimized to yield codes exhibiting errorrate performance superior to the standard codes. The complexity characteristics of the standard codes are maintained in the optimized codes by preserving the extended irregular repeataccumulate constraint.
A variety of optimization techniques exist for improving LDPC code performance. Random permutation matrices, large girth optimization, and column weight optimization have been applied to WiMax standardized codes yielding gains up to dB [5]. Joint design of and LDPC codes under carrier frequency offsets demonstrate the relationship between system properties and code performance [6].
A PNC scheme compensating for correlation among transmitted symbols introduced by channel coding to improve decoder performance at the relay has been considered [7]. In this work it is demonstrated that LDPC codes having different degree distributions are preferred in different SNR regimes. Roughly, smaller degrees are preferable at low SNR and higher degrees at high SNR. A PNC system used as a basis for optimizing LDPC codes via EXIT analysis demonstrated performance dB from capacity [8].
The rest of this work is organized as follows. Section II develops the system model applied for code optimization and errorrate simulation. Section III describes the variable node degree distribution optimization technique in informationtheoretic terms, while Section IV describes the computational procedure followed to generate the optimized codes and demonstrates the performance of the codes versus standard. Section V provides concluding remarks.
Ii System Model
This section describes the model applied for LDPC code optimization and errorrate simulation. The model is depicted in Figure 1.
Iia Transmission by Source Nodes
The source nodes generate binary information sequences having length . A rate LDPC code is applied to each , generating a length channel codeword, denoted by . The codeword is passed through an interleaver, modeled as a permutation matrix having dimensionality . Let denote the set of integer indices corresponding to each FSK tone, where is the modulation order. The number of bits per symbol is . The codeword at each node is divided into sets of bits, each of which is mapped to an ary symbol , where denotes the symbol number, and denotes the source.
The modulated signal transmitted by source during signaling interval is
(1) 
where is the transmitted signal, is the carrier frequency, and is the symbol period.
The continuoustime signals are represented in discretetime by the set of column vectors . The column vector is length , contains a at vector position , and elsewhere. The modulated codeword from source is represented by the matrix of symbols , having dimensionality , where .
IiB Channel Model
All channels are modeled as flatfading channels having independent gains for every symbol period. The gain from node to the relay during a particular signaling interval is denoted by . The gain is represented as , where is the received amplitude and is the phase. The received amplitude has a Rayleigh distribution with parameter , and is the phase having a Uniform distribution over interval .
Consider transmission of a single frame of symbols to the relay. The received frame is
(2) 
where is an diagonal matrix of channel coefficients having value at matrix entry and elsewhere, and is an noise matrix. A single column of represents a single signaling interval, is denoted by , and referred to as a channel observation. In terms of this definition, , where denotes the th channel observation. Denote the th column of by . Each column is composed of zeromean circularly symmetric complex Gaussian random variables having covariance matrix ; i.e., . is the onesided noise spectral density, and is the by identity matrix. The noise spectral density is defined as , where is the bitenergy to noise ratio in decibels . Increasing modulation order decreases the noise spectral density to reflect an increase in energy persymbol when utilizing constant energy perbit.
IiC Relay Reception
The goal of the digitalnetwork network coding (DNC) relay receiver is to detect the networkcoded combination of information bits transmitted by the end nodes, . The relay receiver takes as input the frame of channel observations . The symbols transmitted by the sources are assumed to be received in perfect synchronization at the receiver. Demodulation and iterative channeldecoding are applied to detect . Define the network codeword as
(3) 
where denotes the th bit mapped to the th symbol transmitted by source . Since the channel code considered is a systematic linear code, forms a code from the codebooks used by the source nodes, thus, the channel decoding operation yields a hard decision on the network coded message bits .
The demodulator is implemented using supersymbol probability mapping and DNC soft bitmapping (SOMAP), having formulation described in [3].
The supersymbol probability mapper takes as input the matrix of received symbols and produces estimates of the probability of receiving each super symbol , defined as the tuple
(4) 
to produce estimates , . The cardinality of is , thus, the number of probability estimates computed at the relay during each symbol period grows exponentially with modulation order. This computational complexity is a practical consideration for implementing the relay receiver.
The supersymbol probability estimates are passed to the DNC SOMAP along with with extrinsic decoder feedback to produce extrinsic information , likelihood ratios of the networkcoded bits communicated by the frame.
The extrinsic information is deinterleaved to produce and passed to the decoder. The decoder refines the estimate of , producing new extrinsic information which is interleaved to produce and returned to the SOMAP. After a specified number of iterations, a hard decision is made on the network coded bits .
The transmitted symbol energies and noise spectral density are known at the demodulator. The demodulator may operate under several cases of channel state information (CSI) [3]: the case in which the gain are completely known (full CSI), the case in which only the fading amplitudes are known (partial CSI), and the case in which no information about the gains is known (no CSI).
Since the focus of this work is on the relay reception phase, details of the relaytosource broadcast phase are omitted. Interested readers are referred to [3].
Iii LDPC Code Optimization
This section describes the application of a procedure for optimizing LDPC code errorrate performance based on [4] under a range of receiver configurations to the DNC relay reception phase by varying the LDPC code degree distribution. Relevant LDPC concepts are introduced, followed by a description of metrics relevant to optimization. A theoretical description of the optimization technique is provided followed by the procedure taken.
The goal is to produce LDPC code designs having the same rate and complexity as offtheshelf codes from the DVBS2 and WiMAX standards, while improving error rate performance through optimization of the parity check matrix. The codes are constrained to the class of extended irregular repeat accumulate (eIRA) codes. This constraint ensures computational efficiency by guaranteeing systematic encoding [9]. The codes considered are checkregular, meaning that every check node has the same degree, .
An LDPC code is a linear block code having a sparse parity check matrix , which is an binary matrix. An alternative representation for the parity check matrix is the Tanner Graph, a bipartite graph in which one of the sets contains variable nodes, and the other contains check nodes [10]. Each column of corresponds to a variable node, and each row a check node [10]. A entry in at row and column corresponds to an edge between the th check node and th variable node. The degree of a node is its number of incident edges, thus, the number of ’s in each row and column for each check and variable node, respectively.
The variable and check nodes may be modeled as softinput, softoutput decoders denoted as VND and CND, respectively [4]. The VND decoder takes a channel LLR and apriori information as input and produces aposteriori information as output. The extrinsic information is defined as . The mutual information between the and the corresponding networkcoded bit is , while the mutual information between and the networkcoded bit is .
The CND decoder takes apriori information as input and produces aposteriori information as output. The extrinsic information is . The mutual information between the and the corresponding networkcoded bit is , while the mutual information between and the networkcoded bit is .
An EXIT chart visualizes the transfer characteristic of a decoder by plotting the apriori mutual information against the extrinsic. It has been shown that matching the VND and CND transfer characteristics of a particular LDPC code through selection of variable node degrees improves error rate performance [11].
Iiia Optimization through Selection of Variable Node Degree
This section describes the analytic formulation for degree optimization. The degree distribution is defined as the set containing the degrees assigned to the variable nodes, and the number of variable nodes taking each particular degree. Denote a degree distribution as , where is the number of variable nodes having degree . For a particular code, all check node degrees are the same, thus, the code is checkregular. Denote the check node degree for a particular code as . Define as the fraction of variable nodes having degree
(5) 
each taking a value , and .
A realizable degree distribution satisfies the constraints imposed by the LDPC code parameters. The sum total of edges incident on the variable and check nodes must be equal. The number of edges incident on variable nodes having degree is , thus, the total number of edges incident on all variable nodes is
(6) 
and the total number of edges incident on the check nodes is
(7) 
Equating and and rearranging,
(8) 
A particular degree distribution is selected by choosing parameters and which satisfy (8) and the sum constraint on . The challenge is to select a which optimizes error rate performance for particular channel conditions and relay receiver configurations.
The optimization procedure requires generating EXIT curves corresponding to the variablenode decoder (VND) and checknode decoder (CND). The VND curve is described by the variable node degree distribution and the mutual information computed between the DNCSOMAP output and the network coded bits. The CND curve is completely described by the degree of the check nodes , and is fixed for all variable node degree distributions.
Generation of the VND curve begins by computing the mutual information between the softvalues at the output of the DNCSOMAP and the networkcoded bits, denoted as the detector characteristic . is the mutual information between the DNCSOMAP apriori input and the networkcoded bits, and is the signaltonoise ratio of the supersymbol. The detector characteristic is generated by Monte Carlo simulation under the assumption that the apriori input is conditionally Gaussian. The Gaussian assumption can be verified empirically using histograms of several decoding runs [12].
The VND curve is found by computing a separate curve for each variable node degree and combining each to form the final curve describing the entire variable node detector. The curve corresponding to the  degree is given by
(9) 
where the function is defined in [13] and is computed using the truncatedseries representation presented in [14], and . The VND curve describing the entire detector is given by
(10) 
where is the fraction of edges incident on variable nodes of degree .
The CND curve is computed according to [4]
(11) 
where is the mutual information between the check node inputs and the networkcoded bits. Note that the CND curve is independent of the particular variable node degree distribution .
The curve fit for the VND and CND mutual information metrics corresponding to a particular degree distribution is performed by plotting the curves for a range of values of and noting the value in which the curves barely touch [4]. Specifically, the VND curve is plotted with on the horizontal axis and along the vertical. The CND curve is plotted with along the horizontal and along the vertical. The value of is varied from highest specified point to lowest, and the highest point for which the curves touch is defined as the EXIT threshold. Considering a particular relay receiver and channel configuration, the optimal variable node degree distribution is considered as the distribution having the lowest EXIT threshold.
Iv EXITOptimized LDPC Code Performance
This section presents the errorrate performance of EXIToptimized LDPC codes and the simulation optimization procedure used to generate the codes. Several cases of modulation order and relay receiver CSI are considered. LDPC codes defined by the DVBS2 and WiMax standards are used as reference codes.
Iva Optimization Procedure
The purpose of this subsection is to describe the procedure for generating EXIT curves describing the performance of selected degree distributions.
The procedure involves determining the detector EXIT characteristic through simulation for a chosen receiver configuration and channel model, and using this characteristic to generate the combined variable node/detector and check node characteristics over a range of SNR [4].
The notation following a bold variable denotes a vector element, for instance, denotes the th element of onedimensional vector and denotes the th element of twodimensional vector .
IvA1 Detector Characteristic
This section describes the computation of the detector characteristic for the DNC relay receiver. These values are computed via Monte Carlo simulation as follows

Select source and relay frame size , modulation order , received channel state information (CSI or no CSI) and SNR value.

Select a discrete range of apriori demodulator mutual information values, represented as the vector . In all of our experiments, the value provided visibly smooth detector transfer characteristics.

Simulate a frame of channelcorrupted relayreceived symbols according to the model given in Section II, having length and modulation order .

For every value of ,

Generate a vector of length apriori LLRs representing LDPC decoder feedback under the assumption that the feedback is conditionally Gaussian
(12) where [4] and is a length vector of i.i.d. samples of a standard normal distribution.

Compute the decoder extrinsic information according to the receiver formulation provided in [3].

Compute the simulated value of as [12]
(13) 
Numerically fit a thirdorder polynomial to the sampled detector characteristic using as the abscissa and as the ordinate, yielding coefficients
(14) Define function . This function approximates the detector characteristic and is used in the computation of the combined detector/decoder characteristic.

IvA2 Combined Detector/Decoder Characteristic
This subsection describes the simulation procedure for generating the combined detector/decoder characteristics.

Select a discrete range of apriori demodulator mutual information values, represented as the vector . In all of our experiments, the value provided visibly smooth detector transfer characteristics.

For every value of detector apriori mutual information ,

For every degree compute the apriori information input to the detector according to
(15) 
Compute according to (IIIA)
(16) where
(17)

The check node characteristic is completely determined by the parameters of the code and is not dependent on the properties of the detector output. Noting that compute the apriori CND characteristic according to (11)
(18) 
Base  
Standard  CSI  fit  
DVBS2  4  Partial  { 2:25920, 4:34560, 22:4320 }  11.9 
{ 2:25920, 33:4560, 30:4320 }  12  
{ 2:25920, 4:37152, 49:1728 }  12  
None  { 2:25920, 3:34560, 30:4320 }  12.3  
{ 2:25920, 4:34560, 22:4320 }  12.3  
{ 2:25920, 4:37152, 49:1728 }  12.4  
8  Partial  { 2:25920, 3:34560, 30:4320 }  9.5  
{ 2:25920, 35:3650, 3:35230 }  9.5  
{ 2:25920, 4:34560, 22:4320 }  9.7  
None  { 2:25920, 3:34560, 30:4320 }  9.8  
{ 2:25920, 35:3650, 3:35230 }  9.9  
{ 2:25920, 4:34992, 24:3888 }  10.1  
WiMax  4  Partial  { 2:672, 3:96, 3:1296, 9:240 }  12.9 
{ 2:672, 3:96, 3:1356, 11:180 }  12.9  
{ 2:672, 3:96, 3:1376, 12:160 }  12.9  
None  { 2:672, 3:96, 3:1248, 8:288 }  13.3  
{ 2:672, 3:96, 3:1296, 9:240 }  13.3  
{ 2:672, 3:96, 3:1356, 11:180 }  13.3  
8  Partial  { 2:672, 3:96, 3:1296, 9:240 }  10.4  
{ 2:672, 3:96, 3:1356, 11:180 }  10.4  
{ 2:672, 3:96, 3:1376, 12:160 }  10.4  
None  { 2:672, 3:96, 3:1356, 11:180 }  10.8  
{ 2:672, 3:96, 3:1376, 12:160 }  10.8  
{ 2:672, 3:96, 3:1392, 13:144 }  10.8  
IvB Optimization Results
In this subsection we apply the optimization procedure to improve performance of standardized DVBS2 and WiMax LDPC codes. The variable node degree distributions for two standardized codes are optimized, and their error rate performance is compared under simulation in the DNC relay reception phase.
The first code is taken from the DVBS2 standard [15], having codeword length and rate . The second code is specified by the WiMax standard [16], having length and rate . The parity check matrices for these codes are specified in their respective standards. Note that the WiMax standard specifies two distinct parity check matrices for rate codes, denoted as 2/3A and 2/3B. We consider case 2/3A. These codes satisfy the eIRA constraint and are checkregular, with for the DVBS2 code and for WiMax.
The parity check matrix for each standard codes contains a submatrix satisfying the extended irregular repeataccumulate (eIRA) constraint. This constraint simplifies encoding and decoding complexity. The submatrices comprising are
(19) 
where satisfies the eIRA constraint and has rows and columns. To preserve the complexity advantages of eIRA, we retain exactly as specified by the standards and consider optimizating the variable node degrees specifying . Retaining places constraints on such that codes based on DVBS2 have , and codes based on WiMax have and . All other degrees may be chosen freely.
Variable node degree optimization is performed under several cases of receiver configuration and channel state information, specifically, modulation orders with and without CSI, for a total of four configurations. A range of degree distributions is considered for each code and receiver configuration, and the best performing degree distribution of each is realized and simulated.
Considering the DVBS2 LDPC code, the number of distinct variable node degrees is , and the degrees considered are all combinations of the sets , , and which satisfy the constraints for realizable codes described in Subsection IIIA. Considering the WiMax code, the number of distinct variable node degrees is , and the degrees considered are all possible combinations of , , , and .
For every receiver configuration and degree distribution , the EXIT threshold is computed to discover the the single variable node degree distribution having the best performance. For each distribution, the threshold is defined as the highest SNR value for which the VND and CND decoder EXIT curves touch. The best performing distribution is defined as minimizing this SNR value. An example is shown in Fig. 2.
The optimization results are shown in Table I. For each degree distribution , a code is realized having parity check matrix satisfying the degree distribution and maintaining the eIRA constraint. The parity check matrix is generated randomly. In the interest of space, further details of parity check matrix generation are omitted here.
IvC Errorrate Performance of Optimized Codes
The errorrate performance of the optimal randomly generated codes is compared to the standardized codes via Monte Carlo simulation according to the model described in Section II.
The source nodes generate information sequences having length for the DVBS2 codes and for WiMax. The information sequences are channel encoded to produce codewords having length for DVBS2 and for WiMax. These codewords are mapped to FSK symbols. Modulation orders are considered. The modulated symbols are transmitted over the channel and received at the relay. The relay performs decoding considering the cases of having channel state information and no channel state information.
In all cases the relay receiver performs decoding iterations before making a hard decision on the networkcoded bits. The standard codes are simulated for two cases of decodertodemodulator extrinsic information feedback: BICM, in which no information is fed back from decoder to demodulator, and BICMID, in which information is fed back to the demodulator after each decoding iteration. Simulation of the randomly generated codes always uses BICMID.
Error rate performance for DVBS2 and WiMax codes, , and for partial and no CSI are shown in Figures 3, 4, 5 and 6. The degree distributions for each plotted code are denoted on the legends as , referring to a degree distribution listed in the column in Table I.
Consider the results for DVBS2. For and partial CSI, the best performing optimized code shows an energy efficiency improvement of approximately dB, while the no CSI case shows improvement of roughly dB. For and partial CSI, the improvement is about dB, while no CSI shows dB of improvement.
Now consider WiMax results. For and partial CSI, performance gain of the best optimized code over the standard code is about dB, while for the no CSI case, improvement is about dB. For and partial CSI, improvement is dB, while no CSI shows dB of improvement.
V Conclusion
This work has presented an optimization procedure for LDPC Tanner Graph variable node degree distribution in the physicallayer network coded two way relay channel. The codes are optimized for relay reception in the multiple access phase. LDPC Codes from the DVBS2 and WiMax standards are used to establish baseline errorrate performance. Optimized codes are generated under a variety of system configurations: modulation order and , The optimized codes outperform the standard codes by margins from to dB.
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