In this paper we study how power line communications are affected by different coding techniques. We investigate the case of LDPC coding scheme and its application to the PLC channel. The modulation method applied is BPSK while the transmission technique used is OFDM. We study regular LDPC codes, which is the worst case scenario concerning the performance of LDPC codes. The OFDM symbol size is initially defined at 4096 carriers, since certain characteristics of LDPC codes, like their sparse parity check matrix, imply that a larger block size improves their performance. In this paper, the coding and decoding technique have firstly been applied to a realistic power line channel. The attenuation caused by the PLC channel is derived from actual measurements. We test the performance of LDPC codes of ½ code rate, by means of BER, compared to Reed–Solomon codes of similar code rate and convolutional codes, when they are all applied to the PLC channel and under the same conditions. Additionally, impulsive noise is generated and the system’s performance is tested under the coding schemes mentioned above. Afterwards, we examine how different code rates affect the BER. All the above results are obtained via computer simulations.
Performance Evaluation of LDPC Codes on PLC Channel Compared to Other Coding Schemes Nikoleta Andreadou, Costas Assimakopoulos and Fotini-Niovi Pavlidou Aristotle University of Thessaloniki, Dept of Electrical & Computer Engineering, Telecommunications Division, Panepistimioupolis of Thessaloniki, 54124, Thessaloniki, Greece +302310994192 e-mail: nandread@auth.gr Abstract— In this paper we study how power line communications are affected by different coding techniques We investigate the case of LDPC coding scheme and its application to the PLC channel The modulation method applied is BPSK while the transmission technique used is OFDM We study regular LDPC codes, which is the worst case scenario concerning the performance of LDPC codes The OFDM symbol size is initially defined at 4096 carriers, since certain characteristics of LDPC codes, like their sparse parity check matrix, imply that a larger block size improves their performance In this paper, the coding and decoding technique have firstly been applied to a realistic power line channel The attenuation caused by the PLC channel is derived from actual measurements We test the performance of LDPC codes of ½ code rate, by means of BER, compared to Reed–Solomon codes of similar code rate and convolutional codes, when they are all applied to the PLC channel and under the same conditions Additionally, impulsive noise is generated and the system’s performance is tested under the coding schemes mentioned above Afterwards, we examine how different code rates affect the BER All the above results are obtained via computer simulations Keywords— Power Line Communications (PLC), Low Density Parity Check Codes (LDPC) I INTRODUCTION The past years there has been a great demand for data and voice transmission Alternative ways of communications are emerging This search for new ways of transferring information has introduced Power Line Communications (PLC) as an innovative manner of information exchange The benefits from implementing this emerging technology are remarkable, starting from the fact that there is no need for new infrastructure, which could be both time consuming and expensive In addition, bearing in mind that no new wires are required, power line communications become even more appealing On the contrary, there are several drawbacks concerning this technology, which need to be overcome for better system’s performance One main drawback is that the network was not originally designed for high frequency signals; therefore it introduces great variance to different signal components Interference is another major problem of the power line communication signal Since the load on the network varies with time, there is an unpredictable impedance introduced, which deteriorates the system’s performance Furthermore, due to this changing load on the network, impulsive noise is added to the communication signal, which makes it even harder for the data to be recovered at the receiver All the above mentioned problems imply that there is a lot to be studied in the field of power line communications, so that satisfactory performance is obtained, [1] – [3] In this paper we focus on the coding and decoding techniques of the signal processing Convolutional and Reed-Solomon codes are two coding schemes commonly used, [1] Nevertheless, the past few years there has been a rapid development of Low Density Parity Check (LDPC) codes These codes [4], became the target of research and investigation just a few years ago, after the evolution of decoding technology Their main advantage is that their performance can reach high levels in many channels, [5] Apart from their great error correcting performance, they don’t require an interleaver at the signal’s transmission, because of their random construction However, the encoding procedure should be carefully designed in order to avoid encoding complexity, [6], [7] It should also be mentioned that LDPC codes are divided into two categories, regular and irregular codes, with the latter ones having a better performance In this study we compare LDPC codes on a PLC channel against widespread coding schemes, such as Reed – Solomon and convolutional codes, since –to the best of our knowledge- such a comparison has not been done yet Specifically, regular LDPC codes are investigated in order to obtain the worst case scenario from this set of codes against other coding techniques A problem that arises is that the power line channels are not described specifically by one channel model, [2] The channel characteristics can be time and frequency dependent, whereas the noise added to the telecommunication signal by the channel is related to the network’s load and location, as well as the current frequency, [3] Therefore, actual measurements were performed on a PLC channel The transmission technique we apply is OFDM, since this scheme is commonly used, [1] The rest of the paper is arranged as follows Section II describes the code construction, the coding and decoding technique used In Section III it is described how the measurements on the PLC channel are carried out and the way the results are derived In Section IV the simulation results are displayed, while the conclusions are drawn in Section V ⎡1 ⎢1 ⎢ ⎢0 H =⎢ ⎢1 ⎢0 ⎢ ⎢⎣0 1 1 0 0⎤ 0 0 1 0⎥⎥ 1 1 0 0⎥ ⎥ 0 1 0 1 1⎥ 0 1 0 1 1⎥ ⎥ 1 0 0 1⎥⎦ II LDPC CODING TECHNIQUE A Characteristics of LDPC codes LDPC codes, which are under investigation in this paper belong to the group of linear block codes, meaning that the uncoded word (u) feeding the encoder consists of k bits and is converted to a codeword (c) of n bits via a generator matrix G The two words are connected through equation (1): c=u*G (1) However, when this generator matrix is utilised in the encoding process, the system’s complexity increases; therefore the parity check matrix H, which is derived from G is used instead, [8] A codeword c is valid if the subsequent equation is true [9]: H * cT = (2) It should be mentioned here that the current code rate k/n determines the size of the parity check matrix, which is specified as (n – k) x n, [10] The main characteristic of the parity check matrix is its sparseness, denoting that it constitutes mostly of zeros, whereas the aces are placed in a random way, [11] Depending on whether or not the parity check matrix has a uniform column weight, LDPC codes are distinguished between regular and irregular respectively Regular codes have constant row weight, as well, [8] B Bipartite Graphs An LDPC code can be represented by a bipartite graph, which facilitates their survey Such a graph consists of two groups of nodes, the check nodes and bit nodes Their number is determined by the parity check matrix size, meaning that if its size is mxn, then the graph will have m check nodes and n bit nodes respectively Each bit node represents one codeword bit, while each check node stands for one row in the parity check matrix, [8], [12] The nodes are connected to each other in case there is an ace in the corresponding position of each row in the parity check matrix, [5] For example, if there is an ace in the third column and fifth row of the matrix, then the third bit node will be connected to the fifth check node An example of a bipartite graph and its equivalent parity check matrix is illustrated in the following graph C Encoding Procedure Primarily, the parity check matrix should be constructed In order to minimize the encoding complexity, it is suggested in [6] that the parity check matrix is in almost lower triangular form Figure An example of a parity check matrix and its corresponding bipartite graph Therefore, we design a parity check matrix with aces in random positions and afterwards we perform the necessary transformations so that the matrix gets a lower triangular form while the column and row weight remain constant, since we deal with regular LDPC codes The size of the matrix is determined by the desired code rate, as mentioned above After the proper matrix is designed, we go on with the encoding procedure, where the parity bits are produced by further matrix calculations also taking into account the data entering the encoder, as recommended in [6] D Decoding Procedure Generally, the decoding procedure followed in LDPC codes is relied on an iterative algorithm, the Message Passing Algorithm or Belief Propagation Algorithm Several versions of this iterative algorithm have been presented [9], [13], [14], but its main structure is similar The bipartite graph helps for its comprehension According to this algorithm, at each iteration round a message is sent from each bit node to its neighbouring check nodes, which is an estimation about the exact value of the corresponding codeword bit this node represents Subsequently, all the arriving messages at each check node are processed and messages are sent back to the neighbouring bit nodes, so one iteration round is completed Afterwards, the messages are processed and the bit nodes send information to check nodes, so the algorithm is repeated The main feature of the algorithm is that the messages sent from the nodes, contain extrinsic information As a result, the message sent by a check node to one bit node, is based on the information received by the check node from all the other bit nodes except for this particular bit node The situation is similar when it comes to messages sent from bit nodes to check nodes At characterized by their position in the frequency spectrum, their number and their depth -30 -35 -40 Channel attenuation in dB the end of each iteration round, a codeword is produced after calculations on the bit nodes’ messages Subsequently, a check is carried out about the codeword’s accuracy The algorithm stops if the codeword produced is valid, which is tested via equation (2), or if a predefined number of iterations is completed, [11] The difference between the algorithm versions lies in the nature of the messages sent by the nodes So, the messages can be log-likelihood ratios [14] or not [13] In this paper, after experimenting on both techniques, we reached the conclusion that the first method applies better, so the messages sent at each iteration round of the decoding procedure are loglikelihood ratios -45 -50 -55 -60 -65 -70 III PREPARATION FOR THE SIMULATIONS -75 A Channel measurements The signal’s attenuation through power lines was measured in the laboratory and in-house An Anritsu network analyzer and two high-pass filters were used to protect the analyzer from the mains 50Hz The special filters and isolation transformers were constructed for this purpose in our lab Those appliances introduce an additional attenuation to the signal injected and measured The impact of those auxiliary tools to the measurements procedure has been measured and subtracted later when the data were processed Special care was also taken to reduce the impedance mismatching effect when the analyzer is connected to wall outlets The transfer function measurements revealed the wellknown characteristics of the power line channel The frequency dependence of the attenuation is obvious In figure 2, several powerline transfer functions are presented The overall attenuation is dependent on the physical length between transmitter and receiver and the position of the connected loads The notches of the transfer function are 10 15 Frequency in MHz 20 25 30 Figure Several powerline transfer functions measured in lab and in apartment -10 -20 Noise power density in dBm The goal of this paper is to evaluate the system’s performance in terms of the BER achieved versus the current SNR The noise level is considered to be predefined, thus the SNR defines the desired signal power level However, since the computer routines for coding and modulation techniques, utilise data in the form of “0” or “1”, the coded and modulated bits of information are properly converted into power level values via the given SNR After this conversion, the attenuation introduced by the PLC channel and obtained by the measurements can be applied to the transformed coded and modulated data The background noise introduced by the channel is also taken into account, by adding a proper value of noise to our data From the receiver’s point of view, during our simulations, an equivalent transformation is performed, so as to obtain the corresponding data bits to enter the demodulator and decoder After such a process has been carried out, the BER can be calculated by comparing the output bits to the input data As it is mentioned above, we performed measurements, so as to include the attenuation introduced by the PLC channel and the noise -30 -40 -50 -60 -70 -80 -90 10 15 20 Frequency in MHz 25 30 Figure Power spectrum density of the powerline noise measured in the lab The noise is measured in the frequency domain using a spectrum analyzer Hence, the power spectral density of the power line noise is measured An automatic measurements set up was used Special care was taken to equalize “a-posteriori” the attenuation of the noise samples passing through the filter and the isolation transformer before entering the front end of the spectrum analyzer The frequency range of interest was swept and noise samples were transferred from the spectrum analyzer to a computer’s hard disk The measurements took place in the lab and in an apartment during several different hours per day The power line noise components have been determined in several papers According to the measurement techniques and technical equipment available there were detected three, four or five components in the literature Nevertheless, all of them deduce that there is a background noise component (such as colored Gaussian and narrowband noise due to radio stations transmitting in frequencies less than 30MHz) and impulsive noise either periodic (synchronous or asynchronous to the mains frequency) or non-periodic asynchronous From a telecommunications point of view the power line receiver is affected in the time domain from the superposition of all those noise components A great number of noise samples were collected The noise samples from 1.15MHz up to 30MHz are shown in figure The noise level has a great deviation It is strongly dependent on three parameters, the time of the day that the measurements are carried out, the place of the measurements and the frequency It should be mentioned that the noise measured in the lab was slightly stronger compared to the noise in the apartment When human activity increased simultaneously with the number of computer fans operating in the lab, while performing the measurements, the noise power density also increased The best and worst noise scenario is the lower and upper envelope as depicted in figure Generally, the noise is stronger in the lower part of the frequency band Since the noise level has a great variance and dependence on the time and place, extracting the average values has no practical interest as those values will be dependent on time and place IV SIMULATION RESULTS In this section we show the BER performance of the various coding schemes at the PLC channel Firstly, we compare the performance of LDPC codes versus convolutional and Reed – Solomon codes of 1/2 code rate BPSK modulated OFDM symbols are loaded with bits The input bits constitute a block of 2048 bits, while by using a code rate of 1/2, the output block size is 4096 This is done because it is widely known that LDPC codes work better when large block sizes are utilised, [13] Whereas LDPC and convolutional codes can be applied with a code rate of 1/2, the concept of Reed – Solomon codes forces us to use a similar code rate, for better performance Therefore, Reed-Solomon codes are used with a code rate of 15/31 and dummy carriers are necessary so as the OFDM symbol is completed In any event, these dummy carriers are not taken into account at the calculation of BER Furthermore, the number of OFDM symbols transmitted at our simulations is 512, to induce more accurate results concerning the BER In figure 4, the BER versus signal to noise ratio for LDPC, convolutional and Reed – Solomon codes is presented under the transmission conditions described above Thus, it can be concluded from figure that the three coding schemes have comparable performance for lower SNR values, while there is a small superiority of Reed – Solomon codes for such SNR values However, it is apparent that for higher SNR rates, there is a sharp decrease concerning the BER of LDPC codes, indicating that the latter codes outperform the other coding techniques Reed – Solomon codes are superior to convolutional codes, while both coding methods experience a decline in BER as the SNR increases Figure BER versus SNR for LDPC, convolutional and Reed – Solomon codes of 1/2 code rate at the PLC channel We also compared the three coding schemes under different code rates For this purpose, code rates of 1/3 and 2/3 were employed The input block size remained at 2048 bits, whereas the output bits were 6144 and 3072 respectively Similarly to the previous case of 1/2 code rate, Reed – Solomon codes were used with code rates of 11/31 and 21/31 correspondingly, due to the fact that their implementation is different than the other coding techniques, while dummy symbols were also essential during transmission Since the differences between the code rates are trifling, the conclusions drawn are of vital importance Figures and illustrate the performance of the three coding methods on the PLC channel with different code rates Figure BER versus SNR for LDPC, convolutional and Reed – Solomon codes of 1/3 code rate at the PLC channel As mentioned before, the PLC channel introduces an attenuation that depends on time, location and the load of the network However, by plugging and unplugging several devices, impulsive noise is added Impulsive noise is higher than background noise In order to create this worst case scenario, we inserted extra impulsive noise striking all the OFDM symbols This was easily performed with the simulations, which tested the performance of the several coding techniques Figure shows how the three coding schemes responded under these circumstances Figure BER versus SNR for LDPC, convolutional and Reed – Solomon codes of 2/3 code rate at the PLC channel Likewise the case of 1/2 code rate, figures and follow a similar curve pattern Whereas Reed – Solomon codes are slightly superior to the other coding schemes for lower signalto-noise ratios, the improved SNR values for LDPC codes operate better Furthermore, it is also evident that the deep fall in the BER plot occurs for higher SNR rates in figure and lower ones in figure This is explicable by the fact that for higher code rates, the bits redundancy is decreased at the expense of the signal’s quality Thus, at the presence of more parity bits, better protection is offered to the original data bits This can be proved in figure 7, where the BER performance is plotted for different code rates used during our simulations The case of LDPC codes is the only one pointed out, since these codes are under survey in this paper The other coding methods show a similar trend, which can be further examined in figures 4, and Figure BER versus SNR for LDPC, convolutional and Reed – Solomon codes of 1/2 code rate at the PLC channel, when extra impulsive noise is inserted From figure it is derived again that LDPC codes outperform the other coding schemes for higher SNR values, while for lower ones, the BER achieved is at similar levels By comparing figure to figure 8, we conclude that the extra noise inserted has an impact on the BER obtained The degradation of the system’s performance is inevitable; however, it is kept at acceptable levels In addition, it is observable from all of the graphs, that after the rapid declination of the BER, its value continues decreasing but in a smoother fashion This behaviour is probably introduced by the simulation accuracy The more extensive the simulation runs, the sharper the inclination of the curve is Nevertheless, LDPC codes also prevail in the BER performance V CONCLUSIONS Figure BER versus SNR for LDPC codes of 1/2, 1/3 and 2/3 code rate at the PLC channel In this paper we have investigated the utilization of LDPC codes, which have shown great progress during the last several years, in the Power Line Communications channel We have taken into account different code rates and compared their performance to other coding methods, such as Reed – Solomon and convolutional codes We used actual measurements of the PLC channel as well as computer simulations, to indicate that LDPC codes can achieve better performance than the other coding techniques This is considered of vital importance, since LDPC codes can be considered as a serious candidate for future use in coding techniques REFERENCES [1] HomePlug 1.0 Technology White Paper, HomePlug Powerline Alliance, pp 1- [2] P L Katsis, G D 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LDPC-Coded MIMO OFDM Systems”, IEEE Trans Signal Processing, vol 52, no 2, pp 348-361 Feb 2004 [13] Michael G Luby, Michael Mitzenmacher, M Amin Shokrollahi, and Daniel A Spielman, “Improved LowDensity Parity-Check Codes Using Irregular Graphs”, IEEE Trans Information Theory, vol 47, no 2, pp 585-598, Feb 2001 [14] Hiroki Nakagawa, Daisuke Umehara, Satoshi Denno and Yoshiteru Morihiro, “A Decoding for Low Density Parity Check Codes over Impulsive Noise Channels”, IEEE ISPLC, pp 85-89, 2005 ... decrease concerning the BER of LDPC codes, indicating that the latter codes outperform the other coding techniques Reed – Solomon codes are superior to convolutional codes, while both coding methods... the performance of the three coding methods on the PLC channel with different code rates Figure BER versus SNR for LDPC, convolutional and Reed – Solomon codes of 1/3 code rate at the PLC channel. .. performance to other coding methods, such as Reed – Solomon and convolutional codes We used actual measurements of the PLC channel as well as computer simulations, to indicate that LDPC codes can