Performance of Array Codes on Power Line Communications Channel

Performance of Array Codes on Power Line Communications Channel Nikoleta Andreadou and Fotini-Niovi Pavlidou Aristotle University of Thessaloniki, Dept. of Electrical & Computer Engineering, Telecommunications Division, Panepistimioupolis of Thessaloniki, 54124, Thessaloniki, Greece +302310994192, e-mail: [email protected] Abstract— In this paper we investigate the performance of array codes and in particular Generalised Array Codes (GAC) and Row and Column Array Codes (RAC) as coding schemes in the Power Line Communications (PLC) environment. We apply different code rates and modulation techniques and we examine how these codes perform in the hostile channel of power lines in terms of BER (bit error rate) versus Eb / N0 (signal to noise power). In addition, we propose a variation concerning the construction of GAC codes. As a result we get a code which has the same code rate with the original GAC code; however it is derived from an altered construction. QPSK and 8PSK modulation schemes are being used for the system’s realisation. Furthermore, we compare the codes’ performance with convolutional codes. For the power line channel, we exploit Zimmermann’s model, while Middleton’s class A noise model is used to represent the channel’s noise. For reasons of completeness, we examine the effect of the noise model’s parameters on the channel’s performance, by using different values for them. Finally, the transmission technique of OFDM is also taken under consideration, meaning that data is transmitted in terms of blocks simultaneously. Keywords— Power Line Communications (PLC), Generalised Array Codes (GAC), Row and Column Array Codes (RAC).

I. INTRODUCTION The increasing demand on data transmission the last years, due to the rapid development in the telecommunications area, has introduced new ways of transferring information. One of the emerging new technologies is power line communications (PLC). However, the power line channel was not originally designed for transmission of telecommunication signals, thus making it a hostile environment for high frequency signals. In addition, the load on the network varies with time and as a result it introduces unpredictable impedance. Interference is another major problem of the power line communication network, which deteriorates the system’s performance. On the contrary, there are remarkable benefits from implementing this technology, starting with the fact that there is no need for new wires. It is actually this lack of new infrastructure that makes power line technology appealing, [1-2]. This work was supported by the Greek General Secretariat for Research and Technology (Program PENED 2003).

In this study we focus on array codes and in particular, Generalised Array Codes and Row and Column Array Codes and their effect on the power line system, since –to the best of our knowledge- these codes have never before been used in a power line system. These coding schemes are not complicated and simple to apply. Generally, the information bits are organised in terms of arrays, according to which the parity bits are produced. The modulation scheme following the encoding procedure depends on the number of data bits in each row of the array, meaning that different modulation schemes are suitable for each code, [3 – 5]. What we propose is a variation regarding the construction of the array during the encoding procedure, which results in a varied kind of GAC codes. We then compare the two code versions’ performances. It should also be mentioned here that the decoding procedure becomes feasible via the trellis decoding scheme for all the encoding cases. The results obtained are compared to widespread coding techniques, such as convolutional codes. When approaching the noise of the power line channel, most researchers are based on their own measurements in order to derive a noise model, [6 – 7]. For the realisation of our system, we implement Middleton’s class A model [8], which is found in the literature as the noise model for PLC channels, [9 – 10]. With the aim of checking how the model’s parameters affect the system’s performance, we use a range of values for two basic parameters and we compare the results. Similarly to the case of the noise model, many researchers perform measurements so as to model the power line channel, [11-12]. On the other hand, there are several popular channel models found in the literature. One of the first ones introduced is Philipps’ model, which takes into account the multipath scenario occurring in a power line network, [13]. Later on, Zimmermann’s [14] and Galli and Banwell’s [15] models were built. While Galli and Banwell’s model implies a multiconductor configuration, Zimmermann’s model is a frequency domain model which takes under consideration both the multipath scenario and the signal’s attenuation during its transmission through the power line network. We implement Zimmermann’s model, since it is a straightforward model and it is found a lot in the literature, [16 – 17]. The rest of the paper is arranged as follows. Section II describes the code construction, the coding and decoding technique used for RAC and GAC codes. In Section III the noise and channel models are being analysed. In Section IV

the simulation results are displayed, while the conclusions are drawn in Section V. II. A.

the case of our codes, we get the trellis diagrams illustrated in figures 1, 2 and 3.

RAC CODING AND DECODING TECHNIQUE

Encoding of RAC codes

Row and Column Array codes (RAC), which are under investigation in this paper, are constructed easily provided that the input data bits are formed in terms of arrays. Assuming that the array of information bits consists of k1 rows and k2 columns, the parity bits are produced in such a way [3], so as to result in an array of n1 rows and n2 columns. As a result, the code rate is k / n, with k = k1 * k2 and n = n1 * n2. The code that comes up can be written as (n, k, 4) RAC code. We examine the case of (9, 4, 4), (12, 6, 4) and (16, 9, 4) RAC code, for which the resultant matrices C1, C2 and C3 are shown by equations (1), (2) and (3) correspondingly.

⎡ x1 C1 = ⎢⎢ x3 ⎢⎣ p3

p1 ⎤ p2 ⎥⎥ p5 ⎥⎦

x2 x4 p4

Fig. 1. Trellis diagram of (9, 4, 4) RAC code with number of states Ns = 4 and trellis depth Nc = 4.

(1)

where xi , i = 1, … 4 are the information bits and pj, j = 1,…5 are the parity bits, p1 = x1 ⊕ x2 , p2 = x3 ⊕ x4 , p3 = x1 ⊕ x3 ,

p4 = x2 ⊕ x4 and p5 = p1 ⊕ p2 . ⎡ x1 C2 = ⎢⎢ x4 ⎢⎣ p3

x2 x5 p4

x3 x6 p5

p1 ⎤ p2 ⎥⎥ p6 ⎥⎦


(2)

where xi , i = 1, … 4 are the information bits and pj, j = 1,…5 are the parity bits, p 1 = x 1 ⊕ x 2 ⊕ x 3 , p 2 = x 4 ⊕ x 5 ⊕ x 6 , p3 = x1 ⊕ x4 p4 = x2 ⊕ x5 , p 5 = x 3 ⊕ x 6 and p6 = p1 ⊕ p2 .

⎡ x1 ⎢x C3 = ⎢ 4 ⎢ x7 ⎢ ⎣ p4

x2 x5 x8 p5

x3 x6 x9 p6

p1 ⎤ p2 ⎥⎥ p3 ⎥ ⎥ p7 ⎦

(3)

where xi , i = 1, … 4 are the information bits and pj, j = 1,…5 are the parity bits, derived in a similar way as in the previous cases. It should be mentioned at this point that each row of bits is transferred to one modulation symbol, thus resulting in transmission of bits per row. Consequently, the modulation technique depends on the information bits present per row. The larger the number of columns, the higher the modulation scheme required. In our case, the modulation schemes needed for the symbols’ transmission are QPSK, 8PSK and 8PSK respectively. B.

Decoding of RAC Codes

The decoding of RAC codes can be easily realised via a trellis diagram, which can be constructed following a certain procedure found in [3]. After carrying out this procedure for


In figures 1 the input and output vectors of the trellis diagram are displayed, whereas in figures 2 and 3 they are omitted due to reasons of clarification. C.

Encoding Procedure of GAC Codes

Generalised Array codes share a lot of similarities but also some main differences with Row and Column array codes. Like RAC codes, the coded bits are expressed in terms of n1 * n2 arrays and the code rate is k / n, with n = n1 * n2 and k the number of data bits. However, the main difference between

the two coding schemes is that in GAC codes the number of information bits per matrix row can be unequal, [4]. The GAC code matrix representing the encoded bits can be constructed after following the procedure in [4]. In our case, for the (8, 4, 4), the (12, 7, 4) and the (16, 5, 8) GAC code this procedure results in matrices given by equations (4), (5) and (6) respectively. ⎡ x1 x4 ⊕ p1 ⎤ ⎢x x 4 ⊕ p 2 ⎥⎥ (4) C8, 4 , 4 = ⎢ 2 ⎢ x3 x 4 ⊕ p 3 ⎥ ⎢ ⎥ ⎣ p 4 x4 ⊕ p4 ⎦ where xi, i = 1,…4 are the information bits, pj, j = 1,…4 are the parity bits, pi = xi, i = 1, 2, 3 and p4 = x1 ⊕ x2 ⊕ x3 .

⎡ x1 x4 p1 ⊕ x7 ⎤ ⎢x x5 p2 ⊕ x7 ⎥⎥ 2 (5) ⎢ C12 , 7 , 4 = ⎢ x3 x6 p3 ⊕ x7 ⎥ ⎢ ⎥ ⎣ p 4 p5 p 6 ⊕ x 7 ⎦ where xi, i = 1,…7 are the information bits, pj, j = 1,…6 are the parity bits, p1 = x1 ⊕ x4 , p2 = x2 ⊕ x5 , p3 = x3 ⊕ x6 ,

and k2 the number of information bits in each row of C1) and A a binary n1 * (n2 – k2 – 1) matrix, (n2 the number of columns in C1). The first row of A consists of (n2 – k2 – 1) information bits and all columns are repetition codes. 3) Redesign C = ⎡ 0 ⎤ ⎫⎬n rows with B a repetition row 3 ⎢B⎥ 1 ⎣ ⎦⎭ code containing the last information bit.

Fig. 4. Trellis diagram of (8, 4, 4) GAC code with number of states Ns = 4 and trellis depth Nc = 5.

p6 = p4 ⊕ p5 , p4 = x1 ⊕ x2 ⊕ x3 , p5 = x4 ⊕ x5 ⊕ x6 . ⎡ x1 p1 ⊕ x 4 p1 ⊕ x5 p1 ⊕ x 4 ⊕ x5 ⎤ ⎢x p 2 ⊕ x 4 p 2 ⊕ x5 p 2 ⊕ x 4 ⊕ x5 ⎥⎥ (6) C16 , 5 ,8 = ⎢ 2 ⎢ x3 p 3 ⊕ x 4 p 3 ⊕ x 5 p 3 ⊕ x 4 ⊕ x5 ⎥ ⎥ ⎢ ⎣ p 4 p 4 ⊕ x 4 p 4 ⊕ x5 p 4 ⊕ x 4 ⊕ x5 ⎦ where xi, pi i = 1,…4 are the information and parity bits respectively, pj = xj, j = 1, 2, 3 and p4 = x1 ⊕ x2 ⊕ x3 . Similarly to the case of RAC codes, the bits per row are transferred to one modulation symbol which is further transmitted by the channel. Therefore, the modulation scheme followed in each case depends on the information bits that are present per row. Particularly, QPSK, 8PSK and again 8PSK modulation schemes are needed for the (8, 4, 4), (12, 7, 4) and the (16, 5, 8) GAC code correspondingly. D.


Decoding Procedure of GAC Codes

The decoding of GAC codes is easily realised via a trellis design, which is built according to a procedure described in [4]. For the codes of our interest, the (8, 4, 4), the (12, 7, 4) and the (16, 5, 8) GAC code, the trellis designs are shown in figures 4, 5 and 6.

E. Proposed scheme We propose a scheme according to which a varied version of GAC codes can be constructed. The difference lies in the encoding procedure. Regarding to [4], the resultant matrix of GAC codes is derived from a modulo-2 addition of three matrices. The procedure proposed in this paper is described below: 1) Design matrix C1 and C2 as in [4]. 2) If matrix C3 is a zero matrix, redesign C2 = |PA|, with P a binary n1 * (k2 + 1) matrix (n1 the number of rows in C1


4) Construct the final matrix for the code C = C1 ⊕C2 ⊕C3 . According to this proposed scheme we can get three altered versions of the (8, 4, 4), the (12, 7, 4) and the (16, 5, 8) GAC code. The code rate remains the same; however the encoding matrix is altered. We notice that arrays C2 for the (8, 4, 4) and

the (12, 7, 4) GAC code are zero matrices. Equations (7), (8) and (9) illustrate the resultant matrices. ⎡ ⎢ ′ ⎢ C 8,4,4 = ⎢ ⎢ ⎣ p4

x1

⎤ p 2 ⎥⎥ p3 ⎥ ⎥ p4 ⊕ x4 ⎦ p1

x2 x3 ⊕ x4

(7)

where xi, i = 1,…4 are the information bits, pj, j = 1,…4 are the parity bits, pi = xi, i = 1, 2, 3 and p4 = x1 ⊕ x2 ⊕ x3 .

x4 p1 ⎤ ⎡ x1 ⎢ x x5 p 2 ⎥⎥ (8) ′ 2 C12 , 7 , 4 = ⎢ ⎢ x3 x6 p3 ⎥ ⎢ ⎥ p x p x p ⊕ ⊕ 7 5 7 6 ⊕ x7 ⎦ ⎣ 4 where xi, i = 1,…7 are the information bits, pj, j = 1,…6 are the parity bits, p1 = x1 ⊕ x4 , p2 = x2 ⊕ x5 , p3 = x3 ⊕ x6 ,

p6 = p4 ⊕ p5 , p4 = x1 ⊕ x2 ⊕ x3 , p5 = x4 ⊕ x5 ⊕ x6 . ⎡ x1 ⎢ x ′ 2 C16 , 5,8 = ⎢ ⎢ x3 ⎢ ⎣ p 4 ⊕ x5

p1

p1 ⊕ x4

p2

p 2 ⊕ x4

p3

p3 ⊕ x 4

p 4 ⊕ x5

p 4 ⊕ x 4 ⊕ x5

p1 ⊕ x4

⎤ p 2 ⊕ x4 ⎥⎥ (9) p3 ⊕ x 4 ⎥ ⎥ p 4 ⊕ x 4 ⊕ x5 ⎦

where xi, pi i = 1,…4 are the information and parity bits respectively, pj = xj, j = 1, 2, 3 and p4 = x1 ⊕ x2 ⊕ x3 . The modulation scheme required for each case is BPSK, 8PSK and QPSK, since the potential symbols for transmission are 2, 8 and 4 respectively. The decoding procedure is again processed through a trellis diagram, designed as explained in [4]. The resultant trellis diagrams for the proposed GAC codes are shown in figures 7, 8 and 9.

III. CHANNEL AND NOISE MODELS

A. Channel model For the system’s realization, we utilize a channel model for the PLC channel. For this purpose, we implement Zimmermann’s model [14], which is a frequency channel model, it is found frequently in the literature and is easily applied. This model takes into account the multipath phenomenon by computing the N dominant paths. Thus, it estimates for each path the attenuation factor, the delay portion and a weighting factor indicating the total reflections this particular path undergoes. The transfer function characterizing this channel model is given by equation (10): N

H( f ) = ∑

i =1 weighting factor

− (α 0 +α1 ⋅ f ) ⋅ d i i p ⋅ e1 4243 ⋅ e14243 (10) attenuation portion

delayportion

The factor di / up stands for the time delay τi of each path, where τi is: di ⋅ εr di (11) τi =

=

c0

υ

p

di is the distance of the i-th path, εr is the dielectric constant of the insulating material, a0, a1 and k are constants describing the model and c0 is the speed of light.

B. Noise Model As it has been aforementioned, the noise model used in this paper is Middleton’s class A model [8], which takes under consideration both the background and the impulsive noise and it is suitable for PLC channels. According to this model, the probability density function of the noise amplitude v is given by equation (12): p (v ) =

∞ e − A ⋅ Ak 1 ⋅ k ! 2π ⋅ σ k =0

∑

where σ 2 = P ⋅ (k / A) + Γ , P k

Fig. 7. Trellis diagram of the proposed (8, 4, 4) GAC code with number of states Ns = 4 and trellis depth Nc = 5.

− j 2πf ⋅( d / υ )

k

g {i

1+ Γ

=σ

2 2 +σ G I

⎛ 2 ⎞ ⎜ −v ⎟ ⎟ ⎜⎜ 2 ⋅σ 2 ⎟ k ⎠ ⎝

⋅ exp k

(12)

stands for the total

noise power, σ G2 is the Gaussian noise power, σ I2

is the

Impulsive noise power, Γ = σ 2 / σ 2 (Gaussian to Impulsive G

I

noise power Ratio, GIR) and A is the Impulsive index. The noise sample is given by equation (13): (13) n = xG + K m ⋅ y where xG is a White Gaussian background noise sequence with zero mean and variance σ G2 , y is a White Gaussian Fig. 8. Trellis diagram of the proposed (12, 7, 4) GAC code with number of states Ns = 8 and trellis depth Nc = 5.

sequence with zero mean and variance σ I2 / A , Km is a Poisson distributed sequence whose pdf is characterized by the impulsive index A. In this study, for reasons of completeness, we take several values of the parameters A and Γ (GIR) in order to observe how they affect the system’s performance. IV. SIMULATION RESULTS

Fig. 9. Trellis diagram of the proposed (16, 5, 8) GAC code with number of states Ns = 8 and trellis depth Nc = 5.

In this section we illustrate the results, derived via computer simulations. First of all, the simulations were not an easy task

to perform, since a frequency channel model was utilised, while the OFDM transmission technique implies that a cyclic extension of the symbol is necessary in time domain. Therefore, careful operations of the FFT and IFFT functions were required during the signal’s transmission and reception, so as to transfer our data from frequency to time domain and vice versa. Due to these transformations and to the numerous components the systems consists of, the total simulation time was large. Consequently, there had to be a trade off between the number of transmitted data blocks and the minimum possible BER value the results approach. So, the number of transmitted data blocks was set to 500 in each case, whereas each block was transmitted over an OFDM symbol consisting of 128 sub-carriers. The number of total data bits for the different cases of each coding technique depended on the modulation scheme required. The results are grouped into two categories according to which modulation techniques is applied (QPSK or 8PSK), so that a better comparison is obtained. For reasons of completeness, convolutional codes modulated with QPSK and 8PSK are also examined, so that their performance is compared to the one obtained from GAC and RAC codes. Figures 10 and 11 show the performance of the examined coding schemes in terms of BER versus the Eb / N0 value in the PLC channel. It can be concluded from both figures that GAC codes seem to perform better than their equivalent RAC codes, indicating a stronger coding technique. In addition, it is noticeable that all curves follow the same trend, meaning that when the Eb / N0 value increases, the codes’ performance improves.

In figure 10 it is clearly observed that convolutional codes are inferior to RAC and GAC coding techniques, which becomes more obvious for higher Eb / N0 values. The case is more or less the same regarding the codes requiring an 8PSK modulation scheme. However, as illustrated in figure 11, RAC codes’ performance resembles the one of the convolutional code, with the latter being slightly inferior for higher Eb / N0 values. Furthermore, codes with the QPSK modulation scheme show a better performance than those requiring the 8PSK modulation method, proving that when more symbols are available at the receiver’s demodulating process, the possibility of an error is larger. As it has been mentioned above, we also examine how two of the most important parameters taken into account when designing the noise model affect the system’s performance. In particular, the effect of the impulsive index (A) and the one of the Gaussian to Impulsive noise power Ratio (GIR or Γ) is studied.

Fig. 12. Eb /N0 at BER = 10-3 versus different values of Γ for all the examined RAC and GAC codes.

Fig. 10. Bit error rate versus Eb/N0 for convolutional, the (9, 4, 4) RAC and the (8, 4, 4) GAC codes with QPSK modulation.

Fig. 13. Eb /N0 at BER = 10-3 versus different values of impulsive index for all the examined RAC and GAC codes.

Fig. 11. Bit error rate versus Eb/N0 for convolutional, (12, 6, 4), (16, 9, 4) RAC codes and (12, 7, 4), (16, 5, 8) GAC codes with 8PSK modulation.

Figure 12 shows how different values of Γ influence the system’s performance, while A is stable. On the other hand, in figure 13, Γ remains constant and different values of the impulsive index are used to illustrate the effect on the system’s performance. All of the examined RAC and GAC coding schemes behave similarly. When Γ increases, a smaller Eb/N0 value is required to achieve the same BER. This is in

agreement with the physical meaning of Γ, since when it is increased, the impulsive noise power becomes less, meaning that the negative effects of this kind of noise are constrained. Concerning the effect of the impulsive index, by definition, when it increases, the presence of impulsive noise becomes more intense, therefore, the Eb / N0 value required for the same BER becomes larger. In figure 14 we illustrate the performance of the proposed GAC codes opposed to their equivalent ones, described in section II-C. What we observe is that the proposed (8, 4, 4) and (12, 7, 4) GAC schemes are inferior to their equivalent ones, while the proposed (16, 5, 8) GAC code shows a better performance than the existing one. Concerning the latter case, the performance is mainly improved because of the modulation scheme required, which is QPSK for our technique, whereas it is 8PSK for the (16, 5, 8) GAC code explained in section II-C. Regarding the (12, 7, 4) GAC code, the modulation scheme is 8PSK for both versions of the GAC code. By taking a closer look at the matrices of which the codes are derived, we notice that for our scheme the information reaching the receiver for data bit x7 is entailed only in the last symbol, meaning that if this symbol is in error, this bit will probably not be reproduced correctly. On the contrary, in the original version of the (12, 7, 4) GAC code, all symbols arriving at the receiver enclose some kind of information for this data bit and so its reproduction is more possible. This is mainly the reason why our scheme is inferior to the original one. Finally, as to the proposed (8, 4, 4) GAC code, the reason why its performance is poorer than the one of the original GAC code is explained after examining the matrices from which the codes are derived. In our case, the parity information is exclusively entailed in the last symbol reaching the receiver, which causes trouble in the decoding procedure in case it is in error. Thus, its performance is inferior, even though the modulation scheme is BPSK instead of QPSK.

are simple coding schemes and by altering their parameters, we obtain different codes of this family, thus adjusting in a better way to our system’s demands. After performing a series of computer simulations and being compared to convolutional code, the results show that they could be considered as a candidate coding scheme for the hostile PLC environment. In addition, we propose a new version of GAC codes, which in some cases proves to be better than the original one. It depends on the number of input bits and their arrangement in the matrix from which the code originates, whether or not the resultant performance will be enhanced opposed to the one of the original GAC code. REFERENCES [1] [2] [3] [4] [5] [6] [7]

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Fig. 14. Bit error rate versus Eb/N0 for (8, 4, 4), (12, 7, 4), (16, 5, 8) GAC codes and their equivalent proposed schemes.

V. CONCLUSIONS In this paper we study array codes and in particular GAC and RAC coding techniques and we apply them to power line communications in order to examine their performance. They

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