On
the Performance
of
Typical
Turbo Codes in the Presence Power Line Asynchronous Impulsive of
Noise Joao Luiz Rebelatto DAELN / CEFET-PR Curitiba, PR, BRAZIL joaoluiz@eletrica . eng.br
Richard Demo Souza
CPGEI / CEFET-PR Curitiba, PR, BRAZIL richard@cpgei . cefetpr .br
Abstract-We investigate the performance of turbo codes in an environment with asynchronous impulsive noise present in power lines. The model used for the noise is based on actual measurements, and is built using partitioned Markov chains. Computer simulations exploring the effects of code length, noise energy, transmission rate, and channel interleaving are presented. I. INTRODUCTION
The electric power distribution grid, present in any building and with a capillarity much larger than the telephone network, is a very attractive solution for high-speed data transmission on the last mile [1], [2]. The power grid was projected for energy distribution, and not for communications, and thus it imposes several challenges for the reliable transmission of information. Among these we can cite the fact that the channel is frequency selective and time-varying, with attenuations up to 60 dB [2]-[41, the presence of multipath fading caused by impedance mismatches [2], and many noise types caused by another equipments, switching, among other factors [3]. Within these characteristics, probably the most limiting one is the noise. Contrary to many other communication channels, the noise present in a power line can not be characterized only as additive Gaussian noise (AWGN) [5]. At least five classes of noise can be found in a power line [6], [7]: i) background noise with a relative low power spectral density; ii) narrowband noise caused by broadcast stations; iii) periodic impulsive noise asynchronous to the mains frequency, caused by switched power supplies; iv) periodic impulsive noise synchronous to the mains frequency, caused mainly by the switching of rectifier diodes; and v) asynchronous impulsive noise caused by transients in the network. The noises of types i), ii) and iii) usually remain stationary over long periods and can be summarized as background noise [7]. The noises of types iv) and v) are usually the most harmful, being also the more complicated ones for modeling and combating [7]. In a communication system the effects of noise are usually compensated with the use of error correcting codes 'This work
113969/2004.
was
partially supported by CNPq (Brazil) under grant
1-4244-0000-7/05/$20.00 02005 IEEE.
Marcelo E. Pellenz PPGIA / PUC-PR Curitiba, PR, BRAZIL marcelo@ppgia . pucpr . br
[8]. Commercial schemes for data transmission over power lines [1] have used traditional coding solutions, such as the concatenation of convolutional codes and block codes (ReedSolomon). Recently, some authors [9] started investigating the performance of turbo codes [10] in an environment with impulsive noise. However, the noise has been modeled as Middleton's Class A noise [11], which is an approximation of the noise present in a power line [7], and the effects of some of the code parameters in the performance has not been thoroughly investigated. In this paper, we investigate the performance of turbo codes in the presence of typical power line asynchronous impulsive noise. We use the noise model presented in [7], which is based on actual measurements carried out in Germany and is built using partitioned Markov chains [12]. Moreover, the performance of turbo codes is investigated in terms of interleaver size, impulsive noise energy, transmission rate and channel interleaving. The results of the computer simulations allow us to draw some design rules that can help in the development of new schemes for data transmission over power lines. This paper is organized as follows. In Section 1I we introduce the system model to be considered. In Section III we discuss the impulsive noise model based on a Markov chain, while in Section IV we present and discuss several simulation results. Finally, Section V concludes the paper.
II. SYSTEM MODEL The power line channel is frequency selective [4], with considerably long echoes. In practice [1], OFDM modulation [14] is used for avoiding the necessity of very long and complex equalizers. Then, even in the presence of impulsive noise, if the number of carriers and the extension of the cyclic prefix are adequately designed, the series interconnection of the OFDM modulator, the frequency selective channel, and the OFDM demodulator can be approximated as a flat channel [13] within a reasonable margin2. Then, since our goal is to 2Note that this assumption, -equivalente flat channel, will bound on the system performance in terms of bit error rate.
797
yield
a
lower
IBackground NoiseI
PIJ
Impuive Noise 1
,P22
nk Uk
Turbo
Ck
BPSK
-ncde
Mapper
lxk
P2F1
I
Y
I
Rceve
U
Fig. 2. Two state Markov chain.
Flat Channel
Fig. 1. Block diagram for the equivalent system, with the frequency selective channel and the OFDM modulator/demodulator replaced by a flat channel.
where Pa,b represents the transition probability between states a and b (a,b = 1,2,... ,m). Figure 2 shows a two state Markov chain, with transition probabilities P1,2 and P2,1Consider now a Markov chain with m states, which are divided into two groups, SI (i 1,2,...,v) and S2 (i = v + 1, v + 21 ... m). If we define the following output function for the chain, where 1 represents the occurrence of an error and 0 represents the opposite:
investigate the performance of turbo codes, from now on we consider the equivalent model shown in Figure 1. The signal Yk at the receiver, at time instant k, can be written as: Yk = Xk + nk =Xk
b
n +
I
(1)
where nb represents the background noise and it is a Gaussian random variable with zero mean and variance Gb, e nk represents the impulsive noise, being non-zero only at the time instants of a burst occurrence. At these time instants the impulsive noise is defined as a Gaussian random variable with zero mean and variance a2 > ,2 The symbols xk are obtained by mapping in BPSK the output ck of the turbo encoder, which is of rate 1/3 and includes a systematic bit and two recursive convolutional encoders with 16-states and generator matrix G(D) - 1+D3+D4±D The code interleaver has length L. The information bits Uk are equally likely and the receiver outputs an estimate Lk of these bits.
I(j') >= (sj) = =i U){ °, I E S
then the v states within S I represent the error-free case, and the w = m - v states within S2 represent the case of an err6r occurrence. Introducing transition states between the groups SI and S2, then the two groups can be described by the transition probability matrices F and G, respectively:
F-=
III. ASYNCHRONOUS IMPULSIVE NOISE
A Basic Definitions Consider a Markov chain with m states, where the output function 4 (j) at each time instant j depends only on the current state s (j):
1(j) =
(s(j) = i)'
(2)
i = 1, 2,.. , m, and where the transition probabilities between states are given by the probability matrix: P1, 1
P1,2
[P2,1
P2,2
P2 L
-'~ -
**
P
Pm-i,m Pm,m-i
Pm,m
I,
0
0
f2,2
...
0
fi,v1+
f2,v+1
0
L fv+l,i fv+ 1,2
0
0
fvv
fV+i ,V
(5)
fv1v+ 0
and gi,i 0
0
...
0
'.. 0
0
92,2
gi,w+i 9J2,w+1
0 9w+1,1
gw+1,2
*--
gw,w gw+l,w
(6)
gw,w+1
0
With the above definitions, the probability pfw that an error event exceeds certain duration tw: w
Pfw(i)
-
EZw±+l,k (gk,k)j,
k=l
(7)
and the probability pfd that the period between two error events exceeds a given time td: V
Pfd(j) = E fv+l,k- (fk,k)j,
1m
...
fi,l
.
A simple and efficient method for the modeling of burst events is the use of Markov models such as the GilbertElliot [15], [16]. The Gilbert-Elliot approach models both the interarrival time (IAT) and the width of the impulse events with two states exhibiting exponentially distributed durations. Measurements carried out in Germany [7] showed that the IAT and the impulse widths correspond to a superposition of several exponential distributions. In [7] the authors used a partitioned Markov chain [ 12], with a variable number of states per partition for modelling the impulsive noise. We explore this model in our computer simulations. In the sequel we describe in details the model presented in [7]
(4)
k=l
(8)
can be expressed by the elements of the matrices F and G, which can be deternined through measurements and numerical methods, as in [7].
798
1O0
c
D1
vr
D
Fig. 3. Partitioned Markov chain with 7 states (5 error-free states and 2 states with errors), representing the model for the generation of asynchronous impulsive noise introduced in [7].
0
5
10
15
25 30 interarrivaltime (ms)
20
35
40
45
50
Fig. 4. Probability pfd that the penod between two impulse events exceeds
a given time td, both for the simulation of the Markov chain shown in Figure 3 with the transition probabilities given in Table T, as for the analytical model given by (8). The sampling period used was ta = 80us.
B. The Noise Model
In [7] the authors presented a model based in a Markov chain with in = 7 states divided in two groups, SI with v = 5 states and S2 with w = 2 states, as shown in Figure 3. The ........... ::::o Eb/No imptO1 dB states within group SI represent the error-free case, while ::-:o Eb/No imp= -5 dB, ...Eb/No imp= -3 dB 10'2 ..-le-t the states within group S2 represent the case of an error Impulse fee occurrence (impulse). After a measurement campaign, and .. using numerical methods, the authors in [7] determined that .............................. the values for the elements of the matrices F and G that best ...................... . .. .. approximate the IAT and the impulse width are those shown in Table I. X 10 t3 ............. wU .. .. ... .. .. ... ......................... Figure 4 shows the probability pfd that the period between ............ ............................. two impulse events exceeds a given time td, both for the .... simulation of the Markov chain shown in 3 with the transition 10 probabilities given in Table I, as for the analytical model given by (8). The sampling period used was ta = 8Q0s. Thus, each error event has a duration of 80ts. If the period t5 between the transmission of two symbols over ihe channel is smaller than b 0.6 0.7 ___x 0.8 0.9 100.2 0.3 0.4 0.5 0.1 ta, then an error event will affect more than one symbol. In Eb/No (dB) order to investigate the effect of the power line impulsive noise for different transmission rates over the channel, we defined Fig. 5. Bit error rate (BER) vs. signal to background noise ratio (Eb/No), the factor B as the number of symbols affected by each error for a rate 1/3 turbo code, interleaver length L 16384, error bursts of B 128 symbols per burst, varying the signal to impulsive noise ratio event (burst): = {-3., -5.0, -10.0} dB. .........I...........
........... ............................. ...............
...
...
.................
..
..
...
.......................
........................
........................
.................................
.............
..............................
..............
.....
.....
..
.............
=
=
B
[1ta
(9)
A. Impulsive Noise Energy
IV. COMPUTER SIMULATIONS In this section we investigate the performance of the 16state turbo encoder presented in Section II, in an environment with impulsive noise as defined in Section III. The simulations show the bit error rate (BER) as a function of the signal to background noise ratio (Eb/No, where Eb = 3 x Es and No = 2 x ar ), while varying the following factors: a) impulsive noise energy, b) transmission rate, c) code length (length L of the
code interleaver), and d) channel interleaving.
Eb/NO[imp
L
Figure 5 shows the BER vs. Eb/No for an interleaver length = 16384, error bursts of B 128 symbols per burst, and =
the ratio between the energy per bit and the energy of the impulsive noise (Eb/NO[imp]). Considering a residual BER of 10-4, which can be further eliminated by an extemal code as a Reed-Solomon, the performance of the turbo code in the presence of background and impulsive noise was less than 0.5 dB worse than the case of background noise only.
varying
799
TABLE I TRANSITION PROBABILITY MATRICES FOR THE PARTITIONED MARKOV CHAIN SHOWN IN FIGURE 3.
0 0 0 0.9999775 0 0 0 0.8173416 0.9992129 0 0 0 0.9900302 0 0 0 0 0 0 0 0.4432897 0.0466043 0.0908189 0.1135221
0 0 0 0 0.7202658 0.3057651
0.0000225 1 0.1826584 0.8844900 0.0007871 G 0 0.0099698 0.-0749
0.2797342 0
0.0787479
0 0.3991290 0.9212521
0.39212521
J
0.115510 0.600871
0
I
S
LLw 10 to
0.6 Eb/NO (dB)
0.6 EblNo (dE)
Fig. 6. Bit error rate (BER) vs. signal to background noise ratio (Eb/No), for a rate 1/3 turbo code, interleaver length L = 16384, signal to impulsive noise ratio Eb/NO[[imp = -5.0 dB, varying the length of the error bursts as B = {64, 128, 256 symbols per burst.
Fig. 7. Bit error rate (BER) vs. signal to background noise ratio (Eb/No), for a rate 1/3 turbo code, signal to impulsive noise ratio Eb/Noljmp =--5.0 dB, error bursts of B = 128 symbols per burst, varying the length of the code interleaver as L = {4096, 16384, 65536)
B. Transmission Rate Figure 6 presents the BER vs. Eb/No for the case of an interleaver length L = 16384, but setting the signal to impulsive noise ratio to Eb/NO[imp] = -5.0 dB, and varying the length of the error bursts as B = {64,128, 256} symbols per burst. Again, considering a residual BER of 10-4, we can see that for B 64 symbols per burst the degradation when compared with the case of background noise only is of just 0.1 dB, while for the case of B = 128 symbols per burst this degradation is of less than 0.3 dB. In case of B = 256 symbols per burst the degradation is of approximately 0.9 dB, while for values of B larger than 256 symbols per burst, the irreducible error floor was always larger than 10-4.
For the case of L 16384 the performance is about 0.3 dB worse than in the case of background noise only (where convergence for a BER of 10-4 is around 0.2 dB), while for the case of L = 65536 convergence is achieved around 0.14 dB. Note that, for L = 65536 and background noise only, convergence for a BER of 10-4 is achieved with an Eb/NO of 0.13 dB. Thus, the performance degradation is practically negligible for L = 65536. Moreover, decreasing the signal to impulsive noise ratio to Eb/NO[jmp] =-20.0 dB, and for L -65536, convergence for a BER of 10-4 happens around 0.15 dB, which is only 0.01 dB worse than in the case of Eb/NO[imp =--5.0 dB.
C. Code Length In this section we evaluate the effect of the block length (length L of the code interleaver) in the code performance. Figure 7 shows the BER vs. Eb/No for the case of a signal to impulsive noise ratio Eb/No[impI = -5.0 dB, error bursts of B 128 symbols per burst, and varying the code interleaver length L = {4096, 16384, 65536}. Considering a residual BER of 10-4, we can see in the Figure that for an interleaver length of L = 4096 convergence could not be achieved, and an irreducible error floor appears at a BER of 10-3. =
=
D. Channel Interleaver
Practical schemes for the high speed data transmission over lines [1] utilize a channel interleaver with the goal of spreading the burst errors, and improving the performance of the inner convolutional code. In this section we investigate the effect of a channel interleaver in the performance of a turbo code in an environment with impulsive noise. Consider the case of a code interleaver of length L = 4096, signal to impulsive noise ratio Eb/NO[mpj -5.0 dB, and burst errors of B = 128 symbols per burst. After inserting a channel interleaving of length J 3 x L before power
800
=
noise, a gain of more than 4.0 dB would be obtained. This means a savings of at least 2.5 times in the energy spent in the data transmission for achieving the same BER.
mU w
C03
Eb/No (dB)
Fig. 8. Bit error rate (BER) vs. signal to background noise ratio (Eb/No), for a rate 1/3 turbo code, signal to impulsive noise ratio Eb/NO[imp] = -5.0 dB, burst errors of length B = 128 symbols per burst, for the cases of N = 1 and L = 4096, N = 4 and L = 4096, and N = 1 and L = 16384.
V. FINAL COMMENTS We presented a numerical evaluation of the performance of turbo codes in an environment with asynchronous impulsive noise typical of a power line. The noise model we used was obtained from actual measurements carried out in Germany, and it is based on partitioned Markov chains. The effects of impulsive noise energy, transmission rate, code length and channel interleaving were investigated. Computer simulations showed that the adequate choice of the code length as a function of the transmission rate and the impulsive noise energy, allows that the harmful effects of the impulsive noise to be practically eliminated by the turbo code. We also demonstrated that the inherent interleaving of a turbo code is more efficient than a simple channel interleaving. Moreover, we showed that the utilization of a turbo code in place of a convolutional code can bring large savings in terms of the energy spent in the data transmission.
the transmission of the coded symbols over the channel, we verified an almost negligible difference in perfonnance when compared with the case without channel interleaving. Now, suppose that N blocks of coded symbols, each one of length 3 x L, feed a channel interleaver of length N x 3 x L. Figure 8 shows the BER vs. Eb/No for such a case, where N = 4 and L = 4096. In the Figure we also show the curves for the cases of N = 1 and L = 4096, and for the case of N = I and L = 16384. We can see a considerable improvement when the channel interleaver processes four blocks of coded data (each one of length 3 x L), when compared with the case of N = 1 and same code length L. However, a turbo code with a block length of N x L (code interleaver of length N x L), and without channel interleaving, presents better performance than a code with block length L (code interleaver of length L) concatenated with a channel interleaver of length N x 3 x L. Therefore, once the decoding delay is the same, it is better to explore the interleaving inherent to a turbo code (by increasing the length of the code interleaver), than to use a long channel interleaver. E. Comparison with Convolutional Codes Consider the best rate 1/3 16-states convolutional code known in the literature [17]. In an environment without impulsive noise (background noise only), BPSK transmission, 4.0 dB [17, a BER of 10-4 is achieved with Eb/No Figure 11.22]. Note that, according to Figure 7, a turbo code of same rate, with constituent convolutional encoders with 16 states and an interleaver of length L = 65536, achieves 0.14 dB, but in an the same BER of 10-4 with Eb/NO environment with impulsive noise where the burst errors have length B = 128 symbols per burst, and for an Eb/NO[-mp =l -5.0 dB. Therefore, if the turbo code is used in place of the convolutional code then, in an environment with impulsive -
-
801
REFERENCES [1] http://www.homeplug.com [2] H. Dai and H. V. Poor, "Advanced signal processing for power line communications," IEEE Commun. Magazine, vol. 41, no. 5, pp. 100107, May 2003. [3] M. Gotz, M. Rapp, and K. Dostert, "Power line channel characteristics and their effect on communication system design," IEEE Commun. Magazine, vol. 42, no. 4, pp. 78-86, Apr. 2004. [4] M. Zimmermann and K. Dostert, "A Multipath model for the power line channel," IEEE Trans. Commun., vol. 50, no. 4, pp. 553-559, Apr. 2002. [5] J. G. Proakis, Digital Communications, McGraw-Hill, Third-Edition, 1995. [61 0. Hooijen, "A channel model for the residential power circuit used as a digital communications medium," IEEE Trans. Electromagn. Compat., vol. 40, no. 4, pp. 331-336, Aug. 1998. [7] M. Zimmermann and K. Dosterl, "Analysis and modeling of impulsive noise in broad-band power line communications," IEEE Trans. Electromagn. Compat., vol. 44, no. 1, pp. 249-258, Feb. 2002. [8] S. Lin and D. J. Costello Jr. Error Control Coding: Fundamentals and Applications. Prentice-Hall, 1983. [9] D. Umehara, H. Yamaguchi, and Y. Morhiro, "Turbo decoding in impulsive noise environment," IEEE Globecom '04, vol. 1, pp. 194-198, Dec. 2004.
[10] C. Berrou and A. Glavieux, "Near optimum error correcting coding and decoding: turbo-codes," IEEE Trans. Commun., vol. 44, pp. 1262-1271, Oct. 1996. [11] D. Middleton, "Statistical-physical models of electromagnetic interference," IEEE Trans. Electromagn. Compat., vol. EMC-19, pp. 106-127, Aug. 1977. [12] B. D. Fritchman, 'A binary channel characterization using partitioned binary Markov-chains," IEEE Trans. Inform. Theory, vol. 13, pp. 221227, Apr. 1967. [13] M. Budsabathon and S. Hara, "Robustness of OFDM signal against temporally localized impulsive noise," IEEE VTC'01 Fall, vol. 3, pp. 1672-1676, Oct. 2001. [141 J. A. C. Bingham, "Multicarrier modulation for data transmission: an idea whose time has come," IEEE Commun. Magazine vol. 28, no. 5, pp. 5-14, May 1990. [15] E. N. Gilbert, "Capacity of burst-noise channels," Bell Syst. Tech. .J, vol. 39, pp. 1253-1266, 1960. [161 E. 0. Elliot, "Estimates of error rates for codes on burst-noise channels,," Bell Syst. Tech. J., vol. 42, pp. 1977-1997, 1963. [17] S. Benedetto and E. Biglieri, Principles of Digital Transmission: With Wireless Applications, Kluwer Academic Press, 1999.
