Reed-Solomon Coding to Enhance the Reliability of M ... - IEEE Xplore

Reed-Solomon Coding to Enhance the Reliability of M -FSK in a Power Line Environment D.J.J. Versfeld,

A.J. Han Vinck,

H.C. Ferreira

Department of Electrical and Electronic Engineering, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa Email: [email protected]

Institute for Experimental Mathematics, University of Essen, 45326 Essen, Germany, Email: [email protected]

Department of Electrical and Electronic Engineering, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa Email: [email protected]

Abstract— M -FSK conforms to the CENELEC standard for Power Line Communications. We investigated the performance of three M -FSK detectors with Reed-Solomon coding on the additive white gaussian noise channel (AWGN) and an AWGN channel with wideband noise. Furthermore, we developed two schemes to improve communications in the presence of narrowband noise. The first scheme relies on an external mechanism to detect the presence of narrowband noise and this information is then conveyed to the M -FSK demodulator, where it is used to correctly decode the data. For the second scheme, we define the same symbol weight of a codebook. We show how the same symbol weight can assist us in detecting narrowband noise. We then describe two coding techniques for Reed-Solomon codes which can detect the presence of narrowband noise. When narrowband noise is detected, both schemes try to correctly decode the data.

I. I NTRODUCTION One of the contenders for the “last mile” for telecommunications is Power Line Communications (PLC) [1]. M -FSK is a good candidate for the CENELEC standard. However, the channel is quite harsh and experience attenuation, permanent frequency disturbances, narrowband noise and impulsive noise [1] and [2]. In this paper we investigate the performance of three noncoherent M -FSK detectors for the additive white gaussian noise (AWGN) channel, an AWGN channel with wideband noise and an AWGN channel with narrowband noise. The three detectors that we considered are the envelope detector (theoretical detector found in most textbooks), a threshold detector and a detector based on Viterbi’s ratio test [3]. The envelope detector allows errors-only decoding, while the other two detectors provide side information allowing erasures decoding as well. In Section III we consider the performance of the three decoders for the AWGN channel and the AWGN channel with wideband noise. In Section IV, we develop two new techniques to improve communications in the presence of narrowband noise, and evaluate these techniques by simulation. Matache [4] investigated the capacity and cutoff rate of M ary frequency shift keying (FSK) modulation for noncoherent hard decisions detection in conjunction with erasure insertion using Viterbi’s ratio test for Rayleigh fading channels. Choi

0-7803-8844-5/05/$20.00 c 2005 IEEE.

et al. [5] examined the asymptotic performance of M -FSK in multichannels using Reed-Solomon decoding and Viterbi’s ratio threshold test. II. D ESCRIPTION OF THE S YSTEM M ODEL Due to space limitations, we refer the reader to [4], [5] and [6] for the normal AWGN channel. We use the same one-toone mapping from the field GF (2m ) onto the symbols of the M -FSK modulator as is done in [4] and [5]. The noncoherent detection scheme uses a bank of 2M correlators, and using the square law, a set of metrics is determined for each symbol. The most likely transmitted symbol is determined based on these metrics. For normal envelope detection, the symbol corresponding to the metric with the highest value is chosen as the candidate. With threshold decoding, we choose a threshold λ. Optimally, the value for λ would be the value where the area of probability distribution of the send signal equals the area of the probability distribution of the noise envelopes. However, Shannon showed that a good approximation is the cross-point where the two distributions intersect. When no envelope exceeds the threshold λ, we declare an erasure. When more than one envelopes exceed the threshold, we also declare an erasure. An error occurs when the envelope corresponding to the send signal does not exceed the envelope corresponding to the highest noise present. In his paper [3], Viterbi developed the Viterbi ratio threshold test. The aim of the test is to provide side information on the reliability of the received symbols. The test will indicate whether a symbol can be regarded as reliable, or whether a symbol should be marked as an erasure. The wideband noise channel is modelled by two states. We have a probability ρ that the channel will be in state 1, where the Gaussian noise have a two-sided spectral density N1 /2 and we have a probability 1 − ρ that the channel will be in state 2, where the Gaussian noise have a two-sided spectral density N2 /2. For the scenario where narrow band noise is present, we make a small addition to the AWGN channel. A narrowband noise source has the probability ρn to be present, and the duration will be for n × d symbols, where d ∈ [0, . . . , ∞).

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The assumption that the duration of the narrowband noise is a multiple of the duration of a codeword is to simplify the problem. However, if asynchronous transmission is used, the duration of the narrowband noise disturbance is likely to exceed the duration of the transmission making the above assumption valid. The narrowband noise signal has energy Ens and frequency fns = fi , where fi corresponds to one of the transmission frequencies. III. T HE PERFORMANCE OF VARIOUS M -FSK DETECTORS WITH R EED -S OLOMON DECODING IN THE PRESENCE OF AWGN AND AWGN WITH WIDEBAND NOISE In Fig. 1(a), we compared the three detection schemes for the AWGN channel. The data was encoded using an (15, 9) Reed-Solomon code. The envelope detector’s output was fed into a hard-decision errors-only Reed-Solomon decoder, while the other two detectors inserted erasures into the datastream which was decoded using errors-and-erasures Reed-Solomon decoding. The line denoted by ‘Theoretical’ is the theoretical performance of normal Envelope detection, calculated by [6, p.212, eq. 4.3.32]. The line denoted by ‘ED UC’ is the actual symbol error rate experienced during the simulation. The line ‘ED C’ denotes the performance of the envelope detector after decoding, the line ‘TD-0.7’ is threshold detection with a threshold of 0.7 and ‘VRTT-0.8’ is the performance after Viterbi Ratio Threshold Test was used with a λ = 0.8 in conjunction with Reed-Solomon decoding. In Fig. 1(b), we investigated the use of a low rate Reed-Solomon code over an AWGN channel. In Fig. 2(a) we investigated the performance of the three detection schemes in a wideband noise channel. We had two states. State 1 is a good state where the SNR is equal to the SNR shown. State 2 is a bad state where the SNR is equal to the SNR of state 1 divided by 10. In the simulation, the channel had a probability of ρ = 0.8 to be in state 1, and a probability (1 − ρ) = 0.2 to be in state 2. In Fig. 2(b) we used the same setup as in the above case, except that we used a lower rate RS code. For the AWGN channel only, the best performance when using high rate codes is obtained by using the envelope detector. For the AWGN channel with low rate codes, the detector based on Viterbi’s threshold ratio test outperforms the others. For channels with wideband noise, the best performance is obtained when Viterbi’s threshold ratio test is used. IV. M -FSK IN THE P RESENCE OF NARROWBAND N OISE When narrowband noise is present, and the output energy of the narrowband noise source exceeds the output energy of the modulator, the demodulator will have a symbol that is “always on”. This poses a problem as the all e, e ∈ GF (2m ) vector is a valid codeword for some Reed-Solomon codes, and the presence of narrowband noise disturbances for these codes will result in undetected errors. Thus, some modifications should be made to the detectors or the coding scheme in order to detect the presence of narrowband noise. In this section we consider two schemes. The

scheme of Section IV-A assumes that an external mechanism exists that detect the presence of narrowband disturbances. In Section IV-B, we develop coding schemes to detect the presence of narrowband noise disturbances. A. A Threshold Detector with External Narrowband Noise Detection This scheme assumes that some mechanism exist which can detect when narrowband noise is present. When the threshold detector is notified that narrowband noise is present, the detector ignores the largest metric. A few outcomes are now possible. The second largest metric can correspond to the symbol sent, the narrowband noise disturbance and the sent symbol coincided, or more than 1 of the remaining metrics exceed the threshold. In the first case, the correct output will be given. For the second case, we adapted the detector such that when none of the remaining metrics exceed the threshold, the symbol corresponding to the narrowband noise disturbance is chosen. In the third case, an erasure is declared. In Fig. 3(a) we investigated the performance of an (15,9) RS code and in Fig. 3(b) we used an (15,3) RS code. B. Coding Schemes to Detect Narrowband Noise Narrowband noise can also be detected by coding schemes. By assuring that a codebook doesn’t contain any all e, e ∈ GF (2m ) codewords, narrowband noise can be detected when an all e vector is received. In order to differentiate between coding schemes, we will use the following definitions. Definition 1 (Same Symbol Weight of a Codeword): The same symbol weight of a codeword c is the maximum number of same symbols contained in c. Definition 2 (Same Symbol Weight of a Codebook): The same symbol weight of a codebook C is the number of same symbols of cs , where cs is a codeword in C containing the maximum number of same symbols. As an example, the same symbol weight of the all-zero codeword of an (n,k) RS code is equal to n. Also, the same symbol weight of the all e codeword (e ∈ GF (2m )) is equal to n. Thus the same symbol weight of an (n,k) RS codebook is equal to n. Proposition 1 (The optimal Same Symbol Weight Codebook): A code with the minimum same symbol weight will have the minimum false detections. The codeword cs of a code will be the most vulnerable to noise as it is the codeword which needs the least number of symbols changed to result in the all e codeword. When this happen, a false narrowband noise detection will occur. We will now describe two coding schemes to detect narrowband noise. The first coding scheme has a pseudo same symbol weight of k, and uses an (n, k) Reed-Solomon code. Because the code is a linear code, the all-zero codeword is present. Excluding the all-zero codeword, we show that the code will have a same symbol distance of k. The second coding scheme makes use of a coset Reed-Solomon code. With a proper selection for the coset, this code will have a true same symbol distance of k.

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1) An (n, k) Reed-Solomon Code with Pseudo Same Symbol Distance k: Consider the Reed-Solomon code generated by

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and generator polynomial g(x) = (x − α)(x − α2 )...(x − αn−t−1 ).

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The minimum distance and thus the minimum number of differences between two codewords is N − k + 1 = t. Hence, the maximum number of agreements between any two code words is smaller than or equal to N − t − 1 = k. Since the constant codewords are in < g(x) >, a non constant codeword can never have more than k same symbols. Construct the generator matrix Gs by deleting the first row of G. The subcode < gs (x) > has parity-check polynomial hs (x) = (x − α−1 )(x − α−2 )...(x − α−t )

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and generator polynomial gs (x) = (x − 1)(x − α)(x − α2 )...(x − αn−t−1 ).

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Because the subcode Cs does not contain any constant codewords (except the all-zero codeword), the pseudo same symbol weight of Cs is k. At the decoder, the presence of narrowband noise can be detected. If the received vector is an all e-vector, the decoder assumes that narrowband noise was present. Using the same set of metrics the detector used for the received vector, the detector now ignores all the metrics corresponding to e, and from the remaining metrics chooses the best candidate for each symbol. Normal Reed-Solomon decoding follows. If the received vector is not the all- e vector, normal Reed-Solomon decoding is done. However, if narrowband noise was detected, the narrowband interferer corresponds to the zero frequency and after symbol detection excluding the zero symbols results in a vector with only erasures, the decoder knows that the all-zero codeword was transmitted. In Fig. 4(a) we investigated the performance of an (15,9) RS code and in Fig. 4(b) we used an (15,3) RS code using the scheme described in this section. 2) A Coset (n, k) Reed-Solomon Code with Same Symbol Distance k: In a paper, Choi [7] used a coset code of a binary BCH code to achieve synchronization. With synchronization, i.e. comma-free codes, the all e, e ∈ GF (2m ) vector cannot be in the codebook, otherwise the definition of commafreedom will be violated. Choi then used the coset code of the BCH code, and found that these codes were comma-free. The comma-free coset codes, however, cannot achieve high rates. In the presence of narrowband noise, the strict conditions of comma-freedom do not have to be honoured. However, the all e, e ∈ GF (2m ) vector should be eliminated. By adding a coset vector to the code C, this can be achieved without compromising the rate of the code. At the decoder, the received vector can be tested. If the received vector is not an all e-vector, the coset header can be subtracted, and normal Reed-Solomon decoding can be done. When an all e-vector is detected, the decoder can assume that

narrow band noise is present. Using the same set of metrics the detector used for the received vector, the detector now ignores all the metrics corresponding to e, and from the remaining metrics chooses the best candidate for each symbol. In order to minimize false detections, an optimal coset should be used. Consider the Reed-Solomon codebook C and a polynomial h(x). A coset code is denoted as hi + C, where hi is an arbitrary polynomial. We only have to consider polynomials of the form h(x) = h0 + h1 x + h2 x2 + . . . + hn−k−1 xn−k−1 as candidates for the coset for the coset code. This is true due to the fact that any polynomial p(x) of degree greater than or equal to n − k belongs to a coset hi + C, where hi has degree equal or less than n − k − 1. Due to complexity we could only investigate (7,k) coset codes. The use of a (7,k) code limits us to use 8-FSk. We investigated the optimal coset codes derived from various (7,k) Reed-Solomon codes, using exhaustive computer searches. Some observations that can be made from the searches are: • the same symbol distance of the codebook (h + C) is greater than or equal to k, • optimal coset codes, i.e. codes with a same symbol distance equal to k, are formed by cosets of weight n − k and • for a given Reed-Solomon code C, the optimal cosets are not unique. Using the narrowband noise channel discussed in Section II, we investigated the performance of coset Reed-Solomon codes. A narrowband disturbance had probability ρn = 0.5 to be present with duration d × n, d ≤ 10 and energy Ens = 2 × Es . In Fig. 5(a) we depict the performance of the modified envelope detector (’ED C’), the modified Viterbi Ratio Threshold Test detector with various values for λ (’VRTT’) and the threshold detector (’TD’) discussed in Section IV-A with various threshold values using an (7,4) coset RS code. In Fig. 5(b) we use an (7,2) coset RS code. The (7,4) coset code has a same symbol weight of 4 and was derived by adding the coset α4 +x+x2 to the (7,4) RS code generated by g(x) = (x − α1 )(x − α2 )(x − α3 ). The (7,2) coset code has a same symbol weight of 2 and was derived by adding the coset α2 +α2 x+α5 x2 +α6 x3 +α6 x4 to the (7,2) RS code generated by g(x) = (x − α1 )(x − α2 )(x − α3 )(x − α4 )(x − α5 ). V. C ONCLUSION We compared the performance of envelope detection, threshold detection and an M -FSK decoder based on Viterbi’s threshold ratio test for the AWGN channel and a wideband noise channel. For the AWGN channel, we found that for high rate codes, the errors-only envelope detection scheme outperformed the errors-and-erasures schemes. For low rate codes, the errors-and-erasures schemes, especially the detector deploying Viterbi’s threshold ratio test, outperformed the errors-only envelope detector. The reason for this is that the number of erasures declared in the high rate coding scenario exceeds the number of correctable erasures of the code. For the low rate coding scenario, more of these erasures can be

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corrected, thus the improved performance by the detector using Viterbi’s threshold ratio test. We further developed two schemes to be used in an AWGN channel with narrowband noise disturbances. The first scheme is a modification to the threshold detector, where we take into consideration the effects of narrowband noise disturbances. The modified threshold detector needs an external mechanism to detect the presence of narrowband noise. We also developed two coding schemes capable of detecting narrowband noise with a build in error-correcting capability based on ReedSolomon codes. We defined the same symbol weight of a codebook, and used this weight to determine the effectiveness of the two coding schemes.

[3] A. Viterbi, “A robust ratio-threshold technique to mitigate tone and partial band jamming in coded MFSK systems,” in Proc. 1982 IEEE MILCOM, Boston, MA, USA, Oct. 17 – 20, 1982, pp. 22.4.1 – 22.4.5. [4] A. Matache and J. A. Ritcey, “Optimum code rates for noncoherent MFSK with errors and erasures decoding over Rayleigh fading channels,” in Signals, Systems and Computers, 1997. Conference Record of the ThirtyFirst Asilomar Conference on, Pacific Grove, CA USA, Nov. 2 – 5, 1997, pp. 62 – 66. [5] J. D. Choi, D.-S. Yoo, and W. E. Stark, “Performance limits of M -FSK with Reed-Solomon coding and diversity combining,” IEEE Transactions on Communications, vol. 50, pp. 1787–1797, Nov 2002. [6] J. G. Proakis, Digital Communications. Johannesburg: McGraw-Hill, Inc., 1983. [7] Y. Choi, “Coset codes with maximum capability of synch-correction,” Electronic Letters, vol. 35, pp. 1535 – 1537, Sept 1999.

R EFERENCES [1] A. H. Vinck, “Coded modulation for power line communications,” AEU International Journal of Electronics and Communications, vol. 54, pp. 45 – 49, 2000. [2] N. Pavlidou, A. H. Vinck, J. Yazdani, and B. Honary, “Power line communications: state of the art and future trends,” IEEE Communications Magazine, vol. 41, pp. 34–40, Apr 2003.

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