On Iterative Multistage Decoding of Multilevel Codes for Frequency Selective Channels

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1 On terative Multistage Decoding of Multilevel Codes for Frequency Selective Channels B.Baumgartner, H-Griesser, M.Bossert Department of nformation Technology, University of Ulm, Albert-Einstein-Allee 43, D Ulm Abstract Combined channel coding and digital modulation (multilevel codes) yield power and bandwidth efficient digital transmission schemes. t is known that systems optimised for the Gaussian channel are not optimum for fading channels and vice versa []. We apply iterative multistage decoding of multilevel codes which leads to good prformance for both Gaussian and fading channels at bit error rates (BER) around lo4. n this paper we want to discuss this effect for the power line channel in low voltage networks at high frequencies. Keywords: multilevel codes, multistage decoding, iterative decoding. OFDM, frequency selective channel, power line communication. 1. NTRODUCTON AND SYSTEM OVERVEW Multilevel coding is one of the main approaches in constructing coded modulation schemes and was introduced by mai and Hirakawa [2]. The resulting codes have either a higher rate or an increased minimum distance compared to the case where coding and modulation is separated. The problem that occurs when applying multilevel coding is decoding. An optimum decoding strategy would be an overall maximum likelihood decoding of the multilevel code. This is impractical due to the immense complexity. Already mai and Hirakawa proposed multistage decoding as a practical approach to decode multilevel codes. The decrease in Sfecoding complexity results in an increase in the bit error rate. This loss can be much reduced by doing the process of multistage decoding iteratively. This holds for the channel with additive white Gaussian noise (AWGN) as well as for fading channels. To ensure the gain for both kind of channels the multilevel code has to be designed properly. As underlying fading channel we assumed the power line channel in low voltage networks at frequencies between 500 khz and 20 MHz. Within this frequency range, the power line channel can be modelled as a time invariant frequency selective fading channel 131. To cope with the poor behaviour of fading channels like this, it was reasonable to take OFDM (orthogonal frequency division multiplexing) as digital modulation scheme. The investigated system is a multilevel coded OFDM transmission scheme as depicted in Figure Multilevel Encoder data Figure 1: System overview

2 This paper is organised as follows. n section 2. the shaded part in Figure 1 will be replaced by a discrete time channel model. Section 3 deals with the coding and decoding process. Afterwards there is a short introduction into some design rules for multilevel codes in section 4. Finally.the simulation results are presented. 2. CHANNEL MODEL AND MODULATON SCHEME Most of the power line channel models found in literature deal with a large number of parameters in order to catch all of the effects occurring on the channel. n contrast to these approaches the channel can be just described by a time invariant transfer function [3], pointing out the effect of static multipath propagation. This effect of multipath propagation is due to the numerous reflections of the signal at unmatched input impedances of consumer equipment as well as on branch line connections. We considered the model from [3] based on N=6 paths which leads to the transfer function,depicted in Figure 2. Figure 2: Transfer function and transmission band By dividing the bandwidth into a number of subchannels, the OFDM transmission technique offers the possibility to deal with the impairment of such a frequency selective channel. Because of the time invariant transfer function description of the power line channel the OFDM scheme and the channel (shaded part in Figure 1) can be replaced by the discrete time channel model depicted in Figure 3. Figure 3: Discrete time channel model The received value y, is given by: Here a, represents a complex signal point out of the chosen 16QAM signal constellation. hk denotes the complex channel transfer function values of the subchanneh and nk is the AWGN component. Throughout this paper we assume perfect knowledge of the channel transfer function values hk defined as the channel state information

3 (CS). With the CS, the received values yt can be equalised by just dividing the received value yk with hk. n me addition the CS is used to calculate log likelihood ratios for the purpose of soft decision decoding. on We dimensioned the OFDM part according to the parameters given in Table 1. System Bandwidth 10 MHz Symbol Duration 51.2 ps Guard nterval 5 Ps Centre Frequency 10 MHz Number of Subchannels 512,Used Subchannels 440 Table 1:OFDM parameters The windowed region in Figure 2 represents the transmission bandwidth, and therefore roughly indicates the distribution of the channel transfer function values hk. 3. CODNG AND DECODNG As already mentioned, we applied a multilevel coding scheme for the purpose of forward error correction. The structure of a multilevel encoder is depicted in 4. DEMUX $2) c(2) = l(2) Encoder C(2) L MOD c(ml = 7 Figure 4: Structure of the multilevel encoder The idea of multilevel coding is that m outer encoders protect the iabelling of a partitioned signal constellation. As partitioning strategy we assume the so called mapping by set partitioning introduced by Ungerboeck [4]. Carrying out this strategy for the underlying 16QAM signal constellation, we arrive at the partitioning tree depicted in Figure 5. n each partitioning step, the nearest neighbour signal points are separated into two different subsets. Therefore the minimum Euclidean distances di) in the levels increase from level 1 down to level m: 6(0)<6(1) <... <a(") This effect of increasing distances is used in designing the outer encoders.

4 Figure 5: Mapping by set partitioning for l6qam 1 i n order to ensure a large degree of freedom in choosing the code parameters of the outer codes, we used punctured convolutional codes with memory M=6 as outer encoders. The generator polynomials and proper puncturing schemes can be found in [5][6][7][8]. To distribute deep fades over several symbols, we took separate random interleavers at each level. The length of a codeword is the same as the length of the interleaver and corresponds to one OFDM symbol. For decoding the multilevel code we use iterative multistage decoding similar to [9]. This means that the process of multistage decoding is repeated iteratively. n Figure 6 the structure of a multistage decoder is depicted. The number of outer encoders equals m=4 because of the 4-step partitioning of the 16QAM signal constellation..' n the first iteration, the level 1 code is decoded without any information about the decoding result of the other levels. A symbol by symbol A-posteriori (s/s APP) algorithm is used for decoding in order to provide reliability information and not only hard decision values. This reliability information is passed to the next stage. Stage number 2 can now use the received reliability information of the previous decoding step as a-priori knowledge about the signal points to improve the decoding result. This procedure is carried out until the last code is decoded. ;() d t(l) 321 * Y 4) i(2) i(m) ;, 8 j(m) - MUX. Z- i Figure 6: Structure of the multistage decoder

5 The problem now is that for the first iteration, the first level has to be decoded without a-priori knowledge about signal points at all. The BER is mainly determined by the first level because of the multiple representation of binary symbols by signal points. The degradation from decoding the first stage without a-priori information can be reduced drastically by doing further iterations incorporating the a-priori information from previous decoding steps. 4. CODE DESGN RULES A first practical approach in designing multilevel codes is the error probability rule (EPR). The aim of this rule is that all the levels should equally contribute to the total BER. For example this can be-achieved by doing simulations. t is obvious that this rule is practical for both, the AWGN channel as well as fading channels. A more theoretical rule for designing a multilevel code in the AWGN channel is targeted on achieving a code with preferably high minimum Euclidean distance. This means maximisation of the minimum distance between any two symbol sequences produced by the multilevel code in the Euclidean space. Such a code offers the best achievable asymptotic coding gain. With the constraint of a preferably high code rate of the multilevel code the so called balanced distance rule (BDR) achieves the above mentioned targets. Herethe product distance df) is equal (balanced) over all the levels i [lo]. Because the minimum product distance upper bounds the minimum Euclidean distance of the.multileve1 code. Where the product distance is defined as the product of the minimum Hamming distance d$) and the minimum Euclidean distance 6(') : dg) = dg) -a('). Following the BDR we des~gned the rate QAM multilevel code C-BDR for the AWGN channel. The parameters are given in Table 2. Level i Punc?xtd rate Free distance d$) Product distance dg) Overall rate 1 i S ' ": Table 2: Code CBDR designed for the AWGN n fading channels t: situation changes drastically. The aim under this constraint no longer requires the maximisation of the a;inimum Euclidean distance of the multilevel code. The Euclidean distance is not of great importance in fading channels. n this case the Hamming distances of the level codes are decisive. One can state that the level with the minimum Hamming distance is responsible for the asymptotic performance of the overall multilevel code under fading conditions. So the rule for getting the best asymptotic coding gain with the constraint of preferably high code rate is to choose identical level codes. n the sequel, this rule is referred to Hamming distance rule (HDR). n contrast to the AWGN channel, the simulation results in the power line fading channel showed that a code designed according to the HDR has inferior performance concerning the desired BER range than a code designed according to the EPR. Therefore we designed the rate l/2 QAM multilevel code C-EPR considering the case of several iterations. The parameters of this code are given in Table 3. Level i Punctured rate Free d$ Product distance dg) Overall rate 1!A U / W Table 3: Code C-EPR designed for the power line fading channel

6 According to both rate 1/2 16QAM multilevel codes, we obtain a system with a bandwidth efficiency of 2 bitslsymbol. Therefore we yield a data rate of approximately 17.2 Mbitls. 5. SMULATON RESULTS One of the main issues in investigating the process of iterative multistage decoding is to find a good compromise between decoding complexity together with decoding delay and the BER performance. n Figure 7 the result for the code C-BDR designed according to the BDR for the AWGN channel is depicted. Most of the coding gain is already achieved after the second iteration. All further iterations are not very practical because the coding gain increases only very slightly with the number of iterations, however the decoding complexity and decoding delay increases linearly with the number of iterations. n further simulations we apply a maximum of two iterations. 104L \ SNR [db] Figure 7: terative multistage decoding for the AWGN channel i As pointed out in section 4, the crucial points in multilevel code design differ with the underlying channel 1 model. Whereas for the AWGN channel the performance is guided by the product distance, for fading channels the Hamming distance is the key parameter in designing proper coding schemes. n Figure 8 the two codes C-BDR and C-EPR are compared concerning the power line channel. The inferior 1 performance of C-BDR after two iterations is about 1.5 db although C-EPR has almost the same performance as C-BDR after one iteration. The reason of the worse performance of code C-EPR after the fist iteration is due to the poor performance of the first level. This changes drastically with the second iteration. Code C-EPR reaches a much higher coding gain with the second iteration because the last level code can still improve the first level because of the higher Hamming distance. f we compare the performance of both codes in the AWGN channel as depicted in Figure 9, it points out that the loss with code C-EPR compared to code C-BDR after two iterations is small. There is a gap of approximately \ 0.5 db. The problem of the first level (and therefore the poor performance after the first iteration) of code C-EPR is again reduced by applying a second iteration.

7 PcR d01 Figure 8: Power line channel SNR [del Figure 9: AWGN channel 6. CONCLUSONS n this paper we discussed the effects of iterative multistage decoding with respect to an underlying OFDM transmission scheme over the power line channel. The main point with applying several iterations is the improved performance of the fist level. Because of this, we were able to shift code redundancy from the upper levels to the levels at the bottom in order to improve the performance under fading conditions. The resulting degradation in the upper levels can be overcome with iterative multistage decoding. This kind of improved multistage decoding seems to be a step towards the overall maximum likelihood decoding of the multilevel code. Furthermore only two iterations are enough to exploit almost all coding gain. We conclude that this coding scheme with constraint to a proper code design is able to achieve sufficient performance both for the AWGN channel and the power line fading channel. References [] C.E. Sundberg and N. Seshadri, "Coded modulation for fading channels: An overview", European Trans. on Telecommunications, vo1.4, pp , May-June [2] H. mai and S. Hirakawa, "A New Multilevel Coding Method Using Error-Correcting Codes", EEE Trans. n. Theory, vol. T-23, pp , May [3] M. Zimmermann and K. Dostert, "A Multi-Path Signal Propagation Model for the Power Line Channel in the High Frequency Range", Proc. 3'd nternational Symposium on Power Line Communications (SPLC '99), Lancaster, UK, [4] G. Ungerboeck, "Channel Coding with MultileveVPhase Signals", EEE Trans. nf: Theory, vol. JT-28, pp , January [5] J. Hagenauer, " Rate-Compatible Punctured Covolutional Codes (RCPC Codes) and their Applications", EEE Trans. on Comm.., vol. 36(4), pp , April [6] R. Palazzo Jr., " A Network FLOW Approach to Convolutional Codes", EEE Trans. on Comm., vol. 43(4), pp , April [7] M. G. Kim, "On Systematic Punctured Convolutional Codes". EEE Trans. on Comm., vol. 45(2),pp February 1997.

8 i [S] L. C. Lee, "Convolutional Coding: Fundamentals and Applications", Artech House, 1997 [9] T. Woerz and J. Hagenauer, "Decoding of M-PSK-Multilevel Codes7', European Trans. on Telecommunications, vol. 4, pp , May-June 1993.' [lo] J. Huber, U. Wachsmann and R. Fischer, "Multilevel Codes: Theoretical Concepts and Practical Design Rules", EEE Trans. n. Theory, vol. T45, pp , July 1999.