Bit-Interleaved Coded Modulation with Iterative Decoding in Impulsive Noise

Transcription

1 Bit-Interleaved Coded Modulation with Iterative Decoding in Impulsive Noise Trung Q. Bui and Ha H. Nguyen Department of Electrical Engineering, University of Saskatchewan 57 Campus Drive, Saskatoon, SK, Canada S7N 5A9 Abstract Power line communications (PLC) suffers performance degradation due mainly to impulsive noise interference generated by electrical appliances. This paper considers the application of bit-interleaved coded modulation with iterative decoding (BICM-ID) in the Class-A impulsive noise environment to improve the spectral efficiency and error performance of PLC. In particular, the soft-input soft-output demodulator is developed for an additive Class-A noise (AWAN) channel so that iterative demodulation and decoding can be performed at the receiver. The effect of signal mapping on the error performance of BICM- ID systems in impulsive noise is also thoroughly investigated, both with computer simulations and a tight on the asymptotic performance. I. INTRODUCTION In recent years, the use of existing power lines for transferring voice or data signals has received considerable interest, both in research community and practice. By the early 1940 s, the concept of using the power grid for communication or control purposes was turned into reality with an PLC system over high voltage distribution 1]. Nowadays, PLC can provide a natural communications line for various devices such as alarm sensors, controllers, and even slow scan TV images for security purposes ]. Furthermore, the birth and growth of the Internet has accelerated the demand for digital telecommunication services to every home 3]. If the electricity distribution network can carry such services, every premises, factory, office, and organization will be interconnected and form a truly global information superhighway network. However, electrical distribution grids that constitute a universal wiring system were not built for communication purposes. Varying levels of impedance and attenuation due to electrical hardware configurations are frequent. Such variations and other interferences from various sources lead to a very poor performance of PLC systems. Those interferences, referred to as man-made noise, have statistical characteristics much different from that of classical Gaussian interference. Man-made noise is typically impulsive. A relatively simple model that incorporates background noise and impulsive noise is suggested in 4] and known as Middleton s Class- A noise. This noise model corresponds to an independent and identically distributed (i.i.d.) discrete-time random process whose probability density function is an infinite weighted sum of Gaussian densities, with decreasing weights for Gaussian densities and increasing variance 5]. The Middleton s Class- A noise model has been used widely in performance analysis of PLC systems 6] 8]. In general, an optimal or suboptimal receiver designed for an additive white Gaussian noise (AWGN) channel does not work well for the systems disturbed by impulsive noise. For example, the work in 6] considers the optimal receiver design for an uncoded M-ary quadrature amplitude modulation (M- QAM) system in additive white Class-A impulsive noise (AWAN), where the authors show that using the optimal receiver designed for AWGN in an impulsive noise environment results in 40dB performance degradation at the symbol error probability of. Similarly, the design of a turbo receiver over an AWAN channel in 7] shows that the performance improvement can reach over 11dB by replacing the classical optimal turbo receiver for AWGN with a properly designed receiver for the impulsive noise. The work in 7] is then extended in 9] for low density parity check codes. Other research work on the optimal and suboptimal (coherent or non-coherent) detection in impulsive noise environment can also be found in 4], 8], 10], 11]. This paper is primarily concerned with the application of bit-interleaved coded modulation (BICM) in an impulsive noise environment in general, and in PLC systems in particular. It was first suggested in 5] that BICM could be a fruitful coding option for PLC systems due mainly to the fact that BICM links binary coding and M-ary modulation in a simple way. The structure of BICM is simpler than that of trellis coded modulation (TCM) or multilevel coded modulation (MCM) and it allows a large degree of flexibility (e.g., the choice of the binary code can be made independent of the modulation scheme). The suitability of BICM as a coding option in PLC systems becomes more evident with the success in applying iterative (turbo) decoding for BICM. In particular, it is shown in 1] 14] that using suitable mappings, the error performance of BICM with iterative decoding (BICM- ID) significantly improves over that of the original BICM with. It should also be pointed out that iterative decoding has been proposed in 15] to deal with impulsive noise. However, the system considered in 15] is quite different from the one in this paper as it does not includes classical error control coding. In fact, the complex number codes in 15] refer to a linear transform defined over the complex numbers. Such a concept is better known as modulation diversity. Here the application of BICM-ID over an AWAN channel to improve the error performance as well as spectral efficiency

2 of PLC systems is considered. Specifically, a soft-input softoutput demodulator is developed for iterative decoding over an AWAN channel. The effect of signal mapping on the error performance is also studied, both with computer simulations and analytical evaluation of a tight on the asymptotic performance. SISO decoder II. SYSTEM MODEL uk c k vk sk Encoder π Modulator Channel information bits P( u ; O) Fig. 1. P( c ; I) the extrinsic information 1 π P( v ; O) Demodulator P( c ; O) P( v ; I) π the a priori probability Block diagram of a BICM-ID system. The block diagram of BICM-ID is shown in Fig. 1. The transmitter is built from a serial concatenation of an encoder, a random bit interleaver π and a modulator. The information bits {u k } are first encoded by a convolutional code to produce a coded sequence {c k }. The coded sequence {c k } is then interleaved by a random interleaver. The interleaved sequence {v k } is mapped by the modulator into the symbol sequence {s k } for transmission. Each symbol s k is chosen from a two-dimensional M-ary constellation Ψ according to some mapping rule µ( ). The baseband received signal over the kth symbol period can be written as r k = s k + n k (1) where n k is impulsive noise. As discussed earlier, noise in PLC systems has impulsive features and Middleton s Class- A model is commonly used to characterize impulsive noise 6], 7], 10], 11]. The probability density function (pdf) of additive white complex Class-A noise (AWAN) is: ) A Am 1 p A (n) = e m! πσm exp ( n () where σm = σ m/a+γ 1+Γ is the mth impulsive power, A is known as the impulsive index, σ is total noise power (including the powers of impulsive noise and Gaussian background noise), and Γ=σG /σ I is Gaussian-to-impulsive noise power ratio (GIR) with σg and σ I are the powers of Gaussian and impulsive noise, respectively. When A is increased, the impulsiveness reduces and the noise comes close to the Gaussian noise. Equation () shows that sources of impulsive noise r k have a Poisson distribution, and each impulsive noise source generates a characteristic Gaussian noise with a different variance. Let N 0 = σ be the one-sided power spectral density of the total white noise. The pdf of Class-A impulsive noise can be written as a function of A, Γ and N 0 as follows: p A (n) = e A Am A(1 + Γ) m!πn 0 m + AΓ exp ( A(1 + Γ) m + AΓ n N 0 Due to the presence of bit-based interleaving, the true maximum likelihood decoding of BICM is too complicated to implement in practice. Therefore the receiver in Fig. 1 uses a suboptimal, iterative method with soft-output demodulator and the SISO decoder. Though the two components of the receiver are individually optimal, the demodulation of the received signal and the decoding of the convolutional code act separately. The SISO channel decoder uses the MAP algorithm 1], 13]. Similar to decoding of Turbo codes, here the demodulator and the channel decoder exchange the extrinsic information of the coded bits P (v ; O) and P (c ; O) through an iterative process. The detailed algorithm for the optimal soft-output demodulator, i.e., the maximum a posteriori probability (MAP) demodulator is described in 1] for the case of an AWGN channel. For the channel disturbed by impulsive noise as considered in this paper (i.e., AWAN channel), the optimal soft-output demodulator is developed in the next section. III. ITERATIVE RECEIVER FOR AWAN CHANNELS Let m = log M be the number of coded bits carried by one M-ary symbol. The a posteriori probability for coded bits can be computed as follows: P (vk i = b r k )= P (s k r k ) p(r k s k )P (s k ) (4) s k Ψ i b s k Ψ i b where i = {1,,,m}, b = {0, 1} and the signal subsets Ψ i b = {µ(v1 k,v k,,vm k ]) v = b}. The notation indicates replacement by an equivalent statistic. As in Fig. 1, denote P (q; I) and P (q; O) the aprioriand a posteriori probabilities of random variable q, respectively. In the initial demodulation, assume that the aprioriprobability P (s i ) is equiprobable. With this assumption, the demodulator computes the extrinsic probability P (vk i ; O) for each group of m coded bits per constellation symbol. After being deinterleaved, P (vk i ; O) becomes the aprioriprobability input P (c ; I) to the SISO decoder. The SISO decoder, in turn, calculates the extrinsic information P (c ; O) and P (u ; O). The hard decision is made from the knowledge of P (u ; O). Then P (c ; O) is reinterleaved and fed back as the aprioriprobability P (vk i ; I) for the next iteration. Because an ideal interleaver makes m bits in one symbol independent, the a priori information P (vk 1; I),P(v k ; I),,P(vm k ; I) can be assumed to be ) (3)

3 independent. Therefore, for each constellation symbol, the a priori probability can be computed by: P (s k ) = P ( µ ( v 1 (s k ) v m (s k )] )) m = P (v j k = vj (s k ); I) (5) j=1 where v j (s k ) is the value of the jth bit in the label of symbol s k. From the second iteration, the extrinsic information can be determined as follows: P (vk i = b; O) = P (v = b r k) P (vk i = = b; I) = s k Ψ p(r s k )P (s k ) b P (vk i = b; I) P (v j k = vj (s k ); I) (6) p(r k s k ) s k Ψ i j i b Equation (6) tells that one can calculate the extrinsic information for one bit by using the aprioriprobabilities of the other bits in the same channel symbol. Also, such calculation depends on the conditional probability density function p(r k s k ). Concentrating on the case of two-dimensional constellation Ψ, the received signals and the constellation symbols are separated into imaginary and quadrature components. Let {r k1,r k } and {s k1,s k } denote the two components of the received signal and constellation symbol at time k, then p(r s k ) is given by: A Am 1 A(1 + Γ) p(r k s k ) = p A (r k s k )= e m! πn 0 m + AΓ A(1 + Γ) (r k1 s k1 ) +(r k s k ) ] exp (7) m + AΓ N 0 Note that computing (7) involves infinite series of exponential functions and can be complicated. For easy implementation, such computation can be simplified with an algorithm and Jacobian approximation as in 7]. IV. ASYMPTOTIC PERFORMANCE Owing to the large coding gain produced by the iterative process, one is most interested in the asymptotic performance to which the iterative processing converges. This asymptotic performance can be analyzed with the error-free-feedback bound (). The error-free-feedback bounds for BICM-ID systems under Gaussian noise or Rayleight fading channels were obtained in 1], 13]. Here, the is derived for BICM-ID systems under AWAN. In general, the union bound of the bit error rate () for a BICM-ID with convolutional code of rate k c /n c, constellation Ψ, and labelling ξ is given by: P b 1 C d f(d, Ψ,ξ) (8) k c d=d H where C d is the total information weight of all error events at Hamming distance d H. The function f(d, Ψ,ξ) is the average pairwise error probability, which depends on the mapping ξ, the constellation Ψ and Hamming distance d. Letc and c denote the input and estimated sequence with Hamming distance d between them. Let x and x represent the signal sequences corresponding to c, c respectively. Also without loss of generality, assume that c differs from c in the first d consecutive bits. Thus x and x can be considered to have length of d M-ary symbols. That is, x = x 1,x x d ] and x = x 1, x x d ]. To obtain the function f(d, Ψ,ξ), one needs to compute the pairwise error probability (PEP), P (x x), which is the probability that the receiver makes a decision on x given that x was transmitted. Following the same notation as in 4], define Ω:=w(ˇx, r) w(x, r), where w(x, r) is defined as the following additive decoding metric for a codeword length of d: d α m w(ˇx, r) = ln πσm exp ( r l x l ) ] (9) l=1 A Am with α m = e PEP can then be calculated as: m! and, as before, σm = N0 P (x ˇx) =P (Ω > 0 x) = 0 p(ω = ζ x)dζ m+aγ A(1+Γ).The Using the Chernoff bounding technique, this PEP can be upper-bounded by: d P (x x) min C(x l, x l,λ) (10) λ l=1 where C(x l, x l,λ) is the Chernoff factor, given by: C(x l, x l,λ)= e λ ω( x l,r l ) ω(x l,r l )] p(r l x l )dr l (11) with ω(x l,r l )=lnp A (r l x l ) and ω( x l,r l )=lnp A (r l x l ). Since MAP detection is used in our system, the Chernoff factor can be minimized with λ =1/, and as in 4], it is given by: C(d l ) := C(x l, x l, 1/) = 1 π α m σm exp (x d l/) + y σ m α m σm exp (x + d ] l/) + y dxdy (1) where d l := x l x l denotes the Euclidean distance between the two M-ary symbols. Now the function f(d, Ψ,ξ) can be obtained by averaging the PEP over all possible sequences x and x. Due to the success of iterative decoding, one needs to only consider two signal symbols x l and x l whose labels differ in only 1 bit. Under assumption that d l is i.i.d discrete random variable, f(d, Ψ,ξ) can be bounded as follows: f(d, Ψ,ξ)=E{P (x ˇx)} d E{C(d l )} =E{C(d l )}] d (13) l=1 For each signal symbol s i Ψ, lets j (i, k), where k = 1,...,m, denotes the signal symbol whose label differs at ]

4 position k compared to that of s i. Then a direct and bruteforce way to evaluate E{C(d l )} for any signal constellation and mapping is as follows: 1 m E{C(d l )} = m m s i Ψ k=1 ( ) α m x si sj(i,k) + y πσm exp ( α m x + si sj(i,k) πσm exp ) + y dxdy (14) However, for given signal constellation and mapping scheme, the probability mass function (PMF), P (d l ), of the discrete random variable d l can be easily obtained and E{C(d l )} can be simply obtained by averaging C(d l ) over such a PMF. For example, Table I tabulates the PMFs of d l for different mapping schemes, namely Gray, set partitioning (SP) and semi set partitioning (SSP) mappings, of 8PSK (phase-shift keying) constellation as shown in Fig.. Such constellation and mappings will also be used in the following section to illustrate the error performance of BICM-ID. yields a spectral efficiency of bits/sec/hz. Each information block has a length of 3999 information bits. Three different mapping schemes, namely Gray, the set partitioning (SP) and the semi set partitioning (SSP) mappings, are considered. These mappings are shown in Fig.. 9 iter., standard demodulator, A=10 9 iter., standard demodulator, A= iter., optimal demodulator, A=10 9 iter., optimal demodulator, A= Fig. 3. performance of 8PSK-SSP mapping for BICM-ID over an AWAN channel: comparison of different demodulators with different impulsive parameters, Γ= Fig d 3 d d 001 d Set partitioning (SP) mapping PSK with different mapping schemes Semi set partitioning (SSP) mapping TABLE I PMFS OF RANDOM VARIABLE d l CORRESPONDING TO GRAY, SPAND SSP MAPPINGS OF 8PSK CONSTELLATION. d l SP mapping SSP mapping d 1 /3 1/3 0 d 0 1/3 1/3 d 3 1/3 0 1/3 d 4 0 1/3 1/3 V. RESULTS AND DISCUSSION Although any channel coding and constellation/mapping schemes can be used in the BICM-ID systems under consideration, in the interest of space, this section concentrates on studying the error performance of a BICM-ID system employing 8PSK modulation and a 4-state rate-/3 convolutional code, whose generator polynomials are G = 1001; 0001; 1100]. This combination of channel code and modulation scheme First, Fig. 3 compares the bit error rate () performance of BICM-ID system employing 8PSK-SSP signalling scheme for two different demodulators. The first demodulator, referred to as the standard demodulator, is the optimal demodulator for an AWGN channel. The other demodulator is the one optimally designed for AWAN channels as described in Section III. The comparison is investigated for two impulsive parameters of A = 10 and A = whereas the GIR parameter is fixed at Γ=. At the level of one can observe a huge SNR gain of approximately 38 db by using the optimal demodulator over the standard one for AWAN channel parameters {A =, Γ= }. 1 When AWAN channel parameters are {A =10, Γ= } the SNR gainsaredband3dbatof and respectively. These substantial gains thus clearly illustrate the need of having the optimal demodulator for BICM-ID in impulsive noise environment. One can also expect the reduction of SNR gain when the impulsive index A increases. This is because when A is large enough (i.e., A 1), the impulsive noise statistic comes close to that of Gaussian noise and there is not much difference in performance by using the optimal and standard demodulators. Next, Figs. 4 and 5 show the performance of 8PSK- SSP and 8PSK-SP versus. The parameters of impulsive noise channel are A = and Γ= in both systems, which means that the channel noise is highly impulsive. Also 1 Here SNR is the same as with the energy per information bit.

5 SP mapping, 5 iterations SSP mapping, 9 iterations coded BPSK 1 iteration 3 iterations 5 iterations 7 iterations 9 iterations Fig. 4. performance of BICM-ID over AWAN channel with A = and Γ= : 8PSK and SSP mapping. Fig. 6. performance of 8PSK BICM-ID over an AWAN channel with A =10and Γ= : Comparison of different mappings and coded BPSK. SP mapping, 5 iterations SSP mapping, 9 iterations coded BPSK 1 iteration 3 iterations 5 iterations 7 iterations 9 iterations Fig. 5. performance of BICM-ID over AWAN channel with A = and Γ= : 8PSK and SP mapping. Fig. 7. performance of BICM-ID over an AWAN channel with A = and Γ= : Comparison of different mappings and coded BPSK. shown in each figure is the asymptotic performance (or error bound), calculated as described in Section IV, by using the first 0 Hamming distances of the convolutional code. As can be seen from these two figures, performance improvement due to iterations is very significant with both SSP and SP mapping. In particular, Fig. 4 shows that with 9 iterations the performance of 8PSK-SSP mapping can reach the at SNR of 4.5 db, and starting at SNR of 3.5 db the error performance of 7, 8 and 9 iterations are practically the same and also reach the asymptotic performance. Similar observations can also be made for the performance of 8PSK-SP signalling scheme shown in Fig. 5. However, the iteration process in 8PSK-SP does not improve error performance as much as compared to 8PSK-SSP. It can be seen from Fig. 5 that the largest performance improvement of 8PSK-SP mapping is reached only after 3 iterations and at the high SNR of.5 db. Though not shown here, similar to the cases of AWGN and fading channels, it was observed that iterations do not improve performance for of BICM-ID in impulsive noise. Figs. 6, 7 and 8 explicitly compare the error performance of different mapping schemes for 8PSK. Shown in each figure is the performance for a particular set of impulsive parameters, namely {A = 10, Γ = }, {A =, Γ = } and {A =, Γ = }. As can be observed from Fig. 6, when SNR is larger than 3.75 db (or 6.5 db in

6 SP mapping, 5 iterations SSP mapping, 9 iterations coded PBSK Fig. 8. performance of BICM-ID over an AWAN channel with A = and Γ= : Comparison of different mappings and coded BPSK. Fig. 7 and 16.5 db in Fig. 8) the SSP mapping always gives the best bit error rate () performance. For example, at level of, the SNR gains provided by the SSP mapping over the SP mapping are about 3, 3.5 and 3. db corresponding to channel noise parameters of {A = 10, Γ = }, {A =, Γ = } and {A =, Γ= }, respectively. It can also be seen that performs very poor. In these three figures, besides the performance of each mapping scheme, the corresponding error bound on the asymptotic performance is also plotted. Observe the tightness of the s in all cases, which suggests that the derived in this paper can be used as an effective tool for analyzing the asymptotic performance of different mapping schemes for BICM-ID systems in impulsive noise environment. The error performance of a coded system that employs the same convolutional code and BPSK modulation is also included in Figs. 6, 7 and 8. Note that the spectral efficiency of such a system is only /3 bits/sec/hz, which is only one third of that of the systems employing 8PSK constellation. Observe that there is error performance degradation suffered by 8PSK system that employs Gray or SP mapping. Impressively, the 8PSK system with SSP mapping not only can provide three times higher spectral efficiency but can also achieve error performance gain compared to the coded BPSK system. In particular, at the level of, the SNR gains of 8PSK- SSP over coded BPSK are about 1.6,.7,.5 db from Figs. 6, 7 and 8, respectively. Finally, Figs. 3, 6, 7 and 8 also show the dependence of performance on both the impulsive index A and the Gaussian impulsive ratio Γ. Simple inspection of these figures shows that for all mapping schemes of 8PSK, performance of BICM-ID improves if the channel noise becomes more impulsive (i.e., corresponding to smaller value of A and/or Γ). This observation is similar to the ones made in 6], 7] for uncoded systems and coded systems with BPSK, respectively. VI. CONCLUSIONS Application of BICM-ID to improve error performance and spectral efficiency of PLC systems was considered in this paper. In particular, Middleton s Class-A impulsive noise channel model is adopted and the optimal demodulator for this channel model was developed to facilitate the iterative receiver. The error-free-feedback bound was also derived to accurately predict the asymptotic performance of the systems. Computer simulation results showed that the use of demodulator optimally designed for impulsive noise significantly improves the performance of BICM-ID over the use of the standard demodulator designed for Gaussian noise. Simulation results also confirm the tightness of the bound on the asymptotic performance. It was also shown that using SSP mapping of 8PSK helps to improve the performance with iterations and gives the best performance compared to SP and Gray mappings. REFERENCES 1] J. E. Newbury and K. J. Morris, Power line carrier systems for industrial control applications, IEEE Trans. on Power Delivery, vol. 14, pp , Oct ] H. C. Ferreira, H. M. Grove, O. Hooilen, and A. Han Vinck, Power line communications: overview, in IEEE AFRICON 4th, pp , ] N. Pavlidou, A. J. Han Vinck, J. Yazdani, and B. Honary, Power line communications: state of the art and future trends, IEEE Commun. Mag., pp , Apr ] J. Haring and A. Han Vinck, Performance bounds for optimum and suboptimum reception under Class-A impulsive noise, IEEE Trans. Commun., vol. 50, pp , July 00. 5] E. Biglieri and P. de Torino, Coding and modulation for a horrible channel, IEEE Commun. Mag., vol. 41, pp. 9 98, May ] S. Miyamoto, M. Katayama, and N. Morinaga, Performance analysis of QAM systems under Class-A impulsive noise environment, IEEE Trans. Electromagnetic Compatibility, pp , May ] D. Umehara, H. Yamaguchi, and Y. Morihiro, Turbo decoding in impulsive noise environment, in Proc. IEEE Global Telecommun. Conf., pp , July ] T. Fukami, D. Umehara, and Y. Morihiro, Noncoherent PSK optimum receiver over impulsive noise channel, in Proc. of the 00 ISPLC Conf., pp , 00. 9] H. Nakagawa, D. Umehara, S. Denno, and Y. Morihiro, A decoding for low density parity check codes over impulsive noise channels, in Proc. of the 005 ISPLC Conf., pp , ] A. D. Spaulding and D. Middleton, Optimun reception in an impulsive interference environment - part I: coherent detection, IEEE Trans. Commun., vol. 5, pp , Sept ] D. Middleton, Statistical - physical models of electromagnetic interference, Electromagnetic compatibility; Proceedings of the Second Symposium and Technical Exhibition, vol. EMC-19, pp , June ] X. Li, A. Chidapol, and J. A. Ritcey, Bit-interleaved coded modulation with iterative decoding and 8-PSK, IEEE Trans. Commun., pp , Aug ] N. H. Tran and H. H. Nguyen, Design and performance of BICM-ID systems with hypercube constellations, to appear in IEEE Trans. on Wireless Commun. 14] N. H. Tran and H. H. Nguyen, Signal mappings of 8-ary constellations for bit-interleaved coded modulation with iterative decoding, to appear in IEEE Trans. on Broadcasting. 15] J. Haring and A. Han Vinck, Iterative decoding of codes over complex numbers for impulsive noise channels, IEEE Trans. Inform. Theory, vol. 49, pp , May 003.