On the performance of Turbo Codes over UWB channels at low SNR

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1 On the performance of Turbo Codes over UWB channels at low SNR Ranjan Bose Department of Electrical Engineering, IIT Delhi, Hauz Khas, New Delhi, , INDIA Abstract - In this paper we propose the use of Turbo Codes for M-ary pulse position modulated pulses over UWB communication channels with multipath. The transmitted power of the UWB pulses is required to be very small as they are used in the existing spectrum designated to other uses. This constraint requires that UWB signals have good performance at low SNR s. Turbo codes have been shown to perform well at low SNRs. We present the simulation results of using rate 1/2 and 1/3 turbo codes. Also, the effects of varying the symbols set size, the SNR of the channel and the interleaver size are presented. It has been shown in this paper that for a fixed SNR (4 db), the rate 1/3 code shows an improvement of 2 orders of magnitude when M = 2 and 3 orders of magnitude when M = 4 or M = 8. For the same SNR, the rate 1/2 code shows an improvement of 1 to 2 orders of magnitude. To achieve a BER of 10-5, the rate 1/3 turbo code requires SNR = 6 db when M = 4 and SNR = 3 db when M = 8. Increase in the size of the interleaver from 2 9 to 2 14 results in a decrease in BER from 10-2 to 10-3 for SNR = 4 and M = 2. Key-Words: Turbo Codes, Ultra Wideband Communication 1. Introduction This paper deals with the performance of Turbo Codes [1, 2] for M-ary pulse position modulated pulses over UWB communication channels with multipath [3]. An Ultra-wide band signal is short impulse radio signal whose spectrum is spread over a wide range of frequencies. Due to the recent development of low power emission systems and the low power density of the signals, it seems possible that UWB signals can be used in the existing spectrum already designated to other use by Federal Communication Commissions (FCC) without harmful interference. However, this requires good performance at low SNR for these signals. In this paper we propose the use of turbo codes on UWB communication system due to there superior performance over previous coding schemes. Introduced first in early 1990 s the near Shannon limit performance of turbo codes have made them increasingly popular. Turbo codes employ the use of two or more Recursive Systematic Convolutional (RSC) encoders and a same number of decoders that use an iterative decoding algorithm [4], [5]. Each iteration involves the receiving of a priori information from previous decoder and passing posteriori information to the next decoder by each decoder. The key innovation of turbo coding is the interleaver Π which separates the encoders and permutes the incoming block before feeding it to the other encoders. This allows that input sequences for which one encoder produces low-weight codewords will usually cause the other encoders to produce a high-weight code word. Thus, even though the constituent codes might be individually weak, the combination is surprisingly powerful. For analyzing the performance and design issues of turbo codes in a UWB channel, an indoor wireless communication channel has been considered which offers the prospect of freeing our homes and offices from wiring. In any such channel environment, multipath fading is prominent. Many models and experiments for both indoor and outdoor UWB signal propagation have been discussed in previous works [3], [6-8]. 1

2 This paper has been organized as follows. In Section II, we first describe the system model employed by us for the simulation. The turbo encoder, the communication channel (with multipath fading and AWGN noise ) and the decoder have been discussed in this section. The models used to simulate these blocks have been described. In section III we discuss the simulation environment. Simulations have been carried out for bit error rate versus SNR, with and without turbo coding. Performance for both unpunctured turbo codes (which gives rate 1/2) and for punctured turbo codes (which gives rate 1/3) can be seen. Also the effect of the size of the interleaver and the number of iterations on the bit error probability is obtained. Finally, the results are discussed in section IV. The paper concludes in section V. 2. System Model The system model is shown in Fig. 1. A source generates random binary output bits that are encoded using a turbo encoder. These encoded bits are converted to the UWB pulses that are then passed through the simulated UWB communication channel. A maximum likelihood detector compares the received signal with all the possible input symbols and chooses the input as the one having maximum correlation. These detected symbols are then fed into the turbo decoder as binary bits and the output bits are obtained. The turbo coder and the UWB channel are described below in detail. 2.1 Turbo Encoder The turbo encoder encodes blocks of N bits. It consists of two Recursive Systematic Convolutional (RSC) encoders separated by an N bit interleaver. The interleaver permutes the block of N bits randomly before feeding it to the second encoder. The interleaver permutation can be designed for specific purposes to give better performance [9] than the random interleaver. However, the pseudorandom interleaver performance performs just as well when N is large [4] or when number of iterations is increased [10]. The output bit stream is obtained by multiplexing the input bit stream and the parity bit stream generated by the individual encoders. This gives an overall rate of 1/3. Puncturing of the encoded bit stream can increase the rate to 1/2, which has been incorporated by dropping all even parity bits from one encoder and odd parity bits from the other. The generator matrix for both the encoders are taken to be identical without any loss of generality. For each encoder, the number of states in the trellis is given by 2 m where m is the memory of the encoder. For efficient decoding it is desirable that at least the first encoder ends in an all-zero state. Therefore, the last m bits of the N bit block should force the first encoder to the all zero state by the end of the block. 2.2 UWB Communication Channel The choice of the UWB signal and the modeling of M-ary PPM modulation is based on the work presented in [11]. A UWB pulse has been modeled by the second derivative of a gaussian function exp(-2π[t / t n ] 2 ). The same pulse has been taken as examples in [11]. Thus the transmitted pulse is p TX (t) = t exp(-2π[t / t n ] 2 ) (1) The effect of the antenna system in the transmitted pulse is modeled as a differentiation operation. Thus, the received pulse is p(t) = [1-4π[t / t n ] 2 ] exp(-2π[t / t n ] 2 ) (2) This pulse is plotted in Fig.2(a). Each signal consists of N s intervals of time T f, each containing a time-shifted UWB pulse. The N s shifts for the j th pulse in one signal is given by a i τ min (i = 0,2, N s -1) where a i is an m-sequence consisting of 0,1 pattern and τ min is the value for which the correlation function has the minimum value. The choice of m-sequence is due to their favorable correlation property [12]. The N s time shifts for other symbols are obtained by the M cyclic shifts of a i. These choices of time-shifts give us M-ary PPM signals that are equally correlated. The PPM modulated UWB wave is shown in Fig 2(b). The above parameters are subject to the constraints that T p + τ min < T f where T p is the pulse duration and 1 < M < N s. The noise in the channel is AWGN with variance N o /2. 2

3 The indoor statistical model given in [3] has been used to model the multipath effect of the channel. The model treats the short duration radar like pulses to be arriving in clusters at the receiver, each cluster consisting of a number of multipath signals. The inter-arrival time for the clusters has been modeled as a poisson process with a fixed rate Λ. Inter arrival time of signals within each cluster is modeled as a poisson process with fixed rate λ. If the arrival time of the l th cluster be denoted by T l, l=0,1,. and the k th signal measured from the beginning of the l th cluster be denoted by t kl, k=0,1,2... By definition, for the first cluster, T o =0, and for the first ray within the l th cluster, t 0l =0. Hence, T l and t kl are described by the independent inter-arrival exponential probability density functions, p(t l T l-1 ) = Λ exp[ -Λ (T l T l-1 ) ], l > 0, (4) p(t kl t (k-1)l ) = λ exp[ -λ (t kl t (k-1)l ) ], k > 0. (5) Let β kl be the path amplitude gain of the k th signal of the l th cluster Then the mean square values {(β kl) 2 } are monotonically decreasing functions of {T l } and {t kl }. According to the model in [3], β 2 kl = β 2 (T l, t kl ) = C exp(-t l / Γ) exp(-t kl / γ) (6) where C is the average power gain of the first ray of the first cluster, and Γ and γ are power-delay time constants for the clusters and the rays, respectively. The probability density function for the path power gain is given by p(β 2 kl) = (β 2 kl) -1 exp(-β 2 kl / β 2 kl) (7) and for the path amplitude gain it is given by the Rayleigh probability density function, p(β kl ) = (2β kl / β 2 kl) exp(-β 2 kl / β 2 kl). (8) The above density functions take the assumption that the probability distribution of the normalized power gain is independent of the associated delays or of the location within the building. The value of constant C depends upon the distance between the transmitter and the receiver and on the path loss exponent. 2.3 Turbo Decoder The turbo decoder uses the modified BCJR algorithm [4], [5]. It is an iterative decoder that uses two BCJR-MAP (maximum a posteriori) decoders. Each constituent decoder sends a posteriori likelihood estimates of the decoded bits to the other decoder, and uses the corresponding estimates from the other decoder as a priori likelihood. The uncoded information bits are available to each decoder to initialize the a priori likelihood. If y is the noisy received codeword, then in the modified BCJR algorithm, the decoder decides u k = +1 if P(u k = +1 y) > P(u k =-1 y) and it decodes u k = -1 otherwise. Hence the decision u k is given by u k = sign [L(u k )] (9) Where L(u k ) is the log a posteriori probability (LAPP) ratio defined as L(u k ) = log(p(u k = +1 y) / P(u k = -1 y)) (10) This ratio is calculated recursively using the modified BCJR algorithm. A detailed analysis is presented in [1, 2], [5]. Note here that the initial and boundary conditions require that the encoder starts in state 0. For the iterative MAP decoding, the LAPP ratio for an arbitrary MAP decoder can be written as L(u k ) = log(p( y u k = +1) / P( y u k = -1)) + log(p( u k = +1) / P(u k = -1)) (12) where the second term represents the a priori information. Unlike the conventional decoders in which P(u k = +1) = P(u k = -1), this term is non-zero for iterative decoders. In each iteration, one decoder (say D1) receives extrinsic information from the. other (say D2) for each u k This serves as a priori information for D1. Similarly D2 receives extrinsic information from D1. The decoders calculate the extrinsic information by only using the information not available to the other decoder. The value of L(u k ) for a decoder can be summed up as the channel value, the a priori information about u k provided by a previous decoder and the extrinsic 3

4 information that can be passed on to the subsequent decoder. In case of 2 decoders, the first one uses the interleaver permutations and the second decoder uses the inverse of those permutations to calculate the value of the posteriori information to be passed to the other decoder. Hence, the decoders require knowing the trellis of the encoders and the knowledge of the interleaver and the de interleaver that is employed by the encoder. For details of how to extract extrinsic information, the derivation of the channel value and the derivation for L(u k ), refer to the work done in [2], [4]. 3. Numerical Simulations Simulations were carried out using the model described in the previous section. For the M-ary communication PPM signals, the different parameter values used are τ min =.4073, which can be obtained from the auto-correlation function of the UWB pulse chosen, and t n =.7531 ns, which gives T p = 2 ns [11]. The values of frame duration T f = 4 ns and number of frames in each symbol N s = 100 were chosen arbitrarily to satisfy the constraints mentioned in the previous section (T p + τ min < T f and 1 < M < N s.). For the UWB communication channel, the values of the parameters are the same as were experimentally obtained [3]. These are γ = 20ns, Γ = 60ns, 1/λ = 5ns and 1/Λ = 300ns. The attenuation with distance is taken according to the inverse square law and the distance between the transmitter and the receiver is taken to be 30 m. It has been observed in the experiment conducted in [3] that the rays and clusters outside a roughly 200 ns observation window in general are too small and negligible. Using the poisson distribution of clusters with arrival rate Λ=300ns, it can be seen that P(n > 2) = where n is the number of additional clusters in the 200 ns period. Hence in our simulation we have taken a maximum number of 3 clusters arriving at the receiver which is true with a probability of Simulation was also done for higher number of clusters with no significant changes in the results. Also, the number of signals in each cluster has been limited to a maximum of 5 because for signals received after that, the gain becomes much less than 1 due to the factor of exp (- t kl / γ). The simulation results for the performance of the channel without the use of turbo codes for M = 2, 4 and 8 is shown in Fig. 3. The improvement in the performance by the use of turbo error correction codes can be seen in Fig 4. The generation matrix used for turbo encoder is (g 1, g 2 ) = (7,5). Each encoder, therefore, has a memory of m=2 and the trellis has 2 2 states. The d free obtained for weight 2 and 3 inputs are 9 and 11 respectively for the unpunctured code. For the case of punctured code, these values are 6 and 5. The interleaver size is taken as 1024 bits and the decoder performs 5 iterations before deciding the output. Increasing either the interleaver size or the number of iterations of the decoder improves the performance as shown by Fig 5 and Fig 6 respectively. 4. Discussion Of The Results The improvement in the bit error rate due to turbo error correction codes for the simulated indoor UWB communication channel is most significant for low SNRs. Simulations were also performed for SNR less than 0 db. These values of SNR yielded an improvement in performance for the rate 1/3 code (unpunctured) whereas the rate 1/2 code showed no significant improvement over the channel performance. To achieve a BER of 10-5, the rate 1/3 turbo code requires SNR = 6 db when M = 4. The same BER is achieved for SNR equal to 3 db when M = 8. The SNR value is further expected to decrease with increase in value of M as can be inferred from Fig. 7. The maximum improvement (in terms of BER) due to turbo codes is observed for SNRs in the range of 2-5 db irrespective of the M value as shown in fig 4(a), (b), (c). In this range, the reduction in BER is about 3 orders of magnitude for both M = 4 and M = 8. After 5 db, the BER performance curves for rate 1/3 seem to flatten and do not increase much with increase in SNR. For M = 2, the rate 1/2 code does not perform as well as the rate 1/3 code. This is shown in Fig 4(a). For rate 1/2 code, the performance improves by 2 4

5 orders of magnitude at higher values of SNR (5 to 6 db). As is the case in the rate 1/3 code, there is an improvement in performance for the rate 1/2 code when we move from M = 2 to M = 8. For example, for SNR=4, the rate 1/2 code gives a BER of 10-2 when M = 2, but give BER of 6x and 9x10 5 when M = 4 and 8 respectively as depicted in Fig. 4(b) and 4(c). By varying the interleaver size for the rate 1/2 code, we find that the improvement in the BER is one order of magnitude as the interleaver size is increased from 2 9 to 2 14 as shown in Fig. 5. We next compare the performance of Turbo codes over UWB channels with the performance of chaotic codes over UWB channel [15]. There is definite improvement when turbo codes are used. For example, when M = 2 and SNR = 5, BER performance for the chaotic codes is of the order of 10-3 as compared to an order of 10-4 in case of turbo codes with rate 1/3. The BER for chaotic codes when SNR = 5 is of the order of 10-4 for M = 16, whereas in case of rate 1/3 turbo codes the same is below 10-5 for M = 8. Moreover, the UWB channel used in [15] does not consider multipath fading. 5. Conclusions In this paper we propose the use of turbo codes for indoor mutipath UWB channels at low SNRs. The UWB channel simulated is an indoor one and an appropriate model has been chosen. Simulations have been carried out for both rate 1/2 and rate 1/3 codes, for different values of M. The effect of performance for different interleaver sizes has been analyzed. The number of iterations of the decoder was also varied and the performance improvement was observed. It was seen that the BER performance was better for the rate 1/3 than for rate 1/2. The BER improvement for rate 1/3 was three orders of magnitude (from 10 2 to 2x10-5 ) for M = 4 and SNR = 4. For the same values of M and SNR, the improvement for rate 1/2 was two orders of magnitude (from 10 2 to 6x10 4 ). This is achieved at the price of a reduced bit rate. It was also found that there is a steady improvement in performance for both 1/2 and 1/3 rate codes as we increase M from 2 through 8. For SNR as 3 db, rate 1/3 codes showed a decrease in BER from 2x10-3 to 1.8x10-5 while the rate 1/2 code showed a decrease from 3x10-2 to The effect of increasing the interleaver size from 2 9 to 2 14 was one order of magnitude for SNR = 4. We have demonstrated that there can be significant performance improvement using Turbo codes over multipath UWB channels in low SNR scenarios. References [1] Berrou C., Glavieux A. and Thitimajshima P., Near Shannon limit error correcting coding and decoding: Turbo Codes, Proc Int. Conf. Comm., p.p [2] Berrou C.and Glavieux A, Near optimum error correcting coding and decoding: turbo codes, IEEE Trans. Comm., p.p , Oct [3] Saleh A. M. A. and Valenzuela R. A., A Statistical Model for Indoor Multipath Propagation, IEEE journal on Selected areas in Communication, vol. SAC-5, p.p , Feb [4] Ryan W. E., A Turbo Code Tutorial, [5] Bahl L., Cocke F., Jelinek F. and Raviv J., Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans. Inf. Theory, p.p , Mar [6] Win M. Z., Ramirez-Mireles F. and Scholtz R. A., Ultra-wide Bandwidth Signal Propagation for Outdoor Wireless Communications, Proc. IEEE VTC Conf., May [7] Devasirvatham D. M. J., Multipath time delay jitter measured at 850 Mhz in the portable radio environment, IEEE journal on Selected areas in Communication, vol. SAC-5, p.p , June 1987 [8] Bultitude R. J., Mahmoud S. A. and Sullivan W. A., A comparison of indoor radio characteristics at 910 MHz and 1.75 GHz, IEEE journal on Selected areas in Communication, vol. SAC-7, p.p , Jan [9] Andersen J. D. and Zyablov V. V., Interleaver design for turbo codes, 5

6 Porceedings Int. Symp. On Turbo codes, Brest, Sept [10] Andersen J. D., Selection of codes and interleaver for Turbo coding, First ESA workshop on Tracking, Telemetry and command systems, ESTEC, June [11] Ramirez-Mireless F., On the performance of Ultra-Wide-Band signals in Guassian Noise and Dense Multipath, IEEE transactions on vehicular technology, vol. 50, no. 1, Jan [12] Golomb S. W., Construction of signals with favorable correlation properties, Survey in combinatorics, London, Mathematical Society Lecture Notes Series 166, U.K.:Cambridge Univ. Press, [13] Maggio G. M and Reggiani L., Application of Symbolic Dynamics to UWB Impulse Radio, In Proc. ISCAS 2001, Sydney, Australia, May 6-9, 2001 x s Source RSC Encoder 1 x 1p N-bit Interleaver RSC Encoder 2 x 2p Puncturing Mechanism MUX AWGN Noise x s, x 1p, x 2p Maximum Likelihood Detector Indoor UWB Communication channel with Multipath fading N-bit De- Intrleaver L e 21 y 1p DMUX y s MAP Decoder 1 L e 12 N-bit Interleaver MAP Decoder 2 Output y 2p N-bit Interleaver Fig. 1: The system model. 6

7 (a) (b) Fig 2(a): The UWB pulse p(t (T p / 2)) used for simulations as a function of time t, (b) the UWB Pulse Position Modulated wave for M = 2. Fig 3. Performance of UWB communication channel with multipath fading, without any error correction codes, for M = 2, 4 and 8. 7

8 (a) (b) (c) Fig 4 Improvement in performance of UWB Communication Channel with turbo coding with and without puncturing for M = 2, M = 4 and M = 8. 8

9 Fig 5. Effect of the size of interleaver on the performance of the turbo codes with puncturing. Fig. 6 The BER performance with respect to the number of decoding iterations for rate R = 1/2 code, M =2 and SNR = 4. Fig. 7 The relative BER performance of the communication channel for different M values. 9