A new analytic approach to evaluation of Packet Error Rate in Wireless Networks

A new analytic approach to evaluation of Packet Error Rate in Wireless Networks Ramin Khalili Université Pierre et Marie Curie LIP6-CNRS, Paris, France ramin.khalili@lip6.fr Kavé Salamatian Université Pierre et Marie Curie LIP6-CNRS, Paris, France kave.salamatian@lip6.fr Abstract Bit Error Rate (BER) and Packet Error Rate (PER) are important Quality of Service Parameters for Wireless network. Most of researches in QoS have been devoted to the analysis of BER which gives insight to the mean behavior of the wireless network. However the mean behavior is not sufficient for PER evaluation and more precise characterization of the error process is needed. Residual errors at the output of physical layer are not uniformly distributed. This is due to error correcting mechanism used at physical layer as well as the correlation induced by the memory existing in fading channels. Not taking into account the correlation in the error process and assuming for example that errors are uniformly distributed, as done in most of the published paper in wireless networking, lead to gross overestimation of PER. This over-estimation can go to hundredfold factors as will be shown in this paper. In a previous paper we presented a new analytic formula for predicting Packet Error rate in Wireless networks where convolutional codes are used ointly with Viterbi decoder over an AWGN channel. The approach was based on a precise analysis of the error process at the output of the Viterbi decoder. This formula was shown to precisely predict the PER as a function of convolutional code parameter and SNR over the AWGN channel. In this paper we extend the result obtained for the AWGN channel to the case of fading channel under block fading hypothesis. A closed form formulas approximation for derivation of PER in Fading channel is proposed and it is shown that it give very tight prediction for the PER. Introduction Classical performance analysis of convolutional error correcting codes has been focused on the mean behavior of decoding algorithm summarized in the widely used Bit Error Rate (BER) parameter. However the mean behavior is not sufficient in a lot of scenarios and more precise characterization of the error process is needed. An important example where the mean behavior is not enough is PER derivation. As stated in the nice survey [4], in the most of Error Modeling schemes for wireless channels, the models are based on the assumption that data packet transmissions are i.i.d. In addition, many coding schemes and protocols were initially designed for i.i.d. channels. However residual errors at the output of physical layer are not uniformly distributed. This is due to the error correcting mechanism used at physical layer as well as the correlation induced by the memory existing in fading channels. We will show in the rest of the paper, not taking into account the correlation in the error process and assuming for example that errors are uniformly distributed, as done in most of the published paper in wireless networking, lead to a large overestimation of PER, with proportion which can go to a tenfold factors. However analysis of error event distribution is not easy because of the peculiar interaction of the particular error correcting codes used in the physical layer of wireless networks. In [7] a practical methodology is presented that help in analyzing the distribution of error events at the output of Viterbi decoders that are widely used in wireless networks like UMTS or IEEE 802.a. This method was based on the code termination technique first presented in [5]. This technique relates error probability of a convolutional code to the error probability of a particular block code. The error probability behavior of block codes is a well known problem in information theory and is asymptotically studied through error exponent [2]. This technique enables to bound the probability of long error events. Obtained error event length distribution used ointly with the distribution of errorless period lead to a simple model for describing residual error at the output of a Viterbi decoder applying soft decoding. This model can be easily used in the context of wireless link simulation for generating residual errors. It can also be used to precisely derive the PER at the output of

00 00 00 00 00 Error Event of length 5 containing 3 errors 0 0 0 0 00 0 00 0 0 - Decoded sequence 0 0 Figure. Error event in convolutional code Trellis the Viterbi decoder. In this paper we extend this approach to the more general case of fading channel. We assume a ricean block fading channel. In the next section error event and error exponent are defined and concepts needed in the paper are introduced. Sections III will give a brief overview of the PER derivation presented in [7]. Section IV the result will be extended to the block fading situation. Section V will validate the method through experimental validation and last section will conclude. 2 Concepts Convolutional codes are widely used in wireless network physical layers. In convolutional codes, each block of k bits is mapped into a block of n bits but these n bits are not only determined by the present k information bits but also by the previous information bits. This dependence can be captured by a Finite State Machine (FSM). The number of previous bits that influence any bit is called the constraint length (ν). A longer constraint length means a more powerful code. Convolutional codes are characterized by these three parameters (n, k, ν) and the transitions of the FSM. It is assumed without loss of generality in the performance analysis that the sent sequence is a sequence of 0 and that the sent path was the floor path of the code trellis. An error occurs when a divergence of the decoded path through the trellis from the all 0 path occurs, i.e if a path different from the all 0 path is detected as a shortest path by Viterbi algorithm (see Fig. ). Definition An error event is defined as any divergence of the decoded path at the receiver from the initially followed path in the encoder. The three following simple rules can be applied to parse the error sequence into error events and error free periods: A received symbol is inside an error event or is inside an error free period. Each error event begins with an erroneous received symbol. Each error event ends by a sequence of ν bits received without any errors, where ν is the memory of the code. Probability 0-2 0-3 0-4 Rate=/2,m=8,g()=56,g(2)=753,d free =2 Rate=/2,m=8,g()=75,g(2)=557,d free =2 Rate=/3,m=7,g()=56,g(2)=552,g(3)=656,d free =6 Rate=/3,m=7,g()=452,g(2)=662,g(3)=756,d free =6 Rate=/2,m=5,g()=65,g(2)=57,d free =8 Rate=/2,m=5,g()=75,g(2)=55,d free =8 0 0 20 30 40 50 60 70 80 Error Event Length Figure 2. Distribution of error length at the output of an AWGN channel with SNR=2 db for 6 different convolutional codes The last property results from the fact that for returning to the all zeros state we have to flush the code memories with ν correctly decoded symbol such that the state of the decoder is correctly inferred, i.e ν bits should be received without errors. The two last properties means that error events has at least ν + bits ( erroneous bits followed by at most ν errorless bits). These three properties are used to derive the distribution of error events length from the simulated sequence as described in section 5. One of our obectives in this paper is to derive the distribution of the number of decoded bits that are inside an error event. We will call this distribution Error Event Length (EEL) distribution. It is clear that all bits inside error events are not erroneous (as shown in Fig. ), nevertheless they should be considered as highly unreliable (unless the last ν bits of the event that are surely true). Analysis of error event is much simpler that the study of errors themselves as the derivation is not polluted by specific code structure. Globally the shape of the distribution of error event length seems to be generic between codes sharing same basic characteristics as rate, memory and free distance d free. We show in Fig. 2 the EEL distribution as observed in simulation at the output of an AWGN with a SNR equal to 2 db for different codes (see section V for details). It can be observed that codes sharing the same characteristics have similar EEL distribution regardless of the fact that their structure and even BER can be very different. Asymptotic analysis of error correcting codes through error exponent is a classical approach in information theory. This approach is interesting as it gives closed form bounds on the error behavior of codes. These bounds can be readily used to design and optimize error correcting schemes. For long codes, the error probability of block codes and convolutional codes is upper-bounded in terms of error exponent. The error exponent E(R) is defined as

E(R) = N log P e(n, R) where P e (N, R) is the error probability of a block code of length N and rate R. There are classically two kinds of bounds: lower bound given by the sphere packing bound and upper bound given by the random coding bound. The error behavior of a practical code should be somewhere between these two bounds. As we are interested in upper bound for code behavior we are more interested in random coding bound and error exponents. In the forthcoming we will discuss the lower and upper bound for the particular case of AWGN channel. As we are interesting about the behavior of codes over AWGN and fading channels where soft decoding is applied, we will focus on the error exponent in this context. Random coding error exponents for soft decoding over AWGN channels might be derived using bounds first obtained by Shannon in []. For the sake of simplicity and because of the length constraint we are not going on the details. To find more details about the error exponent and numerical methods to calculate the error exponent you can see [8, 3, 7]. 3 error event length distribution and PER at the output of viterbi decoder This section introduces code termination technique [5] that relates convolutional codes and block code performances. The error event and errorless period analysis and packet error rate derivation results from this idea. The code termination idea consists of terminating a convolutional code after a certain number, let say τ, of input information symbols. The termination is made by forcing the input to zero for at least ν input symbol until the encoder has returned to the null state. This termination creates a block code that can be analyzed using block code error exponents. The performance of a convolutional code is analyzed by considering a sequence of block codes of increasing length and rate constructed using the code termination technique. Let s C be a (n, k, ν) convolutional code with constraint length ν and rate r = k n. Terminating a convolutional code after τ time units will lead to a block code with length N = nτ, M 0 = 2 (τ ν)k codewords and an overall rate R = N log 2 M 0 = r( θ), where θ = ν τ is the ratio of the code constraint length to block length. T heorem (Error event length distribution [5]) Using the code termination technique the probability per unit time of an error event of length τ time units in an (n, k, ν) convolutional code is bounded by : Prob { τ, E } e nτe(( ν τ )r) = e nνe(( θ)r)/θ () where θ = ν τ <, r = k n is the convolutional code rate and E(.) is the block code error exponent One can remark that this result is the same as for a block code of length nτ and rate R = ( θ)r. This theorem gives the distribution of error event length. However the output of a Viterbi decoder contains error events mixed with errorless periods. The characterization of the output of the Viterbi remains incomplete without describing errorless periods. In this widely believed that errorless periods length and error event length are independent. However a more precise analysis of the Viterbi algorithm shows that this assumption is not correct, specially for low SNR. We define a decoding epoch as the interval between the end of two consecutive error events. A decoding epoch consists of an errorless period followed by an error event. In fact the length of an error event and an error epoch can be assumed as independent. We will argue in the forthcoming that the distribution of error epochs can be easily derived using a geometric memoryless distribution. The distribution of errorless event is derived based on these two distributions. Lets α i be the length of the i th decoding epoch, and β i being the length of the i th error event. The value γ i = α i β i gives the length of the errorless period preceding the i th error event. Because of the memoryless property of the AWGN channel error epochs are independent and memoryless events. Moreover the probability that a decoding epoch finishes at any bit can be easily obtained through a classical derivation that can be found in basic coding textbooks [9, 5]. Let s EER(SNR) (Error Event Rate as a function of SNR) be this probability for a given SNR over the AWGN channel. EER(SNR) A dfree e R SNR d free (2) where A dfree is the number of path of length d free in the code trellis and R is the code rate. As the decoding epoch last at least ν bits, α i = ν + α where α follows a Bernoulli process with parameter EER(SN R). The mean length of decoding epochs is therefore equal to E { } α i = ν + EER(SNR). The characterization of error and error epoch events enables the derivation of PER at the output of decoder. Derivation of PER need the distribution of errorless period length. This distribution can be obtained precisely using the fact that the error event length and decoding epoch length are independent as a convolution: f y = f α f β (3) However this formulation does not give a close form. For high value of SNR a decoding epoch consists of a long errorless period followed by a small error event. In this situation we will approximate the distribution of errorless period length with Bernoulli process with mean length that can be derived using the mean length of decoding epoch and error

event length. This assumption will be shown to be valid for the range of SNR that have practical relevance. As it is shown in [], for large SNR the error exponent bound can be approximated by using only the linear part of the random coding error exponent. This linear part is a function of rate and SNR. While this is a rough estimate of error exponent, we will show that it is tight enough to give tight upper bound for PER. The upper bound for random coding exponent for a block code (n b, k b ) of rate r b = k b n b with a Signal to Noise Ratio given by s can be given as following based on the error exponent approximation : E(s, r b ) = s/2 2sr b + r b (4) The average error event length can be approximated as τ(s) = (ν + ) + n c (s/2 2sr c + r c ) which leads to this new derivation of PER : P ER = N i=0 where λ(s) is given by λ(s)[ λ(s)] i = ( λ(s)) N (5) λ(s) = ω = EER(s) [(ν + ) + n ]EER(s) c(s/2 2sr c+r c) (6) (see [7] for a proof). This relationship gives a remarkably simple equation for PER derivation. It is noteworthy to compare this relationship with the classical approach where it is frequently supposed that errors are uniformly distributed in packets with a probability given by BER. With this hypothesis the PER is derived as P ER = ( BER) N. The obtained relation has exactly the same structure however with the BER replaced by the parameter λ defined in Eq. 6. All the parameters of the new formula can be easily obtained using basic hypothesis and we will show in the experimental validation part that this formula gives remarkably tight bounds for PER. 4 Extension to the block fading situation Till now we have focused on AWGN channel. However real world is more complex and we have frequently to deal with fading channels where the effective SNR (γ) fluctuates randomly because of shadowing and/or multipath fading. It is interesting too see how the previous derivation extends to this situation. In general, the error exponents of a fading channel are completely different from the error exponents of the AWGN channel and a full new derivation is needed. Up to now only very rare paper have attacked the rather p, p, p, p, R R p, 2 p, 2 Figure 3. Finite state Markov chain model hard problem of deriving general error exponent for fading channels. However the analysis can be simplified under the block fading hypothesis, i.e. level of SNR being constant at least at the level of one packet transmission. As the coherence time of practical channels is much larger than a block transmission time, this hypothesis is valid for a large scale of practical settings where the sender and receiver are not moving fast. In this case, the average BER (resp. PER) observed at the output of the fading channel might be computed by calculating a mean over fading statistics f(γ) : P = 0 P (γ b )f(γ b )dγ b where P (γ b ) is the AWGN BER (resp. PER). However the calculation of this integral might become complex because of interaction between fading and BER (resp. PER) statistics. One way to deal with this complexity is the use of Finite State Markov Model (FSMC) model. These models that have been widely used in the recent literature, The complex mathematical characterization of flat-fading can be difficult to incorporate into wireless performance analysis such as the packet error probability. Therefore, simpler models that capture the main features of flat-fading channels are needed for these analytical calculations. One such model that has been investigated extensively in the recent literature in the Finite State Markov Model (FSMC) [2, 3]. The FSMC model discretized the range of fading value into regions R defined as R = {γ : A γ < A + }, where the region boundaries A and the total number of fade regions are parameters of the model. Based on the block fading assumption we will suppose that the received SNR γ stays within the same region over the time interval needed for transferring a packet. Given that the channel is in state R, at the next time interval the channel can only transition to a neighboring states where in state, p,+ and p, are the transition probability to the next and previous states. The hypothesis of FSMC model is reasonable under slow and flat fading situations. With these assumptions the steady state probability that the received SNR is in the th region is defined by : π i = p(γ R ) = p(a γ < A + ).

and P can be calculated as P = n π P ( γ ) (7) = where n is the number of states of the FSMC and γ is the average fading envelope in the state that can be calculated as A+ A γ = γ f(γ)dγ In summary equation (7) says that under block fading situation the BER (resp. PER), and every other statistics at the output of decoder, can be expressed as a mixture of the different values of the parameter to calculate P (.) in function of the fading channel statistics. Based on this hypothesis the Packet error rate over fading channel can be approximated by : n P ER = π ( ( λ(γ )) N ) (8) = where λ is defined in equation (6). The distribution of error event length at the output of viterbi decoder is also a mixture of the different values of distribution in function of the fading channel statistics. This simple result enables us to extend the previously developed approaches for Error Event Length and Packet Error Rate to more general situations where block fading exists. In the next section we will validate this approach for a channel with Ricean statistics. π 5 Experimental validation In this section we will validate our proposed approach by simulation. We illustrate the approach with two different convolutional codes have been defined as standard for the UMTS [] and WIFI with the same rate r = /2. UMTS code has the constraint length 9 and generator polynomial in octal format defined as (56,753) while WIFI code hast constraint length 7 and generator polynomial (33,7). These codes have a relative large memory that makes difficult the application of the classical weight distribution approach. We have simulated a communication systems consisting in a convolutional encoder followed by a BPSK modulator. The encoded modulated signal is sent over an block fading channel, and decoded at the receiver by a soft decoding Viterbi algorithm without memory truncation. To validate the approach presented in section 4 we simulated a Ricean physical channel under block fading hypothesis. For these channel the fading envelope has a distribution given by : f(s) = 2s(+K) exp{ K (K+)s2 Ω p I 0 (2s ) K(K+) Ω p, s 0 Ω p } Distribution of Burst Error Length 0-0 -2 0-3 0-4 0 0 2 0 3 Length of Burst Error Simulation Result - K=0.0 Simulation Result - K=0 Simulation Result - K=20 Simulation Result - K=00 Theoretical Bound - K=0.0 Theoretical Bound - K=0 Theoretical Bound - K=20 Theoretical Bound - K=00 Figure 4. Error Event Length for the UMTS (56,753) convolutional code with memory constraint ν = 9 over a Ricean fading channel with SNR=4 db for different value of fading parameter K where the average power in the Ricean fading is given by Ω p and K is the fading parameter, i.e. the ration between the power received from the direct path and the power received by other paths. A larger parameter K means a channel that is close to the AWGN channel. Each simulation tests have been applied over more than 200 Mbits of data. 5. Validation of Error Event Length distribution Applying the rules defined in section II, an error event is detected if an error occurs after a sequence of at least ν bits without error and is terminated after a sequence of ν bits without error. We compared the resulting distribution of error event length observed over the simulated data with the distribution predicted using Eq. (7) and a 0 state FSMC model in Fig. 4. As can be seen the predicted distribution give the global shape of the distribution. However, the fit seems to be looser in the tail of the distribution. This can be explained by the fact that the overall distribution will be a mixture of different component. Each component needs to have enough samples to reach a correct estimate of the tail behavior. Therefore the number of needed simulation for converging to the tail in the fading case might be much more larger than in the AWGN case. 5.2 Validation of Packet Error Rate formula We have used the same simulated set of data to derive PER for different SNR and K parameters over Ricean physical channel. The obtained value are compared with the formula presented in equation (8). We have also compared the value with what have been predicted using the uniform hypothesis. There are two different approach in the uniform hypothesis. The first one which consider that the

2 3 4 5 6 7 8 9 0 0 0 0 - properties of the output of viterbi decoder explained in this paper. It can be seen that in the same situation, our proposed formula lead to almost perfect fit. Packet Error Rate Packet Error Rate 0-2 0-3 0-4 0-5 0 0 0-0 -2 0-3 0-4 Uniform hypothesis, UMTS. Uniform hypothesis, UMTS. Proposed bound, UMTS. Simulation result, UMTS. Uniform hypothesis, WiFi. Uniform hypothesis, WiFi. Proposed bound, WiFi. Simulation result, WiFi. Signal to Noise Ratio (a) UMTS (56,753) convolutional code; Uniform hypothesis, K=0. Uniform hypothesis 2, K=0. Proposed bound, K=0. Simulation result, K=0. Uniform hypothesis, K=20. Uniform hypothesis 2, K=20. Proposed bound, K=20. Simulation result, K=20. 2 3 4 5 6 7 8 9 0 Signal to Noise Ratio (b) WiFi (33,7) convolutional code; Figure 5. Packet Error rate over a Ricean fading channel for different K-factor as a function of SNR distribution of error is uniform even over the fading channel. In this hypothesis P ER(s) = ( BER(s)) N where BER(s) is the bit error rate at the output of the viterbi decoder over the fading channel and can be evaluated through (7). N is the packet length which is fixed to 4000 bits. The second uniform approach takes into account the properties of fading channel as explained at [0]. In this case the fading channel is modeled by a markov chain as explained in section 4. The PER can be then estimated as P ER = n = π P ER( γ ) where P ER( γ ) = ( BER( γ )). In this approach the distribution of error at each markov chain state is still considered to be uniform. Similar to the results of AWGN channel [7], the uniform error hypothesis approaches give results that are very far from realistic values as shown in figure5. For low SNR, the second uniform approach give a better estimation but for large SNR the over-estimation goes up. For large SNR the fading channel is usually in a good state. In this situation the principle reason of correlation of error comes from the 6 Conclusion This paper has presented a new approach derivation of error event lengths at the output of Viterbi decoder for convolutional codes. The approach is based on asymptotic error exponents. It gives tight results for the approximation of EEL distributions. We have also derived the distribution of errorless period length. This last derivation enables us to obtain a precise and tight characterization of Packet Error Rate at the output of Viterbi decoders. In this paper we validate the results for the Ricean block fading channel hypothesis. The simulation results show the accuracy of this approach. A software implementation of the error distribution model has been developed which can be find at www-rp.lip6.fr/ salamat. References [] ETSI. Universal mobile telecommunications system (umts); multiplexing and channel coding (tdd). (3GPP TS 25.222 version 5.2. Release 5), October 2002. [2] R. G. Gallager. Information Theory and reliable Communication. Wiley, 968. [3] Y. L. Guan and L. F. Turner. Generalised fsmc model for radio channels with correlated fading. IEE Proc. Commun., April. 999. [4] H. A. H. Bai and M. Atiquizzaman. Error modeling scheme for fading channels in wireless communications: A survey. IEEE com. Surveys, 5. [5] G. D. F. Jr. Convolutional codes ii. maximum-likelihood decoding. Information and Control, 25(3):407 42, 974. [6] J. Justesen and J. D. Andersen. Critical length of error events in convolutional codes. IEEE Trans. on Information theory, 44(4), July 998. [7] R. Khalili and K. Salamatian. A new analytic approach to evaluation of packet error rate in wireless networks. MSWIM, 2004. [8] R. Khalili and K. Salamatian. A new analytic approach to evaluation of packet error rate in wireless networks. Technical report, LIP6-PERmodel, 2004. [9] S. Lin and D. Costello. Error Correcting Coding: Fundamentals and Applications. Prentice Hall, 983. [0] L. B. M. M. Zorzi, R. R. Rao. On the accuracy of a firstorder markov model for data block transmission on fading channels. IEEEICUP 95, pages pp.2 25, Nov. 2004. [] C. Shannon. Probability of error for optimal codes in a gaussian channel. Bell System Technical Journal, 38, 959. [2] H. Wang and N. Moayeri. Finite-state markov channel - a useful model for radio communication channels. IEEE Trans. Vehic. Technol, Feb. 995. [3] S. Yousefi and A. K. Khandani. A new upper bound on the maximum likelihood decoding error probability of linear binary block codes in awgn interference,.