1 FER Estimation in a Memoryless BSC with Variable Frame

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feedback is analyzed by means of the corresponding CRLB In this setting the ML estimator is obtained. by applying the Expectation Maximization algorithm to jointly estimate the error probabilities of the. data and feedback links Simulation results illustrate the benefits of the proposed estimators. Index Terms, FER Estimation Maximum Likelihood Unreliable Feedback Link Adaptation Expectation Maximization. I I NTRODUCTION, Frame error rate FER estimation is a crucial step in many problems related to the design and. deployment of wireless communication systems FER estimation is used in PHY layer abstraction. for system level simulation 2 4 as well as in rate adaptation algorithms where a modulation. and coding scheme MCS is selected according to the state of the channel 5 8 Some of these. rate adaptation schemes rely only on the estimation of the channel state by exploiting binary. feedback information about the success or failure in the decoding of previously transmitted. packets 9 10 Also network analyzers for different wireless protocols IEEE 802 11 11. 3GPP LTE 12 DVB T 13 provide FER estimates as a result of the observation of the current. The core of any FER estimation method is the observation of the outcome of the transmission. process of several frames over a channel For the case of constant length codewords the FER can. be readily estimated as the sample mean of the error events In many communication standards. however frames have a length that varies with different parameters and depends on the size of. the protocol data units PDU delivered by the MAC layer For instance standards such as those. in the IEEE 802 11 family 14 or 3GPP LTE 15 operate with codewords of variable length as. a particular example in the LTE Physical Downlink and Uplink Shared Channels PDSCH and. PUSCH the PHY layer adds a 24 bit Cyclic Redundancy Check CRC to data blocks whose. sizes may vary between 40 and 6144 bits including CRC bits 16 Estimating the FER for. a given frame length based only on those observations corresponding to received frames with. that same length is clearly suboptimal due to an inefficient usage of available data However. generalizing FER estimation in order to account for observations corresponding to packets with. different lengths is not straightforward as it will be shown The goal of this paper is to study. and build novel FER estimators for this setting based on statistical estimation theory. Previous work on FER estimation dealt with the issue of variable frame length in different. ways In 5 7 17 18 constant frame length was assumed which is not realistic for current. communication systems as discussed above In other examples 19 22 perfect knowledge of. the bit error rate BER was assumed and the corresponding FER is obtained from that value. This approach however has two significant drawbacks First it is not clear how BER estimation. errors would affect the quality of the resulting FER estimate Second in many communication. scenarios link adaptation or network analysis the BER is not directly measurable since it would. require knowledge of the transmitted bits Alternatively if the transmitter makes use of an error. detection code e g CRC as it is usually the case then it becomes feasible to measure packet. error events at the receiver so that FER estimation becomes feasible based on these observations. even if the BER is not available or observable, Our approach hinges on a memoryless binary symmetric channel BSC abstraction for bit. level transmission Previous works have considered estimation of the BSC parameter in several. scenarios e g in 23 24 where estimation is based on the observation of the BSC output and. the assumption of a nonrandom input with finite complexity or in a distributed source coding. or channel coding framework 25 26 based on the availability of the received individual bits. including the syndrome of a single codeword These approaches can be regarded as PHY layer. estimators In contrast the FER estimators considered in this paper are based on the observation. of multiple binary success failure packet error events alone albeit with packets of different. lengths and therefore they can be directly applied at the MAC layer. FER estimation is further complicated when an observer different from the receiver a trans. mitter or a third node acting as a network analyzer tries to estimate the FER experienced by. a receiver In that case the FER related information is obtained from the ACK NAK sequence. reported by the receiver If this feedback channel is unreliable then the estimation procedure. has to be modified since treating the ACK NAK information as true may result in performance. degradation In this paper we will consider unreliable feedback links and develop estimators for. the error probability of the feedback channel as well as the FER of the forward data channel. The contributions of this paper are summarized as follows. 1 FER estimation from perfect measurements We derive estimators for the FER of a given. frame length from ACK NAK observations which may correspond to frames of different. lengths In this first step we assume that the ACK NAK information is correct i e either. the feedback channel can be assumed error free or the estimation process is directly carried. out by the receiver itself We obtain the Maximum Likelihood estimator MLE in this setting. by means of an iterative procedure We also obtain reduced complexity approximations to. the MLE in some particular scenarios e g large number of observations small number of. errors and compare their performance to the Cramer Rao Lower Bound CRLB. 2 FER estimation with an unreliable feedback link In this second setting the ACK NAK obser. vations are no longer assumed to be perfect The feedback link is modeled as a BSC with error. probability The impact of unreliable feedback is analyzed by studying the corresponding. CRLB We obtain joint estimators for the general case in which the error probabilities in both. links are unknown Interestingly these estimators only exist for the case of having codewords. of different length for the constant codeword length the parameters become unidentifiable. The joint estimator is obtained by applying the Expectation Maximization E M algorithm. The remaining of the paper is organized as follows Section II presents the system model. addressing the difficulty of FER estimation with codewords of different lengths The CRLB and. the estimators for the case of reliable feedback are derived in Section III The estimation problem. with an unreliable feedback channel is presented and analyzed in Section IV and conclusions. are presented in Section V Simulation results for the different estimators are included at the. end of each of the corresponding sections,II S YSTEM M ODEL AND P RELIMINARIES. Consider a transmitter receiver pair communicating through a noisy channel The transmitter. builds blocks of bits b1 bL bi 0 1 of variable bit length L that we will refer to. as frames The receiver observes b 1 b L at the output of the channel with b i 0 1. We assume a memoryless BSC with BER p i e p P b i 1 bi 0 P b i 0 bi 1. We also denote q 1 p for the sake of simplicity The transmitter makes use of an error. detection encoder such that the receiver is able to identify any received block with at least one. erroneous bit We also assume that the undetected error probability of the error detection code. is negligible1 If we denote by L the probability of receiving an erroneous block of length L. The undetected error probability of a CRC code is approximately 2 r where r is the number of check bits For example. for the PDSCH and PUSCH of LTE r 24 and 2 r 6 10 8. i e the FER for frame length L then we have that,L 1 P bi b i 1 qL 1.
If the FER for a frame length L was perfectly known then it would be possible to obtain the. FER for a different frame length L as,L 1 1 L L L 2. Unfortunately in practice the exact FER is unlikely to be available for any length and its. value has to be estimated based on the observation over a time period of the success and failures. of the transmission of frames of a certain length Moreover in a realistic environment frames. of different length may be transmitted during the observation window In the following we. formalize the problem of FER estimation from observations of frames of various sizes. Consider a communication system over a BSC with different frame sizes L1 L2. L During an observation period a receiver observes ni transmissions of size Li out of which. mi are received with errors The random variable mi is binomially distributed with parameters. ni and Li so that its probability mass function PMF is given by. f mi Li 1 Li ni mi 3,The mean value of mi is given by. E mi ni 1 q Li 4, The MLE of Li given the observation ni mi is the empirical FER i e. which is unbiased and whose variance is given by,i E Li Li 6. In view of 2 when only observations from equal length frames are available i e 1. the MLE of L can be obtained from that of L1 by virtue of the invariance property of the. MLE 27 Th 7 2 as L 1 1 L1, If 1 however the derivation of the MLE of L is more involved Due again to the.
invariance property the MLE of L could be readily obtained if that of q were available L ML. 1 q ML L We will see that although the MLE cannot be expressed in closed form in general. it is possible to derive some of its properties as well as an efficient numerical method for its. computation We will also analyze some asymptotic cases small p large number of observations. for each length for which closed form approximations will be exposed. III FER ESTIMATION WITH OBSERVATIONS OF DIFFERENT LENGTH. Given a vector of observed errors m m1 m T the goal is to estimate the FER. corresponding to a frame length L Since the channel is assumed memoryless observations are. independent and therefore the PMF of m parameterized by the FER L can be obtained from. 2 and 3 as,Y ni Li Li,f m L 1 1 L L 1 L L ni mi 7,Alternatively we can rewrite 7 in terms of q as. Y ni mi ni mi Li,f m q 1 q Li q 8, Depending on the formulation of the problem we may resort to using 7 or 8 In the following. we derive different estimators for the problem under study First we derive the CRLB which. constitutes a bound on the variance of any unbiased estimator to benchmark the performance. of the different proposed estimators, We now derive the CRLB for the estimation of q from the observations m the CRLB for the. estimation of L can be obtained by applying a suitable transformation to the final result 27. The log likelihood function LLF of q is obtained from 8 as. mi log 1 q Li ni mi Li log q, where a constant term was omitted Its derivative is readily obtained as. ni mi Li mi Li q Li 1, It can be checked that the regularity condition E L0 q 0 necessary and sufficient for the.
application of the CRLB is satisfied just by substituting 4 after taking expectation in 10 It. is also seen that an efficient estimator for q or equivalently p does not exist for this problem. as it is not possible to find functions g m and I q such that L0 q I q g m q 27. Th 3 1 The second derivative of the LLF is,X i i n 1 q mi 1 Li q 1. L00 q 2 11,q 2 1 q Li,Therefore using 4 the Fisher information is found. 1 X ni L2i q Li,I q E L00 q 12,q 2 i 1 1 q Li, As the observations are independent the Fisher information is the sum of the corresponding. contributions for each frame length Li weighted by the number of observations ni The dominant. individual term in 12 depends on the actual value of the parameter for very small p the term. with largest frame length is dominant but eventually the situation is reversed as p increases. Note that I q increases without bounds as p 0 and goes to zero as p 1 This is due to. the fact that we can only observe if a packet is in error or not but not how many errors there. are in an erroneous packet For p Li, multiple bit errors within a packet become unlikely. so that the BER becomes easier to estimate from the observations When the probabilty of. multibit error events is not negligible however estimating the BER based on observation of. packet errors becomes much more difficult, The CRLB which bounds the variance of any unbiased estimator q is Var q I 1 q In.
view of 1 and following 27 Sec 3 6 the Normalized CRLB NCRLB of any unbiased FER. estimate for packets of length L follows,Var L 1 q L. X ni L2 q Li, It is insightful to examine the asymptotic behavior of 13 On the one hand as p 0 using. 1 1 p L pL in 13 yields the following low BER approximation of the NCRLB. which is independent of L On the other hand for p 1 the corresponding asymptote is. 1 p 2L L1 15,L 2 n1 L21, The behavior of 15 clearly depends on the target length L If L 2. then the NCRLB goes,to zero as p 1 it settles to a constant when L 2. and goes to infinity if L 2, that in the extreme case of packets with length L 1 bit the FER equals the BER and from.
the discussion following 12 we know that the Fisher information for the problem of BER. estimation goes to zero as p 1 Thus it is reasonable to observe such behavior for short. As an example Fig 1 shows the NCRLB 13 for a setting with 3 L1 L2 L3. 1000 2000 3000 for different target lengths and with a total number of observations n1. n2 n3 60 distributed in three different ways It is clear that estimating the FER for packets. whose length is much shorter than those for which observations are available is considerably. Previous work on FER estimation dealt with the issue of variable frame length in different ways In 5 7 17 18 constant frame length was assumed which is not realistic for current communication systems as discussed above In other examples 19 22 perfect knowledge of

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