FAILURE
3.4 Variable-length State-of-the-art FS techniques
In many communication systems, packets are of variable-length, this is the case,e.g., in the 802.11/802.16 standards [101; 32] under investigation in this thesis. Variable-length FS is much more complicated, as a proposition to study the time-window cannot be applied due to the variable nature of packets. The Conventional and simple Hard Decision (HD)- based FS using a length field (containing the packet size), assumed present in the header of the variable-length packet, can be used if the noise is moderate. HD-based FS rely on
the correctness of the length field of the previous packet to find the start of the current packet and suffers from limitations if the length field is corrupted with noise.
The most widely used method for providing variable-length FS is to insert a SW or header into the data stream. A simpler and common engineering approach for FS of unknown lengths consists in a sequential detection algorithm: Starting from a given posi- tion, the correlation between the received (continuous valued) samples and the SW/header symbols is computed, and compared with some threshold. If the threshold is exceeded the synchronizer declares a SW/header presence, otherwise the search continues [90]. In the presence of SW, in [19; 20; 21; 22], several hypothesis testing techniques have been presented to perform variable-length FS. These techniques are discussed in Section 3.4.1.
Nevertheless, one can use the 3S automaton as presented for the fixed-length packets, but the misframe time (time taken to wrongly signal FS failure) for variable-length packet is too short as α = 1, thus even with a single random bit error in the header the system may quickly and falsely switch to the HUNT state by signaling FS failure due to missed detection. This problem is addressed in [17], which is briefly reviewed in Section 3.4.2.
3.4.1 NP FS Method
If noa prioriinformation on the packet length is assumed, then an acquisition algorithm based on a step-by-step comparison of a proper metric with a threshold is a preferable solution. Several detection metrics for AWGN channel based on the Likelihood Ratio Test (LRT) are considered by Chiani et al [19; 20; 21; 40; 22] to perform FS through correlation, where Neyman-Pearson (NP) criterion is used to choose the threshold. Authors in [40]
studied the performance of FS for equiprobable data symbols with the metrics derived in [21; 20], while in [22] an optimal metric is derived for non-equiprobable data symbols. It is further extended in [23] to exploit a priori information on the prevalence of ones and zeros in the payload at the price of a small additional computational complexity. This method is denoted as NP FS in what follows. Below a brief description of NP FS method is provided.
Let us consider the case of binary signaling where we have data symbolsdi {+1,−1}, with probabilities P(di = 1) and P(di = −1). Similarly, we have known SW or header symbolshi {+1,−1}.
Letbi {0,1}be theithbit of a burst transmitted using binary antipodal modulation through AWGN channel. At the receiver, we have
yi = (−1)bi +ni,
whereni are independent, identically distributed Gaussian random variables (r.v.s), with zero mean and varianceσ2.
The NP FS algorithm works as follows: Starting from a bit index `, the synchronizer observes a vector of`h subsequent samples; based on the metric evaluated from this vector it decides if the SW/header is at this bit index `; if not, it moves to next bit index`+ 1, repeating the steps until the SW/header is detected. After observing`hsubsequent samples at bit index`, the synchronizer must choose between the following two possible hypotheses, i.e., the data hypothesis
Hd : yi =di+ni, i=`, `+ 1, ..., `+`h−1 (3.1) and SW/header hypothesis
Hh : yi=hi+ni, i=`, `+ 1, ..., `+`h−1. (3.2) Decisions are indicated byDh and Dd, corresponding to the true hypotheses Hh andHd, respectively.
Let the received sequence of `h symbols be denoted by y = [y`, y`+1, ..., y`+`h−1], which is composed of either noisy SW/header symbols or noisy data symbols. Let Y = [Y`, Y`+1, ..., Y`+`h−1] be the r.v.s corresponding to the vectory = [y`, y`+1, ..., y`+`h−1] of received samples, the LRT is represented by
Λ(y) = PY|Hd(y|Hd) PY|Hh(y|Hh)
Dh
≶ Dd
λ. (3.3)
WherePY|Hj(y|Hj)is the probability density function ofYunder hypothesisHj,j {d, h}, andλis the selected threshold. Thus, according to the LRT,Λ(y)< λcorresponds to the decisionDh,i.e., the presence of a SW or header is detected; otherwise, the decision isDd. Now we will provide a brief evaluation of the LR of (3.3) for the AWGN channel. Since the channel is memoryless, we know
PY|Hj(y|Hj) =
`+`h−1
Y
i=`
PYi|Hj(yi|Hj).
GivenHh, the r.v.s Yi are Gaussian-distributed with meanhi and variance σ2,i.e., PYi|Hh(yi|Hh) = 1
√2πσe−(yi−hi)2/2σ2. (3.4) Similarly, givenHdanddi, the r.v.sYiare Gaussian-distributed with meandi and variance σ2,i.e.,
PYi|Hd(yi|Hd, di) = 1
√2πσe−(yi−di)2/2σ2. (3.5)
Equiprobable data symbols
In case of equiprobable data symbols, under hypothesis Hd,di take values +1and −1 with equal probability, one gets [21]
PYi|Hd(yi|Hd) = 1
2PYi|Hd(yi|Hd, di = 1) +1
2PYi|Hd(yi|Hd, di =−1), (3.6) which, using (3.5), becomes
PYi|Hd(yi|Hd) = 1 2
√1
2πσe−(yi−1)2/2σ2 +1
2
√1
2πσe−(yi+1)2/2σ2. (3.7)
Using (3.7) and (3.4) in the LR of (3.3), one finally has
Λ(y) = 1 2`h
`+`h−1
Y
i=`
e−(yi−1)2/2σ2 +e−(yi+1)2/2σ2 e−(yi−hi)2/2σ2
= 1
2`h
`+`h−1
Y
i=`
1 +e−2yihi/σ2
. (3.8)
Non-equiprobable data symbols
In case of non-equiprobable data symbols, under hypothesis Hd,di take values+1and
−1with probabilities p1 and 1−p1 respectively, one gets [22]
PYi|Hd(yi|Hd) = √2πσ1 h
p1e−(yi−1)2/2σ2 + (1−p1)e−(yi+1)2/2σ2i
. (3.9)
Using (3.9) and (3.4) in the LR of (3.3), one finally has
Λ(y) = 1 2`h
`+`h−1
Y
i=`
p1e−(yi−1)2/2σ2+ (1−p1)e−(yi+1)2/2σ2 e−(yi−hi)2/2σ2
= 1
2`h
`+`h−1
Y
i=`
h
P(di=hi) +P(di 6=hi)e−2yihi/σ2i
. (3.10)
NP FS method thus decides Dh (the SW/header is present) if Λ(r) < λ and Dd otherwise. One can observe that neither the LR nor the threshold depends on the a priori probabilities P(Hj), they depend only on the conditional densities PYi|Hj(yi|Hj). Furthermore, the metric for non-equiprobable case is somewhat similar to the one for the equiprobable case, and depends on the channel conditions through σ2 . Thus, to perform optimum FS using the NP FS method the instantaneous knowledge of the SNR is required.
The threshold is chosen according to the NP criterion, i.e., by fixing the maximum tolerable probability of FA (emulation), pem. The probability of emulation, pem,i.e., the probability of choosing hypothesisHh when Hd is true, is
pem =P(Dh|Hd),
while, the probability of missed detection,pmd,i.e., the probability of choosing hypothesis Hdwhen Hh is true, is
pmd=P(Dd|Hh).
NP FS method achieves high gain as compared to commonly used correlation-based FS techniques, but in case SW is short e.g., less than 16 bits, the FAs significantly degrade the FS performance. This can be overcome by exploiting the a priori information on the packet length in order to adapt the threshold, this problem is addressed in Chapter 6 using Bayesian hypothesis test.
CRC:
Error detection mode CRC:
Forced two bit error correction
mode 0, 1 or 2 bit errors
SYNCH HUNT
PRESYNCH
Incorrect CRC
±
Consecutive correct CRC
2 bit errors (while two candidates are already examined)
3 or more bit errors
CRC:
Error detection mode Packet-by-packet
Correct CRC
Packet-by-packet
Bit-by-bit
Figure 3.5: 3S automaton used by the Ueda’s method 3.4.2 Ueda’s Method
In [17], an CRC-based variable-length FS for IP packets is proposed, where a length field is assumed to be present in the packet header. This method uses the 3S automaton, similar to the one used in CD, but instead of distinguishing between FS failures and channel errors, up to two bit error correction (to increase misframe time) is performed before hunting for the correct FS, if required. This method is denoted in what follows asUeda’s method.
More precisely, the automaton presented in [17] consists of three states: SYNC,HUNT, andPRESYNC.Assume that the automaton is in theSYNC state. It remains in this state as long as no FS failure is detected using CRC (as a HEC). If CRC detects a FS failure, one first tries to correct errors in the header. The 3S automaton used by the Ueda’s method is shown in Figure 3.5.
In theSYNC state,CRC can correct one-bit error but not two-bit errors because two- bit error syndromes do not necessarily correspond to a single candidate, thus multiple candidates may exist. For two-bit errors, if a single candidate exists then the error can be corrected immediately. But if dual candidates are possible, the best packet length candidate is searched by evaluating the CRC over the next header, at the position indicated by the corresponding candidate. If no candidate is correct,i.e., corresponds to correct CRC at the potential location of the next header, then the candidate that gives one-bit error syndrome at the next header is selected. In case of failure, the automaton switches to the HUNT state. As few syndromes of more than two-bit errors are similar to one-bit or two-bit error syndromes, so it is assumed that these error syndromes are originating from one-bit or two-bit errors because they are more often.
In the HUNT state, the automaton hunts for the correct FS by searching bit-by- bit for the correct CRC over the assumed header fields. Once an agreement is found, the automaton switches to the PRESYNC state, an intermediate state. In this state, a packet-by-packet checking is performed. In thePRESYNC state, one makes sure that the FS retrieved in theHUNT state is indeed the correct FS by verifying that CRC is correct
for δ > 0 consecutive packets. Onceδ consecutive correct CRCs have been obtained, the automaton returns to theSYNC state and in case of failure it again switches to theHUNT state.
Synchronizer can be implemented in two ways; one is to utilize two synchronizers to select the best suitable candidate out of maximum of two candidates. The second is to sort the two candidates in decreasing order and utilize a single synchronizer to examine the candidates in decreasing order. The second implementation is adopted in Ueda’s method.
Ueda’s method is well adapted for rather long CRC like CRC-16, because in this case there are no more than two candidates for two-bit error syndromes. For low order or short CRC compared to the size of the header,e.g., CRC-8, many more candidates can be found for syndromes with two bits in error, thus Ueda’s method needs some extension to search for the best candidate.
3.4.2.1 Modified Ueda’s (MU) method:
We propose to search for the best candidate (among C candidates) by shifting the bit-stream by the potential packet length and then calculating HEC over the next header sequence at the position indicated by corresponding candidate. If no candidate is correct (i.e., corresponds to correct HEC) at the potential location of the next header sequence, then the candidate that gives one-bit error syndrome at the next header sequence is selected as the best candidate. In case of failure, the FS automaton switches to the HUNT state.
The extended flow diagram for the SYNC state is shown in Figure 3.6.
Ueda’s and MU methods, in the HUNT state, check for no errors in CRC calculations while shifting bit-by-bit or byte-by-byte, a situation impossible to appear at a very low SNR, when there would always be few errors in the header, thus making the method of hunting using HEC unreliable at lowSNR.Furthermore, in case of low order HEC or short HEC meeting a configuration in which random bits emulate the header is more likely.
These problems are addressed by jointly analyzing the successive packets in FS techniques presented in Chapters 4 and 5, and by using soft hunt operation in a robust 3S automaton presented in Chapter 6.