FAILURE
6.2 Robust 3S FS Automaton
In this section we propose several improvements to the HEC-based Ueda’s method to develop a robust 3S automaton, see Figure 6.1. First, instead of performing error correction in the SYNC state, the robust header estimation presented in [8], and briefly recalled in Section 6.2.1, is employed to estimate the length field of a packet. In case of failure to verify HEC of header with estimated length field (HEC verification is performed after replacing the received noisy length field with the estimated length field), the automaton switches to the HUNT state, where Bayesian hypothesis testing is performed to search for the correct FS, see Section 6.2.2. The operation performed in the PRESYNC state remains unchanged.
Note that alternatively, an automaton with a singleHUNT State (HUNT State Alone, HSA) may be considered to perform FS, without using the packet length field present in the header.
6.2.1 SYNC State: Header Recovery
In [8], a MAP estimator is proposed to determine some fields in the headers of the aggregated packets. In case of FS with packets of variable lengths, one is mainly interested
in the length field denoted byunfor then-th packet. Rewriting the notations of Chapter 4, we have all the fields of thenth packet header in a vectorhn= [k,un,on,cn]and its corre- sponding observation at the output of the channel represented by a vector [yk,yu,yo,yc].
Using the representation of Section 4.5.2.1, one obtains the MAP estimator for the length field [8]
ˆ
un= arg max
un
P(un|k,yk,yu,yo,yc). Considering equally likely packet sizes, we have
ˆ
un= arg max
un
P(k,yk,yu,yo,yc|un), which, given that the kfield is well know, becomes
ˆ
un= arg max
un
P(yu|un)P(yo,yc|k,un). (6.1) The term P(yo,yc|k,un) is evaluated as
P(yo,yc|k,un) = X
on,cn
P(yo,yc,on,cn|k,un)
= X
on,cn
P(on)P(yo,yc,cn|k,un,on)
= X
on,cn
P(on)P(yo|on)P(yc,cn|k,un,on)
= X
on,cn
P(on)P(yo|on)P(cn|k,un,on)P(yc|k,un,on), which, using the same reasoning as in Section 4.5.2.1 that HECcn is fully determined by k,un, andon, becomes
P(yo,yc|k,un) =X
on
P(on)P(yo|on)P(yc|cn=f(k,un,o)). (6.2) Finally, using (6.2) in (6.1), the MAP estimator becomes
ˆ
un= arg max
un∈Ωu
P(yu|un) X
on
P(yo|on)P(on)P(yc|cn=f(k,un,on)), (6.3) where Ωu = {`min, ..., `max} is the set of lengths which may be taken by the length field.
P(on)is the a priori probability of on,P(yu|un) is the likelihood of the length field from the channel and f is a generic encoding function to calculate HEC/HCS. The evaluation of (6.3) may be done optimally with a complexity O(`(o) 2`(c)), or sub-optimally with a reduced-complexity algorithm, see [8] for more details.
6.2.2 HUNT State: Bayesian hypothesis test
In NP FS method, discussed in Section 3.4.1, hypothesis tests based on NP criterion are used to determine whether a packet starts at a given bit index. This technique is efficient
when the SW is long, but suffers limitations when it is short.
This section is devoted to the construction of Bayesian hypothesis tests exploiting all sources of redundancy present in the header along with the soft information provided by the channel as evidenced earlier in the trellis-based and the ST-based FS techniques. This allows to build more efficient LRT than the LRT of NP FS method, especially when the SW is short.
Let di and hi be the ith symbol of the modulated data and SW/header grouped in vectors of same sizedandh(i.e.,`(d) =`(h) =`h), respectively. Transmission is assumed to be performed over a Rayleigh fading channel. Letyibe the received sample atithsymbol.
After observing`h subsequent samples, the synchronizer must choose between the following two hypotheses,Hh(header hypothesis) andHd(data hypothesis), representing whether the
`th location is the start of a packet or otherwise, respectively.
Hd : yi = ridi+ni, i=`, `+ 1, ..., `+`h−1 Hh : yi = rihi+ni, i=`, `+ 1, ..., `+`h−1
Where, ri are Rayleigh fading coefficients and ni are independent, identically distributed Gaussian r.v.s, with zero mean and variance σ2. Decisions are indicated by Dd and Dh corresponding to the true hypotheses Hd and Hh, respectively.
Let PHj be thea priori probability of hypothesisHj , wherej {d, h}. Our objective is to select the hypothesis with maximum APPq(Hj),i.e., we chooseDh ifq(Hd)≤q(Hh).
Where,
q(Hj) =P(Hj|y) = P(y|Hj)PHj(`) P(y)
Consider the bit index ` of a burst. Under the hypothesis Hh that a packet header h= [k,u,o,c]starts at`, one may interpret the corresponding channel output starting at
`asy= [yk,yu,yo,yc] = [y`, y`+1, ..., y`+`h−1]and
P(y|Hh) = X
h
P(y|h, Hh)P(h|Hh). (6.4) With the hypotheses of Section 4.5.2, only headers starting withkhave to be considered, thus (6.4) becomes
P(y|Hh) =P(yk|k) X
u∈Ωu
(P(yu|u)P(u) X
o
P(yo|o)P(o)P(yc|c=f(k,u,o))). (6.5)
Under the hypothesis Hd that`does not correspond to the beginning of a packet, now y= [y`, y`+1, ..., y`+`h−1]is the channel output when data bitsdare transmitted. Assuming balanced data symbols, one gets
P(y|Hd) = X
d
P(y|d, Hd)P(d|Hd)
= X
d
P(y|d, Hd) 2−`(d). (6.6)
Bayesian hypothesis test can now be represented as Λ(y) = P(y|Hd)
P(y|Hh) Dh
≶ Dd
Pa(`, Hh)
Pa(`, Hd), (6.7) where Dh or Dd correspond to deciding Hh or Hd respectively. Pa(`, Hh) and Pa(`, Hd) are thea priori probabilities of the hypotheses at the bit index `.
When L−` < `max, ` may also represent the start of a padding packet. Thus, an additional hypothesis Hp, corresponding to the presence of a padding packet, has to be considered. After observingL−`subsequent samples towards the boundary of the burst, the synchronizer must choose between the two hypotheses, i.e., Hp(padding hypothesis) andHd(data hypothesis), representing whether the`thlocation is the start of the padding packet or otherwise, respectively.
Hp : yi = ri1i+ni, i=`, `+ 1, ..., L−1 Hd : yi = ridi+ni, i=`, `+ 1, ..., L−1 The Bayesian hypothesis test for deciding betweenHp andHdis given by
Λ(y) = P(y|Hd) P(y|Hp)
Dp
≶ Dd
Pa(`, Hp)
Pa(`, Hd). (6.8) Under Hp, we haveP(y|Hp) =P(y|1), where 1 is a vector of ones of the same size as of d(i.e.,`(d) =`(1) =L−`). P(y|Hd) is given by (6.6).
First Bayesian hypothesis test of (6.7) is applied to choose between header and data till one reaches the bit index ` = L−`min. If data has been decided (i.e., decision is Dd) for this bit index, then the Bayesian hypothesis test of (6.8) is applied for the bit indexes ` > L−`max starting from the last correct FS bit index, to see whether the data corresponds to the padding packet. Finally, the best bit index ` > L−`max is selected to signal the start of the padding packet.
6.2.2.1 A priori probabilities
To evaluate the a priori probabilitiesPa(`, Hh) andPa(`, Hp), one knows that P(`) =Pa(`, Hh) +Pa(`, Hp)
=P(`)P(Hh|`) +P(`)P(Hp|`),
whereP(`)is thea priori probability that a packet (be it a data and/or a padding packet) starts at a bit index`of a burst ofL bits, and
P(Hh|`) =P(Pn= 0|Sn−1=`) =
0, if 0< L−` < `min
1, if L−`≥`max
L−`
P
λ=`min
πλ, else,
(6.9)
andP(Hp|`) =P(Pn= 1|Sn−1=`) = 1−P(Hh|`)are the conditional a priori probabil- ities ofHh (header hypothesis) and Hp (padding hypothesis), respectively.
To determine P(`) consider again the trellis representation as shown in Figure 4.3 (Chapter 4). One may write
P(`) = X
16n6d`/`mine
P(Sn=`), (6.10)
whereP(Sn=`) is calculated using (4.16). A priori probability Pa(`, Hd) =P(`) corre- sponding to an absence of the start of packet at bit index ` is calculated by Pa(`, Hd) = P(`) = 1−P(`).