FAILURE
5.2 Sliding Trellis-based FS Algorithm
Contrary to the trellis for a CC [29], the trellis considered in Figure 4.3 has a variable number of states for each value of the packet-clock n. One may apply directly the SW- BCJR ideas, but due to the increase of the size of the trellis (at least for small values of n), this would still need large trellises to be manipulated, with an increased computation time. Here, a ST-based approach is introduced: a reduced-size trellis is considered in each decoding window. As in [31], some overlapping between windows is considered to allow better reuse of already computed quantities reaching to a complexity-efficiency trade-off.
Window 1
L1
`
"0 n
`
_
n_
Figure 5.1: ST for the first decoding window, the original trellis is in gray
5.2.1 Sliding Trellis
In the proposed ST-based approach, a burst of L bits is divided into M overlapping windows with sizes Lm,m= 1, ..., M. For each of these windows, the bits fromεm−1+ 1 to εm−1+Lm are considered, where εm−1 is the index of the last bit of the last packet deemed reliably synchronized in the m−1-th window.
A ST moves from window-to-window to perform decoding. One such ST is shown in Figure 5.1. Let n¯ and `¯be the local trellis coordinates. Once P(Snm¯ = ¯`|yεεm−1+Lm
m−1+1 ) is evaluated, one can apply the estimators (4.7), (4.8), and (4.9) to determine the number of packetsNbm in them-th window (including the last truncated packet), the beginning, and the length of each packet. For them-th window,m < M, among theNbm decoded packets, only the first Nbmc packets are considered as reliably synchronized, since enough data and redundancy properties have been taken into account. Truncated packets, especially when the HEC has been truncated, and the packet immediately preceding such packets, are not considered reliable. For this reason, only the Nbmc complete packets ending in the first Lm−`max−`h bits of the window are considered as reliable. The unreliable region towards the boundary of the window is dashed in Figure 5.1.
The initialization of β is performed as in Section 4.5, since no knowledge from the previous window can be exploited. The initialization ofα and the evaluation ofγ towards the window boundary may depend on the location of the window inside a burst. Three types of window locations are considered: the first window at the start of a burst, the intermediate windows in the middle of the burst, and thelast window at the end of the burst.
First Window, m= 1
Consider the first window of Lm < Lbits for which ε0 = 0, shown in Figure 5.1. The ST representing all possible successions of packets within Lm bits is very similar to the trellis in Figure 4.3. The decoding approach, including the initialization ofα for this first window is similar to that presented for the trellis-based approach in the previous chapter.
An exception is the computation ofγ¯n `¯0,`¯
, where two cases have again to be considered corresponding to
1. the normal data packets, leading toγd¯n `¯0, Lm , and
2. the truncated data packets towards the boundary of the window, leading toγnt¯ `¯0, Lm , detailed in Section 5.2.2.
Intermediate Windows, 1< m < M
Consider now them-th window(m >1)containing the bits from εm−1+ 1 to εm−1+ Lm < L, see Figure 5.2. The bit index εm−1, of the last bit of the last packet deemed reliably synchronized (i.e., the Nbm−1c -th packet) in the m−1-th window, corresponds in the local coordinates of the m−1-th ST to `¯= εm−1−εm−2. The m-th ST starts at the local coordinates
¯
n=Nbm−1c ,`¯=εm−1−εm−2
of m−1-th ST. The computation of γn¯ `¯0,`¯
for an intermediate window is identical to that of the first window. A first choice for the initialization ofαmn¯ `¯
in this ST, would be to considerαm0 `¯= 0
= 1 and α0m `¯6= 0
= 0. The drawback of this approach is that all computations of α performed in them−1-th window are not utilized and are lost. Therefore, following the idea of the SW-BCJR decoder [106; 30], up to`max initial values for αm0 `¯
are propagated from the
`max
Lm 1- Window-1m Windowm
Lm
Lo m-1
`
`
_
"m-1
n_
n
Figure 5.2: ST for them-th intermediate decoding window, the original trellis is in gray
m−1-th window to the m-th window, see Section 5.2.3. This allows a better FS in case of erroneous FS in them−1-th window.
Last Window, m=M
Finally, for the last window, the incomplete packets at the end of the window are not to be considered any more: only the presence of a padding packet has to be taken into consideration, see Figure 5.3. For this window, the decoding is performed as in the trellis- based method (Section 4.5), except for the initialization ofαM0 `¯
, which is similar to that of the intermediate windows case.
Them-th andm+1-th windows overlap overLombits, with`h+`max6Lom < `h+2`max.
5.2.2 Evaluation of γn¯
When m < M, transitions corresponding to truncated packets have to be considered at the end of the window. When the size of the truncated packet is larger than `h, the header is entirely contained in the truncated packet. In this case
γn¯ `¯0, Lm
= γnt¯ `¯0, Lm ,
with
γnt¯ `¯0, Lm
=p Snm¯ =Lm|S¯n−1m = ¯`0 ϕt
y`L¯0m+1,xL¯`0m+1
. (5.1)
In (5.1), since truncated packets have to be considered,p Snm¯ =Lm|Sm¯n−1 = ¯`0
is given by (4.4). Moreover, the length of a packet, i.e., the content of the length field u, is now only known to be betweenmax(Lm−`¯0, `min) and `maxbits. Thus
`max
LM1- Window-1M WindowM
LM
Lo M-1
`
n
`
_
"M-1
n_
Figure 5.3: ST for the last decoding window, the original trellis is in gray
ϕt
yL`¯0m+1,xL`¯0m+1
=P(yk|k)X
p
P(yp|p)P(p)
`=`max
X
`=max(Lm−`¯0,`min)
P(u(`))X
o
(P(yu|u(`))P(yo|o)
P(yc|c=f(k,u(`),o))P(o)). (5.2) When the size of the truncated packet is strictly less than`h, for the sake of simplicity, all bits of the truncated header are assumed equally likely. In such case
γn¯ `¯0, Lm
= γne¯ `¯0, Lm
,
with
γ¯ne `¯0, Lm
= p Snm¯ =Lm|S¯n−1m = ¯`0 X
xLm¯
`0+1
P
yL`¯0m+1|xL¯m
`0+1
P
xL`¯0m+1
, (5.3)
wherep Snm¯ =Lm|Sn−1m¯ = ¯`0
is still given by (4.4) and P(xL¯`0m+1) = 2−`(xLm`¯0+1), since all beginning of headers are assumed equally likely.
For m=M,the evaluation of γ¯nM is as in the trellis-based method (Section 4.5.2).
5.2.3 Initialization of α in the sliding trellises
In the SW-BCJR algorithm proposed in [30], the αms evaluated in the m-th window are deduced from those evaluated in them−1-th window. Here, since the number of states Sn evolves with packet-clockn,αm¯n cannot be obtained that easily from αnm−1¯ .
In the m−1-th window, one has evaluated αm−1n¯ `¯
, with 0 6 `¯6 Lm−1 and 0 6
¯
n6dLm−1/`mine. We choose to propagate at most`maxvalues of αfrom the packet-clock
¯
n = Nbm−1c in the m−1-th window to the packet-clock n¯ = 0 in the m-th window (for
`¯= 0, ..., `max−1) as follows
αm0 (¯`) =κ αm−1
Nbm−1c εm−1−εm−2+ ¯`
, (5.4)
whereκ is some normalization factor chosen such that theαm0 (¯`)s sum to one. This allows the first packet of the m-th window to start at any bit index between εm−1 + 1 and εm−1+`max.
5.2.4 Complexity Gain
Now we will compare the complexity of the ST-based FS technique to that of the trellis- based FS technique presented in the previous chapter. For the ST-based FS technique, considerM windows of approximately the same sizeLw, which are overlapping on average on Lo=`h+ 1.5`max bits. For sufficiently largeL, one will have to process
M ≈ L
Lw−Lo
overlapping windows with as many STs, each one with Nnw ≈ (Lw)2
2
`max−`min
`max`min
nodes. The total number of nodes to process is then MNnw ≈ L
Lw−Lo (Lw)2
2
`max−`min
`max`min
. (5.5)
The total number of nodes is much less than that of the trellis-based FS technique. A comparison for different values of burst sizeL is provided in the Table 5.1 for window size Lw = 480 +Lo,in the Table 5.2 for window sizeLw = 600 +Lo,and in the Table 5.3 for window sizeLw = 900 +Lo. For example, when L= 12960 bytes andLw = 480 +Lo, the ST-based technique reduces the number of nodes by a factor 10 as compared to the trellis- based technique. Figure 5.4 compares the number of nodes in the trellis-based technique with the ST-based technique for different values of burst sizeL and window size Lw.
The decoding complexity of the ST-based technique is thus less than that of the trellis- based technique. One can observe that the complexity gain increases with the increase in the burst size L. The graph showing the complexity gain vs. burst size L for different window sizesLw is shown in Figure 5.5. Choosing small values forLwreduces the decoding complexity as well as the latency, see Figure 5.5. The price to be paid is some sub- optimality in the decoding performance. Note thatLwcannot be chosen too small (smaller than`h+ 2`max) to ensure at least one reliable FS in each window.
L (bytes) 1800 4000 8000 12960 16000 24000 Trellis-based (# of Nodes) 24300 120000 480000 1259712 1920000 4320000 ST-based (# of Nodes) 17400 38600 77200 125100 154450 231700
Complexity Gain 1.4 3.1 6.2 10.0 12.5 18.6
Table 5.1: Complexity comparison between the trellis-based and ST-based (Lw = 480+Lo) FS techniques
L (bytes) 1800 4000 8000 12960 16000 24000
Trellis-based (# of Nodes) 24300 120000 480000 1259712 1920000 4320000 ST-based (# of Nodes) 18500 41000 82000 133000 164200 246200
Complexity Gain 1.3 2.9 5.8 9.5 11.7 17.5
Table 5.2: Complexity comparison between the trellis-based and the ST-based (Lw = 600 +Lo) FS techniques
L (bytes) 1800 4000 8000 12960 16000 24000
Trellis-based (# of Nodes) 24300 120000 480000 1259712 1920000 4320000 ST-based (# of Nodes) 21800 48500 96900 157000 193900 290900
Complexity Gain 1.1 2.5 4.9 8.0 9.9 14.9
Table 5.3: Complexity comparison between the trellis-based and the ST-based (Lw = 900 +Lo) FS techniques
Figure 5.4: Total # of Nodes vs. burst sizeLfor different window sizes Lw
Figure 5.5: Complexity Gain vs. burst sizeLfor different window sizes Lw