Conclusions and Future Work
5.4 SISO Channel Decoding of Convolutionally- Convolutionally-Encoded VLCConvolutionally-Encoded VLC
For the transmission scheme with channel coding, a soft-output channel decoder can be used to provide both decoded bits and their reliability information for further pro-cessing to improve the system performance. The commonly used BCJR algorithm is a trellis-based MAP decoding algorithm for both linear block and convolutional codes.
The derivation presented in [23] led to a forward-backward recursive computation on the basis of a bit-level trellis diagram, which has two branches leaving each state and every branch represents a single symbol-bit. Proper sectionalization of a bit-level code trellis may result in useful trellis structural properties [31][32] and allows us to de-vise SISO channel decoding algorithms which incorporate parameter-oriented extrinsic information from the source decoder. To proceed with this, we propose a modified BCJR algorithm which parses the received code-bit sequence into blocks and computes the APP for each parameter symbol on a symbol-by-symbol basis. Unlike classical BCJR algorithm that decodes one bit at a time, our scheme proceeds with decoding the parameter symbols as nonbinary symbols that are matched to the number of bits in an symbol. By parsing the code-bit sequence into l(c(xt))-bit symbols, we are in essence merging l(c(xt)) stages of the original bit-level code trellis into one. In previous work[31][32], the sectionalized trellis originally proposed for fixed length codevector, is not suit for symbol-by-symbol channel decoding of convolutionally-encoded VLC.
In the case of variable-length branch, different paths entering a state have used up a different number of bits from the received sequence and can therefore be extended dif-ferently [48]. For this reason, we extend 1-dimensional state st to 2-dimensional state As an example, we illustrate in Figure 5.3 three stages of the bit-level trellis diagram of a rate 1/2 convolutional encoder with generator polynomial (7, 5)8. The solid lines and dashed lines correspond to the input bits of 0 and 1, respectively. σt= (st, gt). Figure 5.3 also shows the sectionalized trellis diagram when two stages of the original bit-level trellis are merged together. In general, there are 2M branches leaving and entering each state in a l(c(λ))-stage merged trellis diagram. Having defined the trellis structure as such, there will be one symbol APP corresponding to each branch which represents a particular parameter symbol xt = λ. For convenience, we say that the sectional-ized trellis diagram forms a finite-state machine defined by its state transition function Fσ(xt, σt−1) and output function Fp(xt, σt−1). Viewing from this perspective, the code-bit combination associated with the branch from state σt−1 to state σt = Fσ(xt, σt−1) can be written as yt = (xt, zt), where zt = Fp(xt, σt) is the variable-length codevector
00 the convolutional encoder with memory order ν. An example for 3-dimension trellis digram is shown in Figure 5.4 for N = 5, T = 2 and ν = 2.
We next applied this new sectionalized code trellis to compute the APP of a sys-tematic symbol xt= λ given the received code sequence ˜Y1T = {˜y1, ˜y2, ..., ˜yT} in which
0
1/P (˜y1T) is a normalizing factor. For the recursive implementation, the forward and backward recursions are to compute the following metrics:
αxt(λ, σt) = X
and in (5.15)
γλ,qx (˜yt, σt, σt−1) = P (xt= λ, σt, ˜yt|xt−1 = q, σt−1)
= P (st|xt = λ, gt, ˜yt, xt−1 = q, σt−1)
·P (˜yt|xt= λ, gt, xt−1 = q, σt−1)P (xt= λ, gt|xt−1 = q, σt−1)
= P (st|xt = λ, st−1)P (˜yt|xt= λ, gt, σt−1)
·P (xt= λ, gt|xt−1 = q, σt−1).
(5.17)
Having a proper representation of the branch metric γλ,q(˜yt, σt, σt−1) is the critical step in applying symbol decoding to error mitigation and one that conditions all sub-sequent steps of the implementation. As a practical manner, several additional factors must be considered to take advantage of the symbol-level trellis structure and AWGN channel assumption. First, making use of the merged variable length code trellis, the value of P (st|xt−1 = q, st−1) is either one or zero depending on whether symbol q is associated with transition from state σt−1 to state σt = Fσ(xt = λ, σt−1). For AWGN channels, the second term in (5.17) is reduced to
P (˜yt|xt = λ, gt, σt−1) = P (˜xt|xt= λ, ˜zt, gt, σt−1)P (˜zt|xt= λ, σt−1, gt)
= P (˜xt|xt= λ, gt)P (˜zt|zt = Fp(xt= λ, σt−1), gt)
(5.18)
where the conditional pdfs for the received systematic and parity symbols can be computed analogous to (5.6). The third term in (5.17) is reduced to P (xt = λ, gt) under the assumption that xt is uncorrelated with xt−1, which is indeed the case as xt is the interleaved version of parameter symbols. Within the iterations the a priori information P (xt= λ, gt) can be improved by additional a priori information which is provided by the SISO source decoder in terms of its extrinsic probability PSD[ext](xt= λ, gt).
The decoder’s next step is to compute the deinterleaved extrinsic information for each symbol. We first compute APPs for deinterleaved symbol ut = λ which takes VLC trellis states into consideration as follows:
PCD(gt−1, ut= λ, gt|˜y1T) = C′· PCD(ut= λ|˜yT1) (5.19)
where the normalization factor C′ =X
q
X
(gt,gt−1)∈{(gt,gt−1):gt−gt−1=l(q)}
PCD(ut= q|˜y1T)
and PCD(ut= q|˜y1T) = Φ−1(PCD(xt= q|˜y1T)).
An iterative process using the SISO channel decoder as a constituent decoder is realizable, if the reliability of the APPs PCD(gt−1, ut = λ, gt|˜y1T) can be separated into three terms according to Bayes theorem: the a priori probability Pa(ut = λ, gt), the channel-related probability Pc(ut = λ, gt) = P (˜ut|ut = λ, gt), and an extrinsic term resulting from PCD[ext](ut= λ, gt). The equation (5.19) leads to
PCD(gt−1, ut= λ, gt|˜y1T) = P (ut= λ, gt−1, gt, ˜uT1, ˜zT1)/P (˜y1T)
= P (ut= λ, gt−1, gt, ˜uT1) ·P(˜zT1|ut=λ,gP(˜yTt−1,gt,˜uT1) 1)
= P (˜ut|ut= λ, gt−1, gt, ˜ut1−1, ˜uTt+1) · P (xt = λ, gt, gt−1)
·P (˜ut1−1, ˜uTt+1) · P (˜z1T|ut= λ, gt−1, gt, ˜U1T)/P (˜y1T)
= C · Pc(˜ut|ut= λ, gt) · P (ut= λ, gt)
·PCD[ext](ut= λ, gt)
(5.20)
where C is a normalization factor and
PCD[ext](ut= λ, gt) = P (˜zT1|ut = λ, gt−1, gt, ˜uT1). (5.21) With this, the deinterleaved extrinsic probability PCD[ext](ut= λ, gt) can be calulated by
PCD[ext](ut= λ, gt) = PCD(gt−1,ut=λ,gt|˜y1T)
Pc(ut=λ,gt)PSD[ext](ut=λ,gt)P (ut=λ,gt) (5.22) and used as new a priori information for the source decoder. Similarly, the interleaved extrinsic probability PCD[ext](xt= λ, gt) can be calulated by
PCD[ext](xt= λ, gt) = PCD(gt−1,xt=λ,gt|˜yT1)
P(x=λ,g)P[ext](x=λ,g)P (x=λ,g). (5.23)
Finally, we summarize the proposed symbol-based ISCD of convolutionally-encoded VLC as follows:
1. Initialization:
Set the extrinsic information of source decoding to PSD[ext](xt, gt) = 1. Set the iteration counter to n = 0 and define an exit condition nmax.
2. Read series of received sequences ˜yT1 and map all received systematic symbols ˜xt
to channel-related probabilities Pc(xt, gt).
3. Perform MAP channel decoding on each symbol to compute the APP PCD(xt = λ|˜y1T) leaded by substituting (5.15) and (5.16) into (5.14). Then the symbol APP for each interleaved symbol which takes VLC trellis states, P (xt= λ, gt, gt−1|˜y1T), is computed by (5.19). The de-interleaved extrinsic probability and interleaved extrinsic probability PCD[ext](ut, gt), PCD[ext](ut, gt) can be calculated by (5.22) and (5.23), respectively. These extrinsic probabilities can be used as a priori infor-mation for the source decoder.
4. Perform SISO source decoding of VLC by inserting the de-interleaved extrinsic probability PCD[ext](ut, gt) into (5.5) and then de-interleaved extrinsic probability for each symbol is calculated by (5.11). Compute the symbol a posteriori probability PSD(ut, gt, gt−1|˜yT1) by (5.9) and then substitute these probabilities into (5.1) to lead APPs PSD(ut|˜y1T). Interleave these APPs to compute the interleaved symbol APP for each symbol which takes trellis states, P (xt = λ, gt, gt−1|˜y1T) by (5.13).
Then, the interleaved extrinsic probability PSD[ext](xt, gt) is computed by (5.12) and is forwarded to the channel decoder as a priori information.
5. Increase the iteration counter n ← n + 1. If the exit condition n = nmax is fulfilled, then continue with step 6, otherwise proceed with step 3.
6. Using the symbol APPs obtained from step 4 to calculate the decoder output
signals ˆvt by MAP estimation, where the estimated value are given as
ˆ
vt= cλ∗, λ∗ = arg max
λ PSD(ut= λ|˜y1T) (5.24)
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Appendix A
This section gives the detailed computation of R(l)(m, n, DF,i) and S(l)(m, n, DF,i) when (1) a Reed-Solomon code (N, K) is used, (2) packets are sent over a Gilbert channel and (3) the FEC delay of packet i is DF,i. For m = 1, n ≥ 1, R(l)(1, n, DF,i) is the probability that none of the packets are missing in the next n − 1 packets following the network loss of packet i, and is given by
R(l)(1, n, DF,i) = Pr(Wi+1i+n−1= 0n−1|Wi = 1)
= q(l)(1 − p(l))n−2·Qn−1
h=1(1 − e(l)b,i+h).
(A.1)
For 2 ≤ m ≤ n, we compute R(l)(m, n, DF,i) conditionally to the event {Aj, Bj, Cj, j = 0, 1, . . . , n − m} on the arriving states of packets:
Aj = {Wii+j+1 = 10j1}
Bj = {Wii+j+1 = 10j2}
Cj = {m − 2 missing packets in Wi+j+2i+n−1}
(A.2)
where 0j is a shorthand for j successive 0’s. For a Gilbert loss model with parameters p(l) and q(l), we have
Pr(Aj) =
(1 − q(l)), j = 0 q(l)(1 − p(l))j−1p(l)Qj
h=1(1 − e(l)b,i+h), j ≥ 1 (A.3)
Pr(Bj) = q(l)(1 − p(l))jQj
h=1(1 − e(l)b,i+h)e(l)b,i+j+1, j ≥ 1 (A.4)
Pr(Cj|Aj) = Pr(m − 2 missing packets in Wi+j+2i+n−1|Wii+j+1 = 10j1)
From the total probability theorem, R(l)(m, n, DF,i) can be computed as follows:
R(l)(m, n, DF,i) =
Similarly, the probability ˜R(l)(m, n, DF,i) can also be computed by recurrence as R˜(l)(m, n, DF,i) = packets following the late loss of packet i, and is given by
S(l)(1, n, DF EC,i) = Pr(Wi+1i+n−1 = 0n−1|Wi = 2)
= e(l)b,i(1 − p(l))n−1·Qn−1
h=1(1 − e(l)b,i+h).
(A.9)
For 2 ≤ m ≤ n, we compute S(l)(m, n, DF,i) conditionally to the event {Cj, Dj, Ej, j =
For a Gilbert loss model with parameters p(l) and q(l), we have
Pr(Dj) = e(l)b,i(1 − p(l))jp(l)Qj
From the total probability theorem, S(l)(m, n, DF,i) can be computed as follows:
S(l)(m, n, DF,i) =
Similarly, ˜S(l)(m, n, DF,i) can be computed by recurrence as
S˜(l)(m, n, DF,i) =
e(l)b,i(1 − p(l))n−1·Qn−1
h=1(1 − e(l)b,i−h), m = 1, n ≥ 1
n−m
X
j=0
{e(l)b,i(1 − p(l))j
j
Y
h=1
(1 − e(l)b,i−h) · {p(l)R˜(l)(m − 1, n − j − 1, DF,i−j−1) +(1 − p(l))e(l)b,i−j−1S˜(l)(m − 1, n − j − 1, DF,i−j−1)}}, 2 ≤ m ≤ n
(A.16)
Appendix B
In this Appendix B we shall show that the a priori LLR in (4.24) is equal to the de-interleaved sequence of extrinsic information provided by the SISO channel decoder, i.e, La(uI,t = lI) = L[ext]CD(uI,t = lI). The APP of a systematic symbol xD,t = lD, given the received code sequences ˜YD,1T = ( ˜XD,1T , ˜ZD,1T ), can be decomposed by using the Bayes theorem as
P (xD,t = lD| ˜YD,1T )
= P (xD,t= lD, ˜XD,1T ) · P ( ˜ZD,1T |xD,t = lD, ˜XD,1T )/P ( ˜YD,1T )
= P (˜xD,t|xD,t = lD, ˜XD,1t−1, ˜XD,t+1T ) · P (xD,t = lD, ˜XD,1t−1, ˜XD,t+1T )
·P ( ˜ZD,1T |xD,t = lD, ˜XD,1T )/P ( ˜YD,1T )
= C · P (˜xD,t|xD,t = lD) · P (xD,t= lD) · P ( ˜ZD,1T |xD,t = lD, ˜XD,1T )
(B.1)
where C = P ( ˜XD,1t−1, ˜XD,t+1T )/P ( ˜YD,1T ). We rewrite (B.1) in log-likelihood algebra as
L(xD,t = lD| ˜YD,1T )
= La(xD,t = lD) + Lc(xD,t = lD) + L[ext]CD (xD,t = lD)
(B.2)
with
L[ext]CD (xD,t= lD) = logP( ˜PZD,1T |xD,t=lD, ˜XD,1T )
( ˜ZD,1T |xD,t=0, ˜XD,1T ) . (B.3)
Since the de-interleaved sequence of L[ext]CD (xD,t = lD) is used by the source decoder, we have
L[ext]CD (uD,t= lD) = logP( ˜PZD,1T |uD,t=lD, ˜UD,1T )
( ˜ZD,1T |uD,t=0, ˜UD,1T ) (B.4)
where ˜UD,1T = Φ−1( ˜XD,1T ). Once the LLR L[ext]CD (uD,t= lD) has been determined, we can compute the probability as follows
P ( ˜ZD,1T |uD,t= lD, ˜UD,1T ) = eL[ext]CD (uD,t=lD)/P2R−1
j=0 eL[ext]CD (uD,t=j). (B.5)