Aerospace and Electronic Systems Magazine June 2017 - 11
Baldi et al.
ri, computes their correlation only where the symbol signs are different, namely
ΓM =
i :sgn ( pi ) ≠ sgn ( ri )
pi ri
(2)
The performance of these codes depends on the decoding algorithm adopted (the standards do not specify the decoding algorithms, whose choice is left to the Agencies). In this work, we focus on three different algorithm families:
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The simplified Massey detector provides an excellent solution,
with limited complexity and very good performance. A comparison between the different detectors is given in Figure 6: fixed a
false-alarm probability Pfa of 10−6, the behavior of the miss-detection probability Pmd is shown as a function of Es/N0. The simplified
Massey detector has nearly-optimal performance, close to the LRT
and much better than hard or soft correlation.
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DECODER EVALUATION AND OPTIMIZATION
The considered short LDPC codes are based on a protograph-pluscirculant construction [14]. As a result, their parity check matrix
consists of a 4 × 8 array of circulant sub-matrices with size Q × Q,
where Q = n/8. Each submatrix has a number of symbols 1 per row/
column not greater than 2.
Table 5.
Required E s / N 0 [dB] at CER = 10 -5
Quantization Parameters
Code
LDPC(128, 64)
LDPC(512,256)
LDPC(128, 64)
LDPC(512, 256)
LDPC(128, 64)
LDPC(512, 256)
LDPC(128, 64)
LDPC(512, 256)
LDPC(128, 64)
LDPC(512, 256)
LDPC(128, 64)
LDPC(512, 256)
LDPC(128, 64)
LDPC(512, 256)
JUNE 2017
Iterative decoding algorithms. These are classical LDPC softdecision iterative decoding algorithms, like the sum-product
algorithm (SPA) [19], typically implemented using log-likelihood ratios (SPA-LLR), or its simplified versions, e.g., minsum (MS) [20] or normalized min-sum (NMS) [21].
Noniterative MRB decoding algorithms. They exploit a softdecision procedure, potentially able to achieve performance
very close to that of the optimum maximum likelihood (ML)
decoder [22]. Their main drawback is complexity, which is an
important issue in TC links, where decoding is performed O/B.
However, the length of the LDPC codes proposed for TCs is
rather short, and a number of speedup solutions can be adopted
to limit the MRB complexity. Application of the MRB algorithm to the short LDPC codes is reported in [9].
Hybrid decoding algorithms. These decoders perform first
a low complexity decoding attempt
through an iterative algorithm, and
invoke MRB only when the iterafor Various Decoding Algorithms and
tive algorithm is not able to find any
codeword (detected error). Notably,
the hybrid algorithm has been successfully applied even for decoding
SPA-LLR
MS
NMS
MRB
Hybrid
of short nonbinary LDPC codes [23].
Unquantized
Table 5 reports the Es/N0 val2.22
2.38
2.24
0.60
0.55
ues, obtained through simulation,
0.62
1.07
0.59
1.85
0.43
required by the considered decoding algorithms to achieve the target
Linear quantization q = 6
CER = 10−5 with the two codes. Both
2.24
2.38
2.24
0.60
0.55
the unquantized (i.e., ideal) and the
0.70
1.11
0.62
1.85
0.50
quantized cases, for different values of the number q of quantization
Linear quantization q = 5
levels, have been analyzed. We have
2.29
2.46
2.24
0.72
0.64
used two quantization rules: linear
0.78
1.2
0.74
1.93
0.58
and logarithmic (i.e., with denser
quantization levels around the zero
Linear quantization q = 4
signal amplitude). Looking at the un2.64
2.68
2.62
1.20
0.94
quantized case, for the short code the
1.54
1.56
1.42
1.94
1.26
best performance is achieved by the
hybrid algorithm, with a gain in the
Logarithmic quantization q = 6
order of 1.67 dB over the SPA-LLR.
2.22
2.38
2.24
0.60
0.55
The hybrid algorithm is also the best
0.64
1.10
0.59
1.85
0.45
for the long code, but the gain is just
0.16 dB over the NMS algorithm, the
Logarithmic quantization q = 5
latter offering the best performance
2.24
2.38
2.24
0.60
0.55
among the iterative algorithms. This
0.70
1.14
0.64
1.94
0.50
is because the value of l in the "orLogarithmic quantization q = 4
der-l reprocessing" stage of the MRB
algorithm [9] is upper bounded due
2.50
2.58
2.24
0.72
0.68
to complexity issues. In these con1.16
1.20
0.68
1.94
0.88
ditions, for the short code the MRB
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