Aerospace and Electronic Systems Magazine June 2017 - 12

State-of-the-Art Space Mission Telecommand Receivers

Figure 8.

UCER performance of the LDPC(128, 64) code by using the SPA-LLR
decoding algorithm and the MRB algorithm followed by the CRC.

Figure 7.

CER performance of the selected decoding algorithms: (a) unquantized case, (b) assuming q = 3 quantization bits for the channel and/or
decoder messages.

algorithm used alone ensures practically the same performance as
the hybrid algorithm, while for the long code it is characterized by
the worst performance, more than 1.4 dB away from the hybrid
algorithm. This is because, for the MRB algorithm alone, the constraints on the value of l are even stricter for the case of n = 512;
the complexity is relaxed by resorting to the hybrid approach that
however, as we will show afterward, remains rather distant from
the theoretically achievable limits. So, also pondering its reduced
complexity, NMS is preferable for the long code.
Figure 7a summarizes the CER performance of the best decoders for both codes, assuming no quantization. This means that the
soft reliability values that the algorithms involve are represented
with a very high precision (e.g., 32-bit floating point for each reliability value). The union bound (UB) [24] for the short code and
the sphere packing bound (SPB) [25] for the long code are also
plotted as benchmarks. The UB gives an estimate of the ML decoder performance, which becomes more and more reliable for
increasing SNR. The performance of the hybrid algorithm is at a
distance of about 0.6 dB from the UB, thus confirming the goodness of the hybrid approach when applied to the short code. The
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SPB has a meaning similar to the Shannon capacity but, contrary
to the latter, it is able to capture the finiteness of the code length.
The distance of the curve for the NMS decoder (or even the hybrid
decoder) from the SPB is rather large, on the order of 2 dB, thus
confirming that, for the long code, margins exist for performance
improvements.
From Table 5 we also see that using q ≥ 5 with the linear law is
sufficient to have almost negligible losses with respect to the ideal
case; on the other hand, q = 4 is enough with the logarithmic law
that, however, is characterized by slightly higher complexity.
When applied to an LDPC iterative decoder, quantization
involves both channel messages and decoder messages. In view
of ensuring compliance with the current architecture, we should
consider that, in a typical implementation, the soft quantization
logic inside the transponder may be limited to 3 bits. Hence, the
impact of q = 3 must be investigated. With such a small number of quantization bits, it becomes of paramount importance to
choose suitable clipping thresholds to limit the effect of noise.
Moreover, a linear quantization law is adopted. Though the limit
on the number of quantization bits should be set only on the
channel messages (while the decoder messages can be quantized by using q = 6 without problems), in Figure 7b we have
reported the CER curves resulting from different combinations
of the quantization granularity, under NMS decoding. We see
that, when using q = 3 for the channel messages and q = 6 for the
decoder messages, the loss against the unquantized case is very
limited, in the order of 0.15 dB for the short code and 0.2 dB for
the long code.
According to [5], TFs are of variable length, so that each TF
produces a variable number M of codewords. Besides the payload, the TF also includes a 40-bit header and (optionally) a 16-bit
cyclic redundancy check (CRC) that, however, do not need to be
discriminated from the data for the purposes of the CER analysis.
We have 1 ≤ M ≤ 128 for the LDPC(128, 64) code and 1 ≤ M ≤ 32

IEEE A&E SYSTEMS MAGAZINE

JUNE 2017

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine June 2017