Similarly a turbo decoder equipped with a smart

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Unformatted text preview: tection capability of the CRC code is superfluous when the RS code with E = 16 is used. The RS code with E = 8 offers much lower error detection capability, on the same order as that provided by the 16-bit CRC code. Similarly, a turbo decoder equipped with a smart stopping rule that notes whether the decoder’s iterations converge to a valid codeword can achieve some degree of error detectability and somewhat alleviate the need for the 16-bit CRC code. However, in these borderline cases the CRC code is still required. It is also required for uncoded data or convolutionally coded data, which offer absolutely no capability for error detection on their own. If a lower detected error rate is desired than that offered by the recommended 16-bit CRC code, and RS coding is not used, then one option is to use a 32-bit or 48-bit CRC code (not in the CCSDS Recommended Standards). CCSDS 130.1-G-1 Page 8-8 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE 8.5 CODE TRANSPARENCY Rotationally invariant (transparent) coding schemes are used to overcome the phase ambiguity inherent in usual coherent demodulation techniques. Let us consider the transmission over a band-limited channel using phase-coherent demodulation. To estimate the carrier phase, the receiver uses its knowledge of the signal set S, which is the set of points produced by the modulator. By examining the pattern of received signal points, the receiver can infer the carrier phase up to an ambiguity corresponding to a rotational symmetry of S. Let us denote a counterclockwise rotation of x degrees about the origin by ρ. A rotational symmetry of the signal set S is a rotation ρ mapping S into itself. The set of all the rotational symmetry of S is called the rotational symmetry group φ. If φ has n elements then it is a cyclic group generated by the rotation ρ of x = 360/n degrees (the smallest non-zero rotation belonging to it). As an example, an M-PSK constellation has M rotational symmetries. In particular, a 2-PSK constellation has 2 rotational symmetries: φ = {ρ0,ρ180}, while a 4-PSK constellation has 4 rotational symmetries φ = {ρ0,ρ9,ρ180,ρ270}, as a square QAM constellation (16-QAM, 64QAM, 256-QAM). For non-square QAM constellations, φ depends on the signal choice. When used in a modulation scheme with coherent demodulation, the carrier phase is estimated from the ensemble of the received signal points. However, an ambiguity corresponding to a rotation of φ cannot be solved without external reference. For example, if a 2-PSK is used, the demodulator observes the two received points and estimates a carrier phase which can be correct, or wrong by 180 degrees. The receiver can handle the n-way phase ambiguity in several ways. One way to resolve the phase ambiguity is through training. At the start of the transmission, and within it, the transmitter sends a predetermined sequence of signal points which the receiver uses to correct its phase estimation. Another method uses transparent coding schemes to solve the problem. In this case, the receiver does not try to resolve the possible phase error but uses transparent schemes able to cope with it. Let us consider an uncoded signal set S=2-PSK transmitted over a channel without noise. If a 180-degree error occurs at the receiver side, all the transmitted bits are received inverted. We observe that this is equivalent to sum an all-one sequence to the transmitted sequences. A simple differential precoder at the transmitter side, followed by a differential postcoder at the receiver can cope with this situation (see figure 3-1 and figure 8-4 below). In fact, the constant all-one sequence is eliminated by the differential devices. Now, let us consider a binary code C mapped over S=2-PSK. The precoder/postcoder operation could still be applied to cope with possible phase errors. However, it is essential that a rotation of 180 degrees maps the code into itself (otherwise, in case of phase error, the decoder would work over a different set of codewords). In this case we say that C is rotationally invariant (transparent): for any code sequences c∈C, its inverted version still belongs to C: a differential precoder/postcoder pair is able to solve the phase ambiguity of the coded sequences. CCSDS 130.1-G-1 Page 8-9 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE 8.6 REMAPPINGS OF THE BITS In figure 3-1 there is an optional ‘NRZ-L to NRZ-M conversion’ block at the transmitter and, inversely, an ‘NRZ-M to NRZ-L conversion’ block at the receiver. NRZ-L is a modulation format that represents a data ‘1’ by one of two levels, and a data ‘0’ by the other level. On the other hand, NRZ-M represents a data ‘1’ by a change in level and a data ‘0’ by no change in level. The conversion from NRZ-L to NRZ-M is a form of differential precoding that can be used to resolve the ambiguity between true and complemented data. figure 8-4 shows a block diagram for implementing the ‘NRZ-L to NRZ-M conversion’ and its inverse. u ' + D u u (a) D + u ' (b) Figure 8-4: Block Diagrams for Implementing the (Optional) (a) ‘NRZ-L to NRZ-M Conversion’ and (b) Its Inverse When all three...
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