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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 16bit 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 16bit
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 16bit CRC
code, and RS coding is not used, then one option is to use a 32bit or 48bit CRC code (not in
the CCSDS Recommended Standards). CCSDS 130.1G1 Page 88 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 bandlimited channel using phasecoherent 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 nonzero rotation
belonging to it).
As an example, an MPSK constellation has M rotational symmetries. In particular, a 2PSK
constellation has 2 rotational symmetries: φ = {ρ0,ρ180}, while a 4PSK constellation has 4
rotational symmetries φ = {ρ0,ρ9,ρ180,ρ270}, as a square QAM constellation (16QAM, 64QAM, 256QAM). For nonsquare 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 2PSK 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 nway 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=2PSK transmitted over a channel without
noise. If a 180degree error occurs at the receiver side, all the transmitted bits are received
inverted. We observe that this is equivalent to sum an allone 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 31 and figure 84 below). In
fact, the constant allone sequence is eliminated by the differential devices.
Now, let us consider a binary code C mapped over S=2PSK. 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 130.1G1 Page 89 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE 8.6 REMAPPINGS OF THE BITS In figure 31 there is an optional ‘NRZL to NRZM conversion’ block at the transmitter and,
inversely, an ‘NRZM to NRZL conversion’ block at the receiver. NRZL 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, NRZM represents a data ‘1’ by a change in level and a data ‘0’ by no change
in level. The conversion from NRZL to NRZM is a form of differential precoding that can
be used to resolve the ambiguity between true and complemented data. figure 84 shows a
block diagram for implementing the ‘NRZL to NRZM conversion’ and its inverse.
u
' + D u u (a) D + u
' (b) Figure 84: Block Diagrams for Implementing the (Optional) (a) ‘NRZL to NRZM
Conversion’ and (b) Its Inverse
When all three...
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 Spring '14

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