130x1g2 - CCSDS

TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALEThis is the optimal algorithm for synchronization if one must use hard decisions and a singleASM. On the other hand, when soft symbols y iare available, several metrics may beadopted. The soft correlator uses the matching functionf (s, y) = s ⋅ yand simply sums together the symbol-by-symbol products. It requires a good deal ofaddition, so when complexity is particularly constrained, the hard correlator is often used.James Massey proved in reference [53] that the optimal metric for argmax based framesynchronization is not the soft correlation, but instead uses the matching functionN ⎛f ( s,y)= sy − log cosh⎜2 E⎝0 2y E ⎞⎟N0 ⎠where E is the symbol amplitude and N 0 is the one-sided noise spectral density. Hence, E/N 0is the symbol SNR, and E / N0is the combining ratio. The transcendental functions areinconvenient in implementation, so Massey also showed that a good approximation is⎧ 0f ( s,y)= ⎨⎩−yif sign( y)= sotherwiseThat is, only noisy symbols with the wrong sign are counted, and the magnitude of the softsymbol is its penalty. While the exact Massey algorithm requires an estimate of thecombining ratio, this approximate Massey algorithm does not. For each of these algorithms,performance can be improved by matching two (or more) consecutive markers, achievingwhat would be possible with a single marker of twice (or several times) the length.The performance of the argmax based frame synchronizer can be evaluated in a very simpleway using the tight upper bound developed in reference [67]. Accordingly, the probability offalse synchronization of the argmax based synchronizer, assuming AWGN ands i∈ {−1, +1}, may be approximated very accurately by the analytical formulaP FS(E / N 0) ≈ (F −1) P sep(E / N 0)where P sep(E / N 0), called the pairwise synchronization error probability, depends on signalto-noiseratio E / N 0, on the ASM length M , and on the adopted metric (e.g., Massey exact,Massey approximated, soft correlation, hard correlation) but not on the frame length F . Fora given metric and ASM word length, the values of this probability as a function of E / N 0may be easily obtained by numerical integration as detailed in reference [67]. Through theabove formula for P FSit is also easy to recognize that the probability of falsesynchronization of the argmax based synchronizer is affected by an error floor for largeenough E / N 0, as follows:CCSDS 130.1-G-2 Page 9-13 November 2012