130x1g2 - CCSDS

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TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALEIn some cases, some of these error rates are synonymous or uninformative. For example,WER=BER for uncoded data because in this case each ‘codeword’ consists of one bit. Similarly,FER=WER for CCSDS Turbo codes, because in this case the CCSDS Transfer Frame consistsof the information bits from one Turbo codeword. A codeword for unterminated convolutionalcodes is theoretically infinitely long, so WER=1 (except on an error-free channel) and thus WERis not a very interesting measure of performance in this case. It is natural to define WER forterminated convolutional codes. Even for unterminated convolutional codes it is valid tocompute FER on a segment (defining the frame) of the convolutional codeword.3.3.2 FUNDAMENTAL LIMITS ON CODE PERFORMANCEGood channel codes lower the error rate in the data, or equivalently they can achieve desirederror rates more efficiently as a function of the bit-SNR E b /N 0 on the channel. Shannon (seereference [8]) derived fundamental limits on the performance of all codes. There are coderate-dependentchannel capacity limits on the minimum E b /N 0 required for reliablecommunication that are theoretically achievable by codes of a given rate in the limit ofinfinite block sizes. In addition, there are block-size-dependent limits that preclude capacityattainingperformance when the code’s block size is also constrained.Code-Rate-Dependent Capacity Limits — Figure 3-3 shows the Shannon-limit performancecurves for a binary-input AWGN channel for rates r=1/6, 1/4, 1/3, and 1/2. These curves showthe lowest possible bit-energy-to-noise ratio E b /N 0 required to achieve a given BER over thebinary-input AWGN channel using codes of these rates.10 -110 -2RATE 1/210 -3RATE 1/3RATE 1/4RATE 1/6BER10 -410 -510 -6-1.5-1.0-0.5Eb/No (dB)0.00.5Figure 3-3: Capacity Limits on the BER Performance for Codes with Rates 1/2, 1/3,1/4 and 1/6 Operating over a Binary Input AWGN ChannelCCSDS 130.1-G-2 Page 3-4 November 2012
TM SYNCHRONIZATION AND CHANNEL CODING—SUMMARY OF CONCEPT AND RATIONALEFor low BER, each of these capacity-limited performance curves approaches a verticalasymptote dependent on the code rate. The asymptotes are at 0.2 dB for rate 1/2, -0.5 dB forrate 1/3, -0.8 dB for rate 1/4, and -1.1 dB for rate 1/6. The vertical asymptote for the ultimateShannon limit on performance (i.e., rate →0) is -1.6 dB. A comparison of these limits showsthe improvement that is theoretically possible as a result of lowering the code rate. Forexample, for a binary-input AWGN channel, rate-1/2 codes suffer an inherent 0.7 dBdisadvantage relative to rate-1/3 codes, a 1.0 dB disadvantage relative to rate-1/4 codes, anda 1.8 dB disadvantage relative to the ultimate limit (rate →0).Codeword-Length-Dependent Limits on Code Performance — Just as a constraint oncode rate raises the minimum threshold for reliable communication above the ultimateunconstrained capacity limit, so does a constraint on codeword length. The theoretical limitsshown in figure 3-3 assume no constraint on block size. Approaching these limits requiresthat block sizes grow arbitrarily large.Figure 3-4 shows some classic Shannon sphere packing lower bounds on the performance ofarbitrary codes of a given block size and code rate on the additive white Gaussian noise channelwith unconstrained input (i.e., not necessarily binary-input as in figure 3-3). The curves labeled‘bound’ are the block-size-dependent bounds for each code rate. The horizontal asymptoteslabeled ‘capacity’ are the rate-dependent capacity limits. These asymptotes are slightlydifferent from the vertical asymptotes in figure 3-3 because they represent capacity limits for anunconstrained-input channel instead of a binary-input channel.5Bound r=1/2Capacity r=1/24Bound r=1/3Bound r=1/4Capacity r=1/3Capacity r=1/4Minimum Eb/No (dB) for WER = 1E-43210Bound r=1/6Capacity r=1/6Capacity r=0-1-210 100 1000 10000 100000Information Block Size k (bits)Figure 3-4: Shannon Sphere-Packing Lower Bounds on the WER Performance forCodes with Varying Information Block Length k and Rates 1/6, 1/4, 1/3,1/2, Operating over an Unconstrained-Input AWGN ChannelCCSDS 130.1-G-2 Page 3-5 November 2012
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