# For example for a binary input awgn channel rate 12

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Unformatted text preview: mate Shannon limit on performance (i.e., rate →0) is -1.6 dB. A comparison of these limits shows the improvement that is theoretically possible as a result of lowering the code rate. For example, for a binary-input AWGN channel, rate-1/2 codes suffer an inherent 0.7 dB disadvantage relative to rate-1/3 codes, a 1.0 dB disadvantage relative to rate-1/4 codes, and a 1.8 dB disadvantage relative to the ultimate limit (rate →0). Block-Size-Dependent Limits on Code Performance — Just as a constraint on code rate raises the minimum threshold for reliable communication above the ultimate unconstrained capacity limit, so does a constraint on codeblock length. The theoretical limits shown in figure 3-3 assume no constraint on block size. Approaching these limits requires that block sizes grow arbitrarily large. Figure 3-4 shows some classic Shannon sphere packing lower bounds on the performance of arbitrary codes of a given block size and code rate on the additive white Gaussian noise channel with 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 asymptotes labeled ‘capacity’ are the rate-dependent capacity limits. These asymptotes are slightly different from the vertical asymptotes in figure 3-3 because they represent capacity limits for an unconstrained-input channel instead of a binary-input channel. 5 Bound r=1/2 Bound r=1/3 Minimum Eb/No (dB) for WER = 1E-4 Capacity r=1/4 Bound r=1/6 3 Capacity r=1/3 Bound r=1/4 4 Capacity r=1/2 Capacity r=1/6 Capacity r=0 2 1 0 -1 -2 10 100 1000 10000 100000 Information Block Size k (bits) Figure 3-4: Shannon Sphere-Packing Lower Bounds on the WER Performance for Codes with Varying Information Block Length k and Rates 1/6, 1/4, 1/3, 1/2, Operating over an Unconstrained-Input AWGN Channel CCSDS 130.1-G-1 Page 3-5 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE This figure shows that, for any given code rate, the minimum threshold for reliable communication is significantly higher than the corresponding ultimate limit for that code rate, if the codeblock length is constrained to a given finite size. For example, 1000-bit blocks have an inherent advantage of about 1.3 dB compared to 100-bit blocks for each of the four code rates plotted. An additional gain of just over 0.5 dB is potentially obtained by going from 1000-bit blocks to 10000-bit blocks, and another 0.2 dB by going to 100000-bit blocks. After that, there is less than another 0.1 dB of improvement available before the ultimate capacity limit for unlimited block sizes is reached. 3.4.3 EXAMPLES OF PERFORMANCE OF RECOMMENDED CODES The relative performance of various recommend (non-punctured, non-shortened) codes on a Gaussian channel is shown in figure 3-5. Here, the input is constrained to be chosen from between two levels, because biphase modulation is assumed throughout the Recommended Standard. 3 These performance data were obtained by software simulation and assume that there are no synchronization losses (see reference [10] for a discussion on the effect of receiver tracking losses). The channel symbol errors were assumed to be independent: this is a good assumption for the deep space channel, and an approximation for near-Earth links which ignores impulsive noise and RFI. In this introductory comparison of code performance, infinite interleaving is assumed in the concatenated code and bit error rate (BER) only is used. Specific results with finite interleaving depth are given in 6.3; results for frame error rate (FER) are given in later Sections discussing specific codes. It is clear from the figure that the convolutional code offers a coding gain of about 5.5 dB over an uncoded system at decoded bit error rate of 10-5. Concatenation of this code with the outer ReedSolomon code results in an additional 2.0 dB of coding gain. Turbo codes can provide even higher coding gains, as illustrated in the figure for the turbo code with rate 1/2 and block size 8920 bits. This code approaches within 1 dB the ultimate Shannon limit for codes with rate 1/2 and improves on the recommended concatenated code’s performance by about 1.5 dB. These codes are included in the CCSDS Recommended Standard because they provide substantial coding gain over an uncoded system. They have already been incorporated, or are planned to be incorporated, into nearly all missions of member agencies of the CCSDS. The next four sections describe the parameters and the performance of each recommended code in more detail, along with brief descriptions of their encoder and decoder realizations. 3 Biphase modulation is appropriate for power-limited links, where bandwidth efficiency is not particularly important. CCSDS 130.1-G-1 Page 3-6 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE 10 -1 CONCATENATED CONVOLUTIONAL AND REED-SOLOMON (Ideal interleaver) 10 -2 B it E r r o r R a t e UNCODED (7,1/2) CONVOLUTIONAL 10 -3 10 -4 TURBO Rate 1/2 Block size 8920 bits 10 -5 (255,223) REED-SOLOMON CAPACITY Rate 1/2 Binary Input AWGN Channel 10 -6 -...
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## This document was uploaded on 03/06/2014.

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