TRANSIENT BEHAVIOR OF PARALLEL ENCODING / DECODING in .NET

Generation Quick Response Code in .NET TRANSIENT BEHAVIOR OF PARALLEL ENCODING / DECODING
TRANSIENT BEHAVIOR OF PARALLEL ENCODING / DECODING
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d7 (T) d6 (T) d5 (N) d4 (N) d3 (N) d2 (T) d1 (T) d0 (N) WT = 4 TAG = (1+1) +
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T T N
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N T N
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circuits c0 ; c1 ; c2 ; c3 , and c4 , respectively. Obviously a different construction of the H matrix will lead to a very different encoding circuit design and thus different transient behavior. Since even the transient patterns with same weight may result in different TAG numbers, we count the frequency in which the same number of TAG is produced when the transient patterns have the same weight. For instance, the one transient pattern of WT 4 shown in Figure 8.15 results in TAG 9. There are total 8 70 transient patterns of 4 WT 4 and actually only 19 of them (shown as freq: 19 in Table 8.3) will result in TAG 9, as shown in Table 8.3. In fact there are ve transient patterns of WT 4 that
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TABLE 8.3 TAG Frequencies for all Transient Weights (WT) of the SEC-DED Encoding Circuit in Figure 8.15 WT < 2 TAG 0 freq. 9 WT 2 TAG 1 2 3 4 5 6 freq. 4 9 8 3 2 2 WT 3 TAG 4 5 6 7 8 9 10 11 freq. 6 12 13 11 9 3 0 2 WT 4 TAG 7 8 9 10 11 12 13 14 15 freq. 5 9 19 6 18 9 1 2 1 WT 5 TAG 11 12 13 14 15 16 17 freq. 2 16 11 9 7 2 9 WT 6 TAG 16 17 18 19 20 21 22 freq. 5 10 4 2 4 2 1 WT 7 TAG 22 24 28 freq. 6 1 1
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Source: [LO05]. 2005 IEEE.
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(1) T + (1+2+1+1+1) = 9 T+G T T G
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Figure 8.15 Encoding circuit of the (13, 8) odd-weight column SEC-DED code. Source: [LO05]. IEEE.
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WT 8 TAG 28 freq. 1
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Figure 8.16 Histogram of the frequency distribution and accumulation given inTable 8.3. Source: [LO05]. IEEE.
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produce only TAG 7, and one transient pattern of WT 4 that produces TAG 15. Figure 8.16 shows the histogram of the results shown in Table 8.3. In Figure 8.16, the total number of transient patterns that may produce the same number of TAG are added. For instance, TAG 9 may be produced by 3 transient patterns of WT 3 and 19 patterns of WT 4. Thus the total number of transient patterns that may result in TAG 9 is 22. Since we assume that all inputs have equal arrival rates, we derive the probability for a particular TAG number as follows: Pr TAG a Number of transient pairs that produce TAG a : Total Number of possible transient pairs
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For example, from Table 8.3 we nd that two patterns of WT 3, 18 patterns of WT 4, and two pairs of WT 5 can all produce TAG 11. Therefore there are 22 out of the total 256 transient pairs that will result in TAG 11. Accordingly, we say Pr TAG 11 22=256 0:086. Figure 8.17 shows the distribution of the TAG probabilities for the (13, 8) SEC-DED code. The x-axis represents the TAG number and the y-axis denotes the probability. Next we perform an exhaustive examination of the following (22, 16) odd-weightcolumn SEC-DED code. In this case the number of 1 s per row is exactly 8 for all rows. This means that all six XOR circuits have 8 inputs, a power of two. The previous (13, 8)
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0.1 0.08 Probability 0.06 0.04 0.02 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 TAG number
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Figure 8.17 Distribution of TAG probabilities of the (13, 8) SEC-DED code. Source: [LO05]. 2005 IEEE.
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encoding circuits did not have such perfectly even distribution of H row weights and the XOR circuits did not have power of two inputs. d0 2 0 60 6 60 H 6 61 6 41 1 d1 0 0 1 0 1 1 d2 0 0 1 1 0 1 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 0 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 d13 1 0 0 1 0 1 d14 1 0 1 0 0 1 d15 c0 1 1 1 0 0 0 0 0 0 0 1 0 c1 0 1 0 0 0 0 c2 0 0 1 0 0 0 c3 0 0 0 1 0 0 c4 0 0 0 0 1 0 c5 3 0 07 7 07 7 07 7 05 1 8:7 Table 8.4 shows the TAG frequencies for the (22, 16) SEC-DED code with all possible 216 input transient pairs. Since the H row weights are evenly distributed, the TAG
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TABLE 8.4 TAG Frequencies for All Transient Weights (WT) of the SEC-DED Encoding Circuit of the (22, 16) Odd-Weight-Column SEC-DED Code WT < 2 TAG 0 freq. 17 WT 3 TAG 3 4 5 6 7 8 9 10 11 12 freq. 67 143 142 97 55 36 14 3 2 1 WT 4 TAG 6 7 8 9 10 11 12 13 14 15 freq. 152 310 460 385 261 143 74 23 9 3 WT 5 TAG 9 10 11 12 13 14 15 16 17 18 19 20 21 freq. 87 404 914 1033 867 563 292 135 51 15 4 2 1 WT 6 TAG 12 13 14 15 16 17 18 19 20 21 22 23 24 freq. 8 138 734 1552 1937 1650 1080 518 253 96 30 10 2 WT 7 TAG 17 18 19 20 21 22 23 24 25 26 27 28 30 freq. 90 743 1924 2779 2532 1697 1015 421 164 53 19 2 1 WT 8 TAG 21 22 23 24 25 26 27 28 29 30 31 32 33 freq. 10 418 1556 2800 3194 2368 1386 712 272 106 42 2 4
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WT 2 TAG freq. 0 7 1 38 2 33 3 24 4 8 5 6 6 3 8 1 WT 9 TAG 26 27 28 29 30 31 32 33 34 35 36 37 39 freq. 90 743 1924 2779 2532 1697 1015 421 164 53 19 2 1
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WT 10 TAG 30 31 32 33 34 35 36 37 38 39 40 41 42 freq. 8 138 734 1552 1937 1650 1080 518 253 96 30 10 2