35 lines
2.2 KiB
Typst
35 lines
2.2 KiB
Typst
#import "@preview/glossarium:0.4.1": *
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= Comparison of @smhdt and @bach
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During the course of this work, we took a look at two distinct helper-data algorithms: the S-Metric Helper Data method and the newly presented method of optimization through Boundary Adaptive Clustering with helper-data.
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The S-Metric method will always outperform BACH considering the amount of helper data needed for operation.
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This comes from the nature of S-Metric quickly approaching an optimal @ber for a certain bit width and not improving any further for higher amounts of metrics.
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Comparing both formulas for the extracted-bits to helper-data-bits ratio for both methods we can quickly see that S-Metric will always yield more extracted bits per helper-data bit than BACH.
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Considering #glspl("ber"), S-Metric does outperform BACH for smaller symbol widths.
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But while the error rate for higher order quantization rises exponentially for higher-order bit quantizations, the #glspl("ber") of BACH do seem to rise rather linear than exponentially for higher-order bit quantizations.
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This behaviour might be attributed to the general procedure of shaping the input values for the quantizer in such a way that they are clustered around the center of a quantizer step, which is a property that carries on for higher order bit quantizations.
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We can now compare both the S-Metric Helper Data method and the newly presented method of optimization through Boundary Adaptive Clustering with Helper data based on the @ber and the amount of helper data bits, more specifically the extracted bit to helper data bit ratio $cal(r)$.
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The ratios $cal(r)$ for both methods are defined as:
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$
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cal(r)_"SMHD" = frac(M, log_2(S))\
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cal(r)_"BACH" = frac(M, N-1)
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$
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A good outline to compare both performances of @bach and @smhdt is if $cal(r) = 1$ for varying values of $M$.
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#figure(
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table(
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columns: 7,
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[*M*], [$1$], [$2$], [$3$], [$4$], [$5$], [$6$],
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[*@ber @smhdt*], [$8 dot 10^(-6)$], [$4 dot 10^(-5)$], [$1 dot 10^(-3)$], [$0.02$], [$0.08$], [$0.15$],
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[*@ber @bach*], [$0.09$], [$0.05$], [$0.05$], [$0.22$], [$0.23$], [$0.23$]
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),
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caption: [#glspl("ber") for @bach and @smhdt configurations that equal the bit width of the quantized symbol and the amount of helper data bits]
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)
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