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@ -295,10 +295,27 @@ Here, we multiply @eq:offset with the maximum offsetting index $i_"max" = s/2$.
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Now, if we want to find the maximum offset for a odd number of metrics, we need to modify @eq:max_offset_even, more specifically its numerator.
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We know, that we need to keep the original quantizer for a odd number of metrics, besides that, the method stays the same.
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For that reason, we will decrease the parameter $m$ by $1$:
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For that reason, we will decrease the parameter $m$ by $1$, that way we will still perform a division without remainder:
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$
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phi_"max_odd" &= frac(frac(m-1, 2), 2^n dot m dot 2)\
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phi_"max,odd" &= frac(frac(m-1, 2), 2^n dot m dot 2)\
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&= lr(frac(m-1, 2^n dot m dot 4)mid(|))_(n=2, m=3) = 1/24
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$
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It is important to note, that $phi_"max,odd"$, unlike $phi_"max,even"$ is dependent on the parameter $m$.
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#figure(
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table(
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columns: (5),
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align: center + horizon,
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inset: 7pt,
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[*m*],[3],[5],[7],[9],
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[$bold(phi_"max,odd")$],[$1/24$],[$1/20$],[$3/56$],[$1/18$]
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),
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)<tb:>
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$
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lim_(m arrow.r infinity) phi_"max,odd" &= frac(m-1, 2^n dot m dot 4)\
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&= frac(1, 2^n dot 4) = phi_"max,even"
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$
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