InformationTheory.Shannon.BroadcastChannel.Superposition.Region
Broadcast channel — the superposition inner bound #
Superposition coding sends a cloud center U to both receivers and a satellite X to the
first one only. This file takes the union of its inner bound over the auxiliary alphabets, so
that it can be compared with the outer bounds as a set.
Two unions are taken, over the same full-support indices and differing only in the constraints
cutting out each member. Keeping the sum-rate constraint gives the general superposition bound,
achievable over any broadcast channel; dropping it gives a superset, achievable as soon as the
first receiver decodes the cloud at least as well as the second one does — which is exactly what
IsBCLessNoisy asks, physical degradedness is not needed — because I(U; Y₁) ≥ I(U; Y₂) makes
the omitted inequality follow from the two kept ones.
Main definitions #
bcSuperpositionRegionNoSumRate W— the superposition inner bound without its sum-rate constraint, as a union over auxiliary alphabets, restricted to the full-support indices.bcSuperpositionRegionSumRate W— the same union with the sum-rate constraint kept.
Main statements #
bc_lessNoisy_achievability— the superposition rate pairs of a less noisy channel are achievable.bcSuperpositionRegionNoSumRate_subset_capacity— the two-constraint inner bound sits inside the operational capacity region of a less noisy channel.bcSuperpositionRegionSumRate_subset_capacity— the three-constraint inner bound sits inside the operational capacity region, for any broadcast channel.
Implementation notes #
The auxiliary alphabets range over Marton.bcAuxAlphabet, one cardinality at a time in the
universe of the input alphabet. That universe is forced: IsBCLessNoisy quantifies its
auxiliary variable over the universe of the input alphabet, so a member of the union has to live
there for the class hypothesis to apply to it.
Nonnegativity of the satellite information #
InformationTheory.Shannon.BroadcastChannel.bcInfo₁_nonneg
sourceUsed by
The three informations as (conditional) mutual informations #
InformationTheory.Shannon.BroadcastChannel.bcInfo₂_eq_mutualInfo_toReal
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InformationTheory.Shannon.BroadcastChannel.bcInfoJoint_eq_mutualInfo_toReal
sourceUsed by
InformationTheory.Shannon.BroadcastChannel.bcInfo₁_eq_condMutualInfo_toReal
sourceUsed by
Achievability over a less noisy channel #
InformationTheory.Shannon.BroadcastChannel.bc_lessNoisy_achievability
sourceAchievability half of the superposition inner bound over a less noisy broadcast channel.
The two-tier random-coding argument is shared with the degraded channel through
bc_achievability_of_infoJoint_ge, which takes the rate-sum inequality
I(X; Y₁ ∣ U) + I(U; Y₂) ≤ I((U, X); Y₁) as a hypothesis instead of deriving it;
bc_lessNoisy_infoJoint_ge supplies it from the weaker class hypothesis.
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The superposition inner bound as a region #
InformationTheory.Shannon.BroadcastChannel.bcSuperpositionRegionNoSumRate
sourceThe superposition inner bound of a broadcast channel: the closure of the union, over the
full-support auxiliary laws on Marton.bcAuxAlphabet, of the rectangles cut out by
R₁ ≤ I(X; Y₁ ∣ U) and R₂ ≤ I(U; Y₂).
The general superposition bound also constrains the rate sum by R₁ + R₂ ≤ I((U, X); Y₁), and
that constraint is dropped here. Dropping it is exact over a less noisy channel, where
I(U; Y₁) ≥ I(U; Y₂) forces I(X; Y₁ ∣ U) + I(U; Y₂) ≤ I((U, X); Y₁)
(bc_lessNoisy_infoJoint_ge), so the omitted inequality already follows from the two kept ones.
Outside that class this set is only a superset of the superposition bound.
The full-support indices are the ones the achievability theorem applies to, so this is the form
of the union that is achievable. As for bcCapacityRegion and the outer bounds, no sign
constraint is imposed: a nonpositive rate asks only for a single message.
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Equations
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Instances For
Used by
InformationTheory.Shannon.BroadcastChannel.bcSuperpositionRegionNoSumRate_isClosed
sourceUsed by
InformationTheory.Shannon.BroadcastChannel.bcSuperpositionRegionNoSumRate_subset_capacity
sourceThe superposition inner bound of a less noisy broadcast channel is achievable: it is contained in the operational capacity region. @audit:ok
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The superposition inner bound with the sum-rate constraint kept #
InformationTheory.Shannon.BroadcastChannel.bcSuperpositionRegionSumRate
sourceThe superposition inner bound of a broadcast channel with the sum-rate constraint kept: the
closure of the union, over the full-support auxiliary laws on Marton.bcAuxAlphabet, of the
regions cut out by R₁ ≤ I(X; Y₁ ∣ U), R₂ ≤ I(U; Y₂) and max R₁ 0 + R₂ ≤ I((U, X); Y₁).
The sum constraint is written with the first rate clamped at zero because that is the form the
achievability theorem takes: a nonpositive first rate asks for a single satellite codeword, so the
wrong-cloud slack it costs is measured at max R₁ 0. With that shape the whole set is achievable
with no comparison-class hypothesis, whereas the plain sum R₁ + R₂ would need one on the branch
where the first rate is negative.
bcSuperpositionRegionNoSumRate drops the sum constraint, which is exact over a less noisy
channel but not in general; this set is the general superposition bound and is contained in
it.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Used by
InformationTheory.Shannon.BroadcastChannel.bcSuperpositionRegionSumRate_isClosed
sourceUsed by
InformationTheory.Shannon.BroadcastChannel.bcSuperpositionRegionSumRate_subset_capacity
sourceThe three-constraint superposition inner bound of a broadcast channel is achievable: it is contained in the operational capacity region. No comparison between the two receivers is needed, because the region carries the sum constraint the achievability theorem asks for.