Blackwell channel

The Blackwell channel is a deterministic broadcast channel model used in coding theory and information theory. It was first proposed by mathematician David Blackwell.[1] In this model, a transmitter transmits one of three symbols to two receivers. For two of the symbols, both receivers receive exactly what was sent; the third symbol, however, is received differently at each of the receivers. This is one of the simplest examples of a non-trivial capacity result for a non-stochastic channel.

Definition

The Blackwell channel is composed of one input (transmitter) and two outputs (receivers). The channel input is ternary (three symbols) and is selected from {0, 1, 2}. This symbol is broadcast to the receivers; that is, the transmitter sends one symbol simultaneously to both receivers. Each of the channel outputs is binary (two symbols), labeled {0, 1}.

Whenever a 0 is sent, both outputs receive a 0. Whenever a 1 is sent, both outputs receive a 1. When a 2 is sent, however, the first output is 0 and the second output is 1. Therefore, the symbol 2 is confused by each of the receivers in a different way.

The operation of the channel is memoryless and completely deterministic.

Capacity of the Blackwell channel

The capacity of the channel was found by S. I. Gel'fand.[2][3] It is defined by the region:

1. R1 = 1, 0 ≤ R2 ≤ ½
2. R1 = H(a), R2 = 1 − a, for ⅓ ≤ a ≤ ½
3. R1 + R2 = log2 3, log2 3 - ⅔ ≤ R1  ≤ ⅔
4. R1 = 1 − a, R2 = H(a), for ⅓ ≤ a ≤ ½
5. 0 ≤ R1 ≤ ½, R2 = 1

A solution was also found by Pinkser et al. (1995).[4]

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gollark: Indeed. And it doesn't scale well either.
gollark: ...
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References
  1. L Breiman; D Blackwell; A J Thomasian (1958). "Proof of shannon's transmission theorem for finite-state indecomposable channels". The Annals of Mathematical Statistics. United States: Institute of Mathematical Statistics. 29 (4): 1209–2220. doi:10.1214/aoms/1177706452.
  2. S I Gel'fand (1977). "Capacity of one broadcast channel". Problemy Peredachi Informatsii. Moscow, Russia: Russian Academy of Sciences, Branch of Informatics, Computer Equipment and Automatization. 13 (3): 106–108.
  3. E van der Meulen (1977). "A survey of multi-way channels in information theory: 1961-1976". IEEE Transactions on Information Theory. New York City, New York, United States: Institute of Electrical and Electronics Engineers. 23 (1): 1–37. doi:10.1109/tit.1977.1055652.
  4. M Pinsker; S. Prelov; S. Verdú (November 1995). "Sensitivity of Channel Capacity". IEEE Transactions on Information Theory. New York City, New York, United States: Institute of Electrical and Electronics Engineers. 41 (6): 1877–1888. doi:10.1109/18.476313. S2CID 9687919.
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