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
[edit]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
[edit]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 ≤ 1/2
- 2. R1 = H(a), R2 = 1 − a, for 1/3 ≤ a ≤ 1/2
- 3. R1 + R2 = log2 3, log2 3 - 2/3 ≤ R1 ≤ 2/3
- 4. R1 = 1 − a, R2 = H(a), for 1/3 ≤ a ≤ 1/2
- 5. 0 ≤ R1 ≤ 1/2, R2 = 1
A solution was also found by Pinkser et al. (1995).[4]
References
[edit]- ^ L Breiman; D Blackwell; A J Thomasian (1958). "Proof of shannon's transmission theorem for finite-state indecomposable channels". The Annals of Mathematical Statistics. 29 (4). United States: Institute of Mathematical Statistics: 1209–2220. doi:10.1214/aoms/1177706452.
- ^ S I Gel'fand (1977). "Capacity of one broadcast channel". Problemy Peredachi Informatsii. 13 (3). Moscow, Russia: Russian Academy of Sciences, Branch of Informatics, Computer Equipment and Automatization: 106–108.
- ^ E van der Meulen (1977). "A survey of multi-way channels in information theory: 1961-1976". IEEE Transactions on Information Theory. 23 (1). New York City, New York, United States: Institute of Electrical and Electronics Engineers: 1–37. doi:10.1109/tit.1977.1055652.
- ^ M Pinsker; S. Prelov; S. Verdú (November 1995). "Sensitivity of Channel Capacity". IEEE Transactions on Information Theory. 41 (6). New York City, New York, United States: Institute of Electrical and Electronics Engineers: 1877–1888. doi:10.1109/18.476313. S2CID 9687919.
