Advances in Electronics and Electron Physics, Vol. 79 by Peter W. Hawkes (Ed.)

By Peter W. Hawkes (Ed.)

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Extra info for Advances in Electronics and Electron Physics, Vol. 79

Sample text

Proof: Given any two distinct states x = (x,,, . . ,x,- , ) a n d z = (zo,. , . ) from Eqs. (54)-(57) reveals that c(u’(x)) = x j and c(aJ(z))= zj. It follows from Eq. (51) and x and z are distinguishable at time k = j . Unfortunately, FGHs do not. Deleting noncommunicating and indistinguishable states may result in a structure that isn’t an FGH. Although nonminimal, the SO-realization is observable; a brute-force method of determining the size of an FGH trellis code is to iterate the SO-realization to find all reachable states.

There must be I UI branches entering each node. 3. The trellis must be highly symmetric. 4. If two trellis paths diverge, they must not be able to remerge for several time steps. Rules 1 and 2 insure that all states in the trellis are visited with equal frequency, while rules 3 and 4 tend to insure that the closest two trellis paths have a large dfree. Selecting the constellation is more difficult. The size of the constellation is usually chosen as ICI = 21UI. The factor of 2 is arbitrary, although Ungerboeck justifies this expansion factor using information theoretic arguments.

The former possibility increases the required bandwidth of the transmitted signal, which is unacceptable in a bandlimited environment. The latter choice requires a higher average energy to maintain interpoint distance, since there are more points per dimension. However, the energy efficiency (see Eq. (28)) added by the encoded more than compensates for the loss. Codes that use the latter method of constellation expansion are termedalphabet redundant. All codes add redundancy. With alphabet redundant codes, the redundancy is absorbed by increasing the number of points in the constellation (the channel alphabet), and not by transmitting more channel symbols (Biglieri, 1984).

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