By Peter W. Hawkes (Ed.)
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Extra info for Advances in Electronics and Electron Physics, Vol. 79
An encoder with as few states as any other output equivalent encoder is termed output minimal. ) and g ( - ) describe an inherently input-output relationship. Intuitively, an output minimal code is at least controllable, observable, and zero delay. For example, the delay-1 trellis in Fig. 11 is input-output minimal but not output minimal. A more difficult case is shown in Fig. 13. The three-state code in Fig. 13a is inputoutput minimal because all states are distinguishable, but a one-state output minimal realization is shown in Fig.
Theorem 2 shows that the identity state of an F G H has no special properties; any state can be transformed into the identity via Eqs. (58)-(61). This allows us to assume that the initial state is the identity element when we enumerate FGH trellis codes in Section VII. In contrast, an LSC in the zero state is “unexcited”: A zero input will always produce a zero output and the system will remain in the zero state. For LSCs, however, the concept of initial state can be entirely avoided by using minimal realizations and by considering two-sided rather than one-sided infinite input sequences (Forney, 1970).
The following example demonstrates the immediate realization of an LSC as an FGH. Example: Ungerboeck's eight-state 8-PSK rate scribed as an LSC in the Galois field GF: trellis code can be de- (44) (45) where sk = [ S , k S l k S O k l T , u k = [Ulkfd()k]T, yk = [y2ky,ky,klT. The output yk is mapped to the constellation in Fig. 19. 19. 8-PSK constellation. B. Weighting Patterns The performance of a trellis code is determined by its input-output behavior, not by the particular realization that achieves this behavior.
Advances in Electronics and Electron Physics, Vol. 79 by Peter W. Hawkes (Ed.)