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### Abstract

For a general class of constant-energy trellis-coded modulation schemes with $2^{\nu}$ states, necessary and sufficient conditions to guarantee that a maximum-likelihood sequence estimator can decode each symbol with a fixed delay of $\nu$ symbols are derived. Additive white Gaussian noise is assumed. {MSK} is a special case that belongs to the family of modulation schemes with $\nu=1$. It is shown that when these conditions are met, the minimum squared Euclidean distance is upper bounded by $4E_s$, where $E_s$ is the signal's energy per interval. Necessary and sufficient conditions to achieve the upper bound are given and it it shown that these conditions are met if and only if the trellis-coded modulation scheme can be implemented as pulse amplitude modulation using a pulse that extends over $\nu+1$ symbols. Signals that achieve this upper bound and maximize the power within a given bandwidth are found. The bandwidth efficiency of such schemes is significantly higher than that of MSK.