Asymptotic Coding Gain

For a linear, time-invariant, discrete-time channel with transfer function H(f), and information rate R bits/T, where T is the symbol interval, an optimal signal set of length K is defined to be a set of 2RK inputs of length K that maximizes the minimum 1, distance between pairs of outputs. This paper studies the minimum distance between outputs, or equivalently, the coding gain of optimal signal sets as K +m. For large K this coding gain, relative to single-step detection, can approximately be decomposed into the coding gain of an optimal signal set of length K for the identity channel, plus the gain of a “baseline” coding scheme for the channel H(f). The baseline signal set is selected from the multidimensional integer lattice, where the basis vectors of the space are taken to be the eigenvectors of H‘H, and H is the Toeplitz matrix that maps channel inputs to channel outputs. The coding gain of the baseline scheme can be computed explicitly as K +CO in terms of IH(f)l and R. The minimum distance between channel outputs for optimal signal sets as K --$ m is determined by the €-rate of the channel. Existing upper and lower bounds on the e-rate are used to compute bounds on the maximum asymptotic coding gains achievable for some partial response channels. These asymptotic coding gains are compared with the coding gains corresponding to signal sets found by numerical optimization techniques. A comparison of bounds on r-rates for the identity and 1 D channels indicates that for a given large K, the squared minimum distance of an optimal signal set for the 1 D channel is 2 dB more than the squared minimum distance of an optimal signal set for the identity channel at a rate of 1 bit/ T. For rates greater than 2 bits/ T, however, this comparison indicates that optimal signal sets of length K for these two channels have nearly the same minimum distance. Index Terms -Coding, partial-response channels, intersymbol interference, c -rate, multidimensional signal sets.