A new technique is proposed for constructing trellis codes. which provides an alternative to Ungerboeck's method of "set partitioning." The new codes use a signal constellation consisting of points from an n -dimensional lattice \Lambda , with an equal number of points from each coset of a sublattice \Lambda ' . One part of the input stream drives a generalized convolutional code whose outputs are cosets of \Lambda ' , while the other part selects points from these cosets. Several of the new codes are better than those previously known.

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New trellis codes based on lattices and cosets

Channel coding combined with expanded signal sets has been shown [1] to improve error performance over uncoded modulation without expanding the bandwidth of the transmitted signals. In this paper, new coded modulation formats defined over an expanded set of signals varying both in phase and frequency are presented. The new schemes combine FSK and PSK modulation and make use of trellis coding and Viterbi decoding to improve error performance over uncoded modulation. The free Euclidean distance is calculated for several classes of codes, and upper bounds and simulation results are also presented for some simple codes. The spectral characteristics of the proposed coded modulation formats are evaluated and compared to conventional two-dimensional modulation formats. Differential encoding and various extensions of the basic scheme are also discussed.

Coded Phase/Frequency Modulation

A trellis code is a “sliding window” method for encoding a binary data stream {ai}, ai = 0, 1, as a sequence of signal points drawn from Rn. The rule for assigning signal points depends on the state of the encoder. In this paper n = 4, and the signal points are 4-tuples of odd integers. We describe an infinite family of eight-state trellis codes. For k = 3, 4, 5, … we construct a trellis encoder with a rate of k bits/four-dimensional signal. We propose that the codes with rates k = 8 and 12 be considered for use in modems designed to achieve data rates of 9.6 kb/s and 14.4 kb/s, respectively.

/pdf/four-dimensional-modulation-with-an-eight-state-trellis-code-2womsfz5ha.pdf

Four-dimensional modulation with an eight-state trellis code

A family of multi-dimensional convolutionally coded modulation systems achieves enlarged minimum distance between possible sequences of signal points, reduced number of error events with the minimum distance, acceptable peak-to-average power ratio, reduced number of signal points in each constituent two-dimensional constellation, immunity to rapid carrier phase changes, and reduced complexity, resulting in a reduced error probability when maximum likelihood decoding is used. These advantages are achieved by the construction and by the partitioning into subsets of the multi-dimensional constellation, by the design of convolutional codes using those multi-dimensional subsets, by using a bit converter and a block encoder to convert a multi-dimensional constellation mapping into multiple two-dimensional constellations mappings, and by a simplified decoding technique.

Multidimensional, convolutionally coded communication systems

An 8-state trellis code is described that uses a signal constellation from the 8-dimensional Gosset lattice E 8 . It can be used for example to transmit data at 9.6, 14.4, and 19.2 kbits/s with a nominal coding gain of close to 6 dB.

An eight-dimensional trellis code

Four dimensional modulation as a means of improving communication efficiency on the band-limited Gaussian channel, with the four dimensions of signal space constituted by phase orthogonal carriers (cos omega sub c t and sin omega sub c t) simultaneously on space orthogonal electromagnetic waves are discussed. "Frequency reuse' techniques use such polarization orthogonality to reuse the same frequency slot, but the modulation is not treated as four dimensional, rather a product of two-d modulations, e.g., QPSK. It is well known that, higher dimensionality signalling affords possible improvements in the power bandwidth sense. Four-D modulations based upon subsets of lattice-packings in four-D, which afford simplification of encoding and decoding are described. Sets of up to 1024 signals are constructed in four-D, providing a (Nyquist) spectral efficiency of up to 10 bps/Hz. Energy gains over the reuse technique are in the one to three dB range t equal bandwidth.

https://ntrs.nasa.gov/api/citations/19830028029/downloads/19830028029.pdf

S. G. Wilson

Papers

Four-dimensional modulation and coding: An alternate to frequency-reuse