On SVQ Shaping of Multidimensional Constellations - High-Rate Large-Dimensional Constellations

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An optimal shaping scheme for multidimensional constellations, motivated by some ideas from a fixed-rate structured vector quantizer (SVQ), was recently proposed by Laroia. It was shown that optimal shaping could be performed subject to a constraint on the CER2 or PAR2 by expressing the (optimally shaped) constellation as the codebook of an SVQ and using the SVQ encoding/decoding algorithms to index the constellation points. Further, compatibility with trellis coded modulation was demonstrated. The complexity of the proposed scheme was reasonable but dependent on the data transmission rate. In this paper, we use recent results due to Calderbank and Ozarow to show that complexity of this scheme can be reduced and made independent of the data rate with essentially no effect on the shaping gain. Also, we modify the SVQ encoding/decoding algorithms to reduce the implementation complexity even further. It is shown that SVQ shaping can achieve a shaping gain of about 1.20 dB with a PAR2 of 3.75 at a very reasonable complexity (about 15 multiply-adds/baud and a memory requirement of 1.5 kbytes). Further, a shaping gain of 1 dB results in a PAR2 of less than 3. This is considerable less than a PAR2 of 3.75 for Forney's trellis shaping scheme that gives about 1 dB shaping gain.