Betts, W; Calderbank, AR; Laroia, R, *Performance of nonuniform constellations on the Gaussian channel*,
IEEE Transactions on Information Theory, vol. 40 no. 5
(1994),
pp. 1633-1638 [doi] .
**Abstract:**

*Testing of high-speed voiceband modems has revealed a significant increase in distortion for points near the perimeter of a QAM signal constellation. This distortion increases with distance from the center of the constellation and limits performance at data rates above 19.2 kb/s. The perimeter distortion can be reduced by transforming the signal constellation so that points near the center are closer together, and points near the perimeter are further apart. When the channel SNR is high, such a transformation reduces immunity to Gaussian noise because points near the center of the transformed constellation are closer together than in a uniformly spaced constellation with the same average power. This paper demonstrates theoretically that for channel SNR's of practical interest, there is actually a small gain in immunity to Gaussian noise. In fact, an appropriate coded modulation scheme can produce gains of about 0.25 dB.*