Address decomposition for the shaping of multi-dimensional signal constellations

An efficient addressing scheme to realize points near to the knee of the tradeoff curves of an optimally shaped constellation is introduced. This scheme, called the address decomposition, is based on decomposing the addressing into a hierarchy of addressing steps, each of a low dimensionality. As the memory size associated with a direct addressing scheme has an exponential growth with the dimensionality, this decomposition of addressing results in a substantial decrease in the complexity. In this case, by using a memory of a practical size, one can move along a tradeoff curve which is nearly optimum. For example, in a space of dimensionality N=32, a block of memory of 2.5 kbytes per N dimensions is used to achieve a shape gain of 0.92 dB with a constellation expansion ratio of 1.25 and a peak to average power ratio of 2.95. This scheme has no associated computation, is straightforward to implement, and is adaptable to the structure of a general coset coding scheme