RoUlet Transform With Roots of Unity

초록

We study unitary transforms whose entries are roots of unity and whose implementations are sparse. The question is whether a dense N × N unitary transform with N-th roots of-unity entries can be realized through a sequence of sparse structured factors. We show that this is possible when N admits a coprime decomposition. In particular, the dense transform can be written as a product of sparse unitary factors whose nonzero entries belong to smaller roots-of-unity sets. These factors com mute, so the order of multiplication does not matter. This leads to the RoUlet transform: a structured unitary transform that preserves the algebraic simplicity of roots of unity while reducing arithmetic and interconnection costs. We then use this transform to construct RoUlet transform-based multiplexing (RUTM). In doubly dispersive channels, RUTM induces a more structured effective channel than conventional alternatives in the examples considered, which improves robustness under a common receiver. The same commutative factorization also enables order-free distributed relaying, where different nodes can perform partial processing without a predetermined order. © 1967-2012 IEEE.

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