Maximal Spectral Efficiency of OFDM with Index Modulation under Polynomial Space Complexity
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In this letter we demonstrate a mapper that enables all waveforms of OFDM with Index Modulation (OFDM-IM) while preserving polynomial time and space computational complexities. Enabling all OFDM-IM waveforms maximizes the spectral efficiency (SE) gain over the classic OFDM but, as far as we know, the computational overhead of the resulting mapper remains conjectured as prohibitive across the OFDM-IM literature. For an N-subcarrier symbol, we show that OFDM-IM needs a Θ(2^N/√(N))-entry LUT to ensure that any sequence of N/2+_2N N/2 bits can be mapped in the same asymptotic time of the classic OFDM mapper i.e., O(N). We demonstrate this trade-off between SE and computational complexity can be improved if one stores Θ(N^2) binomial coefficients (the so-called Pascal's triangle) instead of the classic LUT. With this, we show the OFDM-IM mapper can achieve its maximal SE in the same time complexity of the OFDM's mapper under polynomial (rather than exponential) space resources.
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