Maximal Spectral Efficiency of OFDM With Index Modulation Under Polynomial Space Complexity

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. We show that the largest number of binomial coefficient calculations performed by the original OFDM-IM mapper is polynomial on the number of subcarriers, even under the setup that maximizes the SE gain over OFDM. Also, such coefficients match the entries of the so-called Pascal's triangle (PT). Thus, by assisting the OFDM-IM mapper with a PT table, we show that the maximum SE gain over OFDM can be achieved under polynomial (rather than exponential) time and space complexities.

I. Introduction

Look-up table (LUT) is a fundamental technique for the efficient implementation of OFDM mappers. In an -subcarrier OFDM system, the mapper translates bits into complex baseband samples chosen from an -ary constellation diagram. To achieve better spectral efficiency (SE), novel physical layer techniques are expected to map more bits in the same amount of spectrum. In this context, index modulation (IM) has gained increasing attention in the literature [1], [2]. In the OFDM with IM (OFDM-IM) reference design [3], only subcarriers of the symbol are active, which enables waveforms. Of these, OFDM-IM employs to map bits. Further, by employing an -ary constellation for the active subcarriers, more log bits can be modulated, yielding a total of bits per OFDM-IM symbol.