What is the Cost of the Index Selector Task for OFDM with Index Modulation?

Index Modulation (IM) is a technique that activate k out of n subcarriers of an OFDM symbol to transmit p(1) = right perpendicularlog(2) (n k)left perpendicular bits in symbol's indexes. Since both the symbol's spectrum width and transmission air-time duration remain the same, OFDM-IM outperforms OFDM's Spectral Efficiency (SE) for larger values of (n k). However, OFDM-IM requires an extra step called Index Selector (IxS) which takes T-alpha time units to map a given p(1)-bit input to its corresponding pattern of active subcarriers. This extra overhead virtually enlarges the symbol duration, which is not captured by the classic SE definition. To fulfill this gap, in this work we present the Spectro-Computational Efficiency (SCE) metric. SCE parameterizes either the absolute runtime of T-alpha on a reference hardware or its computational complexity T-alpha(n; k) as function of n and k. Based on SCE, we present theoretical case studies to identify the asymptotic bounds for T-alpha(n, k) across different choices of k. if T-alpha(n, n=2) is at most linear on n the resulting overhead is asymptotically negligible and IxS can handle an arbitrarily large OFDM symbol. Otherwise, OFDM-IM's SCE tends to zero regardless of the hardware processor speed. Also, we situate the inflection-point values for OFDM-IM's SCE between (6 3) and (14 7) in some practical case studies.