recently the indexed modulation (IM) technique in conjunction with the multi-carrier modulation gains an increasing attention. It conveys additional information on the subcarrier indices by activating specific subcarriers in the frequency domain besides the conventional amplitude-phase modulation of the activated subcarriers. Orthogonal frequency division multiplexing (OFDM) with IM (OFDM-IM) is deeply compared with the classical OFDM. It leads to an attractive trade-off between the spectral efficiency (SE) and the energy efficiency (EE). In this paper, the concept of the combinatorial modulation is introduced from a new point of view. The sparsity mapping is suggested intentionally to enable the compressive sensing (CS) concept in the data recovery process to provide further performance and EE enhancement without SE loss. Generating artificial data sparsity in the frequency domain along with naturally embedded channel sparsity in the time domain allows joint data recovery and channel estimation in a double sparsity framework. Based on simulation results, the performance of the proposed approach agrees with the predicted CS superiority even under low signal-to-noise ratio without channel coding. Moreover, the proposed sparsely indexed modulation system outperforms the conventional OFDM system and the OFDM-IM system in terms of error performance, peak-to-average power ratio (PAPR) and energy efficiency under the same spectral efficiency.