Cyclic prefix (CP) deploying techniques such as orthogonal frequency division multiplexing (OFDM) and single carrier frequency domain equalization (SC-FDE) offer considerable advantages in terms of equalizing time dispersive effect of wireless channel at the expense of a reasonable spectral redundancy. However, CP introduces cyclic features to the signal which can also be exploited for signal interception, blind parameter estimation and synchronization, and therefore, compromises the security of the signal against eavesdropping attacks. In order to provide a covert communication against such attacks, in this paper, we present two novel techniques that suppress the cyclic features of the CP utilizing signals while maintaining their advantages in equalization without reducing spectral efficiency. The first technique is built on a CP selection strategy while the second one is based on randomizing the symbol time. We also performed peak-to-average power ratio mitigation and out-of-band leakage suppression along with the cyclic feature concealing in the second technique at the expense of a reasonable complexity and signaling. Subsequent to the presentation of the proposed techniques, their performances are discussed and compared for OFDM and SC-FDE in terms of complexity and bit-error-rate along with cyclic feature suppression.11A part of this study is presented in IEEE Military Communication Conference (MILCOM), 2014.