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Recent years have witnessed some interest in Spectrally Efficient Frequency Division Multiplexing (SEFDM) communications systems, where subcarrier orthogonality is intentionally violated to improve the spectral efficiency at the expense of system complexity. This paper investigates reliable polynomial-time hard detection techniques for SEFDM systems, by relaxing the optimal combinatorial Maximum Likelihood (ML) detection to a Semidefinite Program (SDP). SDP can be solved in almost cubic complexity over the number of the SEFDM subcarriers, N. However, the relaxation results into a degradation of the system error performance. In particular, we study the effect of the number of SEFDM subcarriers, N, and the subcarrier separation, Â¿f, on the SDP relaxation gap in the presence of Additive White Gaussian Noise (AWGN). We find that as N increases and/or Â¿f decreases, the SDP estimate gradually diverges from the optimal solution. To overcome this problem, we propose the use of a boxed ML procedure around the SDP estimate. We show by simulation that the SDP-ML combination approximates the optimum detection for N Â¿ 32 subcarriers and up to 20% of bandwidth reduction with respect to an equivalent Orthogonal FDM (OFDM). Our SDP results show a small error penalty when compared to optimal Sphere Decoders (SD), whose computational effort is random and noise dependant, and thereby indicate that our proposed technique is useable in practical SEFDM systems with a moderate number of subcarriers.