The purpose of this article is to investigate the viability of Multi-Carrier Modulation (MCM) systems based on the Fast Walsh Hadamard Transform (FWHT). In addition, a nonlinear Joint Low-Complexity Optimized Zero Forcing Successive Interference Cancellation (JLCOZF-SIC) equalizer is proposed. To that end, general equations for the number of flops of the proposed equalizer and various other equalizers are given. This article discusses the use of Banded Matrix Approximation (BMA) as a technique for reducing complexity. The proposed equalizer uses BMA to accomplish both equalization and co-Carrier Frequency Offset (co-CFO) corrections. In addition, three cases involving the proposed equalizer were investigated. In the first case, diagonal compensation is used. In the second case, BMA compensation is used. In the third case, complete matrix compensation is used. In the presence of frequency offset, noise, and frequency-selective Rayleigh fading environments, analysis and simulation results show that the OFDM-FWHT system with the proposed equalizer outperforms the conventional OFDM system with various linear and nonlinear equalizers.