In this paper, we address the problem of noise reduction and speech enhancement by adaptive filtering algorithms using the forward blind source separation structure (FBSS), which is often combined with adaptive algorithms to efficiently cancel the acoustic noise at the output. In this paper, we propose to combine the FBSS with the Simplified Fast Transversal Filter (SFTF) algorithm, where the adaptation gain is obtained only from the forward prediction. The performances of the proposed SFTF algorithm are compared with the Normalized Least Mean Square (NLMS) algorithm in different noisy conditions. This comparison is evaluated in terms of Cepstral Distance (CD), the System Mismatch (SM) and the Segmental Signal to Noise Ratio (SegSNR) criteria.