This paper proposes a novel compressed sensing (CS) framework for reconstructing electroencephalogram (EEG) signals. A feature of this framework is the application of independent component analysis (ICA) to remove the interference from artifacts after undersampling in a data processing unit. Therefore, we can remove the ICA processing block from the sensing unit. In this framework, we used a random undersampling measurement matrix to suppress the Gaussian. The developed framework, in which the discrete cosine transform basis and orthogonal matching pursuit were used, was evaluated using raw EEG signals with a pseudo-model of an eye-blink artifact. The normalized mean square error (NMSE) and correlation coefficient (CC), obtained as the average of 2,000 results, were compared to quantitatively demonstrate the effectiveness of the proposed framework. The evaluation results of the NMSE and CC showed that the proposed framework could remove the interference from the artifacts under a high compression ratio.