Improving the temporal resolution of ground-penetrating radar (GPR) data is a fundamental factor in presenting the characteristics of the underground structures. The advantages of sparse signal processing using the majorization-minimization (MM) method in GPR signal compression are investigated. In this method, minimizing the cost function is determined with L1 and L2 norms; also, the banded structures of matrices resulting from the sparse deconvolution problem are regarded. Then, the MM algorithm has been implemented with least-squares deconvolution (LSQR) on the synthetic and real data collected by a system with dual-frequency antennas of 300 and 800 MHz. The compression process has resulted in a high-resolution image from the subsurface layers and anomalies. Analysis of the outputs reported that the reflection coefficient improved significantly by application of the MM algorithm to the synthetic and real data compared with the least-squares deconvolution which only filters the data. The power spectrum after using the MM algorithm shows acceptable compression. Moreover, this algorithm leads to a considerable improvement on the amplitudes so that the hidden anomalies are better restored.