The MAPbI$_3$ halide perovskite single crystals are studied at 5 K temperature using the photoluminescence excitation spectroscopy. Two non-interacting types of states are determined: bound excitons and defect-related states. Excitation of the crystal with light energy below the bound exciton resonance reveals the complex low-density defects emission, otherwise hidden by the emission of bound excitons. A way to separate these defect-related luminescence spectra is proposed, and the thorough study of this emission regime is carried out. The results of this study opens an area of low-density defects and dopants exploration in halide perovskite semiconductors.