On uniqueness of sparse signal recovery.

•The l0-norm-based condition is proposed for the uniqueness of sparse signal recovery.•The l1-norm-based condition is proposed for the uniqueness of sparse signal recovery.•The analysis of the l1-norm-based condition is given using the geometric explanation.•The minimal uniqueness conditions for l0 and l1-based frameworks are analyzed.•The simulation verifies the effectiveness of the proposed conditions.