Subarray-based hybrid beamforming communication systems are a cost-and power-efficient architectural solution to realize massive multiple-input multiple-output (MIMO) systems. To estimate the required channel state information (CSI) current research focuses on beam training algorithms, which suffer from long estimation times and require precise system calibration. In order to overcome these problems, two channel estimation algorithms in combination with suitable beamforming algorithms are proposed. The presented algorithms are based on sparse array measurements, where only one antenna per subarray is active during the estimation process. This allows for the reconstruction of the complex MIMO channel matrix by performing multiple sparse array measurements. Channel estimation algorithms, which drastically reduce the channel estimation time are proposed in this letter. Their high performance is proven in small cell communication measurements around 28 GHz.