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A method of estimating spatial degrees of freedom (DoF) from measured multipath propagation channels in the multiple-input single-output regime is presented. The DoF of the multipath channels on the transmit (Tx) side is derived by means of the spherical-wave expansion of electromagnetic fields radiated from a Tx antenna array having a certain aperture size. The DoF estimates are independent of particular realization of antenna elements on the Tx aperture. For a given aperture size, the DoF provides the number of the Tx antenna elements for efficient improvement of the system performance by utilizing the spatial diversity. Having confirmed the soundness of our DoF estimation method by channel measurements in an anechoic chamber, the DoF of indoor multipath channels is analyzed. The DoF on the Tx side of the considered multipath channels reveals larger values when the antenna aperture size is increased at a fixed frequency, and when the frequency increases for a fixed antenna aperture size. For a fixed frequency, increasing the antenna aperture size is more effective in observing extra DoF in the obstructed and non-line-of-sight channels than in the line-of-sight channels. Furthermore, for a fixed antenna aperture size, the use of the higher frequency brings larger DoFs in many propagation scenarios. The results also show that electrically smaller antennas are more efficient in observing the DoF. Finally, the solid angle of multipath is derived as a Tx antenna-independent metric of the multipath richness. It is defined as an angular range of dominant multipath clusters subtended on a unit sphere. Our analysis method is extendable to the multiple-input multiple-output regime in a straightforward manner.