This work aims at characterizing the second-order statistics of indoor ultra-wideband (UWB) channels using channel sounding techniques. We present measurement results for different scenarios conducted in a laboratory setting at Institut Eurécom. These are based on an eigendecomposition of the channel autocovariance matrix, which allows for the analysis of the growth in the number of significant degrees of freedom of the channel process as a function of the signaling bandwidth as well as the statistical correlation between different propagation paths. We show empirical eigenvalue distributions as a function of the signal bandwidth for both line-of-sight and non-line-of-sight situations. Furthermore, we give examples where paths from different propagation clusters (possibly arising from reflection or diffraction) show strong statistical dependence.