In this paper, we use several degrees of freedom (central frequency and polarization of the radar waveform) in order to characterize the clutter signature. We study the properties of the covariance matrix of the clutter returns using a multi-dimensional representation of the signal inside a specific clutter cell. We measure clutter returns with an S-band ground radar using multiple waveforms in order to obtain this multidimensional representation. We study how the nature of the degrees of freedom included in the signature affects the properties of the covariance matrix of the clutter cell using an effective rank analysis. Eventually, we address the false alarm problem with the following question: how long is the estimation of the clutter covariance matrix valid. To do so, we empirically estimate the covariance matrices of the clutter at different instants, and generate synthetic clutter returns following these distributions. A detector based on a previous covariance matrix evaluates the impact on the false alarm rate of this mismatch.