Linear dependence of steering vectors associated with tripole arrays

We are concerned with the linear independence of steering vectors associated with tripoles, each of which provides measurements of the three components of electric field induced by electromagnetic signals. We first establish that for a single tripole, any steering vector is linearly dependent on at least one other steering vector corresponding to a different direction-of-arrival (DOA) for a general problem where signals may arrive from anywhere in a three-dimensional (3-D) space, but every two steering vectors with distinct DOAs are linearly independent if the signals are nonlinearly polarized and arrive from a strictly hemispherical space. We then obtain a series of upper bounds for the number of linearly independent steering vectors associated with a tripole array with general sensor configurations. We also show that for applications where signals are known to be linearly polarized in the same direction, the ability to estimate DOAs using a tripole array is identical to that using a scalar-sensor array if both of them have identical sensor configurations.