A fast low-cost through-the-wall radar imaging (TWRI) system, based on a vector network analyzer (VNA), a couple of switches and an array of Vivaldi antennas, has been designed, realized, and tested. To solve the TWRI inversion problem, an original theoretical modeling for a class of TWRI techniques whose basic functions are the cross-range Fourier transform (FT) of the scattered field and its covariance operator has been proposed. Using these functions, four conventional algorithms, namely the delay and sum (DAS), the FT, the multiple signal classification (MUSIC), the hybrid DAS-MUSIC and a new algorithm, the hybrid FT-MUSIC, have been derived. All these techniques have been implemented and their accuracy and field of view have been tested on canonical scatterers. Then, the algorithms have been applied to measured data collected in different scenarios constituted by a metallic bar or a human subject in the absence and in the presence of a wall between the antenna and the considered targets. Using the proposed TWRI system, it has been possible to detect a subject located up to 5-m away from the radar antenna array through a tuff wall. The proposed FT-MUSIC algorithm has evidenced performances similar to those of the DAS-MUSIC but with significantly lower execution times. Finally, FT-MUSIC performances in terms of field of view and immunity to disturbances are better compared to those of the MUSIC algorithm.