Recently, this journal has published a paper which dealt with basis expansion model (BEM) based least-squares (LS) channel estimation in doubly-selective orthogonal frequency-division multiplexing (DS-OFDM) systems. The least-squares channel estimator computes the pseudo-inverse of a channel estimation matrix. For the existence of the pseudo-inverse, it is necessary that the channel estimation matrix be of full column rank. In this paper, we investigate the conditions that need to be satisfied that ensures the full column-rank condition of the channel estimation matrix. In particular, we derive conditions that the BEM and pilot pattern designs should satisfy to ensure that the channel estimation matrix is of full column rank. We explore the polynomial BEM (P-BEM), complex exponential BEM (CE-BEM), Slepian BEM (S-BEM) and generalized complex exponential BEM (GCE-BEM). We present one possible way to design the pilot patterns which satisfy the full column-rank conditions. Furthermore, the proposed method is extended to the case of multiple-input multiple-output (MIMO) DS-OFDM systems as well. Examples of pilot pattern designs are presented, that ensure the channel estimation matrix is of full column rank for a large DS-MIMO-OFDM system with as many as six transmitters, six receivers and 1024 subcarriers.