  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.
