Channel and frequency offsets estimation in MIMO time-selective channels

We address the frequency offsets and channel gains estimation problem for a MIMO time-selective channel using a training sequence. The general case where the frequency offsets are possibly different for each transmit antenna is considered. The Crame´r-Rao bound (CRB) for the problem at hand is derived. Additionally, we present a simple, closed-form expression for the large-sample CRB and show that it depends in a simple way on the channel parameters. Next, the parameters estimation issue is investigated. First, the maximum likelihood estimator (MLE) is derived which entails solving an n-dimensional maximization problem with n being the number of transmit antennas. Then, we show that the likelihood function can be written as the product of n 1D functions if a suitable choice of the training sequence is made. Based on this fact, we suggest two computationally simpler methods. Numerical examples are provided which illustrate the performance of the estimators and compare it with the CRB.