is known. Instantaneous CSI Ideally, the channel matrix is known perfectly. Due to channel estimation errors, the channel information can be represented as where is the channel estimate and is the estimation error covariance matrix. The vectorization was used to achieve the column stacking of , as multivariate random variables are usually defined as vectors. Statistical CSI In this case, the statistics of are known. In a Rayleigh fading channel, this corresponds to knowing that for some known channel covariance matrix . Estimation of CSI Since the channel conditions vary, instantaneous CSI needs to be estimated on a short-term basis. A popular approach is so-called training sequence (or pilot sequence), where a known signal is transmitted and the channel matrix is estimated using the combined knowledge of the transmitted and received signal. Let the training sequence be denoted , where the vector is transmitted over the channel as By combining the received training signals for , the total training signalling becomes with the training matrix and the noise matrix . With this notation, channel estimation means that should be recovered from the knowledge of and . Least-square estimation If the channel and noise distributions are unknown, then the least-square estimator (also known as the minimum-variance unbiased estimator) is where denotes the conjugate transpose. The estimation Mean Square Error (MSE) is proportional to where denotes the trace. The error is minimized when is a scaled identity matrix. This can only be achieved when is equal to (or larger than) the number of transmit antennas. The simplest example of an optimal training matrix is to select as a (scaled) identity matrix of the same size that the number of transmit antennas. MMSE estimation If the channel and noise distributions are known, then this a priori information can be exploited to decrease the estimation error. This approach is known as Bayesian estimation and for Rayleigh fading channels it exploits that The MMSE estimator is the Bayesian counterpart to the least-square estimator and becomes where denotes the Kronecker product and the identity matrix has the dimension of the number of receive antennas. The estimation Mean Square Error (MSE) is and is minimized by a training matrix that in general can only be derived through numerical optimization. But there exist heuristic solutions with good performance based on waterfilling. As opposed to least-square estimation, the estimation error for spatially correlated channels can be minimized even if