In a frequency-selective slow-fading channel in a multiple-input multiple-output (MIMO) system, the channel matrix is of the form of a block matrix. A method is proposed to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. Asymptotic eigenvalue distribution of$HH^*$is also calculated, where the entries of$H$are jointly Gaussian, with a correlation of the form$E[h_{pj}\bar h_{qk}]= \sum_{s=1}^t \Psi^{(s)}_{jk}\mathhat\Psi^{(s)}_{pq}$(where$t$is fixed and does not increase with the size of the matrix). An operator-valued free probability approach is used to achieve this goal. Using this method, a system of equations is derived, which can be solved numerically to compute the desired eigenvalue distribution.

Considering a real signal as the sum of a number of sinusoidal signals in the presence of additive noise, maximum windowed likelihood (MWL) criterion is introduced and applied to construct an adaptive algorithm in order to estimate the amplitude and frequency of these components. The amplitudes, phases and frequencies are assumed to be slowly time varying. Employing MWL an adaptive algorithm is obtained in two steps. First, assuming some initial values for the frequency of each component, a closed form is derived to estimate the amplitudes. Then, the gradient of MWL is used to adaptively track the frequencies, using the latter values of amplitudes. The proposed algorithm has a parallel structure in which each branch estimates parameters of one of the components. The proposed multicomponent phase locked loop (MPLL) algorithm is implemented employing low complexity blocks. It is adjustable to be used in different conditions. The mean squared error of the algorithm is studied to analyze the effect of the window length and type and the step size. Simulations have been conducted to illustrate the efficiency and the performance of the algorithm in different conditions including: the effect of the initialization, the frequency resolution, for chirp components, for components during frequency crossover and for speech signals. Simulations illustrate that the method efficiently tracks slowly time-varying components of the signals such as voiced speech segments.