The paper presents a subspace iteration based eigensystem solution algorithm for solving the minimum eigenpair (eigenvalue and associated eigenvector) of a Hermitian matrix. Specifically, the focus is on the class of covariance matrices which have near-Toeplitz structures. First, a modified Rayleigh quotient iteration (MRQI) method developed earlier is generalised to handle the near-Toeplitz structures. Next, a classical Rayleigh-Ritz (RR) subspace approximation procedure is employed to further enhance the performance. Extensive simulation is carried out to compare the new RR method, the (generalised) MRQI method and the classical bisection method. Favourable results are observed. With parallel processing taken into account, it is estimated that this novel covariance eigensystem solver, withO(N) processors, is able to solve the minimum eigenpair of a covariance matrix inO(kN) time units. It is also observed that the number of iterationskis relatively independent of the dimension of the covariance matrix, and thus may be considered as a constant.